Jan 20, 2015 - resistivity) and water storage capacity to electrical capacity. ...... Houston, and Houston (2005) and Scheuermann and Bieberstein (2007), ..... the first time for connecting self-made STDR to CR1000 and SDM X50 Multiplexer. ...... water content distributions along flat ribbon cables in a fullâscale levee model.
Dynamic multiphase flow in granular porous media
by
Guanxi Yan School of Civil Engineering The University of Queensland
Principal Supervisor: Dr. Alexander Scheuermann Associate Supervisor: Prof. Ling Li
A dissertation submitted for Confirmation of Candidature for Doctor of Philosophy of Geotechnical Engineering in School of Civil Engineering, The University of Queensland.
I.
Abstract
Two-phase flow in porous media is an important physical phenomenon in geotechnical engineering, agricultural hydrogeology, petroleum engineering and in many other engineering disciplines in need of understanding of interfacial physics. In macroscale continuum mechanics, the mathematical model of two-phase seepage is built upon a nonlinear diffusion theory, where the diffusivity is a nonlinear function of state variables. To physically determine this nonlinear function, the concept of capillary pressure-wetting phase saturation and wetting phase hydraulic conductivity-saturation were conceived with appending experiments such as Axis Translation Technique, Evaporation Method and Instantaneous Profile Method. Although the mathematical model of entire capillary diffusion theory is rigorously derived for simulation of the dynamic process of multiphase flow in porous media, the transient process is simply achieved by assuming the applicability of two-phase Darcy’law. Recent physical studies already demonstrated that there are multi-flow regimes in the single capillary tube during a transient process in which viscous-dominated flow regime is only third of four. Also, macroscale experimental studies in the 1970s confirmed that there is a significant difference between static and transient state. Many experimental studies have been conducted for both demonstrations of dynamic behaviour and validation of advanced theories. However, most of them merely focused on studying the dynamic process on primary drying path. Few studies provided experimental results on main drainage, main imbibition process and the hysteresis loops. There is still a lack of knowledge on an overview of the temporal effect of constitutive relationship for immiscible phases seepage problem. In this experimental study, three experimental stages will be achieved to investigate the dynamic process. First, a large size one-dimensional instantaneous profile method will be set up to investigate the dynamic air-water flow in a series of compacted sand columns. It is expected that the experimental result can be used to both validate advanced theories and provide the first dynamic capillary pressuresaturation curves for hysteresis loops. Second, A quasi-two-dimensional model will be set up to study the uniqueness of capillary pressure-saturation-specific interfacial area constitutive surface under both steady and transient state. Pixel counting and meniscus curvature fitting increase the measurement resolution of each state variable. In the final stage, this 2D experiment will be used to calibrate and validate the Shan-Chen Multiphase Lattice Boltzmann Model. It is expected that a well-calibrated SCLBM can be used to study transient two-phase flow under complex boundary conditions, which cannot be experimentally achieved. Through these three experimental stages, the understanding of the dynamic effect of capillary flow in macroscale will be further improved. With more experimental and numerical modelling result revealing and quantifying the dynamic effect, there is hope that the missing physical phenomenon can be well-mathematically formalised and incorporated back to theory to enhance the performance of prediction.
I
II.
Content
Contents I.
Abstract ............................................................................................................................................I
II.
Content ........................................................................................................................................... II
III. Figure ............................................................................................................................................. V IV. Table................................................................................................................................................ 1 1.
2.
3.
Introduction ..................................................................................................................................... 2 1.1
Background.............................................................................................................................. 2
1.2
Problem Statement................................................................................................................... 2
1.3
Research Questions ................................................................................................................. 3
1.4
Research Objectives and Contribution .................................................................................... 4
Literature Review ............................................................................................................................ 6 2.1
Unsaturated soil mechanics overview ..................................................................................... 6
2.2
Advantages and disadvantages of unsaturated soil experiment ............................................... 8
2.3
Dynamic multiphase flow in porous media ........................................................................... 10
2.4
Summary of literature review ................................................................................................ 14
Methodology ................................................................................................................................. 16 3.1
Standard geotechnical experiment for unsaturated soil ......................................................... 16
3.1.1
Material selection........................................................................................................... 16
3.1.2
Sieving analysis and Specimen density control ............................................................. 17
3.1.3
Hydraulic conductivity test by Constant Head Method ................................................. 19
3.1.4
Determination of SWRC by Hanging Column Method ................................................. 21
3.1.5
Relative hydraulic conductivity test by unsaturated soil standard triaxial cell .............. 23
3.2
Experimental design of one-dimensional Instantaneous Profile Method (IPM) ................... 25
3.2.1
Overview of experiment setup ....................................................................................... 25
3.2.2
T5 micro tensiometer and automatic logging system in dynamic suction measurement........ .......................................................................................................... 28
3.2.3
Spatial Time Domain Reflectometry application in dynamic moisture distribution measurement .................................................................................................................. 29
3.2.4
Boundary conditions for one/multi-step pressure boundary and cycling boundary ...... 31
3.2.5
Direct and inverse modelling of Richards’ model ......................................................... 32
3.3
Two-dimensional transparent soil model .............................................................................. 33
3.3.1
Overview of experimental setup .................................................................................... 33
3.3.2
Imaging technique for analyzing pore structure and identification of fluid-fluid interface... ...................................................................................................................... 35
3.3.3
Upscaling state variables ............................................................................................... 36
3.4
Numerical experiment using Multiphase Lattice Boltzmann simulation .............................. 37
II
4.
3.4.1
Shan-Chen Lattice Boltzmann method (SCLBM) ......................................................... 37
3.4.2
LBM simulation setup ................................................................................................... 40
3.4.3
Upscaling state variables and data post-process ............................................................ 41
3.4.4
Scaling between physical space and lattice domain....................................................... 41
3.4.5
Experimental calibration of SCLBM ............................................................................. 42
Preliminary Result and Discussion ............................................................................................... 44 4.1
4.1.1
Grain size distribution.................................................................................................... 44
4.1.2
Range of packing condition and density control............................................................ 45
4.2
Static Soil Water Retention Curve (SWRC) and Predicted SWRC by Pedo Transfer Functions …………………………………………………………………………....…………………47
4.2.1
Testifying available SWRFs of non-deformable soil..................................................... 47
4.2.2
Primary and main SWRC loops of Bribe Island Grey Sand and Budget bricky’s Sandy Loam .............................................................................................................................. 49
4.2.3
Calibration of Fredlund and Wilson Pedo-Transfer Function (PTF) by experimental data for determination of SWRC under different porosity .................................................... 51
4.3
Hydraulic conductivity and unsaturated hydraulic conductivity by Hydraulic Conductivity Functions (HCF) .................................................................................................................... 53
4.3.1
Saturated hydraulic conductivity ................................................................................... 53
4.3.2
Calibration of Kozeny-Carman equation against experimental result for prediction of hydraulic conductivity in different porosities ................................................................ 54
4.3.3
Unsaturated hydraulic conductivity of Bribe Island Grey Sand and Budget bricky’s Sandy Loam predicted by Hydraulic Conductivity Function (HCF) models ................ 55
4.4
1-D soil column experiment preparation ............................................................................... 56
4.4.1
Calibration of T5 tensiometer to data logger and sensor sensitivity analysis ................ 56
4.4.2
Setup of STDR, CR1000 logger and SDM X50 Multiplexer ....................................... 57
4.4.3
Setup of Accumulative bottom flux measurement ......................................................... 57
4.5
5.
Fundamental geotechnical parameters and soil classification ............................................... 44
2-D Multiphase Lattice Boltzmann Model (LBM) of drainage and imbibition .................... 58
4.5.1
Grain size distribution and pore size distribution of virtual beads package in SCLBM simulation domain.......................................................................................................... 58
4.5.2
Bubble simulation results for determination of lattice surface tension .......................... 59
4.5.3
Static SWRC and dynamic SWRC ................................................................................ 60
4.5.4
Pc-S-Anw constitutive surface in static and dynamic conditions .................................. 64
Summary of preliminary progress and the entire Research Schedule ........................................... 69
Reference .............................................................................................................................................. 72 Appendix 1: Full Version of Literature Review ................................................................................... 84 A1.1 Overview of unsaturated soil mechanics ................................................................................ 84 A1.1.1
Soil Suction and Soil Water Retention Curve................................................................ 84
A1.1.2
Steady state and unsteady state multiphase flow theory ................................................ 88
III
A1.1.3
Mechanical behaviour of unsaturated soil ..................................................................... 92
A1.2 Review of unsaturated soil experiment ................................................................................. 96 A1.2.1
Experimental techniques for Soil Water Retention Curve ............................................. 96
A1.2.2
Unsaturated soil hydraulic conductivity experiments .................................................... 99
A1.3 Dynamic water retention behaviour in porous media .......................................................... 102 A1.3.1
Original findings of dynamic effects in Soil Water Retention Curve .......................... 102
A1.3.2
Paradoxes in Theory of Dynamic Multiphase flow in porous media .......................... 103
A1.3.3
Microscale dynamic capillary behaviour in interfacial physics ................................... 104
A1.3.4
Advanced Theory of Dynamic Multiphase flows in porous media ............................. 105
A1.3.5
State of art of Dynamic Multiphase flows in porous media ........................................ 107
Appendix 2: Calibration of 21 T5 tensiometers to DT85G Datalogger .............................................. 113 Appendix 3: Program for CR1000 Logger logging 4 STDR Flat Ribbon Cable ................................ 122 Appendix 4: Grain Size Distribution and Pore Size Distribution of virtual soil in SCLBM simulation domain ......................................................................................................................................... 127 Appendix 5: SCLBM C++ code for simulation two phase displacement in virtual soil and ‘h5’ file data post process Matlab code ............................................................................................................ 131 Appendix 6: Capillary pressure, Wetting phase saturation and Specific Interfacial Area against time ..................................................................................................................................................... 146 Appendix 7: Visualization of 2D LBM simulation of drainage and imbibition ................................. 150 Appendix 8: Fitting performance of Fredlund & Xing SWRC model into dynamic and static SWRC dataset generated by SCLBM ...................................................................................................... 153 Appendix 9: Time derivative of each macroscale state varibles for drainage and imbibition ............ 157
IV
III.
Figure
Figure 1 Setup of sieve analysis and sieve arrangement specification ................................................. 17 Figure 2 Experiment for density control (a) Gas Pycnometer for Specific unit weight (b) Mechanical shaking table for maximum dry density (c) Funnel pouring for minimum dry density ...... 18 Figure 3 Pre-test for checking sample packing condition using 1D soil column .................................. 19 Figure 4 (a) Hydraulic conductivity test using Constant head method (b) flux measurement (c) temperature calibration of dynamic viscosity ........................................................................ 20 Figure 5 (a) Water reservoir (b) diagram of experiment setup (c) hanging columns inside fridge ....... 21 Figure 6 Unsaturated soil triaxial cell: WILLE Product available in Geotechnical Engineering Centre of the University of Queensland ......................................................................................... 24 Figure 7 (a) Overview of experimental setup (b) One-step outflow setp (c) One-step inflow setup (d) Multistep inflow/outflow setup ........................................................................................... 28 Figure 8 (a) T5 micro-tensiometer (b) Infield7 hand reader (c) DT85G data logger with CEM .......... 29 Figure 9 (a) Spatial Time Domain Reflectometry (STDR) (b) Electrical model of SPTDR sensor in an infinitesimal section (c) Electrical model of capacitance for soil covering STDR (Scheuermann, Huebner et al. 2009)................................................................................... 30 Figure 10 (a) A front view of 2-D physical model of mineral oil-water flow in transparent granular glass beads (b) a side view of the 2-D physical model with state variables extracting setup ...... 35 Figure 11 A schematic diagram of idealized interfacial geometry ....................................................... 36 Figure 12 D2Q9 scheme of LBM ......................................................................................................... 38 Figure 13 A package of circular disk in 500*500 2D lattice domain ................................................... 40 Figure 14 Grain Size Distribution of four potential sample used for IPM test ..................................... 44 Figure 15 Density profile of 1D IMP soil column pre-test (a) Bribe Island Sand dry density profile (b) Budget Bricky’s Loam dry density profile (c) Bribe Island Sand porosity profile (d) Budget Bricky’s Loam porosity profile (e) Bribe Island Sand relative density index profile (f) Budget Bricky’s Loam relative density index profile ......................................................... 46 Figure 16 Non-deformable soil SWRFs fitting performance on primary drainage curve (a) Bribe Island Sand (b) Budget bricky's Loam........................................................................................... 49 Figure 17 Primary and main drainage-imbibition curves of Bribe Island Sand.................................... 50 Figure 18 Primary and main drainage-imbibition curves of Budget bricky's Loam ............................. 51 Figure 19 Comparison between Experimental data and Fredlund & Wilson and Arya & Paris PedoTransfer Function on Bribe Island Sand ............................................................................. 52 Figure 20 Comparison between Experimental data and Fredlund & Wilson and Arya & Paris PedoTransfer Function on Budget bricky's Loam....................................................................... 53 Figure 21 Saturated hydraulic conductivity against porosity (a) Bribe Island Sand (b) Budget brick Loam ................................................................................................................................... 55 Figure 22 Hydraulic conductivity of unsaturated Bibe Island Sand (a) Suction basis (b) Water content basis .................................................................................................................................... 56 Figure 23 Hydraulic conductivity of unsaturated Budget bricky's Loam (a) Suction basis (b) Water content basis ........................................................................................................................ 56 Figure 24 Pore size distribution and Grain size distribution of virtual grains in 2D LBM simulation . 58 Figure 25 Information of virtual soil grain and pore size in LBM simulation (a) output image of particle size measurement (b) output image of pore size measurement (c) Histogram of Grain size distribution (d) Histogram of Pore size distribution ........................................................... 59 Figure 26 Lattice surface tension calibration: Capillary pressure vs Radius of single bubble ............. 59
V
Figure 27 (a) Static Capillary Pressure-Saturation Curve (SWRC) (b) Dynamic Capillary PressureSaturation-Time data with FX model fitting curves in 3D (c) 3D contour plot of evolution of Capillary Pressure-Saturation-Time step (d) Dynamic Capillary Pressure-Saturation with FX model fitting curves in 2D (e) Dynamic Capillary Pressure-Saturation fitting curves in 2D........................................................................................................................................ 63 Figure 28 (a) 3D contour plot of Saturation-Interfacial area-Time step (b) a projection of 3D contour plot of Saturation-Interfacial Area-Time step on Saturation-Interfacial area 2D plane (c) 3D contour plot of Capillary pressure-Interfacial Area-Time step (d) a projection of 3D contour plot of Capillary pressure-Interfacial area-Time on 2D Capillary pressure-Interfacial area plane (e) 3D contour plot of Capillary pressure-Saturation-Interfacial area with contour colorbar representing Time step (d) a projection of 3D contour plot of Capillary pressureSaturation-Interfacial area on 2D Capillary pressure-Saturation plane with contour colorbar marking the Time step ........................................................................................................ 67 Figure 29 A flow chart of experimental methods on studying dynamic two phase flow in granular porous media ................................................................................................................................... 70 Figure 30 Research schedule Gant Chart .............................................................................................. 71 Figure 31 (a) capillary suction; (b) ink-bottle effect (Jotisankasa 2005) .............................................. 84 Figure 32 A sketch of Soil Water Retention Curve with first scanning curves inside for a uniform graded sandy soil ............................................................................................................................ 85 Figure 33 Sketch of 1-D vertical Green-Ampt infiltration model (Kale and Sahoo 2011) ................... 92 Figure 34 3-D Mohr-Coulomb failure envelope surface for unsaturated soils (Fredlund and Rahardjo 1993) ................................................................................................................................... 94 Figure 35 An example of 3-D unsaturated soil constitutive surface for studying soil deformation (Zhang and Li 2010) ........................................................................................................................ 95 Figure 36 Standard Axis Translation Techniques: (a) A sketch of Hanging Column Method; (b) the sketch of Pressure Chamber Method (Vanapalli, Nicotera et al. 2008) .............................. 97 Figure 37 An example of tensiometer (a) UMS T5 Micro-Tensiometer (b) A sketch of T5 structure (UMS 2009) ........................................................................................................................ 99 Figure 38 kunsat measurement by ATT (a) flexible wall ATT (ASTM D7664-10 2010) (b) Constant head method by ATT(Richards 1931) ....................................................................................... 100
VI
IV. Table Table 1 Specification of T5 micro-tensiometer .................................................................................... 29 Table 2 Formularisation of direct and inverse Richards' model (Simunek, Huang et al. 2013) ........... 32 Table 3 Equations for upscaling microscale variables .......................................................................... 36 Table 4 The construction of D2Q9 Shan-Chen LBM simulation engine (Sukop 2006) ....................... 38 Table 5 Lattice-scaled parameter values used in numerical experiment ............................................... 40 Table 6 Soil Classification and Sieving analysis .................................................................................. 44 Table 7 Density of Sandy soil and density control parameters ............................................................. 45 Table 8 Density control in a pre-test of 1D sand column ..................................................................... 47 Table 9 Density control of sand specimen for Hanging Column Method ............................................ 47 Table 10 Fitting parameters for Bribe Island Grey Sand ...................................................................... 49 Table 11 Fitting parameters for Budget bricky's Sandy Loam ............................................................. 49 Table 12 Parameters of PTF compared to SWRF for Bribe Island Sand.............................................. 52 Table 13 Parameters of PTF compared to SWRF for Budget bricky's Loam ....................................... 53 Table 14 Hydraulic conductivity and packing condition for two sandy soil ........................................ 53 Table 15 Soil Water Retention Functions ............................................................................................. 86 Table 16 Richards' model forms on different state variable basis (Celia, Bouloutas et al. 1990)......... 89 Table 17 Hydraulic Conductivity Functions ......................................................................................... 90 Table 18 Bishop effective stress parameter equations .......................................................................... 94 Table 19 A simplified system of advanced theory for dynamic multiphase flow in porous media (Hassanizadeh and Gray 1993) ......................................................................................... 107 Table 20 The list of dimensionless voltage-water pressure calibration for 21 T5 tensiometers ......... 114 Table 21 Systematic error of T5 sensors in reading by INFIELD7 .................................................... 117 Table 22 Bias and Standard deviation of logging pressure ................................................................. 118 Table 24 (a) DT85G logging code for instantaneously logging 21 T5 sensors (b) voltage-pressure transformation Matlab script (c) Logging programme activating command lines ............ 119 Table 25 CR1000 Logging code for the case of logging 4 STDR on two terminal on recording waveforms for analysing moisture and eletrical conductivity (Note has been clearly attached by each line of code) ......................................................................................................... 122 Table 26 Matlab code for measuring GSD and PSD of Virtural Beads in LBM domain ................... 127 Table 27 State variables changing with time for primary drainage simulation using 2D LBM ......... 146 Table 28 State variables changing with time for primary imbibition simulation using 2D LBM ...... 148 Table 29 Visualization of 2D LBM simulation of drainage ............................................................... 150 Table 30 Visualization of 2D LBM simulation of imbibition............................................................. 151 Table 31 Fitting performance of Fredlund & Xing SWRC model into dynamic and static SWRC dataset generated by SCLBM .......................................................................................................... 153 Table 32 First order time derivative of each macroscale state variables for drainage and imbibition 157
1
1.
Introduction
1.1
Background
Multiphase flow in porous media is a complex engineering problem. It covers a broad range of disciplines, including agriculture, hydrogeology, geotechnical engineering, petroleum engineering, biological engineering and medical engineering. All of these disciplines require a sound understanding of multiphase motion in porous media, such as air-water moving into the soil, gas-oil-brine moving in the fractured rock, blood flowing into tissue, solution flowing into porous synthetics and ink dipping on the paper. Every phenomenon of fluid anti-gravitational motion in porous media is dominated by the capillary effect on the interface between two immiscible fluids. From the geotechnical engineering perspective, most of the concerns are given in two cases. The first is to study air-water motion in unsaturated soil. In this case, air is the non-wetting phase and water is wetting phase due to hydrophilic character of the natural soil. The air-water interface does not only govern the two-phase flow in soil matrix but also impacts soil stiffness and deformability. The second instance is a system consisting of gas, oil and brine in the oil/gas reservoir. A large amount of crude and natural gas are stored in natural geological formation under overpressure. With the ongoing oil production, overpressure decrease leads to a reduction of oil production. To enhance oil production, gas injection or water injection are often utilised by petroleum engineers. Due to the incapability of measurement of multiphase fluids seepage in the oil reservoir, the theory of multiphase flow in porous media offer engineers a solution to predict the oil recovery efficiency.
1.2
Problem Statement
To a complex immiscible multiphase flow in porous media in large scale, there is always a need of applicable constitutive relationships that governs the main characteristics of multiphase behaviour rather than looking into every trivial detail of multiphase physics. This constitutive relationship has to be established because the continuity assumption is used in every macroscale model. In another way, the macroscale model based on continuity assumption is highly demanded by the constitutive relationship for each representative elementary volume in the modelling domain if the boundary condition and initial condition are determined. Therefore, the key for solving the complex multiphase flow in porous media is to explore and establish a simple and effective constitutive relationship. The first pioneer of studying multiphase flow in porous media is the famous soil physicist, Edgar Buckingham, who invented the very useful fluid mechanics tool-Buckingham π-theory in dimensional analysis. Narasimhan (2005) summarised and appreciated his work in laying the foundation of moisture retention constitutive relationship and capillary flow conductivity formularization. Although his work did not involve scientific measurement of soil suction due to constraining of measuring method in that period, his finding from the experimental result and scientific intuition has been significantly influencing the orientation of multiphase flow study. In general, during the period between the end of the 19th centre and beginning of 20th centre, inspired by studies in heat transfer (Fourier’s Law) and electrical conduction, soil physicists commenced to construct seepage theory for both saturated soil and partially saturated soil based on the diffusion theory. This diffusion theory was also applied to study solute transportation in both surface water and groundwater seepage. Therefore, the state variables and constant parameters selected for groundwater modelling are somehow similar with the diffusion theory’s state variables and constants in both heat transfer and electricity transfer, such as water head to an electrical potential, hydraulic conductivity to electrical conductivity (reverse of electrical resistivity) and water storage capacity to electrical capacity. Since the entire single phase flow and multiphase flow theory is based on diffusion form, two challenges are left to both engineers and applied mathematicians. The first one is the determination of diffusion coefficient, which was later decomposed by chain rule of calculus into different constants including permeability, specific storage and capillary specific storage. To reveal the nature of the assumed constants, diligent engineering experiment are required. Under the single phase flow seepage theory, with incompressible fluid and rigid soil matrix assumption, this diffusion coefficient can be treated as
2
a constant. However, as for the multiphase flow seepage in porous media, this coefficient is a nonconstant parameter, which depends on phase energy or phase saturation. The same circumstance also happens for the single phase flow if the porous media deformation or fluid compressibility are accounted. This is usually identified as the hydro-mechanical coupling model for fully saturated porous media. The second challenge is to derive the solution of diffusion partial differential equation (PDE) under the circumstance that the diffusion coefficient is a constant or a parameter continually changing with the state of phase. With the increasing complexity of the constitutive relationship to determine the diffusion coefficient, the original diffusion PDE becomes non-linear. The analytical solution can only be derived based on simplifying constitutive relationships into exponential form. Provided a complex and flexible constitutive relationships, the non-linear PDE can only be solved using numerical methods. Nonlinear PDE is usually solved by computational mechanical engineers and applied mathematicians. Under the state of the art of the multiphase flow constitutive relationship formulation, different solutions have been given and are reasonable. After checking through the entire map of macro-scale multiphase flow in porous media, the uncertain problem, which might still exist, is in the constitutive relationship, which can only be given by carefully designed experimental set-up and rigorous mathematical description. Therefore, to reveal the dynamic multiphase flow in porous media, the dynamic capillary seepage experiment is the most helpful tool that truly measures real phase energy and saturation. Owing the first-hand experimental result under various hydraulic loading history is the key to validating previous constitutive relationships, verifying forward modelling and even facilitating fitting parameter approximation using inverse modelling. From a microscale physical perspective, when the two-phase displacement is initiated, there is no existence of equality between static capillary pressure and dynamic capillary pressure due to the hydrodynamic process of meniscus advancing or receding. Two phase displacement in a single tube from one state to next is a classical varying acceleration problem and should follow Newton Second Law. Nevertheless, such process is just simply expressed by a viscous-dominated flow regime where pressure gradient balances with a velocity determined viscous friction. After averaging to macroscale, it is finally given by Darcy’s Law. Even neglecting the acceleration terms in a momentum balance, due to the meniscus deformation during a dynamic process, there might be more force terms in the total net force. However, there is no clear analysis of flow regimes dominated by different force terms. Compared to single phase flow theory, built upon experimental observation and validation, the conventional multiphase flow theory is a mere mathematical extension of single phase flow without careful experimental investigation. Therefore, with the development of measuring technique, the more neglected physical phenomenon has been revealed in order to improve the theory.
1.3
Research Questions
Based on previous problem statements, the method to investigate the problems in dynamic multiphase flow in porous media can be summarised into two main streams:
In macroscale, determination of dynamic constitutive relationship from spatial-temporal experimental measurements and inverse modelling of nonlinear diffusion model; In microscale, determination of dynamic constitutive relationship by upscaling the microscale state variables using image processing technique or microscale numerical simulation.
For macroscale study, the experiment should be conducted in a way to replicate the state of natural porous media, which is naturally heterogeneous but can be assumed to be homogeneous in the largescale experimental domain. The state variables, such as capillary pressure, phase saturation and boundary flux, are ought to be measured as an average value in the local section. These state variables can not only be used to plot constitutive relationship locally and temporally but also provide output for a dynamic model to determine this relationship inversely. For microscale study, many approaches are available. Microscale study needs visualisation technique to determine the saturation configuration and advancing wetting front in each pore. In this way, the difference between equilibrium and dynamic can be truly observed instead of theoretical assumptions. The state variables extracted from visualised phenomenon should have more reality than from sensors in non-observed porous media column.
3
Numerical analysis, such as Pore Network Model (PNM), Multiphase Lattice Boltzmann Model (LBM) and Direct Numerical Simulation (DNS), offers another opportunity to visualize dynamic multiphase behaviour in porous media. After careful model calibration for LBM and physical parameter selection for either PNM or DNS, these well-developed numerical models become the second visualisation technique. Compared to physical experiment, it requires less physical effort to implement. Upscaling can be accomplished by weight averaging microscale state variables. In this experimental study, the macroscale experiment, microscale experiment and the numerical experiment will be conducted to explore the comprehensive domain of constitutive relationship of dynamic multiphase flow in porous media. The research aims at answering the following research questions; 1. What is the entire plot of capillary pressure-phase saturation if the temporal effect is considered? What is the dynamic hysteresis (under both dynamic capillary and non-wetting phase trapping effect)? 2. How are the constitutive relationships approximated by inverse modelling located in the entire plot of capillary pressure-phase saturation domain? How does the modeller select parameters from this domain? 3. How significant is the dynamic capillary effect impact on the macroscale model? Is there any extreme transient condition that encloses this domain as an attraction of the process? 4. What is the macroscale manifestation of microscale dynamic capillary flow in porous media? What is the advantage and shortage of advanced theory of dynamic multiphase flow in porous media? 5. For engineering application, which degree of dynamics needs too be taken into account and what can be neglected under certain circumstances?
1.4
Research Objectives and Contribution
The research aim of this experimental study is to enhance the understanding of dynamic multiphase flow in porous media using available experimental techniques. In this project, following issues will be addressed:
To investigate limitation of conventional multiphase flow testing methods by comparing experimental data to Instantaneous Profile Method (IPM); To investigate the dynamic effect in capillary pressure-saturation (Pc-S) relationship using IPM to provide an overview of dynamic Pc-S domain including primary, main and several hysteresis curves under transient process; To investigate the advanced theory of two-phase flow in porous media using both IPM and quasi-two-dimensional visualization; To calibrate and validate the SCLBM using quasi-two-dimensional experiment; To investigate the transient process of multiphase flow using SCLBM in order to characterise the suitable constitutive relationship on the macroscale; Based on the experimental result, to determine the most reliable theoretical method for dynamic multiphase flow in porous media.
The outcome of this experimental study is expected to contribute to following engineering problems:
Macroscale Model prediction of groundwater dynamics Macroscale Model prediction of unsaturated zone solute transport; Macroscale Model prediction of gas, crude and brine displacement in oil/gas reservoir; Hydromechanical coupling model prediction on slope stability, earth pressure and consolidation of unsaturated soil under flooding or intensive rainfall;
Above mentioned engineering problems are all strongly correlated to dynamic multiphase flow in porous media. Capillary fringe and fingers own a capping effect on groundwater table dynamics. Conventional ground water table prediction using Boussinesq equation neglects capping effect from the capillary fringe. With a better understanding of capillary zone dynamics, there will be a better prediction of groundwater table dynamics by accounting surcharge and discharge from the capillary zone. For
4
geotechnical engineering, the shear strength, stiffness and compressibility of unsaturated soil are highly governed by the suction profile vertically along the earth material. The better the suction profile predicted from macroscale model, the more accurate can the action of soil suction on unsaturated soil matrix be determined for safety or economic analysis on earth structure design. Also, a better understanding of dynamic multiphase flow in porous reservoir contributes to a more accurate prediction of CO2 sequestration, oil recovery efficiency and estimation of gas/oil well production. The coefficient of permeability of crude or natural gas is affected by phase trapping mechanism, and such irreducible phase trapping will be highly impacted by the dynamic process of two-phase displacement. Hence, the outcome of this experimental study will facilitate a better understand of any engineering problems involving multiphase flow seepage in porous media.
5
2.
Literature Review
2.1
Unsaturated soil mechanics overview
The overview of unsaturated soil mechanics is reviewed in details in Appendix A1.1 Overview of unsaturated soil mechanics. The entire framework of unsaturated soil includes the core-Soil Water Retention Curve, the application of SWRC on steady-state or transient multiphase flow in unsaturated soil with the addition of hydraulic conductivity function, and the suction induced effective stress enhancement using either classical Bishop effective stress or independent stress variable and suction. Here the important understanding of literature review will be summarised instead of listing knowledge of unsaturated soil mechanics in details. Unsaturated soil effective stress condition is totally opposite to the saturated soil effective stress because the pore pressure between each pair of soil grains instead of neutralising skeleton stress actually pulls them together. To account the cohesion increasing due to such effect, the effective stress has to be reconstructed as a total stress including net stress and suction stress (D. G. Fredlund & Rahardjo, 1993; Lu & Likos, 2004). The stress-deformation constitutive relationship and shear strength of unsaturated soil could be finally quantified. In this case, due to the environmental effect on moisture distribution and negative pore pressure, the effective stress in the unsaturated soil becomes a stress state variable varying with the hydrodynamic process in a multiphase porous structure. Conventional soil mechanics only solve the soil mechanical problems for saturated soil where pore pressure can be simply taken as the hydrostatic pressure even for a dynamic groundwater flow diffusion equation. There is no need of studying environmental impacts on soil stress-strain-failure in exception of dynamic loading like seismic activities. As for unsaturated zone, the real estimation of unsaturated soil consolidation, earth pressure calculation for retaining structure design, bearing capacity for foundation design and wetting induced slope stability analysis are highly dependent on dynamic air-water seepage. Moreover, studying dynamic gas-oil-brine flow in oil reservoir is also another case like the two-phase flow in unsaturated soil with different fluid-fluid viscosity ratio. It does not only benefit the understanding of gas or water injection on oil recovery but also provide a platform for prediction of contamination transport and soil remediation. Therefore, there is a clear logic that mechanical behaviour of unsaturated soil is governed by dynamic multiphase flow and the prediction of this transient process is determined by the fitting parameters in the governing equation. To understand the dynamic multiphase seepage, everything should be initiated from the core (Buckingham, 1914) and the origin who firstly formularised the theoretical framework (L. A. Richards, 1931). Soil suction is also named negative pressure or capillary pressure in the unsaturated zone. The concept is simply assuming the atmosphere pressure as zero reference pressure so the pressure difference between air and water can be transformed into a negative water pressure. It can also be transferred into the suction head as a water head rising above the groundwater table. Due to soil pore throat heterogeneous, capillary pressure encountered a higher raising impedance from a smaller pore to a larger. This is the famous capillary physical phenomenon-ink bottle effect. Originally, with a highly uniform pore size network, once the capillary breakthrough pressure achieved, the wetting fluid will be immediately drained out. There will be a monotonic relationship between saturation and capillary pressure, and the suction profile could be seen as an extremely sharp wetting front advancing through pore network. However, in natural porous media, there is no such ideally uniform pore network. Because of a wider pore size distribution, such slope of the monotonic relationship is slightly gradual for homogenous sand and much more gradual for a highly compacted fine soil. The study of air-water flow in compacted clay is beyond the scope of this study because this work stresses on the dynamic process under an assumption of the rigid porous structure. As for deformable soil SWRC, a threedimensional constitutive surface of suction-saturation-void ratio with hysteresis loops between two boundary surfaces should be considered only for static capillary equilibrium (Domenico Gallipoli, 2012; Hu, Chen, Liu, & Zhou, 2013; Tsiampousi, ZDRAVKOVI, & Potts, 2013). Such a framework is
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already complicated for the application. Further adding dynamic SWRC on this framework will not only lead to a more mathematically complex but also a very challenge work on experimental design to separately investigating non-uniqueness of SWRC resulting from either dynamic capillary effect or pore size distribution shifting by deformation. A review of single-modal Soil Water Retention Functions has been given in Appendix 1: A1.1.1 Soil Suction and Soil Water Retention Curve, involving classic SWRC models from Willard Gardner, Israelsen, Edlefsen, and Clyde (1922), Brooks (1964), Van Genuchten (1980) and D. G. Fredlund and Xing (1994). Because the Van Genuchten model and Fredlund and Xing model owns better fitting performance and continuity around Air Entry Value (Bubbling Pressure), they will be applied to fit experimental result in this work. Compared to Van Genuchten model, Fredlund and Xing SWRC function has a correction factor for the high suction range. When experimental result shows such a gradually decreasing slope above the suction of residual saturation, Van Genuchten model gives a poor fitting performance because this model has no saturation reduction above this threshold suction. Although each model can be used for fitting, the selection of each model has to be carefully taken based on the experimental data and soil types. The application of Soil Water Retention Curves on the prediction of two-phase flow seepage can be separated into two parts. First, the famous Richards’ equation required a water storage coefficient. For unsaturated zone, it is called capillary specific storage that is exactly the first-order derivation of SWRC. Second, due to the application of two-phase Darcy on flux gradient in 3D space, a relative fraction effect has to be taken into account. Hence, under each static condition or steady state, there will be a certain pore network filled up with wetting phase or non-wetting phase. A fluid fraction-determined conductivity terms (coefficient of permeability in geotechnical engineering) are simply added into the Darcy law that is in fact only experimentally validated for single phase flow. Therefore, to model a dynamic moisture varying process, the hydraulic conductivity is ideally assumed to reduce with moisture content reduction and suction increases. To provide such relationship, the concept of hydraulic conductivity function was conceived (L. A. Richards, 1931). The hydraulic conductivity of unsaturated soil can be tested by applying a constant head or constant flux on the top boundary of the specimen under a controlled suction. The Darcy law will be applied to both drainage and imbibition paths on different controlled suctions, and the hydraulic conductivity-phase saturation relationship experimentally turns out to collapse into one curve including primary, secondary and hysteresis SWRCs (D. G. Fredlund & Rahardjo, 1993). For sending this relationship into Richards PDE, an analytical form has to be fitted into experimental data points. It is named Hydraulic Conductivity Function (HCF) model. Famous HCF models are reviewed in Appendix 1: A1.1.2 Steady state and unsteady state multiphase flow theory. For compacted homogenous non-swelling soil, Mualem (1976) statistical model is most accepted in Richards PDE solvers. Due to well-developed inverse modelling Richard equation in HYDRUS1D, van Genucthen SWRC combined with Mualem HCF will be applied to fitting and prediction in this study. Richards equation or Buckley and Leverett (1942) equation is a nonlinear diffusion equation, in which the diffusivity is a function of state variables (either suction head or moisture content) in the governing equation. It can be rearranged into three forms: suction head based form, water content based form and mixing form having both variables. Equations are listed in Appendix1: A1.1.2 Steady state and unsteady state multiphase flow theory. Many numerical and analytical studies on solving this nonlinear diffusion equation with different types of SWRC and HCF has been given (Celia, Bouloutas, & Zarba, 1990; Tracy, 2006). However, as a review of dynamic capillary effect from S. M. Hassanizadeh, Celia, and Dahle (2002), there is no equilibrium maintained in a dynamic process, and the original concept built upon static or steady state cannot actually hold during a transient process. The misusing concept into modelling dynamic process caused significant disagreement between numerical simulation and experimental result (Calvo, Paterson, Chertcoff, Rosen, & Hulin, 1991; Das & Mirzaei, 2012; S. M. Hassanizadeh et al., 2002; Karadimitriou, Hassanizadeh, Joekar‐Niasar, & Kleingeld, 2014; O'Carroll, Phelan, & Abriola, 2005; Topp, Klute, & Peters, 1967; Yang, Krasowska, Priest, Popescu, & Ralston,
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2011). Therefore, in order to investigate the fact of transient flow, conventional experiments of unsaturated soil hydraulic conductivity and SWRC measurement is questionable, because both the concept of theory and experimental techniques are completely design for determination of SWRC and HCF only under a quasi-steady-state condition. Before the invention of Tensiometer (Willard Gardner et al., 1922; Klute & Gardner, 1962), there is no instantaneous measurement of suction combined with moisture profile measurement to check whether the so-called constitutive relationship has no significant temporal effect. In this experimental study, the entire experimental design aims to the investigation of transient multiphase flow in unsaturated sand and transparent artificial pore network.
2.2
Advantages and disadvantages of unsaturated soil experiment
Unsaturated soil hydraulic testing method mainly can be separated into two categories: quasi-steady state test and unsteady state test. As for the quasi-steady state test, suction is applied using standard Axis Translation Technique (ATT) including Hanging Column Method, Pressure Chamber Method (ASTM D6836-02, 2003). The advantage of ATT is that the suction can be easily controlled by lowering the wetting phase pressure under the specimen or increasing non-wetting phase pressure on top of the specimen. With a High Air Entry (HAE) value ceramic porous disk located under the specimen, the wetting phase disconnection and non-wetting phase breaking through specimen can be avoided. A good contact between HAE disk and the specimen is therefore guaranteed. In fact, the maximum suction controlled in ATT system is the Air Entry Value (AEV) of the HAE disk. To generate such a small size distribution of pore structure, corresponding permeability is also inevitably reduced. Therefore, ATT technique is suitable for determination of SWRC under static condition. A review of ATT application in measurement of SWRC and HCF is given in Appendix1: Review of unsaturated soil experiment. The limitation of ATT technique and usage of HAE disk is not firstly announced in this work. In fact, it has been confirmed by not only the experimental review studies (Lu & Likos, 2004; S. K. Vanapalli, Nicotera, & Sharma, 2008), but also the ASTM testing standard (ASTM D7664-10, 2010). Other quasisteady testing technique for SWRC Includes Contact and non-Contact filter paper method, Centrifuge Method and evaporation method. Filter paper method can accurately measure the suction, but it takes more than a week to reach the equilibrium between ambient environmental and filter paper itself. Even this method allows a larger measurement range from 0 to 106 kPa, it is impossible to apply a consistently varying pressure boundary condition using this method. Centrifuge method is used to generate different centripetal accelerations in order to apply different body force on invading fluid. To avoid invading fluid breaking through specimen, HAE material has to be added on both sides of the specimen. The suction range applied by centrifuge is even smaller than the pressure chamber method. It is a steadystate method, but flow rate dependent cannot be avoided under large centrifuge force (Schembre & Kovscek, 2006). On the other hand, unsteady state testing method includes multistep outflow method (WR Gardner, 1956), infiltration method (Bruce & Klute, 1956) and Instantaneous Profile Method (IPM)(S. J. Richards & Weeks, 1953). Multistep outflow method and horizontal infiltration method do not directly measure the hydraulic conductivity of the unsaturated soil. Compared to steady state method that hydraulic conductivity is calculated using Darcy law and each conductivity is matched to a mean suction, unsteady state test actually fit one-dimensional analytical solution of linear diffusion equation into the experimental data. By changing the state of suction within a small increment but large enough for flux measurement, the volume change of pore water expelled for applied suction increment can be recorded with time. One dimensional analytical solution is fitted into expelled water volume (water content variation) with time for a transient process to determine the diffusivity. Diffusivity timed by capillary specific storage (slope of SWRC) indirectly gives the hydraulic conductivity. The calculation for horizontal infiltration shares the same concept-inversely calculating diffusivity but is in need of different recording variables. Instead of fitting 1D analytical solution of linear diffusion equation into experimental data, this method transforms 1D nonlinear diffusion equation into a diffusivity function of both Boltzmann variable and water content using Boltzmann transformation (Philip, 1957). By recording wetting front advancing distance with time and water content, the water content changing
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with Boltzmann variables can be plotted in order to calculate diffusivity function for a transient process. For conducting these type of unsteady state test, many assumptions have to be made to satisfy the diffusion theory (Lu & Likos, 2006): (1) (2) (3) (4) (5) (6)
Each suction increase interval must be small enough, so unsaturated soil hydraulic conductivity can be assumed to be a constant in this interval, which requires very careful suction control; The relation between soil suction and water content is linear, while in fact it is not only nonlinear but also hysteresis; HAE disk does not cause any hydraulic resistivity, but it is actually a large hydraulic resistance, especially for high permeable porous media; Flow is just one dimension; Gravity effect can be ignored; The testing specimen is homogeneous and non-deformable.
Therefore, it is not hard to find that transient flow experiment based on ATT is just a method for approximation. Masrouri, Bicalho et al. (2009) comment on this method is that it owns simplicity and is good at mass control, but there is few reliable result for comparison with other methods. ASTM D7664-10 (2010) also records this method as one transient method in the standard but clearly mentioned its limitation as well due to the usage of HAE disk. Checking through both steady state and unsteady state experiment, there is a common paradox in sequence of the theory and experiment investigation. Single phase Darcy law and Poisseuille law were both experimentally founded and latter theoretically backed up by averaging Navier-Stokes (NS) equation for multi pore channels or non-slipping boundary on channel wall under an assumption that viscous friction dominates flow regime in order to neglect inertial acceleration terms in NS equation (Jacob, 1972). Also, later a large number of experiments were conducted to characterise the upper and lower limit of viscous-dominated seepage (Jacob, 1972). In contrast, flow motion equation for multiphase flow is just a simple extension of single phase flow by adding relative conductivity term to account the mechanism caused by phase fraction in pore structure. The theoretical framework was found before any proper experimental design on discovery. Compared to single phase flow, the multiphase flow has more forces including static capillary force, the viscous force on the solid surface, gravitational force and velocity dependent viscous drag on interface meniscus. After neglecting inertial term because of low-speed capillary seepage, there are still many velocity dependent forces dominating different flow regimes during a transient process. However, for nonlinear diffusion theory, the diffusivity is just a function of the state variable and can be decomposed into hydraulic conductivity and capillary specific storage. In another clear way, storage coefficient is determined by static SWRC measurement and dynamic two-phase seepage is approximated by assuming negative water pressure only balancing laminar viscous resistant friction. To satisfy the nonlinear diffusion theory, experiment were designed under these six assumptions. After fitting analytical solution of linear diffusion equation into the experimental data for a small transient process to extract diffusivity, inserting these diffusivities corresponding varying state variables back to nonlinear diffusion equation can provide a solution, which highly agrees with experimental data generated in such experimental condition. This could be one of the reasons why there is a difference between field measurement and model simulation using hydraulic parameters determined in a laboratory experiment. In the field, Instantaneous Profile Method is often implemented. Many tensiometer and moisture sensors are inserted into soil strata to capture the dynamic response of suction profile and moisture profile under environmental influences. However, there are many impact factors like temperature, atmosphere pressure variation, intensive rainfall, solute-induced osmotic pressure and evapotranspiration, changing the experimental conditions. Hence, it is hard to control them in order to independently investigate dynamic effect. Therefore, 1D IPM test should be replicated into laboratory scale under well-controlled conditions to purely investigate transient twophase seepage without any other impact factors.
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2.3
Dynamic multiphase flow in porous media
With the development of moisture sensor and suction sensors, unsaturated soil experiment did not only belong to the laboratory, where an assumed zero-dimensional REV scale is measured using ATT. Since the early 1960s, the dynamic effect in SWRC was experimentally confirmed by Topp et al. (1967). This finding strongly questioned the validity of Richard’s model that can be used to simulate dynamic multiphase flow in porous media whereas it uses a constitutive relationship having a dynamic effect. This issue is not only discovered by experimental studies on SWRC but some other studies on the experimental validation of Richard’s Model. S. M. Hassanizadeh et al. (2002) reviewed some early studies of Richard’s equation validation to find that the diffusivity depends on the speed of wetting front. Some literature reviewed from the early study clearly draw the conclusion that diffusion equation cannot be verified for moisture transport, and the application of Darcy’s Law is questionable (D. Nielsen, Biggar, & Davidson, 1962). Also, the relationship between diffusivity and moisture content loses uniqueness for transient flow condition (S. M. Hassanizadeh et al., 2002). The wetting front speed determined diffusivity raised the concern that both SWRC and HCF have a dynamic effect. However, in early studies, the study of dynamic effect merely focus on drainage path. The dynamic effect in drainage SWRC founded by Topp et al. (1967) was also experimentally confirmed from later studies (Elzeftawy & Mansell, 1975; Smiles, Vachaud, & Vauclin, 1971; Stauffer, 1978; Vachaud, Vauclin, & Wakil, 1972; Wana-Etyem, 1982). All of these works used a similar setup but in exception of Wana-Etyem (1982) most of them solely focused on drainage curve. Despite dynamic effect in SWRC, Stauffer (1978) also checked the HCF and found that it also owns dynamic behaviour. Smiles et al. (1971) confirmed that dynamic drainage curves have higher suction than static drainage curve for the same saturation, but they concluded that imbibition curve had no observable dynamic effects. Nevertheless, this was later challenged by Wana-Etyem (1982) that dynamic effects happen for both drainage and imbibition SWRC. S. M. Hassanizadeh et al. (2002) summarised their findings into following points: (1) (2) (3) (4)
The dynamic effect is not significant in fine-textured soil; The higher rate of water content variation, the larger the dynamic effect; The dynamic effect is more significant in coarse textured sand; The dynamic effect in primary drainage curves is more significant than the dynamic effect in main drainage curves.
In the original findings of dynamic effect in SWRC, experimental results were the only evidence for demonstration. Because of lacking understanding of multiphase thermodynamic and hydrodynamic, there was no theoretical mathematical formulation to capture this dynamic response. Most of the works were a completely experimental discovery with the discussion of possible causes. Therefore, to theoretical derive a mathematical equation accounting this dynamic effect, the conventional theory has to be critically reviewed to identify the missing part so that the nonlinear diffusion theory can be further improved for a better simulation of dynamic multiphase flow in porous media. Gray and Hassanizadeh (1991) critically reviewed the conventional theory of multiphase flow dynamics in porous media and found four paradoxes in the theory. It can be briefly summarised into following points:
The concept of negative water pressure, which is equal to zero reference atmosphere pressure minus water pressure, should be a pressure calculated from Equation of State while in SWRC concept, it is a purely related to saturation; Due to the absolute zero pressure, negative water pressure physically cannot be above -1 bar, and before approaching to -1 bar, the water should already become vapour;
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Simply applying hydrostatic pressure concept into suction head might have problems regarding uneven distribution of soil moisture density because of neglecting water molecule attraction, binding and electrical charge over surface of mineral particles; Interface meniscus has a significant dynamic response of geometry during a transient flow while this has not been considered in conventional theory. However, the dynamic meniscus was only observed in single capillary tube experiment with some empirical models in function of Capillary Number (Ca) fitted into data.
Looking into microscale pore channels, the fourth paradoxes have been being continuously investigated by multiphase physicists until now and is still ongoing (Baver et al., 2014; Calvo et al., 1991; Hoffman, 1975, 1983; Kim & Kim, 2012; Sheng & Zhou, 1992; Weitz, Stokes, Ball, & Kushnick, 1987; Yang et al., 2011; Zhmud, Tiberg, & Hallstensson, 2000). The objective is still on finding the best model in the physical background to precisely predict the wetting front advancing through the single capillary tube. The original foundation of two-phase displacement in a single tube is derived by Lucas (1918) and Washburn (1921) as the famous Lucas-Washburn (LW) equation. It is not only applied for a single tube but also for imbibition in porous media. However, this equation is also an extension of viscousdominated Poiseuille law without consideration of dynamic capillary contact angles or any velocity dependent driving terms. In latter capillary tube raising studies, a large amount of disagreement between experimental data and model were found. Zhmud et al. (2000) reviewed and discussed incapability of Lucas-Washburn (LW) equation due to unphysical assumptions used for equation derivation. . Starting from Newton dynamics equation assisted with capillary, gravity, viscous and turbulent drag terms, Zhmud et al. (2000) gave experimental, analytical and numerical analysis of dynamic capillary flow in single capillary tube and proved that LW equation fails for prediction under short timescales, small viscosity limit and turbulent drag induced meniscus damping oscillations. As for the damping effect, they concluded that it is required for long capillaries while it can be neglected for short capillaries like head zone is a few of radius. Zhmud et al. (2000) gave an explanation of dynamic capillary flow process: at the beginning of capillary flow, wetting phase absorbed into tube is dominated by capillary force which violates LW equation (x~t1/2) instead gives a x~t2 relation; after viscous drag balancing capillary force, flow reach to quasi-steady state obeying the LW equation; finally flow is eased by gravity. Kim and Kim (2012) reviewed recent physic studies on the dynamic capillary rise to find that the power number of advancing time gives different values for different packing beads. They suspected that this is because pores are not fully filled with wetting phase fluids. Also, not only for a short instant timescale, but LW equation also fails to match experimental result for a large enough time (Kim & Kim, 2012). Therefore, except the dynamic interface, beads packing condition, initial moisture content and local heterogeneous also contribute to a macroscale dynamic capillary effect. Simply introducing a model for single capillary tube might not be able to predict the dynamic in a porous media REV. From the engineering perspective, there are two types of macroscale advanced theories for simulation of dynamic multiphase flow in porous media. The first one proposed by G. I. Barenblatt and Vinnichenko (1980) is a concept of water reconfiguration during a transient process from petroleum engineering system. They did not derive the dynamic state variables from any multiphase hydrodynamics or thermodynamics, but merely characterised the difference between actual saturation and aiming saturation using a saturation relaxation term. The other one is a thermodynamics-based theory developed by F.-M. Kalaydjian (1992) and S. M. Hassanizadeh and Gray (1993a). The simplified version of this theory is that there is a relaxation behaviour regarding saturation between instant dynamic capillary pressure and aiming static capillary pressure. For both two theories, a dynamic coefficient has to be experimentally determined to quantify the difference between capillary dynamic and static. A details of the continuity equation, momentum balance equations and dynamic capillary relaxation equation are given in Appendix 1: Advanced Theory of Dynamic Multiphase flow in porous media. In the simplified version, relaxation equation is the only one should be involved in conventional Richards equation (S. M. Hassanizadeh et al., 2002). However, as for the comprehensive version, there
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are more driving terms in Darcian form momentum balance equation to represent the Helmholtz free energy regarding interfacial area gradient and saturation gradient with two additional material coefficients referring to wettability. According to advanced theory from S. M. Hassanizadeh and Gray (1993b), except the mass balance and momentum balance for each phase, there are an additional mass momentum balance equations for the interfacial area. There is no exact mass balance between two fluid and interface, but a production term accounting the rate of exchange of interface. This additional term accounts the interfacial Helmholtz free energy in Gibbs energy which can be transferred to the pure energy of pressure for single-phase flow (S. M. Hassanizadeh & Gray, 1993b). This production term and two additional material coefficients are difficult to be experimentally determined using 1D IPM test because there is a challenge to measure the local and global interfacial area variation. Hence, the 1D IPM tests having been conducted so far is most used to determine the relaxation coefficient in simplified version of advanced theory, validate the new model performance, and study the relationship between this coefficient and other impact factors, such as fluid viscosity effect, permeability effect, pore structure effect, temperature effect, scale effect, wettability effect and heterogeneous effect (Abidoye & Das, 2014; Das & Mirzaei, 2012; Hanspal & Das, 2012; Mirzaei & Das, 2007; O'Carroll, Mumford, Abriola, & Gerhard, 2010; O'Carroll et al., 2005). However, there is few study on the investigation of imbibition and hysteresis of SWRC and HCF in the transient process. Most of them merely focus on multistep and one step drainage and imbibition test to study the dynamic effect or saturation redistribution effect on primary SWRC. Although the dynamic coefficient has been found a constant for high saturation, whether it is unique for different hydraulic loading paths are still questionable. Scheuermann, GalindoTorres, Pedroso, Williams, and Li (2014) and SA Galindo-Torres, Scheuermann, Li, Pedroso, and Williams (2013) might be the first group of researchers paid attention to the transient effect on hysteresis SWRC and hydraulic ratcheting process in unsaturated soil. These work motivated the geotechnical engineering researchers on studying dynamic multiphase flow in porous media without limitation of hydraulic loading paths. There is so far no comprehensive framework demonstrating a complete plot, including dynamic SWRC for both primary, main and hysteresis SWRC and HCF. One of the great experimental studies on dynamic hysteresis behaviour of SWRC and HCF were given by Chen (2006), in which relaxation coefficient and capillary pressure-saturation-interfacial area were experimentally tested using a 1D IPM test. Even the non-uniqueness of relaxation coefficient was experimentally found in this study, primary drainage and main imbibition dynamic curves were only provided. The difference between dynamic hysteresis and static hysteresis SWRC are still missed in experimental studies so far. Additionally, as for data post process on the calculation of dynamic HCF, Chen (2006) simply applied Darcy law and HCF model of Brooks (1964) to determine the relationship between the dynamic coefficient and effective hydraulic conductivity of wetting phase fluid, while both concepts are only physically suitable for steady-state, which was admitted by the author. To still conduct this method, the author just assumed that there is no difference between transient process and steady state on using Darcy law and Brooks HCF model. Similar experimental investigations were also implemented by Mirzaei and Das (2013) and Sakaki, O'Carroll, and Illangasekare (2010). Their experimental result also confirmed the non-uniqueness of the dynamic coefficient. However, as same as Chen (2006), they only investigated one hydraulic loading cycle on primary drying-main wetting. Due to missing other dynamic SWRC, further experimental investigation on dynamic hysteresis SWRC and HCF are still important for a comprehensive understanding multiphase flow dynamic in porous media. Also, a question could be raised if the mathematical equation on saturation relaxation (G. I. Barenblatt & Vinnichenko, 1980) and capillary pressure relaxation (S. M. Hassanizadeh & Gray, 1993a) is universal to experimental measurement and practical to numerically solving advanced theory. The relaxation coefficient and saturation redistribution need to be further experimentally determined in order to either validate the advanced theory or even help to conceive a new theory to rigorously capture the dynamic effect of SWRC and HCF in governing equations. Because 1D IPM test has a natural constraint on measurement of meniscus motion, to investigate nonhysteresis constitutive surface of capillary pressure-saturation-interfacial area and Darcian form flow
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motion equation proposed by S. M. Hassanizadeh and Gray (1993b) (Appendix 1, Table 5, Equation (36), (38)), both physical microscale pore network model associated with image visualization and numerical dynamic pore network model provide a great vision to investigate this new constitutive surface, its uniqueness under transient process and production term of mass exchange rate between phase and interface (Joekar-Niasar, Hassanizadeh, & Leijnse, 2008; Joekar-Niasar, Hassanizadeh, & Dahle, 2010; Joekar Niasar, Hassanizadeh, Pyrak‐Nolte, & Berentsen, 2009; Karadimitriou et al., 2014; Karadimitriou, Joekar-Niasar, Hassanizadeh, Kleingeld, & Pyrak-Nolte, 2012; Karadimitriou et al., 2013; Pyrak‐Nolte, Nolte, Chen, & Giordano, 2008). These series of studies confirmed the uniqueness of new proposed constitutive surface fitted by a parabolic surface when there is no loss of phase connection. When disconnection of each phase happens, such constitutive surface loses the uniqueness (Joekar Niasar et al., 2009; Karadimitriou et al., 2013). The nonuniqueness due to boundary effect was also investigated by S Galindo-Torres, Scheuermann, Pedroso, and Li (2013) using ShanChen Lattice Boltzmann Method (SCLBM) simulation. However, due to the expensive computational effort, 3D virtual beads package might not achieve the statistical stability of void space in an REV scale. Also, there is still a debate on the parabolic shape of this constitutive surface. When the saturation reduces to very low value, the interfacial area might increase to almost the specific surface of solid particles when film water is not neglected, and the relation between saturation and the interfacial area is experimentally determined as monotonic using interfacial tracer detected by a fibre-optical spectrometer (Chen, 2006). However, in physical pore network model with image technique visualization, due to the fact that image analysis does not have sufficiently high resolution to detect film water in several pixels, a reduction of interfacial area is therefore measured for the saturation under 20%-30%. On the other hand for the transient process, Joekar-Niasar et al. (2010), Joekar‐Niasar and Hassanizadeh (2011) confirmed advanced Darcian form flow motion equation and interface flow motion equation in advanced theory by S. M. Hassanizadeh and Gray (1993b) using the dynamic pore network numerical simulation. However, checking through the structure of governing equation in dynamic pore network in Appendix of Joekar-Niasar et al. (2010), the local capillary pressure was calculated based on a pore-scale SWRC concept, which is derived by applying static meniscus geometry from Young-Laplace equation. This raised concerns that how can a dynamic effect be generated from local static capillary pressure-geometry equation. Also, the linear relationship between fitting coefficient in production term and saturation was later challenged by Karadimitriou et al. (2014) who fitted a parabolic relationship into the experimental data given by a physical pore network model. The production term in interface motion equation is still a challenge for physical experiment and worth to be investigated using the physical pore structure in different material intrinsic wettability. The new 3D constitutive surface in dynamic condition was experimentally found varying with Capillary number (velocity dependent), implying that this new constitutive concept only reduces hysteresis on static or pseudo-steady-state condition (Karadimitriou et al., 2014). Nevertheless, the physical pore network used in this study had a phase saturation dependent wettability (Karadimitriou et al., 2014). In the future, a constant wettability material should be used to more rigorously conduct the physical experiment. Except numerical Pore Network Modelling and a physical one, Multiphase Shan-Chen Lattice Boltzmann Method (SCLBM) also offers another chance to visualise dynamic phase redistribution and measure dynamic capillary pressure directly using LBM Equation of State (EOS). Compared to conventional Computational Fluid Dynamics numerical solving Navier-Stokes equation, SCLBM works in a more fundamental level, on which the solution of Boltzmann equation is approximated by applying probability concept of fluid particles on lattice domain (M. Sukop, DT Thorne, Jr., , 2006). Through calculating the density and velocity of particle cloud on each node associating with surrounding lattice node in a relaxation behaviour, fluid flow can be simulated as solving Navier-Stokes equation for single phase flow. Shan and Chen (1993) applied three potential functions to set the pseudo repulsive force between two phase fluids, the pseudo-attractive force between solid and wetting phase fluid, and the pseudo repulsive force between solid and non-wetting phase fluid. In this way, a phase separation and meniscus phenomenon can be generated in LBM domain to simulate bubble flow and
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two-phase displacement in porous media. However, the intensity constant of each pseudo force is not a physical fluid property, but a virtual potential constant. In order to study multiphase hydrodynamics, a case specific experiment has to be carried out to calibrate these three intensity constants. SCLBM was used to validate the new constitutive surface, and experimentally calibrated by Porter, Schaap, and Wildenschild (2009). Nevertheless, the calibration is case specific, but a generic constant used for another case. SA Galindo-Torres et al. (2013) studied hydraulic ratcheting effect using LBM. However, this study stressed on groundwater table dynamics varying with sinusoidal pressure boundary in water head. However, there is no information of dynamic capillary pressure, due to only using a single component multiphase LBM. Until now, there is no investigation of dynamic effect in SWRC and HCF using SCLBM in either 2D or 3D. Implementation of a numerical experiment using a well-calibrating SCLBM will become a more powerful tool in the investigation of dynamic two-phase flow in granular porous media.
2.4
Summary of literature review
In this chapter, the literature review is given to three main aspects: an overview of unsaturated soil mechanics, advantages and disadvantages of unsaturated soil experiment and dynamic effects in soil water retention behaviour. A more comprehensive literature review is given in Appendix 1: Full Version of Literature Review. The conclusion of this review can be summarised into following points:
Soil Water Retention Curve governs both hydraulic and mechanical behaviour of unsaturated soil but its application to practical engineering is constrained by lack of understanding of SWRC impacted by deformation, hydrodynamics effect, thermodynamics effects and other chemical effects; Conventional theory and steady state unsaturated soil hydraulic experiments leads to overlook of dynamic capillary effect or dynamic water reconfiguration and inaccuracy of model prediction; Instant Profile Method is currently macroscale experiment for studying dynamic multiphase flow in non-deformable porous media and each state variable can be continuously measured with time during fast variation of boundary condition; Conventional multiphase flow theory contains several paradoxes against the physics of thermodynamic, and, therefore, advanced theory was developed based on thermodynamic which demonstrates the Helmholtz free energy, determined by interfacial area, should be considered for dynamic conditions; Dynamic SWRC has been quantified by two available types of theory: thermodynamically based theory and fluid redistribution theory, each of which requires more experimental results to validate; Early experimental studies merely focused on the discovery of dynamic behaviour and recent experimental studies investigated dynamic effect associated with other impact factors. So far, there is no a comprehensive diagram of SWRC including static SWRC, dynamic SWRC and dynamic hysteretic SWRC; Physical and numerical pore-scaled model provide another chance to visualise non-equilibrium effect of multiphase flow and some powerful tools, such as Hele-Shaw cell model, transparent soil model, LBM and DNS, are worth to be implemented, so the non-observable state variables like interfacial area can be measured in order to validate advanced theories;
Soil scientist and hydrogeologist developed Soil Water Retention Curve and Hydraulic Conductivity Function with Richards' model. Unsaturated soil mechanics just introduces this knowledge into Soil Mechanics to fill the gap for the unsaturated condition. The application of SWRC on unsaturated soil shear strength and hardening behaviour are still using the static SWRC. Under transient flow condition, such as flooding and intensive rainfall, SWRC will highly deviate from static SWRC. How the dynamic capillary pressure is impacting unsaturated soil mechanical features, and how significant it is are
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somehow never considered in the geotechnical engineering application for unsaturated soil mechanics. Although this study does not provide any vision into the multiphase hydro-mechanical coupling part, it is still an innovative study of multiphase flow seepage in the discipline of environmental, geotechnical engineering. Both Hydrogeology and Petroleum Engineering already own their theories for dynamic multiphase flow in porous media, while the dynamic effect in SWRC is still on the state of experimental discovery for geotechnical engineering. Through laboratory-scale physical experiment and numerical experiment, it is expected that this study will contribute to a better understanding of dynamic multiphase flow in porous media for geotechnical engineering.
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3.
Methodology
In this chapter, the method of solving the series of research gaps, mentioned in the research objective of introduction and the summary of literature review, will be given in sequence of standard geotechnical experiment for unsaturated soil, a proposed Instantaneous Profile Method (IPM), a proposed twodimensional transparent soil physical model and numerical experiment using Shan-Chen Lattice Boltzmann simulation.
3.1
Standard geotechnical experiment for unsaturated soil
3.1.1 Material selection As previous literature review mentioned that pore matrix and transient effect both affect the uniqueness of SWRC, it is better to study each research question independently. Otherwise, it will be very difficult to identify which one is causing non-uniqueness of SWRC. Therefore, non-deformable soil should be selected as the sample for an experimental study on transient effect. In this study, three types of sand and one silty sized silica powder are selected. Sandy soil includes grey sand collected from Bribie Island, Budget Bricky’s Loam collected from Centenary Landscaping Supplies in Jacob’s Well, 30/60 sand (SAN745-3LA) and silica powder (SIL220-3LA) supplied by Sibelco Australia Limited. One dimensional IPM will be firstly conducted to grey sand and loam. In case of shortage of grey sand, the commercial 30/60 sand is used as a backup, and it has similar Grain Size Distribution (GSD) with grey sand. Loam is selected as a finer sandy soil used for comparison with coarser grey sand. For the purpose of producing homogenous well-graded soil sample, the silt fraction is a critical part for consistency of grain size lower than 75 μm. When the first IPM successfully implemented for two sandy soil, a mixing of sand and silt might be considered for future investigation of transient effect in capillary flow on the well-graded sample. However, this is only the original plan for this IPM test. Due to the deformable behaviour of compacted silty soil (Jotisankasa, 2005), mixing sample might not guarantee insignificant deformation. Hence, this will not be a compulsory experiment in this study if the schedule is insufficient and deformation significantly influences the independence of studying transient effect. On the other hand, the potential types of transparent soil for the two-dimensional model are separately glass beads, silica gel and fused quartz. As for transparent soil application in geotechnical research, Iskander (2010) summarised three families: transparent silica powder, silica gels and Aquabeads. Here, Aquabeads is excluded out of our consideration. Even Aquabeads have similar hydraulic properties with fine sand and silty clay, and its refractive index perfectly matching with that of water, its shear stress is in the range for super soft natural clay, which makes them easily crushed and deformed during the hydraulic testing (Tabe, 2009). The disadvantage of using amorphous silica’s powder is that its consolidation behaviour is similar to natural clay with significant secondary consolidation (Iskander, 2010). Therefore, these easily deformed options will not be used. In contrast, transparent glass beads, silica gel and fused quartz have the high yield strength that promises less deformation on beads package, subsequently maintaining the porosity and pore matrix constancy during a hydraulic test. Iskander (2010) tested the compressibility of silica gel. Their results show that its non-linear stress-strain behaviour is like most sands, and its hydraulic conductivity is in the range for sand (Iskander, 2010). Fused quarts was used by Oldroyd (2011) in an experimental investigation of the capillary barrier as a transparent soil agent. Although the compressibility of fused quartz was not reported in this thesis, due to his success of experiments, fused quarts might still be stiff particles that can produce rigid pore matrix after the certain effort of compaction. The pore fluids used for IPM test are natural sources: water for wetting phase and ambient air for nonwetting phase. As for transparent soil model, the refractive index matching has to be considered so that the interfacial length (interfacial area in 2D) can be captured by a camera without any blurry. As the general knowledge that water refractive index is 1.333, the refractive index of stiff transparent soil cannot perfectly match with the water. The pore fluids is ought to have a similar refractive index with
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such soil, ranging from 1.41 to 1.46 (Iskander, 2010). These pore fluids are usually produced by mixing two mineral oils. The Portable Refractometer should be used to measure this index until the mixing ratio of two minerals allows this index perfectly matched between transparent soil and produced fluids. So far, the experimental preparation is still on IPM test. The index matching pore fluids will be generated in regard of which type of transparent soil finally decided for the 2D model.
3.1.2 Sieving analysis and Specimen density control Two testing methods usually derive the Grain Size Distribution of a sample: Sieving analysis for soil particle size above 75 μm and hydrometer test for particle size lower than it. In this study, most of the sample are sandy soil, so the sieving analysis is given to each of them by ASTM D422-63 (2007). The sieve arrangement is shown in Figure 1 with a photo of series of sieves setting on a mechanical shaker. Each sample was prepared by oven dried method in the case of aggregate. Silty Silica powder is the only sample in need of hydrometer test. This is completed by following ASTM D422-63 (2007).
No. 8 (2.36 mm) No. 16 (1.18 mm) No. 30 (600 μm) No. 50 (300 μm) No. 100 (150 μm) No. 200 (75 μm) Bottom pan
Figure 1 Setup of sieve analysis and sieve arrangement specification
The density control of sandy soil is usually quantified by relative density. The maximum dry density and minimum dry density are determined following AS 1289.5.5.1 (1998) using a mechanical shaking table for determination of maximum dry density (ρd,max) and a funnel gently pouring into mould for minimum dry density (ρd,min). The relative density can be calculated by
Dr
d d ,min emax e d ,max (1) emax emin d ,min d ,max d ,min
where Dr = relative density (dimensionless); emax = maximum void ratio; emin = minimum void ratio; e = void ratio of prepared sample, ρd = dry density (g/cm3). The fundamental equation for calculation of void ratio from experimentally measured dry density is given as
e 1
wGs (2) d
where ρw = density of water (usually 1g/cm3); Gs = specific unit weight of sand (sand particle density, g/cm3), ρd = dry density of specimen (g/cm3). After oven drying sample and cooling down to ambient temperature without absorbing ambient moisture, the mass of soil particles (Ms) are measured by scale and the specific unit weight is measured using a Gas Pycnometer by following ASTM D5550-06 (2006).
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In this experiment, the volume of oven dried sand particles (Vs) can be measured by Gas Pycnometer. Then the specific unit weight can be given by
Gs
Ms (3) Vs w
A series of pictures of density control experiment are shown in Figure 2 (a-c). Because the sample preparation of sand is different depending on testing equipment. There is no guarantee that there will be exactly same relative density for specimens prepared in each test. As long as the relative deviation of porosity is within an acceptable range, we accept this packing condition as its mean dry density or mean void ratio.
(a)
(b)
(c)
Figure 2 Experiment for density control (a) Gas Pycnometer for Specific unit weight (b) Mechanical shaking table for maximum dry density (c) Funnel pouring for minimum dry density
Also to the standard tests for studying density variation of sand, a non-standard test is given to both bribe island sand and budget brick’s loam. Each sand was filled into an acrylic tube that has the same diameter with soil column used for IPM test at 14 cm (Figure 3). The mass of each layer of sand was firstly measured, and is then filled into the tube. The elevation of each layer of sand was marked by a small ring of the white silica powder to identify the volume. A 25 kg surcharge is added upon seven layers of sand. With simultaneously compaction by taping on the tube, the density for each layer can be finally determined. A density profile, therefore, can be plotted along the vertical axis to study the density distribution of sample preparation. Although the vertical length of this experiment is only 45 cm that is much smaller than 245 cm tube used in IPM test, it still provide the first vision to look into probable packing condition in a real IPM test. Figure 11 shows an example of this density profile test.
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Figure 3 Pre-test for checking sample packing condition using 1D soil column
3.1.3 Hydraulic conductivity test by Constant Head Method Saturated Hydraulic Conductivity of sand is measured using Constant Head Method. Two standards ASTM D2434-68 (2006) and AS 1289.6.7.1 (2001) are referred for this test. However, the mould used for the test does not comply with each standard. Therefore, sample preparation procedure is, therefore, different with standard sample preparation. According to ASTM D2434-68 (2006) and AS 1289.6.7.1 (2001), the specimen should be compacted by either standard compaction or mechanical shaking to achieve aiming density ratio and then compacted specimen can be saturated by vacuum. The mould used in this test only allows compaction but does not have an isolated cell for vacuum. To ensure the specimen saturation, the sample is only able to be filled into the mould with a water table above the top of the specimen. Until the filled sample goes above the mould and is contained in the sleeve, water is drained out a little to consolidate the specimen. Due to the full saturation can be maintained for a fine sand package under smaller suction (0-20cm), there should be no worry about drainage induced air invasion into the specimen. After taking out the sleeve and scratching extra sand by a straightedge, the mass and volume of the prepared specimen can be determined to calculate the dry density. This packing condition calculation can be either done before the test by measuring dry sample adding into a mould or afterwards using oven drying specimen in mould. The only shortage is that the density cannot be previously controlled while calculated after consolidation by draining water out of sample in mould. Moreover, because this device uses a surcharge loading on top of the specimen rather than a fluid sealed cap with a porous disk on top, internal erosion will occur under the slightly larger hydraulic gradient. To avoid losing fine particles and porosity increasing during test, a gravel filter as thick as half of sleeve length is added upon the prepared specimen. This gravel filter is assumed no influence on specimen hydraulic conductivity because some pre-test shows that the hydraulic gradient does not change by adding such small layer of gravel filter. A sketch of hydraulic conductivity test is shown in Figure 4. 1-D Darcy Law gives the calculation of hydraulic conductivity of saturated sand
kT
Q h (4) At L
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where kT = the saturated hydraulic conductivity at temperature T (cm/s), Q = volume of water seepage through specimen in period t (ml), A = area of cross-section of specimen (cm2), t = period for bottom flux collection (s), Δh = hydraulic head difference (cm), ΔL = specimen thickness (cm), i = Δh/ΔL hydraulic gradient. Two standards recommend using hydraulic conductivity at 20⁰C for report, so a temperature calibration for dynamic viscosity is required by
k20C kT
T (5) 20C
where k20⁰C = hydraulic conductivity at 20⁰C (cm/s), μT = dynamic viscosity of water at temperature T (Pa·s or kg/m·s), μ20⁰C = dynamic viscosity of water at 20⁰C (Pa·s or kg/m·s). If the laboratory temperature is controlled within a small deviation ± 5⁰C, dynamic viscosity only varies within ±10%. The fluid temperature is constantly within a range 25-30⁰C. In this study, k will be reported in this range of temperature. The temperature of pore fluids will be measured in IPM test. Unless there is significant temperature variation of pore fluid (larger than 5⁰C), this calibration will not be considered because for most of the cases k is usually determined on the level of magnitude. A table of dynamic viscosity varying with temperature can be checked from AS 1289.6.7.1 (2001).
(a)
(b)
(c)
Figure 4 (a) Hydraulic conductivity test using Constant head method (b) flux measurement (c) temperature calibration of dynamic viscosity
The last term can be calculated from this experiment is intrinsic permeability (K, unit cm2). To determine K, there are two extra properties of pore fluids has to be measured, which are density and dynamic viscosity. This is not necessary for the water-air system but should not be neglected if other pore fluids are used. The intrinsic permeability is given by
K
T kT T kT (6) g T g
where ν = kinematic viscosity of pore fluid at temperature T (m2/s), μT = dynamic viscosity of pore fluid at temperature T (kg/m·s), ρT = density of pore fluids of pore fluid at temperature T (kg/m3, constant for incompressible fluids, changed with temperature for compressible fluids), kT = hydraulic conductivity of distilled water at temperature T (m/s). The conductivity for other fluids like mineral oil or even mixture can be calculated by changing the viscosity and density with a known intrinsic permeability. A very important criteria of applicability of Darcy Law is that flow regime in porous media should be predominated by viscous force and inertial force can be neglected, which ensure the derivation of Darcy Law from averaging Navier-Stokes equation (Jacob, 1972). This is less considered in most of geotechnical testing standard. ASTM D2434-68 (2006) consider this non-linear Darcy flow by a recommendation that the hydraulic head should be increased every 0.5 cm to check the non-linear
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relationship between discharge velocity and hydraulic gradient. However, it does not mention using dimensionless Reynold number (Re) as an index to distinguish linear laminar flow and non-linear laminar flow. As for AS 1289.6.7.1 (2001), there is totally no information about the upper limit of Darcy Law. Instead, this standard suggests a head difference of 117 mm, which is a smaller head difference to ensure the flow regime under laminar. The Reynold number is
Re
vd
(7)
where ρ = fluid density (g/cm3), v = discharge velocity (cm/s), d = length dimension representing flow channel of pore matrix (cm), μ = dynamic viscosity of pore fluid (Pa·s). In geotechnical engineering, the hydraulic diameter (d) is often set at D10 of Grain Size Distribution. The literature review on critical Re from Jacob (1972) suggests that the Darcy Law is holding if Re does not exceed some value between 1 and 10. For a future study on constant head method determination of saturated hydraulic conductivity of coarse soil, there should be a multi-manometers vertically connected along specimen column with adjustable back pore pressure system. However, the size of current mould firmly following AS 1289.6.7.1 (2001) so that there is less vertical space for multi-manometer connection. Also simply loading on a sample by a surcharge cannot prevent erosion and loss of effective stress. High hydraulic gradient induced nonlinear laminar flow and inhomogeneous of sand packing induced hydraulic conductivity distribution in a prepared specimen cannot be studied based on the temporary device.
3.1.4 Determination of SWRC by Hanging Column Method The hanging column method is used to determine the static Soil Water Retention Curve (SWRC) in this work. Such a static SWRC can be used to approximate two-phase flow seepage through proposed IPM test using Richards’ model and to be compared with dynamic SWRC measured from IPM as well. ASTM D6836-02 (2003) is taken as a reference for the hanging column test, but there is some difference in details. A picture of the experimental setup and the diagram is given in Figure 5. Rubber cover with pine hole Fridge Measuring Cylinder
Specimen
(a)
Water increment
Suction head (cm)
High Air Entry Disk
(b)
(c)
Figure 5(a) Water reservoir (b) diagram of experiment setup (c) hanging columns inside fridge
In Figure 5, four glass funnels with a High Air Entry (HAE) ceramic disk fixed inside each are located into a fridge. The bottom of each funnel is connected with a measuring cylinder where water is fully saturated. The water table in the measuring cylinder can be seen as the water table in nature. HAE disk
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prevents air invasion into the suction control system. The measuring cylinder can also be used to measure the water expelled from the specimen by checking the water table increment. The funnel is capped by a rubber cover to alleviate evapotranspiration. There is a small pine hole on the cover to keep the standard atmosphere pressure on top of the specimen. Similar evaporation prevention setup is given to the top of the water reservoir. To ensure the accuracy of measurement, two reference cylinders filled with water are set inside and outside the fridge as an indication of evaporation. This setup compared with the requirement of Method A in ASTM D6836-02 (2003) is simpler in the suction control system. However, the isolated vacuum horizontal tube to avoid evaporation is replaced by a reference waterfilled cylinder to calibrate the expelled water volume. Suction head (hc) measurement is given by the length difference between the bottom of HAE disk and water table in the cylinder. This suction measuring method might be rougher than indication of air vacuum pressure using manometer in ASTM D6836-02 (2003), but the reading deviation for suction head is about ±1 cm in this setup, which is only ±0.1kPa and acceptable. The specimen preparation of ASTM D6836-02 (2003) seems only suitable for cohesive soil because it requires compaction in mould and sampling specimen by a retaining ring. The density can be calculated from the dimension of retaining ring or mould. However, this procedure cannot be followed for cohesionless sand. The sand specimen preparation procedure for this setup is different to standard. First, the water table is raised above the top of HAE disk for 2-3 cm. Second, the sample is poured into funnel fully mixed with water. Leaving a certain water above the top of the specimen, the sample in the funnel is gently tapped on the side to reach aiming thickness. This could be rough, and the thickness of specimen might not be totally same around the funnel, so an average is taken. With known specimen thickness, funnel diameter and mass of oven dried sample poured inside, the dry density or 100% volumetric water content (porosity) can be determined following equation (1) and (2). After cycling drainage and imbibition test, the wet specimen can be sampled by a smaller cutting ring to check the packing condition. Whether specimen volume change can be verified in this way. When the fully saturated specimen is prepared, the water table is dropped down to the bottom of HAE disk by adjusting the position of measuring cylinder. With water expelled out of the specimen, the water table might rise above HAE disk. So cylinder position should be continually adjusted until the water table is same to the HAE disk bottom. This thickness of specimen before test commencement should be taken for the calculation. The suction can be increased by lowering cylinder and decreased by lifting cylinder. When an aiming suction is applied, regarding drainage SWRC in ASTM D6836-02 (2003), 48 hours has to wait for capillary equilibrium. Then the head difference between the water table and HAE disk is the suction head. ASTM D6836-02 (2003) does not give any information about time interval for reaching next equilibrium in imbibition test. Here, 48 hours firstly waits, and the water increment in the cylinder is daily checked. Until no further change, the suction head and water increment will be recorded. The volumetric water content for each suction head on drainage SWRC can be calculated by
i i 1
Vdi (8) VT
Moreover, for imbibition SWRC, the equation is
i i 1
Vdi (9) VT
where ϴi = volumetric water content of specimen for suction point i, ϴi-1 = volumetric water content of specimen for last suction point i-1, Vdi = volume of water expelled from specimen after suction i equilibrium approached, VT = the total volume of specimen. The equation used here is different with that of hanging column method in ASTM D6836-02 (2003) but same with that of pressure chamber
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method because of difference in suction control setup. The referencing cylinder records water evaporated from measuring cylinder each day. This amount of water is added back to Vdi on that day measurement taken. As for the referencing cylinder in the fridge, we could not observe significant evaporation. The evaporation is assumed to be no influence on such hanging column test.
3.1.5 Relative hydraulic conductivity test by unsaturated soil standard triaxial cell Although the hydraulic conductivity of unsaturated soil under various suction or water content can be predicted from drainage SWRC using HCFs (Equation (15)-(20)), the prediction performance of each HCF has less been checked in recent studies of numerically solving Richards’ model, because a large amount of early experimental studies supports these HCFs. On the purpose of studying multiphase flow in porous media, HCF should be independently measured by standard unsaturated soil triaxial cell test, not only for validation of available HCFs but also for a better prediction performance from Richards’ model. On the other hand, there is another interesting paradox that how the hydraulic resistivity (inverse of hydraulic conductivity) can be physically predicted from a static SWRC on which there should be no flow condition on each equilibrium state at all. As HCFs mentioned in the literature review, they are either empirical functions or statistical basis functions. For statistical basis function, the relation between SWRC and HCF is derived using the statistical pore structure assumption (E. C. Childs & Collis-George, 1950; Mualem, 1976). There is no any physical explanation of the association between two-phase flow condition and static drainage SWRC but a fraction occupation assumption. For an airwater flow in a specimen of the unsaturated soil triaxial cell, the hydraulic conductivity is tested corresponding to applied suction controlled by ATT. There should be no doubt that air bubble or continue air paths inside the specimen. However, in standard ATT test for static drainage SWRC, water are pushed out of pressure cell by high air pressure. It is impossible to find any similar flow condition between two experiments so this also challenges the performance of Richards’ model using the concept of HCFs. Air Pressure/Volume controller (APC)
Pressure cell
Specimen covered by a membrane
Top water pressure line
Cell pressure line Water Pressure/Volume controller (VPC)
Bottom water pressure line
Load frame
Air pressure line (should be connected with APC)
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Figure 6 Unsaturated soil triaxial cell: WILLE Product available in Geotechnical Engineering Centre of the University of Queensland
With these concerns, the hydraulic conductivity of unsaturated sand will be tested using unsaturated soil triaxial cell. The unsaturated soil triaxial cell is originally designed for testing mechanical parameters of unsaturated cohesive soil to study the stress-suction-strain constitutive relationship. Nevertheless, it can also be used to implement a flexible wall seepage test to study permeability reduction with isotropic loading increase. ASTM D7664-10 (2010) introduces this method (Method B2) as a standard test. The unsaturated soil triaxial cell is larger than a conventional triaxial cell for saturated soil (Figure 6) and has several improvements: (1) the bottom porous stone is replaced by a HAE disk for suction control. The back pressure is applied in the bottom of HAE disk to control the pore pressure using a water Pressure/Volume controller (VPC); (2) the top cap is added with a spiral water flushing path to prevent blockage by bubble accumulated under the top cap. Air pressure is controlled by a gas Pressure/Volume controller (APC); (3) the suction head gradient can be automatically controlled by three Pressure/Volume controllers: two water VPCs are used to produce constant head water infiltration and measure seepage flux, and each one of them coupled with APC to determine suction pressure on the top of specimen (h1) or bottom of specimen (h2); (4) the suction of testing specimen is approximated by averaging two boundary suctions as
h h Pc,mean (ua uw )mean 1 2 w g (10) 2 where ua = air pressure (kPa), uw = water pressure (kPa), ρw = water density (1000 kg /m3), g = gravitational accelerator (9.8 N/kg). The discharge flux (Q, m3/s) can be measured by VPC connected to the bottom. The hydraulic conductivity for a controlled suction (Pc,mean) can be calculated by Darcy Law
kPc ,mean
Q L (11) At (h1 h2 ) L
where A = cross-section of specimen (diameter of large cell is 10 cm), t = time interval of capillary seepage (unit: s, the volume of water should be sufficient depends on type of soil (ASTM D7664-10, 2010)), L = specimen thickness (unit: m, usually 20 cm). The suction reduction can be done by lowering the bottom back pressure vice versa. In this way, hydraulic conductivity for unsaturated sand on each suction, no matter either wetting or drying, can be determined the point by point on HCF in suction basis. With previous primary drying and wetting SWRCs, the k-ϴ function can be also plotted. So far, this experiment is a part of the entire plan, and the author has just been trained. Temporarily, the determination of hydraulic conductivity is still approximated by HCFs. This experiment will be carried out in future for comparison with HCFs. There are few key points for this experiment in accordance of ASTM D7664-10 (2010): (1) prior to any testing procedure, a setup should be done without specimen filled inside to measure the coefficient of permeability of HAE ceramic disk/cellulose membrane (kHAE). Because the entire system determine the hydraulic conductivity of both specimen and bottom HAE disk, this kHAE need to be eliminated to calibrated Pc,mean; (2) Vacuum should be given to the top of specimen at -80 kPa to saturated specimen and saturation should be checked by Skempton B value using B-test procedure (100% saturation achieved for B equal or above 0.9). (3) This standard does not use Darcy Law to calculate k, but the capillary conductivity calculation procedure suggested by WR Gardner (1956). The analytical solution of 1-D water content diffusion equation (1-D Equation (13)) can be given as
VT V 8 D 2t ln( ) ln( 2 ) (12) VT 4 L2 24
where VT = outflow at equilibrium state (m3), V = outflow in time interval to a suction increment (m3), D = water content diffusivity (m2/s), t = time interval to suction increment (s), L = specimen thickness (m). A linear regression can be fitted into the relationship between the left-hand side term of Equation (54) and second right-hand side term including time interval (t). The fitting parameter (a) for t can be used to calculate the diffusivity (D). Also of the notations in Equation (13), the hydraulic conductivity can be calculated by
kPc ,mean
4 L2 a 2 C (hc ) (13)
where a = the fitting parameter timed by t (Dπ2/4L2), C(hc) = slope of SWRC for average suction Pc,mean (m-1). Finally, the hydraulic conductivity can be plotted against either average suction or corresponding water content to plot the k-Pc curve or k-ϴ curve. In this study, sand deformation and deformation induced permeability variation will not be considered. Also, the saturated hydraulic conductivity varying with deformation can be independently determined by a saturated soil triaxial cell test. The cell pressure or isotropic loading will use K0-test (lateral constraint controlled by APC), or the flexible membrane covering the sand will be fixed with a steel cover around to constrain the lateral deformation.
3.2
Experimental design of one-dimensional Instantaneous Profile Method (IPM)
3.2.1 Overview of experiment setup The 1-D soil column is designed to study the instantaneous profile of both volumetric water content and suction. A schematic diagram is shown in Figure 7. There are four soil columns, each of which has a length of 240 cm with a diameter of 14.35 cm. The flat ribbon cable of Spatial Time Domain Reflectometer (STDR) is embedded in the soil specimen filled into each acrylic column. The bottom terminal of STDR is fixed with an artificial gravel filter that does not only fix the end of STDR but is also a high pervious porous media under the specimen being tested. Similar experiments have been implemented by Scheuermann, Montenegro, and Bieberstein (2005), Hui et al. (2010), Schultze, Ippisch, Huwe, and Durner (1997), O'Carroll et al. (2005), Wildenschild, Hopmans, and Simunek (2001), Das and Mirzaei (2012), Sakaki et al. (2010), etc. However, in their experiment, the size of soil column is smaller than 100 cm. For applying high suction by hanging column method or pressure cell, they have to use an HAE disk (a Teflon membrane or a sintered glass plate) under the specimen to ensure no continuity of air path. This increase the hydraulic resistivity of the entire system. Even calibration can be done by independently measuring permeability of setup without specimen filled inside, and their experiment proves dynamic effect in SWRC or water redistribution, in fact, this low permeable disk or membrane reduce the flow, additionally causing more insignificant dynamic effect in soil moisture redistribution. Therefore, a gravel filter is used to replace any porous plate, and a large size soil column is used to replicate somehow a smaller scale of natural soil condition, where there are unsaturated zone, saturated zone and a bottom boundary layer without hydraulic resistivity. In this condition, there is a chance to see that if the dynamic effect of moisture movement in the unsaturated zone is more significant, and no artificial quasi-steady state generated due to the usage of HAE material. The soil moisture content is measured using time domain reflectometer (TDR) technique. Each sensor is exact 200 cm in length, and all of them are connected to a TDR multiplexer. The signal of each TDR will be transferred between TDR 100 and each cable. Using empirical model between water content and dielectric constant (Topp, Davis, & Annan, 1980), the water content profile can be calculated from the dielectric constant (permittivity) profile that can be given by a fast inversion technique developed by Schlaeger (2005). The data will be transfer to a computer (PC1) for the post process. The details of STDR technique will be introduced in the third part of this section.
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As for suction measurement, five T5 tensiometers (UMS Umweltanalytische Mess-Systeme GmbH) are installed on each soil column. There are totally 20 negative pressure sensors on four soil columns, and all of them are connected to one Data logger for automatically logging suction by time. The distance between each sensor is 40 cm. Four sensors will be used to measure the unsaturated zone above the water table, and another one will be used to measure the positive water pressure in the saturated zone. The specification of T5 sensors and logging system is given in following part of this section. Constant head and vibrating head overflow system will be connected to the bottom of each column to conduct one step drainage/imbibition or multistep/drying-wetting cycling paths separately. The bottom flux for one step drainage/imbibition will be measured using electrical bench scales (Ohaus Ranger 3000 R31P30). Each electrical bench scale has an RS232 port to transfer logging of mass into a computer (PC2). The maximum load capacity is 30kg and resolution is 1g. There are four E-scales shown in Figure 15. Each of them is connected to an RS232-USB cable, and a USB Hub is used as a multiplexer to connect four scales with one computer (PC2). Totally, there are two computers, PC1 and PC2 separately. PC1 is used for moisture content profile logging, and PC2 is used for both suction profile and accumulative bottom flux logging. For multistep and cycling paths, Vibrating Height Overflow (VHO) system is used. As shown in Figure 15(c), the overflow drainage pipe can be controlled by a motor engine. With clockwise and counter-clock rotation, the bottom pressure head can be increased or decreased step by step. A sinusoidal vibrating bottom head can also be generated by periodically losing and tightening line holding overflow drainage pipe. The specimen preparation will be filled with soil column layer by layer. Within each layer, the density control will be achieved by tapping on the side of each column until the desired volume approached. Water will be added to the column before the sand. During specimen installation, the surface of the free water should always be at the top of each compacted layer. Regarding the pre-test of density profile in shorter sand column, a very thin plastic O-ring is located between each layer to mark the vertical deformation of each layer, because an O-ring of silica powder cannot be rigidly set on each layer under free water. Although this density control method cannot ensure maximum dry density achievement, as long as the density or relative density index is same as the dry density achieved in standard tests, the packing condition will be accepted for comparison between dynamic data set and static data set. Three dry densities between the minimum and maximum will be prepared to study the dynamic effect influenced by packing conditions. ASTM D7664-10 (2010) introduces three methods for specimen preparation: compaction, wet tamping and pluviatioin. Because the soil column used in this study is 240 cm, compaction method can only be done by taping or vibration. Vertical compaction might lead to column damage due to high earth pressure (240m sand in a dry density of 1.35-1.6 g/cm3) in the column bottom. Mechanical shaking table cannot support and effective shake 240 m soil column. Lins, Schanz, and Fredlund (2009) successfully prepared soil specimen for 1-D soil column IPM test using pluviation method following Vaid and Negussey (1988). However, there is no specification of funnel size and dropping length corresponding to desired density. Most of pluviation method is designed for triaxial cell test of loosely compacted sand (Miura & Toki, 1983). Pluviation method will be considered as the second approach of sand preparation in future. Wetting fluid and non-wetting fluid are temporarily deionized water and ambient air. For the purpose of studying unstable wetting front like capillary fingers and viscosity fingers, invading wetting or nonwetting fluid might be replaced by mineral oil, so that the dynamic effect in gas-oil and oil-water two phase system or even gas-oil-water three phase system can also be studied using same setup.
26
T5 Tensiometer
TDR multiplexer PC1 TDR 100
USB Hub PC2 Datalogger
E-Scale 1
E-Scale 2
E-Scale 4
E-Scale 3
(a) Atmosphere TDR multiplexer
STDR
TDR multiplexer
STDR
TDR 100
TDR 100
T5 Tensiometer
T5 Tensiometer
PC1
Datalogger
240 cm
PC2
PC2
40 cm
40 cm
240 cm
Datalogger
PC1
E-Scale
20 cm
20 cm
Valve
E-Scale
14.35 cm Gravel filter
14.35 cm
Constant head tank
Gravel filter
(b)
(c)
27
Constant head tank
Atmosphere TDR multiplexer
STDR
TDR 100
T5 Tensiometer
40 cm
240 cm
Datalogger
PC1
PC2
Vibrating Head Overflow system
Pulley
Overflow Reservoir 20 cm
M Motor Overflow drainage pipe
14.35 cm Gravel filter
Water supply
(d) Figure 7 (a) Overview of experimental setup (b) One-step outflow setp (c) One-step inflow setup (d) Multistep inflow/outflow setup
3.2.2 T5 micro tensiometer and automatic logging system in dynamic suction measurement The suction transducer used in the experimental setup is T5 micro-tensiometer (UMS) in Figure 7 (a). The specification of this commercial product is listed in Table 6. The suction can be read out using a hand holding reader Infield7 (UMS) shown in Figure 7(b). Infield7 has only one connector for each sensor and is used for calibration. Another commercial data logger, DT85G Datataker (Figure 7(c)), is used for automatically logging 20 T5 sensors with a certain time interval (2-10 mins depending on the speed of varying boundary conditions). This data logger only has 16 analog channels that are compatible with T5 sensor plug (Wheatstone bridge circuit, four pins). The output signal of T5 is in voltage, whereas the output signal of the data logger is a manufacture-specified dimensionless voltage (Bout, Equation (A1) in Appendix 2). Therefore, a calibration between the water pressure and the dimensionless voltage has to be carried out before experiment implementation, and the result of calibration will be given in next chapter as a preliminary result. Our pre-test shows that it will take around 30 seconds to accurately response pressure after a suction or positive pressure applied to sensor horizontally. Klute and Gardner (1962) discussed the response time of tensiometer. As the pioneer working on the development of tensiometer, they pointed out that any tensiometer has a response time because of permeability of ceramic cup and conductivity of surrounding soil. According to UMS official information of response time of T5 tensiometer, it takes only 5 seconds to give accurate suction. Due to lack of suction sensitivity of the T5 sensor, currently, there is no error analysis of T5 sensor in response time. Nevertheless, low permeability of sensor tip only contributes to alleviation of dynamic response but an exaggeration. So far tensiometer is the only sensor that can detect dynamic capillary pressure in macroscale experiment. The response time of a tensiometer is given as
T tensiometer soil exp(
T ) soil (14) TR 28
where T = response time of tensiometer (s), ψtensiometer = initial pressure inside the shaft of tensiometer (kPa), ψsoil = suction of surrounding soil (kPa), TR = S/K, S = dq/dψtensiometer gauge sensitivity, K = intrinsic permeability of ceramic cup (cm2) (Klute & Gardner, 1962). It is expected that the response time can be further studied if the gauge sensitivity of T5 can be provided from manufacture UMS. Table 1 Specification of T5 micro-tensiometer
Measuring range Precision Shaft diameter Shaft length Output signal
+100 kPa - -85 kPa ±0.5 kPa 5 mm 20 cm -100 mV - +85 mV
(a)
(b)
(c)
Figure 8 (a) T5 micro-tensiometer (b) Infield7 hand reader (c) DT85G data logger with CEM
3.2.3 Spatial Time Domain Reflectometry application in dynamic moisture distribution measurement The moisture content of the soil column is measured using Time Domain Reflectometry (TDR). Conventional TDR sensors with three uncoated metal rods have a limitation of transmission line length up to maximum 100 cm (Scheuermann et al., 2009). Three pin TDR can only be used for point measurement. To determine the moisture distribution, a large amount of TDR sensors has to be installed along the specimen or in situ. In this work, Spatial Time Domain Reflectometry (STDR) developed by Scheuermann et al. (2009) and Schlaeger (2005) is applied. This technique only has one coated flat ribbon cable buried inside the specimen and water distribution along this cable can be calculated from distribution of dielectric constant profile using empirical model of Topp et al. (1980). A detail of development and application of SPTDR can be sourced out from Scheuermann et al. (2009). Here, for the purpose of using SPTDR to extract spatial-temporal moisture content distribution, the understanding of SPTDR is summarised.
29
6cm
(b)
(c)
(a)
Figure 9 (a) Spatial Time Domain Reflectometry (STDR) (b) Electrical model of SPTDR sensor in an infinitesimal section (c) Electrical model of capacitance for soil covering STDR (Scheuermann et al., 2009)
A photo of STDR sensor, a schematic diagram and electrical capacitance model of flat ribbon cable are given in Figure 9. STDR replaces three rods on TDR by a polyethylene coated cable in which there are three copper transmission lines. The middle line of three is connected with the central line of coaxial cable to input voltage and rest two lines are connected with outer lines. The equivalent circuit of SPTDR is depicted in Figure 17(b). The STDR cable covered by specimen can be seen as such a circuit consisting of Inductance (L), Resistance (R) connected in series and Conductance (G), Capacitance (C) in parallel connection within an increment of infinitesimal section. Scheuermann et al. (2009) discussed these four parameters impacting on determination of permittivity and identified that L and R can be assumed to be constant. So the permittivity can be determined from conceptual model of capacitance shown in Figure 9(c). After a pulse generated from TDR100 (a pulse generator and an oscilloscope produced by Campbell Scientific), the reflected voltage will be recorded by TDR100 against time to characterise the one-way travel time of wave (ttravel). With the known length of cable (Lcable) and constant impedance (L), the total capacitance (C(εm)) can be determined from the wave velocity (vwave) as
vwave
Lcable ttravel
1 (15) L C ( m )
Moreover, the relative permittivity of the specimen (εm) can be given by
m
C2 (C3 C ( m )) (16) C1C ( m ) C1C2 C1C3
where C1 = the capacitance of surrounding specimen (3.4 pF/m), C2 = the capacitance of isolated coating (323 pF/m), and C3 = the capacitance between positive and negative transmission line (14.8 pF/m). This capacitance concept model of STDR-surrounding soil system neglects the dielectric loss in coating material and it is only suitable for common soil without high electrical conductivity (low salinity in pore water) (Scheuermann et al., 2009). A sensitivity analysis of STDR sensor shows that the accuracy of measurement is highly dependent on contact between flat ribbon cable and surrounding soil, because
30
the area of electrical magnetic field around sensor determine the relaxation behaviour of surrounding soil (Scheuermann et al., 2009). Prior to reconstruct the distribution of permittivity, the distribution of Capacitance (C) has to be characterised. Schlaeger (2005) developed inversely modelling telegraph equations along STDR transmission line to calculate the distribution of C using the conjugate gradient optimization method. With the known distribution of C, the distribution of relative permittivity εm can be given by Equation (16). Eventually, the volumetric water content (ϴ) can be calculated from permittivity (ε) using model from Topp et al. (1980) 3
ai i (17) i 0
where ai = fitting parameter for series of i, εi = i power of permittivity, Topp’s fitting series: a1 = 5.3*10-2, a2=2.92*10-2, a3=-5.5*10-4, a4= 4.3*10-6. This fitting parameters requires soil specific calibration, while for general loosely compacted sand without high salinity in pore fluid, which will be the case of this work, the fitting parameters of Topp et al. (1980) can be taken. Inversing modelling telegraph equation on determination of C distribution needs two-way TDR trace to decrease calculation effort in optimization. Therefore, STDR has two connectors on each terminal, and the pulse has to be generated and recorded from each side. The TDR logging system is shown in Figure 15. Four STDR sensors are embedded into four sand columns. Two male connectors on each sensor are plugged into a TDR multiplexer that has exact eight female connectors and is chained with the pulse generator/receiver (TDR100). The TDR waveform recorded in each time interval is finally transferred to a CR1000 datalogger controlled by a PC (PC1 in Figure 15). The reconstruction of Capacitance distribution will be determined by Matlab code developed by Schlaeger (2005). Topp’s model will be considered as the first option into the calculation of volumetric water content distribution. Unless any unforeseen circumstance appears on the testing specimen, soil specific calibration will not be taken.
3.2.4 Boundary conditions for one/multi-step pressure boundary and cycling boundary There are three type of setup for pressure boundary conditions in this experiment. As shown in Figure 7(b), a constant head tank is attached into the bottom output of the sand column. The overflow of this constant head tank is collected by a water reservoir on the electrical bench scale to measure the bottom flux against time. Two fully saturated soil columns use this setup for studying one step gravitational drainage, which is one of the extreme conditions in the natural earth under fast drainage of groundwater. The other two dry soil columns take the setup shown in Figure 7(c), where there is a constant head attaching to the input of soil column with a water supplying reservoir maintaining its water table. The water table in this tank is manually controlled using the value. The bottom flux for imbibition test is measured by logging the bench scale under the supplying reservoir. The one step drainage and imbibition test is designed for tuning and checking automatically data logging system as well as the acquisition of the dynamic and static SWRC. After the verification of complex data logging system and the success in one step inflow and outflow test, the multi-step test will be applied using a Vibrating Height Overflow (VHO) system. As shown in Figure 7(d), the VHO consists of an overflow reservoir, overflow drainage pipe, water supply reservoir and a motor controlling the cycling movement of overflow drainage pipe. The constant head can be generated by pumping water from supplying tank to the overflow reservoir. Through lifting and dropping the cables holding the overflow pipe, multistep head increment or sinusoidal varying head is generated by the counter-clock or clockwise rotation of the motor. After four soil columns reaching capillary equilibrium condition in previous tests, each column will be sealed by cling wrap to prevent any evaporation. Then each of them will be connected with VHO to study dynamic hysteresis SWRC
31
in multistep drainage/imbibition. After data acquisition of multistep test, finally the periodically cycling head will be applied to the static profile of soil moisture to study the hydraulic ratcheting effect that the water storage capacity of unsaturated soil monotonically increase after several cyclic wetting-drying (Scheuermann et al., 2014). However, this effect was found by an experimental setup using HAE disk under the sand column, and its hydraulic conductivity is 100 times smaller than the sand specimen (Scheuermann et al., 2014). In our new experiment, a combination of saturated zone and high previous gravel filter replaces HAE disk. This is a further exploration of dynamic effect in SWRC and hydraulic ratcheting effect in vadose zone following steps from Dr. Scheuermann.
3.2.5 Direct and inverse modelling of Richards’ model One dimensional soil column test provides an instantaneous profile of matrix suction head, volumetric water content, and boundary flux. This profile cannot only spatially give the hydraulic properties of specimen column, but also the spatial-temporal discretized solution of 1-D Richards’ model. To assist studying dynamic effect of hydraulic parameters, the inversed Richards’ model is a powerful tool, because the boundary flux or suction head variation of time is exactly the dynamic response of the twophase flow in porous matrix. In this study, one dimensional Richards’ model solver-HYDRUS-1D (J. Simunek, 2008) will be used to conduct both direct and inverse solving Richards’ model. HYDRUS1D is a public domain Windows-based modelling environment for analysis of water flow and solute transport in variably saturated porous media (J. Simunek, 2008). One-dimensional Richards’ PDE is solved using Finite Element Method (FEM) in HYDRUS-1D, and a graphic interface is offered to users to reduce time and effort on coding and debugging. As a standard user, the concept and mathematical formularisation should be understood as well as the solving scheme while numerically solving PDE in details can be completed by a computer programme. A summary of understanding of direct and inverse numerically solving Richards’ PDE is given in Table 2. Table 2 Formularisation of direct and inverse Richards' model (Simunek, Huang, & Van Genuchten, 2013)
Direct solving 1-D Richards’ model Water Retention ( ) (1 nVG ) mVG (18) r s r VG Model Hydraulic Conductivity Function Flow Motion Equation
r r k ( ) ks [1 (1 s r s r
Discretised PDE of Governing Equation
j 1,n h j 1,n 1 hi j 1,n 1 h j 1,n 1 hi 1 j 1,n 1 1 j 1, n ) i 1 (ki j 1,n ki 1 j 1,n ) i (k i1 ki zi 1 zi 1 zi 1 zi zi zi 1
0.5
1/ mVG
) mVG ] (19)
h q k ( ) c 1 (20) z Transient Flow h k ( ) c k ( ) Governing (21) z z t Equation
2
ki 1/2
j 1, n
ki 1/2 zi 1 zi 1
j 1, n
i
i hi d j 1,n hi 1 ( ) t j 1 t j dhc t j 1 t j
j 1, n 1
j 1, n 1
j
j 1, n
(22)
H .O.T t j 1 t j j 1, n
j
Boundary constant or vibrating head boundary using Dirichlet boundary – prescribed h0 Condition pressure head on the bottom of the soil column Inverse solving Richards’ model 2 2 2 m n n n m Objective ^ ^ ^ (q, p, l ) a j bi , j q j ( z, t ) q j ( z, ti , l ) c j di , j p j (i ) q j (i , l ) ei , j l j l j Function i 1 j 1 i 1 j 1 i 1 (23) Minimization l l v1 l v (J T WJ I)J T Wr (24) ^ Method: r q q (67) q
q, j
P
32
P, j
P, j
LevenbergMarquardt
J ij
qi ( x, t , l ) (25) l j
The direct model consist of governing equation (21) with constitutive functions Equation (18) and (19). To solve Equation (21) without simplified exponential constitutive functions, this continue PDE has to be rewritten into discretised form as Equation (22) to be numerically solved as a combination of linear equations. Equation (22) can be further transformed into a matrix form, which is given in details in Simunek et al. (2013). After inputting boundary pressure head (h0) into the global matrix equation, the solution on each position (zi) is given by iteratively calculating solution on each time step (j) if the convergence is accepted after iterative step (k). Simunek et al. (2013) also introduced the time control for avoiding numerical diffusion and mass balance checking in the manual of HYDRUS 1-D. However, for analysing experimental data, this study merely focus on the usage of an available software package. Studying different numerical methods on solving PDE is far beyond the scope of this study. After all, the dynamic effect in a constitutive relationship is overlooked due to lacking of mathematical formularization of this physical fact. Thus, solving conventional theory using various numerical methods will not contribute to unveiling dynamic effect. When the effective and applicable mathematical formularization of the dynamic effect of SWRC or HCF are determined, the improved theory might be worth to be numerically solved for model validation. On the other hand, with known cumulative boundary flux, suction head and water content measurement on certain points of soil column, the fitting parameters in SWRC or HCF (α,n,m) can be inversely calculated using optimization methods such as Gauss-Newton method and Levenberg-Marquardt (LM) method. Through minimizing the difference between measured state variables and predicted variables from direct Richards’ model (Equation (18)-(21)), the input fitting parameters can be optimised to the best selection, which is the fitting parameters from a real dynamic process and can also be compared to the parameters of static SWRC, experimentally given by standard unsaturated soil testing methods. A detail of implementation of LM method on the inverse model is given in J. Simunek (2008). Few points, which should be aware, are that initial guess affects the speed of optimization and whether global minimum is achieved. Therefore, the fitting parameters of SWRC from Hanging Column test will be chosen as the initial guess.
3.3
Two-dimensional transparent soil model
3.3.1 Overview of experimental setup One dimensional IPM test does not clearly visualize the variation of the wetting front and interfacial area. It only provides a vision to inspect the dynamic effect in macroscale. To link the macroscale REV with microscale pore channels, a two-dimensional transparent soil physical model should be set up. Such a physical model does not only visualize microscale meniscus, suction and localized saturation of each phase, but is also a macroscale model, of which macroscale state variables can be calculated by averaging microscale variables in selected region. As the experimental design is shown in Figure 10 (a), the entire 2-D physical model is comprised of three components: a large cell filled with a package of glass beads (Width = 70 cm, height = 100 cm, thickness = 3 cm), a bottom wetting phase reservoir controlled by a Vibrating Height Overflow (VHO) system and a non-wetting phase reservoir attached to the top boundary of cell. This cell is originally designed by Dr. Ye Ma on studying discretized bubbly flow in granular porous media made up of hydro-jellybeads (Ye Ma et al., 2014; Y Ma et al., 2014). Here, for different usage, this cell is modified. In Y Ma et al. (2014), there is a gas sparger located in the center-bottom of cell to generate bubbles in size of hundreds of micrometer and eleven gas burettes are evenly distributed along the top of cell to detect soil gas distribution after bubble seepage through saturated porous media.
33
In this study, a package of glass beads will be filled into the cell. To evenly distribute inflow and outflow from two reservoirs, a gravel filter (10 cm in thickness) is filled on both tops and under the bottom of the glass beads package. A manometer is attached to cell bottom as an indication of the groundwater table, above which there is a zone for the multiphase system in glass beads package. The bottom head is controlled using VHO, and the top reservoir is filled up with non-wetting phase fluids, such as mineral oil, paraffinic solvent, glycerol and air (directly connect to atmosphere). Water flow meter (Glass Style 0.6-6L/h Water Flow Measuring Flowmeter for 39.5mm Tube) will be set between VHO and cell to measure the flow rate in the bottom boundary. Here, flowmeter used instead of bench scale is because the permeability and relative permeability for a package of glass beads is much higher than compacted fine sand. Micro-tensiometer could be installed in the back of the cell to give the 2-D spatial distribution of suction head. In front of the cell, the meniscus of each pendular ring or continue phase paths can be captured using a digital camera. This experiment is still in the step of design so the selection of suitable camera will be fixed in a future attempt. The only shortage of 2-D experiment is that it cannot truly represent the 3-D facts due to wall effect (Karadimitriou & Hassanizadeh, 2012). Nevertheless, this is so far the only option to visualize physical phenomenon than point measurement using any pressure sensor. Non-wetting phase Overflow reservoir
Gravel filter Pulley wheel
Wetting phase Overflow reservoir
Overflow drainage pipe
Saturated glass beads
Water pumping
Gravel filter
M 0.6L/hour Flow meter
(a)
34
Gravel filter
Dark background Glassbeads
PC
Datalogger
Camera
Gravel filter
(b) Figure 10 (a) A front view of 2-D physical model of mineral oil-water flow in transparent granular glass beads (b) a side view of the 2-D physical model with state variables extracting setup
3.3.2 Imaging technique for analyzing pore structure and identification of fluid-fluid interface Except that the suction of local region can be measured using T5 tensiometer, any other state variables are measured using imaging technique. As Figure 18 (b) shown, a digital camera is set in front of the cell. This camera will be used to instantaneously capture the distribution of two-phase in the glass beads package. The wetting fluid is water dyed with green color to indicate the wetting phase while the nonwetting fluid is a mineral oil having a same refractive index with glass beads (1.51-1.52). The refractive index matching ensures the identification of residual wetting phase in the non-wetting phase region or trapped nonwetting phase in the wetting phase region. The calibration of length scale will be done by capturing a ruler fixed on the cell front to give the ratio between one pixel and real length. The resolution of camera should at least be able to identify the smallest meniscus. The picture collection of dynamic configuration of two phase will be analyzed using Image Processing Toolbox in Matlab® (The MathWorks Inc., 2014). The two phase fraction can be separated using a threshold of gray scale value. With an appropriate selection of gray threshold, the shape and area of each phase in the 2D image can be accurately extracted. The area of each phase is marked using codebwlabel and measured using code-regionprops (The MathWorks Inc., 2014). Thus, the phase reconfiguration during a transient state can be measured globally in the entire domain or locally in the particular region around installation position of the T5 sensor. The shape of each phase can determine the microscale capillary pressure in equilibrium stage and also offer an opportunity in comparison between the suction measured by the T5 sensor and the average of microscale suctions calculated from the static meniscus. As the previous review on the loss of static meniscus in a transient flow condition, directly calculating suction pressure from dynamic meniscus using Young-Laplace equation is inaccurate, while micro-tensiometers can measure such dynamic capillary pressure. There is an expectation to correlate the dynamic capillary pressure with the dynamic meniscus in each frame of transient flow to characterize if there is a unique association between two variables as Young’s Equation. Also, with speed of advancing front or phase fingers estimated from each frames, this experiment will be very useful to quantify capillary number (Ca) and viscosity ratio (M) on different finger regimes (capillary finger and viscosity finger) and further on validation of dynamic capillary flow motion equation developed by Løvoll et al. (2011).
35
3.3.3 Upscaling state variables The variables directly calculated from counting continue pixels in one image are microscale state variables in the local region while the practical engineering theory is built upon macroscale with the assumption of REV. To determine the macroscale state variables, spatially averaging each microscale variables is ought to be done on selected region or an entire domain. For validation of both phase reconfiguration theory (G. Barenblatt, Patzek, & Silin, 2003) and thermodynamic basis theory (S. M. Hassanizadeh & Gray, 1993b), the saturation, specific interfacial area (interfacial are to volume of REV, specific contact line in 2D case) and static capillary pressure should be determined. A list of upscaling equations for each variable is given in Table 8. A schematic diagram of idealized interfacial geometry is plotted in Figure 19.
Glassbeads grain
Routside
Wetting phase fluid Non-wetting phase fluid
Rinside
Figure 11 A schematic diagram of idealized interfacial geometry
Table 3 Equations for upscaling microscale variables
Macroscale Upscaling Equations state variables (REV scale) Saturation (S) wetting phase: S w
Nw
A i 1 nT
i,w
(26)
A i 1
i
Nn
non-wetting phase: S nw
A i 1 nT
Interfacial area: Anw
(27)
A i 1
Interfacial Area (Anw)
i ,n
i
Nw
N nw
nnw
i 1
i 1
i 1
Ai,w Ai,n Ai,s 2
36
(28)
Nw
Specific interfacial area: anw
A i 1
i ,w
N nw
nnw
i 1 nT
i 1
Ai ,n Ai , s (29)
2 Ai i 1
Capillary Pressure (Pc)
1 1 Microscale capillary pressure: Pc ,i s (30) Routside Rinside n
Macroscale suction pressure: Pc
P
c ,i
i 1
Ai , w
n
A i 1
(31)
i,w
The image analysis (Matlab code: bwlabel) labels the each connected pixels as one objective. The area of single objective (i) accounting fraction of wetting phase and non-wetting phase are individually Ai,w and Ai,nw. The saturation of target domain can be given by the sum of one phase fraction to the total porous area (Equation (26) and (27)). In 2D domain, the interfacial area (Anw) is actually a contact line calculated by Equation (28) and specific interfacial are (anw) is the pixel number of the contact line to pixel number of REV given by Equation (29). The microscale static capillary pressure is calculated by Equation (30) using an idealized model of meniscus geometry given in Lu and Likos (2004). Finally, the macroscale suction is given by spatial averaging method as Equation (31). These equations are frequently used for upscaling microscale variables given by both physical pore network model and Lattice Boltzmann simulation (S Galindo-Torres et al., 2013; Karadimitriou et al., 2014; Landry, Karpyn, & Ayala, 2014). With additional dynamic capillary pressure, the constitutive surface of PcSw-Anw will be rechecked, and dynamic constitutive surface Pc,dyn-Sw-Anw varying with Capillary number will be further investigated. This experimental design has been pre-tested using simply setup in a class gasket. Whereas the fluidfluid interface can be clearly distinguished, there is still a challenge on capturing separation between solid phase and fluid phase by a digital camera. Also, there is a clear observation of meniscus on the contact between 2D glass plane and glass beads, which leads to an error on meniscus measurement using image analysis. In the future, this experiment size will be reduced to a smaller dimension 4 by 5 cm with a rectangular pore network inside domain. Three Dimension Print is a powerful technique to generate transparent pore network. This will be a backup plan for 2D experimental investigation if there is no progress in original design.
3.4
Numerical experiment using Multiphase Lattice Boltzmann simulation
3.4.1 Shan-Chen Lattice Boltzmann method (SCLBM) Lattice Boltzmann Method (LBM) is a method to numerically approximating solution of Boltzmann equation by assuming statistical mechanics of microscale particles inside the single lattice node. Compared to computational fluid dynamics, which solves the problem in control volume (Naiver-Stoke equation), LBM is more fundamental because it directly provide a solution from a level of particle clouds described by statistics. LBM cannot clarify the density and momentum of single fluid particle (molecule in most case) but assign a statistical distribution between central node and surrounding nodes to describe the probability of the density of a cloud of particles inside a specified lattice. A D2Q9 lattice cell is shown in Figure 12. The microscale velocities of a single node is shown in the centre of each lattice. This is like upscaling molecule scale to the specified lattice scale, and then approximate Boltzmann equation on a domain made up of these lattices, using computational iteration to determine the density and velocity of central node in single lattice on each time step. To reconstruct the particles
37
streaming and collision mechanism in a domain, the interaction between each lattice is given by a relaxation behaviour towards an equilibrium distribution of particles (Equation (34)). In this experimental study, LBM is just taken as a numerical experiment to explore the dynamic effect. The D2Q9 SCLBM engine is available in Mechsys (Multi-physics Simulation Library, developed by Dr. Sergio Galindo Torres in School of Civil Engineering at the University of Queensland). The construction of this engine is listed in Table 4. The explanation of LBM in details can be sourced from M. Sukop, DT Thorne, Jr., (2006). As for a model user than a developer, understanding the structure of engine can facilitate parameter adjustment, model calibration by physical experiment and flexibly setting up different cases in the exploration of dynamic capillary flow.
e
6
e
e
e
3
0
e
2
e
5
1 lattice unit
1
e
7
e
4
8
Figure 12 D2Q9 scheme of LBM Table 4 The construction of D2Q9 Shan-Chen LBM simulation engine (M. Sukop, DT Thorne, Jr., , 2006)
D2Q19 Cell
8
Macroscale Fluid Density:
fi (32) i 0 8
fe
Macroscale Fluid Velocity: u Governing Equation
i 0
i i
(33) Collision
2 2 wi (x) 1 3 ei u 4.5 (e u4 ) 1.5 u 2 fi x, t C C C Streaming fi x + ei t , t t f i x, t 1 3 cs 2 Bhatnagar-Gross-Krook(BGK) approximation of fieq
Relaxiation time 0.5
(34) Body Force Equation Newton Second Law: Particles interaction Force (Shan & Chen, 1993) LBM EOS
ueq u
F (35)
8
Fa Ga j (x, t ) wi j (x + ei t , t )ei (36) i 1
P
2 0 G 2 0 exp( ) 3 6
(37)
38
8 Immiscible Multiphase Immiscible fluids repulsive force: Fr Gr j (x, t ) wi j ' (x + ei t , t )ei (38) i 1 Equations 8 (Shan & Chen, 1993) Solid-Fluids interaction force: Fs Gs j (x, t ) wi s(x + ei t , t )ei (39)
i 1
Correction Equation for fieq (Shan & Velocity Correction: u Chen, 1993)
8
1
f e i 0
i
i
(40)
8
Density for each phase:
fi (84) i 0
Pressure of j ( x) j ' ( x) Gr j ( x) j ' ( x) single cell (S P( x) (41) 3 GalindoTorres et al., 2013) Young2 s Capillary pressure: Pc Pj ' Pj (42) Laplace R Equation for G j ' Gsj SCLBM Contact angle: cos s (43)
Gr
where fi = distribution function for direction i projecting from central node of single lattice, ei = discretised velocity shooting from central node to surrounding nodes of single lattice, fieq = equilibrium distribution function, x = the position vector of single node, t = time, τ = relaxation time (better close to 1), C = Δt/Δx (Δt=1 ts, Δx =1 lu, v = μ/ρ kinematic viscosity, cs = speed of sound of lattice (C/30.5), ueq = macroscale fluid Velocity in equilibrium, Fa = body force (it could be Gravity only if there is no other action), Ga = attractive intensity between fluid particles of single phase, ψj = interaction potential of phase j, wj = weight factor of phase j for each node in single cell (w0=4/9, w14=1/9, w5-8=1/36), s = a ‘switch’ value usually to be density but is switched to 1 if zero velocity solid boundary are encountered, ψ0 = constant initial interaction potential, ρ0 = constant initial macroscale density, Fr = immiscible phase repulsive force, Gr = two phase repulsive intensity (positive value), Fs = fluid-solid interaction force (attraction for wetting phase j and repulsion for non-wetting phase j’), Gs = fluid-solid attractive/repulsive intensity, ρj = macroscale density of wetting phase j, ρj’ = macroscale density of non-wetting phase j’, σs = surface tension, ϴ = contact angle, Gsj = wetting phase fluid-solid attractive intensity (negative value), Gsj’ = non-wetting phase fluid-solid repulsive intensity (positive value). Equation (32)-(35) gives the standard LBM using BGK equilibrium distribution function. Shan-Chen LBM added inter-particle pseudo forces (Equation (36)), multiphase repulsive force (Equation 38) and particle-solid interaction force (Equation (39)) to simulate compressibility of each fluid phase and interface. Due to introducing these lattice interaction forces, velocity for equilibrium distribution function has to be locally averaged using Equation (40). In the single component multiphase system, there is only in need of interparticle attractive force (Equation (36). For the multicomponent multiphase system, Equation (36) is modified to Equation (39) to conduct attraction between wetting phase and solid particles and repulsion between non-wetting phase and solid particles. When the lattice cell contains solid nodes on one side of four edges, to generate the solid adsorption behaviour, an imbalance force between fluid-fluid and fluid-solid is given by setting function distributions to unity instead of density. The C++ code for executing SCLBM drainage/imbibition simulation is given in Appendix 5: SCLBM C++ code for simulation two phase displacement in virtual soil. The data post process Matlab script for upscaling state variables are also attached after the SCLBM code.
39
3.4.2 LBM simulation setup For simulating two-phase flow in a bulk of granular porous media, a package of perfect circular disk is filled into a 500*500 square of lattice 2D domain with a small amount of particles cut by the edges of 2D domain as shown in Figure 13. The dark background is the void space, where the two-phase flow percolating through the white disk represents the soil particles in this domain. This setup aims to replicate the phase displacement phenomenon, so the right and left boundary conditions is fixed to solid as same as the outline of each grain using bounce back boundary conditions in M. Sukop, DT Thorne, Jr., (2006)
fi f i (44) where –i represents the distribution function symmetrically swapped, after a fluid collision with solid boundary. The top and bottom boundary conditions are directly controlled by density difference without any buffer zone, which represents pressure difference using LBM EOS transformation (Equation (37)). The conventional pressure and flux boundary condition proposed by Zou and He (1997) fails to be applied into multiphase LBM, because the density significantly increase after few time steps, subsequently causing very high particle interactive force compressing each phase and kinematic viscosity (relaxation time as given in Equation (34)) decreasing in regard of the input of constant dynamic viscosity. These leads to a final numerical instability. This does not happen to single phase flow because there is no interparticle attractive force introduced to increase the fluid compressibility. Wetting or non-wetting phase fluid is fully filled into void space as the initial condition for drainage or imbibition simulation. The parameters for this setup is given in Table 5. 500 lu
500 lu
Figure 13 A package of circular disk in 500*500 2D lattice domain Table 5 Lattice-scaled parameter values used in numerical experiment
Lattice-scaled Parameters
Value
Wetting phase fluid density
2.0
Non-wetting phase fluid density
2.0
Dynamic viscosity of wetting phase 0.16 fluid Dynamic viscosity of non-wetting 0.16 phase fluid Lattice unit length 1.0
40
Lattice time step
1.0
Two phase repulsive intensity
1.0
Wetting fluid attractive intensity
-0.5
Non-wetting fluid attractive intensity 0.5 Gross simulation time
500000
Time interval
5000
Density difference/Initial density
0%, 1%, 2%, 4%, 5%, 10%, 15%, 20%, 25%, 30%,35%, 40%, 45%, 50%, 60%, 70%, 80% 2.0
Initial density for two-phase fluids
Infinitesimal density for each other 0.0001 phase for mixing
3.4.3 Upscaling state variables and data post-process Once the simulation initiates, non-wetting phase or wetting phase drives into void space by density difference boundary between top and bottom, until there is an equilibrium achieved between such density difference and interfacial meniscus generated by Equation (38) and (39). In this state, the single data point on static SWRC is derived. By applying density difference from zero to a value that can drain each phase out of void domain, various capillary equilibrium condition can be produced to the corresponding saturation of wetting phase measured by counting pixels occupied by it. The result of the simulation is given in four arrays: wetting phase density, wetting phase velocity, nonwetting phase density and non-wetting phase velocity. The macroscale pressure of each fluid phase is determined using Equation (41) in a consideration of the small amount of the other phase mixing inside (S Galindo-Torres et al., 2013). Finally, the capillary pressure can be given by the difference between pressures of two mixing areas (two fluid phases) using Equation (42). The saturation of each fluid phase can be determined by counting the lattice nodes (Equation (26)), whose density is larger than a prescribed critical density. In this study, the critical density is set to 0.5 mu/lu2. The selection criterion of this critical density is on clearly distinguishing the density of each phase from infinitesimal density for the other phase. As for the measurement of interfacial area, the area of each fluid phase and solid phase can be given by counting the pixel occupied by each of them as well. Finally the interfacial are is calculated using Equation (29). With known capillary pressure, saturation for both two-phase fluids and interfacial area of the twophase meniscus on the end of each time interval, the static Pc-S-Anw or dynamic Pc-S-Anw can be extracted from equilibrium approached or transient conditions.
3.4.4 Scaling between physical space and lattice domain The state variables in SCLBM are virtual variables that have physical meaning but are not in the physical dimension. For experimental calibration, such variables need to be transformed into a dimension in a real physical field where human being are living and sensoring. Two dimension transforming equations (Equation (45) and (46)) should be used to conduct length dimension and time dimension transformation using similarity theory proposed by Buckingham (1914) as
41
Pc phyisc Lphysic
s physic
Re physic
Pc lattice Llattice
s lattice
(45)
physic Lphysic physic 1 x L Relattice lattice x (46) physic physic t t
where Pcphysic = physical capillary pressure, Pclattice = virtual capillary pressure, σsphysic = fluid surface tension, σslattice = virtual fluid surface tension in lattice unit, Rephysic = Reynold number of physical field, Relattice = Reynold number of lattice domain, ρphysic = density of fluid, μphysic = dynamic viscosity of fluid, νlattice = kinematic viscosity of virtual fluid, Lphysic = resolution of length in physical field, Δx = lattice unit length, Δtphysic = real time interval, Δt = unit lattice time. The dimension transformation can be given by temporarily assuming a ratio between Lphysic and lattice unit length as same as purely numerical studies (SA Galindo-Torres et al., 2013; S Galindo-Torres et al., 2013). For calibration against real physical experiment, this ratio needs to be adjusted, depending on glass beads size or 3D printed pore network, the domain size of REV, fluid viscosity, fluid surface tension and material wettability of beads or pore network.
3.4.5 Experimental calibration of SCLBM Shan-Chen multiphase LBM (SCLBM) is a powerful tool in the simulation of multiphase flow, which allows application of high-density ratio. However, the lattice scale parameters are all clustered into a relationship network as shown in Table 5. According to Table 5, the adjustment of parameters in SCLBM cannot be done independently (Vogel, Tölke, Schulz, Krafczyk, & Roth, 2005). As the collision step of Equation (34), no matter how user tuning density ratio or dynamic viscosity, the kinematic viscosity has to be kept in a range to ensure a suitable relaxation time. Otherwise, the numerical solution will be unstable (SA Galindo-Torres et al., 2013). Also, the interactive forces (Fr, Fs) are determined by not only three interactive intensities but also the density on each lattice node. For tuning the desired interface to match a physical multiphase experiment as well as maintaining a stable solution, initial density, kinematic viscosity and three interaction intensities have to be considered into tunning together. Moreover, the LBM domain is solved on a thermodynamic basis in an isothermal condition. Therefore, the pressure of each fluid is in the form of density. Equation of State (EOS) of LBM (Equation (37)) actually cannot provide a linear relationship between density and pressure over a high density 100 mu/lu2, because the LBM EOS does not have switch between repulsive and attractive force to replicate physical molecular interaction, but only attractive force to simulate meniscus phenomenon (Equation (38) or (39)), consequently leading to high compressibility of fluid phase (M. Sukop, DT Thorne, Jr., , 2006). Last but not least, those LBM parameters are not physical parameters, which can be determined in any physical experiment, but a series of constants applied on pseudo forces. Therefore, for using a well-developed SCLBM model engine to replicate real physical behaviour, case specific calibration between SCLBM and physical experiment is extremely necessary. Otherwise, it can only be used for qualitative analysis in virtual space made up of virtual lattices but a quantitative analysis for practical engineering cases. In tunning the fluid-fluid repulsion intensity (Gr), Martys and Douglas (2001) have some philosophy on relationship between density for each fluid and Gr as
Gr
1 (47) 36( j j ' )
where notations are given in Table 5 (original equation rearranged by Schaap, Porter, Christensen, and Wildenschild (2007)). On the purpose of studying capillary pressure - saturation relationship using SCLBM, Schaap et al. (2007) conducted a parameter study on lattice density and repulsion intensity
42
influencing surface tension by simulating single bubble. A monotonic relationship was found between the scaled fluid-fluid interactions (Grρj) and the scaled surface tension (σs/ρj) on four densities (1, 1.5, 2, 3 mu/lu2), and also the limitation of Grρj is given from 0.028 to 0.09 (Schaap et al., 2007). On the other hand, their study on relationship between scaled density (ρ(x)/ρj) and Grρj shows that when Grρj is over 0.028, two-phase mixing start to be alleviated until 0.06, at which two phase are not completely dissolved to each other (Schaap et al., 2007). Therefore, the window for adjusting Gr and ρj to generate immiscible two-phase flow with a stable solution is 0.6-0.9 in their product. It should be noted that the density (ρj) is not initial density for each phase. When an unstable solution observed in the visualization of each simulation, initial density has to be tuned to rerun the simulation until the stable solution generated. Schaap et al. (2007) also studied contact angle changing by fluid-solid attractive/repulsive intensity (Gs) using tube simulations to identify that when Gr is set to 0.0522, Gs varied between 0 and 0.02. However, due to the wall effects and lattice resolution, the scaled surface tension between tube and bubble simulation will not be identical (Schaap et al., 2007). Also, Gs does govern not only the solid adsorption but also the fluid compressibility. The total aim of this calibration is to tuning the parameters mentioned above so that the simulation from SCLBM can truly replicate the physical phenomenon given by two-dimensional transparent soil model. Once such a calibration is accomplished, the numerical experiments running on a supercomputer can produce a large amount of experimental data to validate macroscale theories in a faster speed than implementing the 2D physical model. Schaap et al. (2007) successfully calibrated D3Q19 SCLBM against water-air two phase system in a 1-D column of glass beads using X-ray tomography. Later, Porter et al. (2009) studied the uniqueness of static Pc-S-Anw taking advantage of this calibrated SCLBM. However, there is no effort into studying dynamics, which is the goal of this study. Also, due to a shortage of methods in physical calibration of SCLBM, to the best of author’s knowledge, there is so far no study on studying the relationship between real interfacial physical variables and lattice parameters. Inspired by a study from Behraftar S (2015), who attempted to build up the function of microscale parameters in Discrete Element Modelling rock propagation and macroscale material mechanical parameters using Buckingham (1914) π theory, a relationship (or a function) between parameters of interfacial physics and virtual parameters employed in SCLBM might be a new way to reduce physical calibration effort in future.
43
4.
Preliminary Result and Discussion
In this chapter, preliminary result will be given in sequence of fundamental geotechnical properties, soil water retention and hydraulic properties, preparation of 1D IPM test and finally the result from SCLBM to study the dynamic Pc-S variation with time.
4.1
Fundamental geotechnical parameters and soil classification
4.1.1 Grain size distribution
Figure 14 Grain Size Distribution of four potential sample used for IPM test
Four type of soil is selected as potential options for 1D IPM test. Three of them are sand, so the sieving analysis was applied to determine the Grain Size Distribution by following ASTM D6913-04 (2009). The other silica silty powder was tested by Hydrometer following ASTM D422-63 (2007). The Grain Size Distribution for each soil is plotted in Figure 14. According to Unified Soil Classification System (USCS), Bribe Island Sand, Budget Bricky’s Loam and 30/60 sand are poorly graded sand with their coefficient of uniformity less than 3. The Bribe Island Sand and 30/60 sand are medium sand while the Loam belongs to fine sand with a slightly higher coefficient of uniformity. All sand have fine content lower than 0.02%. Silica powder is fully in the fine content 98.14% with only 1.86% clay particles. The GSD slope of silica powder is more gradual than poorly graded sand with the coefficient of uniformity up to 33.87. According to USCS, it is classified as purely silt. A details of four soil classification is listed in Table 6. Table 6 Soil Classification and Sieving analysis
Soil Classification
Bribe Island Sand
Budget Bricky’s Loam
D10 (mm) D30 (mm) D60 (mm) Mean Grain Size D50 (mm) Sand content (%) Fines content (%) Clay content (%)
0.2232 0.2912 0.3826 0.3492 99.88% 0.02% -
Silica powder (SIL220-3LA)
0.1299 0.2112 0.3103 0.2747
30/60 sand (SAN7453LA) 0.3717 0.4081 0.4446 0.4326
99.99% 0.01% -
99.99% 0.01% -
98.14% 1.86%
44
0.0005 0.0062 0.0155 0.0122
Coefficient of Uniformity (Cu) Coefficient of gradation (Cc) USDA Texture USCS (ASTM) Texture
1.71
2.39
1.20
33.87
0.99
1.11
1.00
5.51
Sand (Sa) Poorly graded sand (SP)
Sand (Sa) Poorly graded sand (SP)
Sand (Sa) Poorly graded sand (SP)
Silt (Si) Silt (ML)
4.1.2 Range of packing condition and density control The soil pore structure is not only dependent on the grain size distribution and coarse/fine mixing, but also the compaction effort on it, which is quantified by density index like dry density, void ratio, porosity and relative density index. With different compaction effort, the density of testing specimen will achieve different porosity. To quantify the standard deviation of porosity, for sandy soil, the maximum and minimum dry density is usually tested in order to calculate the relative density index for certain packing condition. The Australian Standard AS 1289.5.5.1 (1998) is followed to determine the maximum dry density using a vibrating table and the minimum dry density using funnel method for Bribe Island Sand and Budget Bricky’s Loam. The testing result is given in Table 7. For the medium sand (Bribe Island Grey Sand), the porosity can be controlled from 41%-48%. For the loamy sand (Budget Bricky’s Loam), the porosity can be controlled from 39%-52%. Other sample is not taken into consideration in the first stage because 30/60 sand is a backup for medium sand, and silty silica powder is used for mixing with sand to study dynamic SWRC in the well-graded soil. With the success of the first stage on both medium and fine sand, the well-graded soil will be applied in 1D IMP. Table 7 Density of Sandy soil and density control parameters
Density Parameters Specific Gravity Gs Maximum dry density ρd,max (g/cm3) Minimum dry density ρd,min (g/cm3) Maximum void ratio emax Minimum void ratio emin Maximum porosity nmax Minimum porosity nmin
Bribe Island Sand 2.655 1.58
Budget Bricky’s Loam 2.655 1.61
1.39
1.28
0.91
1.08
0.68
0.64
48%
52%
41%
39%
In addition to the standard geotechnical test on density control index, another non-standard 1D soil column test was conducted by compacting each sand layer by layer to study the porosity variation along sand strata. The dry density, porosity and relative density index profile are given in Figure 15. For the medium sand, the porosity can be controlled at 44.5%±3%; dry density is at 1.40±0.08 g/cm3. The relative density index seems not a good parameter to quantify density control. For the fine loamy sand, the porosity was controlled at 47.2%±2.6%; dry density is at 1.47±0.07 g/cm3. Chen (2006) studied dynamic SWRC on four sandy soil and a porosity standard deviation for 2.6% is accepted for an REV scale experiment on the medium sand. For a soil column in 45 cm, a density control with only 2.6% is hence accepted for large scale 1D IPM test in this experimental study. The details of sand packing condition study are summarised in Table 8.
45
Soil specimen Height (cm)
Soil specimen Height (cm)
45 40 35 30 25 20 15 10 5 0
Density profile Mean density
1.2
1.4
1.6
50 40
Density profile
30
Mean density
20 10 0
1.8
1.2
1.4
Dry Density (g/cm3)
(b)
50
40
40
Soil specimen Height (cm)
Porosity profile
45
Soil specimen Height (cm)
1.8
Dry Density (g/cm3)
(a)
Mean porosity
35 30 25 20 15 10 5 0
Porosity profile
35
Mean porosity
30 25 20 15 10 5 0
0.3 0.35 0.4 0.45 0.5 0.55 0.6
0.3 0.35 0.4 0.45 0.5 0.55 0.6
Porosity
Porosity
(c)
(d)
50 45 40 35 30 25 20 15 10 5 0
Dr profile
Soil specimen Height (cm)
Soil specimen Height (cm)
1.6
Dr mean
40
Dr profile
35
Dr mean
30 25 20 15 10 5 0
0
0.5
1
0
Relative density index Dr
0.5
1
Relative density index Dr
(e)
(f)
Figure 15 Density profile of 1D IMP soil column pre-test (a) Bribe Island Sand dry density profile (b) Budget Bricky’s Loam dry density profile (c) Bribe Island Sand porosity profile (d) Budget Bricky’s Loam porosity profile (e) Bribe Island Sand relative density index profile (f) Budget Bricky’s Loam relative density index profile
46
Table 8 Density control in a pre-test of 1D sand column
Packing condition for 1D soil column pre-test Mean density (g/cm3) Dry density standard deviation (g/cm3) Dry density relative deviation Mean Porosity (%) Porosity standard deviation (%) Porosity relative deviation Mean relative density index (Dr) Relative density index standard deviation Relative density index relative deviation
4.2
Bribe Island Sand
Budget Bricky’s Loam
1.40 0.08
1.47 0.07
5.57% 44.48 3.09
4.90% 47.28 2.59
6.96% 0.473
5.47% 0.408
0.412
0.211
87.11%
51.87%
Static Soil Water Retention Curve (SWRC) and Predicted SWRC by Pedo Transfer Functions
4.2.1 Testifying available SWRFs of non-deformable soil Standard Hanging Column Method was applied to test the SWRC for two sand. The density control is given in Table 9. Since this is the first test after experiment setup, two sand were just loosely compacted so both specimen only achieve low relative density index with a high porosity closer to minimum density. This procedure can generate a very loose pore matrix with much large pore throat inside. Thus, this specimen preparation will lead to a lower Air Entry Value (AEV, also named bubbling pressure) and more ink-bottle effect inducing more bubble trapping inside during the wetting process. Nevertheless, loose pore structure needs less time to achieving equilibrium condition due to high permeability. It speeds up the experimental time consumption. As for practical engineering test but experimental academic discovery, density should be controlled to a value identical to field condition. Table 9 Density control of sand specimen for Hanging Column Method
Density condition for static SWRC Saturation S Volumetric water content ϴ Gravimetric water content w Porosity n Void ratio e Dry density (g/cm3) ρd Total density (g/cm3) ρT Specific gravity (g/cm3) Relative density index Dr
Bribe Island Sand
Budget Bricky’s Loam
100% 0.472 33.60% 47.15% 0.892 1.403 1.874 2.655 0.078
100% 0.485 35.54% 48.55% 0.944 1.366 1.816 2.655 0.309
Primary drainage SWRC of Bribe Island Sand and Budget Bricky’s Loam are individually depicted in Figure 16(a) and (b). Four classical SWRC models (Brooks (1964) BC, Van Genuchten (1980) three parameters VG, D. G. Fredlund and Xing (1994) FX and Van Genuchten-Mualem two parameter G) are fitted into the primary drainage SWRC. The drainage curve data were fitted using SoilVision
47
Database (2005)©. According to Figure 16, it can be seen that FX model offer the best-fitting performance among all options. It accurately calculates the AEV and residual volumetric water content. This accuracy is not able to be achieved by Van Genuchten model. However, FX model has a much complex mathematical expression, which is not often available in the hydrogeological software package. Its usage with E. C. Childs and Collis-George (1950) HCF requires more numerical calculation on the determination of relative hydraulic conductivity (D. Fredlund, Xing, & Huang, 1994). Comparing all model performance, the two parameters Van Genuchten model (G) is the second best. Thus, it is the initial guess for future direct modelling in inverse modelling. Brooks and Corey (BC) model also give a good fitting. However, there is a natural discontinuity in BC model at AEV. It cannot capture the small amount of moisture loss around bubble pressure that might lead to inaccuracy and numerical challenges in solving nonlinear diffusion equation. On the other hand, sandy soil, especially loose compacted sand, should have a low residual water content. Further increasing soil suction above the threshold at residual water content should not have significant moisture reduction. Comparing FX to VG model, VG might be a better option to account suction above this threshold because the residual moisture content can be maintained in high suction range. In contrast, FX model with high suction correction factor mathematically generates a gradually reducing slope of SWRC, which is not often observed in experimental data of sandy soil.
(a)
(b)
48
Figure 16 Non-deformable soil SWRFs fitting performance on primary drainage curve (a) Bribe Island Sand (b) Budget bricky's Loam
4.2.2 Primary and main SWRC loops of Bribe Island Grey Sand and Budget bricky’s Sandy Loam In Figure 17 and 18, the primary and main SWRC for medium sand and fine sand are both plotted. The important hydraulic fitting parameters are summarised in Table 10 and 11. Experimental data were fitted using HYPROP® Data Evaluation Software. Regarding the requirement from this software, suction is transformed in logarithmic form for fitting so fitting parameters are only can give a relationship between volumetric water content and logarithmic suction name pF in HYPROP®. Table 10 Fitting parameters for Bribe Island Grey Sand
SWRC
VG
nVG
s
r
RSME
Primary Drainage Primary Imbibition Main Drainage Main Imbibition
0.0259 0.1760 0.0424 0.1193
6.358 2.254 4.855 1.683
3.5% 0% 2.5% 0%
46.8% 25.3% 25.2% 24%
0.0070 0.0062 0.0009 0.0108
Table 11 Fitting parameters for Budget bricky's Sandy Loam
SWRC
VG
nVG
s
r
RSME
Primary Drainage Primary Imbibition Main Drainage Main Imbibition
0.0142 0.188 0.0341 0.0289
8.455 1.473 2.724 2.524
7.8% 0% 6.8% 4.9%
48.1% 28.3% 28.7% 25.7%
0.0131 0.0183 0.0070 0.0044
Checking through both Table 10 and 11, and Figure 17 and 18, it is clear to find that the air trapping content is much higher than residual water content (ϴr). The bubble trapping content is about six times of residual water content for medium sand. For fine loamy sand, the bubble trapping content is about three times of residual water content. This anomaly is not consistent with previous experimental data that the trapping non-wetting phase is usually twice of trapping wetting phase or even less (Bottero, Hassanizadeh, & Kleingeld, 2011; Gallage, Kodikara, & Uchimura, 2013). One possible reason for so large bubble trapping could be losely compacted sample having more fraction of large void space in pore matrix, subsequently causing more ink-bottle effect, and wetting phase snap off during the wetting process. Another issue is the water content at zero suction for primary and main imbibition curves. When conducting an experiment to achieve a data point at exact zero suction, there is never ending of water absorption so data point can only be very close to zero suction but cannot be exactly on it. Therefore, there is a little different water content at zero suction for primary imbibition and main curves. Some other problems are the intersection between main curves in Figure 17 and main curves collapsing into one in Figure 18. These problems might refer to the heterogeneity of sand packing condition. When density is controlled to achieve same value, it is only a macroscale index controlled by compacting certain mass sample into the aimed volume. Although the macroscale density is controlled to be same for two specimens, there is no guarantee that the microscale pore structure will be identical to each other. However, this is only one explanation for such anomaly. Regarding testing procedure and suction calculation, there are also many other problems in the standard method.
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Reviewing both experimental setup, testing procedure and data post process, there might be some reasons regarding sample setup and suction calculation. The endless water absorption for imbibition curves near zero suction indicated that zero suction might not be really achieved when water table in the outlet reservoir is as same level as HAE disk. ASTM D6836-02 (2003) recommend this reference level as zero suction head and 48 hours as waiting period for capillary equilibrium. Nevertheless, it is only sensible for primary drainage SWRC of a specimen sampled in an O-ring less than 2 cm in thickness. As for imbibition and main drainage or even hysteresis loops, there is no recommendation for the testing procedure and data calculation in details. This raise a question that the specimen thickness in a column might impact on the testing result. The thicker the sample loaded inside the hanging column, the more suction deviation appears for the finite section of specimen laying above HAE disk. The smaller suction profile, therefore, is generated above the HAE disk leading to a spatial distribution of suction value vertically along the specimen. If the standard suction determination is still followed, the Air Entry Value (AEV) for drainage curves and Water Entry Value (WEV) for imbibition curves will be shifted to a smaller value because the water in thicker specimen is more easily drained out. This phenomenon has been confirmed by several experimental studies using ATT (Bottero et al., 2011; Kang et al., 2014; Sakaki & Illangasekare, 2007). The original design of hanging column method aims to characterise the SWRC of a zero-dimensional soil REV. For 1D soil specimen, it assumed the thickness of the specimen is large enough to achieve the statistical stability of porosity, while the thickness is also small enough, so the matrix suction head counting between water table and HAE disk has no spatial effect on the specimen. However, in sample setup, this is a problem referring to experimental sensitivity analysis of spatial effect. The hanging column test was re-set up for different specimen thickness. The test is still ongoing to investigate the sensitivity of specimen thickness on SWRC and hysteresis loops. It is expected that the conclusion can be drawn more clearly when sufficient data is collected. On the other hand, endless water adsorption around zero suction referencing to HAE disk might also indicate that there is more information between positive water pressure and water content in the saturated zone. This experiment is the first try after the first setup, therefore, regarding suggestion from the standard test procedure, there was no attempt on applying positive pressure. However, the hysteresis between positive pressure and saturation has been both experimentally and numerically found in studies from Scheuermann et al. (2009) and Scheuermann et al. (2014). In the test ongoing, zero suction will not be simply applied by the same level with HAE disk. Instead, Water table above HAE disk will be applied to check the SWRC on the range of positive pressure.
Figure 17 Primary and main drainage-imbibition curves of Bribe Island Sand
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Figure 18 Primary and main drainage-imbibition curves of Budget bricky's Loam
Although this preliminary result only provides a reliable SWRC for primary drainage process, it raises more potential problems and motivates the investigation of specimen thickness sensitivity analysis and SWRC on positive water pressure range. Repeating more tests in both different sample scale and pressure range will benefit the understanding of the phase displacement behaviour and constitutive relationship determination. Regarding these series of primary and main curves, it is still hard to draw any conclusion on increasing capillary storage coefficient (slope of SWRC) on the main curves, because of the questionable waiting period for approaching equilibrium. But the trend clearly demonstrates that after rewetting and redrying process, the AEV and WEV significantly shifted to a smaller threshold, implying that simply applying primary drainage SWRC might not be a good option for Richards’ equation. Capillary fringe impacting on groundwater table dynamics were investigated by P. Nielsen and Perrochet (2000) and Cartwright, Nielsen, and Perrochet (2005). In their experimental result on SWRC and non-hysteretic Richards’ equation simulation, a shifted SWRC with higher capillary specific storage and lower AEV applied to non-hysteretic model gave a better prediction and agreed with the experimental result. This also confirmed that primary drainage SWRC inserted back to non-hysteretic nonlinear diffusion model will jeopardise the model prediction. AEV and slope of SWRC for real dynamic conditions will be totally different with static primary drainage curve. However, in these series of study, the imbibition curves are not experimentally determined but generated by shifting AEV of primary drainage curve without even characterising the bubble trapping content. The information for analysing the shifted curve, primary and main curves are not given in most last these series of groundwater table dynamics studies (Cartwright, 2014; Shoushtari, Nielsen, Cartwright, & Perrochet, 2015). The static SWRC of their sand is still sourced from the previous study of P. Nielsen and Perrochet (2000).
4.2.3 Calibration of Fredlund and Wilson Pedo-Transfer Function (PTF) by experimental data for determination of SWRC under different porosity With an aim on reducing time and effort for the determination of static SWRC in various pacing condition, Pedo-Transfer function offers a chance on predicting static primary drainage SWRC. Once the PTF is calibrated by the experimental result, through changing the density factor, the SWRC under different packing conditions can approximate. Popular PTFs are programmed in SoilVision Database© (2005). Here the experimental data of primary drainage curve for both sand are used to calibrate two
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models available in SoilVision (Arya & Paris, 1981; M. D. Fredlund, Wilson, & Fredlund, 2002). Other models in software package were tried but only these two give a reasonable prediction. The prediction against data for each sand is separately shown in Figure 19 and 20. The statistical difference between PTF prediction and experimental data are quantified by R-square value. The R-square value and predicted AEV are listed in Table 12 and 14. For the medium sand, two PTF gives an acceptable prediction with the R-square values above 90%. However, each of them has a shortage on prediction. Arya and Paris (1981) PTF shows a better prediction for AEV than M. D. Fredlund et al. (2002) PTF, but the prediction on residual water content is really poor. The packing factor in M. D. Fredlund et al. (2002) PTF is a packing porosity assumed to each mass fraction of soil grains. In natural soil packing condition, the fine and coarse particles are perfectly mixed. Therefore, this packing factor should be a function of particle size in each fraction but be same for each mass fraction. This explained why there was still a significant discrepancy between PTF prediction and experimental data, even after fitting PTF into experimental data to acquire the optimised packing factor. After all, none of these PTFs can be compared to SWRC models like D. G. Fredlund and Xing (1994) with an R-square value 99.87%. Comparing the AEV given from PTFs and SWRC fitting model, Arya and Paris (1981) seems the best option so far in SoilVision Database package. For the fine sand, both two PTFs fail to predict not only the AEV but also the slope of SWRC and residual water content. Even the porosity factor was manually adjusted to fit into data points, there is no better agreement achieved between data and PTFs. Therefore, the available PTFs predicting primary drainage SWRC based on grain size distribution and packing condition in SoilVision Database cannot satisfy the requirement for the cases in these experimental data. In the future, more recent developed PTFs such as Perera, Zapata, Houston, and Houston (2005) and Scheuermann and Bieberstein (2007), will be applied into validation against the experimental data generated from the this result and future result in ongoing test.
Figure 19 Comparison between Experimental data and Fredlund & Wilson and Arya & Paris Pedo-Transfer Function on Bribe Island Sand Table 12 Parameters of PTF compared to SWRF for Bribe Island Sand
Parameters R2 Air entry value (kPa)
M. D. Fredlund et al. Arya and Paris (1981) (2002) 0.9364 0.9418 1.75 2.39
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D. G. Fredlund and Xing (1994) 0.9987 2.54
Figure 20 Comparison between Experimental data and Fredlund & Wilson and Arya & Paris Pedo-Transfer Function on Budget bricky's Loam Table 13 Parameters of PTF compared to SWRF for Budget bricky's Loam
Parameters R2 Air entry value (kPa)
M. D. Fredlund et al. Arya and Paris (1981) (2002) 0.7987 0.9045 1.71 2.88
D. G. Fredlund and Xing (1994) 0.9986 5.07
4.3 Hydraulic conductivity and unsaturated hydraulic conductivity by Hydraulic Conductivity Functions (HCF) 4.3.1 Saturated hydraulic conductivity The saturated hydraulic conductivity for each sand, also named coefficient of permeability in geotechnical engineering, was tested following AS 1289.6.7.1 (2001). The result is summarised in Table 14, in which packing condition and intrinsic permeability are included with a permeability classification according to Jacob (1972). It is noted that there is only one packing condition achieved for each sand in Constant Head Method. The experimental result is so used to calibrate the Kozeny-Carman (KC) equation (Chapuis & Aubertin, 2003) in order to estimate the hydraulic conductivity for different porosity within the maximum and minimum porosity tested by Standard Method (AS 1289.5.5.1, 1998). To confirm this estimation, more Constant Head Method should be conducted in different packing condition in the future. However, such test is very fundamental and basic. It could be a separated out from this experimental study as an independent experimental study for undergraduate students on acquiring a more practical understanding of single phase seepage through the saturated soil. Table 14 Hydraulic conductivity and packing condition for two sandy soil
Density condition for constant head permeability test Saturation S Volumetric water content ϴ Gravimetric water content w Porosity n Void ratio e
Bribe Island Sand
Budget Bricky’s Loam
100% 0.434 28.86% 43.38% 0.766
100% 0.480 34.73% 47.97% 0.922
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Dry density (g/m3) ρd Total density (g/m3) ρT Specific gravity (g/cm3) Relative density index Dr Saturated hydraulic conductivity ks (cm/s) Intrinsic permeability Ks (cm2) Permeability Classification according to Jacob (1972)
1.500 1.937 2.655 0.625 3.64*10-2
1.389 1.861 2.655 0.359 3.91*10-3
3.64*10-7
3.91*10-8
Well Sorted Sand
Well Sorted Sand
According to Table 14, even the porosity of medium sand is 5% lower than fine sand, the permeability of medium sand package is one magnitude over the finer. This confirmed the contribution of tortuosity to hydraulic resistivity that is not only contributed by porosity and viscous friction on solid particles. As for a finer particle package, although a higher porosity is achieved, due to finer particles packing into unit REV, more twisted flow channels are generated than a coarse beads package. There is also a higher beads specific surface further increasing the frictional solid-fluid interfacial area. Based on this experimental result, it can be presumed that tortuosity has more influence on the permeability of sand package than porosity. When taking consideration of packing condition influencing the permeability of an REV specimen, simply studying the relation between permeability and porosity might not be sufficient.
4.3.2 Calibration of Kozeny-Carman equation against experimental result for prediction of hydraulic conductivity in different porosities Density control of sandy soil sometimes is hardly achieved to completely identical density for different geotechnical tests. Also, for making any comparison between standard tests and novel testing method, there should be an identical density of specimen in each test. To reduce errors resulting from different packing conditions, the hydraulic conductivity-void ratio (or porosity) relation is ought to be calibrated by experimental data to determine the shape and tortuosity constant, so that the saturated hydraulic conductivity under a different porosity can be predicted using this relation. Kozeny-Carman (KC) equation provides such a solution by assuming that clean sand package is similar to a package of spherical beads having similar GSD. One popular form of KC equation is given as
C g e3 ks de 2 (48) 36 w w (1 e) where ks = saturated hydraulic conductivity (m/s), C = a factor to account shape and tortuosity of pore channels, de2/36 = specific surface of spherical package of clean sand, de = particle diameter (m), g = gravitational accelerator (m/s2), μw= dynamic viscosity of water (kg/m·s), ρw = density of water (kg/m3) and e = void ratio of sand package (Chapuis & Aubertin, 2003). Based on the prediction from Equation (48), the saturated hydraulic conductivity of two sandy soil are plotted against various porosity in Figure 21. Here an assumption is made that the shape and tortuosity factor will not be varied significantly so it can maintain the range of porosity from maximum to minimum. In fact, this might not be practical. It needs more fundamental single phase seepage experiment to testify if this assumption is acceptable.
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Figure 21 Saturated hydraulic conductivity against porosity (a) Bribe Island Sand (b) Budget brick Loam
4.3.3 Unsaturated hydraulic conductivity of Bribe Island Grey Sand and Budget bricky’s Sandy Loam predicted by Hydraulic Conductivity Function (HCF) models The unsaturated hydraulic conductivity of two sand were not experimentally determined by unsaturated large triaxial test under steady state condition yet. Until now, the HCF can only be approximated using three classical HCF models: Brooks (1964) and Van Genuchten (1980) SWRC models inserted into Mualem (1976) statistical HCF model or D. Fredlund et al. (1994) SWRC model inserted into continuous form of E. C. Childs and Collis-George (1950) statistical HCF model. Each model equation can be simply sourced from any HCF review papers (D. Fredlund et al., 1994; Leij, Russell, & Lesch, 1997; Van Genuchten, 1980). Three model predictions are plotted for medium sand in Figure 22 and fine sand in Figure 23 in both suction basis and volumetric water content basis. For plotting each model result, there is only a need of inserting fitting parameters of SWRC models into HCF model equations. For the unsaturated hydraulic conductivity-suction prediction, BC-M model agrees with FX-CCG model while VG-M model predict the hydraulic conductivity lower than other two models around AEV as shown in Figure 22 (a). This disagreement is even more obvious for finer sand in Figure 23 (a). This phenomenon was also found in a previous experimental study on validation of these equations (Gallage et al., 2013). In their experiment against three models, FX-CCG and BC-M model prediction are more consistent with steady-state experimental data but VG-M underestimated hydraulic conductivity between residual water content and AEV for fine sand (Edosaki soil), and this underestimation becomes even larger for finer soil (Chiba soil) having fine content up to 30% (Gallage et al., 2013). This is also consistent with the HCF prediction in this study, where there is smaller underestimation for medium sand in Figure 22(a) and larger underestimation for fine sand in Figure 23(a)). For the unsaturated hydraulic conductivity-volumetric water content prediction, there is less difference between BC-M and VG-M models, while compared to previous two models the FX-CGG overestimates the hydraulic conductivity between residual water content and AEV. However, due to hydraulic conductivity-suction pressure or suction head is more popular used in solving suction head of Richards’ equation. Most of the studies do not provide a comparison and contrast between test result and model performance on water content basis. Here, we cannot judge which one is more accurate because we have not conducted steady state two phase seepage through the unsaturated sand sample. Learing how to use unsaturated soil triaxial is a very useful for geotechnical engineering. Even steady state hydraulic conductivity test for unsaturated sand has few contribution to understanding the transient multiphase flow, it is still worth to be studied for undergraduate level education. In future, this test can be set as a project for summer research or undergraduate student research assignment. It will not only provide data on validation of steady-state HCF models but also educate a student on practically using high-level geotechnical testing equipment.
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(a)
(b)
Figure 22 Hydraulic conductivity of unsaturated Bibe Island Sand (a) Suction basis (b) Water content basis
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(b)
Figure 23 Hydraulic conductivity of unsaturated Budget bricky's Loam (a) Suction basis (b) Water content basis
4.4
1-D soil column experiment preparation
4.4.1 Calibration of T5 tensiometer to data logger and sensor sensitivity analysis Twenty-one T5 tensiometers are connected to the Geodatalogger DT85G®. This data logger is not custom-manufactured for logging T5 tensiometers, so the output from logger is not the water pressure calibrated by the manufacturer of T5 (UMS). Therefore, to study how voltage response of 21 T5 sensors logged by DT85G logger, a specific logger-sensor calibration has to be done prior to any measurement. The voltage output of DT85G is not in real voltage but a dimensionless ratio between excite and output voltage calculated by Equation A43 in Appendix A2: Calibration of 21 T4 tensiometers to DT85G Datalogger. Each of 21 sensors is logged for more than 2 hours for each pressure value applied by the inserting sensor under water table certain depth. Five positive pressure values are applied by setting sensor at a different depth below the water table. Linear regression can be fitted into data point between real water pressure and voltage ratio, depicted in Appendix A2. The regression equations are summarised in Table 20. After logging the pore pressure, these equations will be used for each sensor that is specifically corresponding to each analog channel on DT85G logger. Mixing channels numbers with sensor numbers will alter the resistivity. Thus, the linear regression fitting equation will lose their accuracy on voltage-pressure transformation. Therefore, to replicate the suction logging setup, sensorchannel number should be absolutely matched. Otherwise, the calibration has to be reconducted for
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mismatching cases. Furthermore, although UMS recalibrated seven old sensors in the storage of Geotechnical Laboratory, most of them already have an offset value, so the reading from any UMS logger or hand reader are not accurate. The offset for these old sensors is summarised in Table 21 in Appendix 2. These offset has been considered in logger-sensor regression equations. It only needs to be considered when using UMS logger or Infield7 hand-reader device. Sensor sensitivity analysis was also checked by logging hydrostatic pressure for 2-3 days. An example of logging noise sensitivity diagram is shown in Table 22 in Appendix 2. Other sensors are also checked using the same method. According to the data of sensitivity and bias analysis, we found that our new suction logging setup can give a suction value with standard deviation 0.1 kPa due to electrical noise within logging system. Most of the inaccuracy is due to the sensor body dropping off the elevation of the ceramic tip. This bias error is ranging from 0.2-0.5 kPa depending on how much sensor body is dropping off the level and variation of sensor body during logging period. To automatically logging 21 sensors and post-process data, a DT85G Control programme is coded and simple Matlab Script is written. They are also listed in Table 23 separately.
4.4.2 Setup of STDR, CR1000 logger and SDM X50 Multiplexer The setup of four STDR connected to one SDM X50 Multiplexer is controlled by TDR100, and the waveform will be recorded by CR100 logger. To logging waveform for inverse calculating moisture distribution along the sensor, a CR1000 specific code is written for automatically logging four Flat Ribbon Cables for both moisture and electrical conductivity measurement. This code is not verified yet. It will be tested with water pumping into empty IPM column to further determine some parameters in logging code in terms of logging period, points on wave and delay time between two end waveform recording. The pre-test in a 2 meters saturated sand column was implemented to determine the suitable window length of the waveform for analysis of moisture profile and electric conductivity. Each sensor is 2 m long, and window length for moisture profile is 5.5 m. Each cable is 5 m long, and the apparent cable length is 5.7 m. The Vp value is set 0.67 according to the instruction of TDR100. The electrical shock is added on both ends of the sensor to characterise the reflection coefficient peak to determine this window length. In future, more time has to be spent on setting up this logging system, because it is the first time for connecting self-made STDR to CR1000 and SDM X50 Multiplexer. So far SDM X50 was only used for conventional TDR logging. There is no any previous example on building up STDR logging based on a commercial product without technique support. The progress of this logging system is summarised in Appendix 3: Program for CR1000 Logger logging 4 STDR Flat Ribbon Cable.
4.4.3 Setup of Accumulative bottom flux measurement The accumulative flux in the bottom of soil column is measured using Ohus RangerTM Count 3000 Series electrical balance. Four E-scales having RS232 interface are connected to one USB hub and then to the logging computer. There are four bench scales corresponding to four soil columns. The appending software is free to access online to automatically logging many scales simultaneously. This scale is set under the constant head tank attaching to the bottom inlet of the soil column. The weight variation can be logging every 10 seconds which is enough resolution of time step for both one-step outflow/inflow test. Each scale has a maximum measuring capacity 30 kg with a resolution of 1 g. As for multistep inflow/outflow test, Variable Height Overflow was borrowed from Dr. Nick Cartwright already. It is available in Geotechnical Engineering Teaching Laboratory. However, to generate sinusoidal varying water table, in the future, more effort has to be spent on electrical motor purchase and transmission gear box design, which relates to a mechanical engineering discipline. This has not been finalised yet.
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4.5
2-D Multiphase Lattice Boltzmann Model (LBM) of drainage and imbibition
4.5.1 Grain size distribution and pore size distribution of virtual beads package in SCLBM simulation domain
Figure 24 Pore size distribution and Grain size distribution of virtual grains in 2D LBM simulation Grain Size Distribution (GSD) is measured using the Image Toolbox built in MATLAB© R2014b. The GSD and PSD of virtual soil are shown in Figure 24. Pore Size Distribution (PSD) is measured using the image processing algorithm developed by Rabbani, Jamshidi, and Salehi (2014). The output images of virtual soil particle, void space segregation and the histogram of both GSD and PSD are shown in Figure 25 (a)-(d). This algorithm applies Watershed segmentation on pore constriction size to segregate the void space into discrete sections. Then the size is calculated by determining the diameter of the radius fitted into each void section. The Matlab code for measuring GSD and PSD are given in Appendix 4: Grain Size Distribution and Pore Size Distribution of virtual soil in SCLBM simulation domain. According to Figure 24 and Figure 25, the PSD is wider than GSD and pore size is larger than grain size, so the packing condition is not identical to natural soil but similar to glass beads package with a lot of interconnected pore channels.
(b)
(a)
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(c) (d) Figure 25 Information of virtual soil grain and pore size in LBM simulation (a) output image of particle size measurement (b) output image of pore size measurement (c) Histogram of Grain size distribution (d) Histogram of Pore size distribution
4.5.2 Bubble simulation results for determination of lattice surface tension The lattice surface tension is calculated using Equation (42). This study does not stress on wettability, so the as the setting parameters in Table 5, the force intensity (G) for each phase is set to generate a zero contact angle calculated using Equation (43). Because all physical variables in SCLBM are based on lattice unit, dimensionless number in Equation (45) and Reynold number in Equation (46) have to be used to bridge virtual to real. The primary step for unit transformation is surface tension ratio between real and virtual. Therefore, bubble simulation was implemented to characterise the virtual lattice surface tension based on Equation (43). By generating a bubble in various size, the linear regression can be fitted between bubble radius and pressure difference. Finally, the slope of the linear fitting equation is exact the lattice surface tension. The linear regression fitting into bubble simulation is shown in Figure 26. Fitting performance is perfect at the R-square value of 99.96%. The lattice surface tension is 0.1818 mu/Δtl2.
Pc = 0.1818*(1/R) R2 = 0.9996
Figure 26 Lattice surface tension calibration: Capillary pressure vs Radius of single bubble
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4.5.3 Static SWRC and dynamic SWRC The details of each state variables corresponding to the time are given in Appendix 6-9. The visualization of 2D SCLBM simulation are shown in Appendix 7: Visualization of 2D LBM simulation of drainage and imbibition. According to the Table 28 in Appendix 7, there is an observation that with the pressure difference increase between top and bottom boundary, more wetting phase is pushed out the beads package, and more non-wetting phase invaded the domain. When the density difference on two boundaries achieve 0.4 (lattice density), a preferential path is generated in the centre of virtual soil. The wetting phase fluid are separated into two portions on two corners of bottom separately. In this simulation, to reduce computational effort, the initial density and viscosity ratio are both set at one as listed in Table 5. Therefore, such a preferential path is not caused by different flow patterns based on phase diagram of Capillary Number and Bond Number (Lenormand, Touboul, & Zarcone, 1988), but geometrical determined, where the larger pore throat, where the non-wetting phase breaks through the entry pressure first. According to the Table 29 in Appendix, the imbibition process shows more preferential paths on the left side of the domain because the pore throats on the left are smaller than the right side. Also, a small amount of wetting phase trapping and non-wetting phase trapping can be found. However, this trapping is only a little in the domain. When the final equilibrium achieved, most of the trapping phase slowly evaporates causing a slight decrease of each trapping phase fluid, which is consistent with Schaap et al. (2007). The author suspects that this is caused by density difference boundary condition applied on top and bottom of domain and fluid compressibility due to the application of three interactive forces of SCLBM. However, Schaap et al. (2007) claimed their confidence on their results having such issues. Here, we follow their recommendations on neglecting the disappearance of phase trapping. The visualisation in Table 28 and 29 in Appendix 7 clearly demonstrates that the SCLBM physically replicates capillary flow in a pore network, and the observation is consistent with two phase displacement in natural porous media. The Static SWRC and dynamic SWRC simulated using SCLBM are plotted in Figure 27. Each data point is taken for plot after equilibrium achievement. The SWRC model of D. G. Fredlund and Xing (1994) is fitted into dataset because this model has more parameters, subsequently allowing more flexibility for fitting into a dataset in no perfect uniform S-shape. As can be seen from the imbibition data points in Figure 27 (a), there is a gradual saturation decrease of wetting phase after -0.1 kPa. Other models are tried to fit in but FX model is the only one capturing this behaviour due to high suction correction factor. Therefore, the FX SWRC model is fitted into both static SWRC and dynamic SWRC with R-square value all above 90% on the wetting process. The R-square value on drying process is only over 70%, so poor fitting is given to dynamic drainage SWRC before 3.5*105 time step. Afterwards, the FX fitting can reach 90%. The fitting performance of each dataset can be checked in Appendix 8: Fitting performance of Fredlund & Xing SWRC model into dynamic and static SWRC dataset generated by SCLBM. The Equilibrium of static SWRC is ensured by checking the SWRC varying with lattice time step shown in Figure 27 (b). To more clearly see the first order time derivative of each state variables against time, figures are given in Table 31 of Appendix 9: Time-derivative of each macroscale state variables for drainage and imbibition. After 2*105 time steps, the equilibrium is approached already with a slightly fluctuation caused by numerical artifact on meniscus instability of wetting front. The dynamic response of capillary seepage is more significantly within a first 5*104 time step. Within this period, the dynamic SWRC can be plotted, as can be checked from Figure (b)-(c). Because the data points are not in continuity, FX model are fitted into them in order to analyse the pressure difference between transient and static condition for each wetting phase saturation in a continues domain. Except some anomaly for AEV and REV of drainage SWRC, the result shown in Figure 27 replicate the primary static drainage and imbibition SWRC and dynamic SWRC. Also, having the plot of SWRC corresponding to each time frame of simulation, the dynamic SWRC can be checked through a simulation of transient two-phase displacement in the virtual soil. This is a demonstration of dynamic
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capillary effect in SWRC for a transient two-phase seepage, which is missed in our conventional theory just requiring instantaneously achievement of equilibrium on every time step. This result is the first effort on studying dynamic multiphase flow in porous media using 2D SCLBM. Although the model setup is simply, it is still a good start of learning SCLBM. Due to the dimensional constraints, the simulation result temporarily can only be seen as a simple proof. 2D setup leads to less phase trapping, and large pore throats, as shown in Figure 24 and 25, result into low capillary pressure or even positive pore pressure for imbibition process. In the future, quasi-2D or even 3D setup are worth to be constructed to get a better simulation result. Due to the low fitting performance on drainage dynamic SWRC, here, the dynamic capillary pressure theory proposed by S. M. Hassanizadeh et al. (2002) can only be validated by our dynamic imbibition curves shown in Figure 27 (d). Validating this simply theory requires lots of fitting effort on this dataset. The FX model fitting has been accomplished by assistance from Dr. Theirry Bore because the Matlab inbuilt fitting function cannot guarantee a high Rsquare value for fitting FX model into this abnormal dataset generated by 2D SCLBM. The relationship on fitting first order time derivative of saturation against each saturation is important to correlate dynamic capillary pressure difference with saturation variation. However, simply centrally differentiating the saturation to time and then correlate this time derivative to average saturation give destitute dataset for fitting. Better fitting method and data analysis procedure need to be developed. Here, so far the best can be provided an evidence of SCLBM on the simulation of dynamic effect. It is expected that the dynamic coefficient can be determined for this simulation result after an accurate dS/dt in function of each continuous saturation.
(a)
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(b)
(c)
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(d)
(e) Figure 27 (a) Static Capillary Pressure-Saturation Curve (SWRC) (b) Dynamic Capillary Pressure-Saturation-Time data with FX model fitting curves in 3D (c) 3D contour plot of evolution of Capillary Pressure-Saturation-Time step (d) Dynamic
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Capillary Pressure-Saturation with FX model fitting curves in 2D (e) Dynamic Capillary Pressure-Saturation fitting curves in 2D
4.5.4 Pc-S-Anw constitutive surface in static and dynamic conditions The other interesting theoretical framework in Advanced theory of multiphase flow in porous media is the uniqueness of Pc-S-Anw constitutive surface (S. M. Hassanizadeh & Gray, 1993b). Porter et al. (2009) confirmed the uniqueness of the static Pc-S-Anw surface using 3D SCLBM. However, there is no any information about the dynamic effect for this constitutive surface using SCLBM. Hence, the 2D SCLBM is applied here to discover the uniqueness of surface under transient condition. Karadimitriou et al. (2014) experimentally investigated this constitutive surface under transient condition using an elongated PDMS micromodel and find that this surface is not unique in transient condition and also Capillary number dependent, which means that it depends on velocity. Joekar-Niasar et al. (2010) also studied this surface under transient condition using numerical dynamic pore network model, and concluded that due to the each phase fluid-losing continuity in transient condition (different velocity dependent viscous relaxation), there is a significant discrepancy between equilibrium and noequilibrium constitutive surface. In this numerical experiment, the evolution of these three state variables are plotted in Figure 28 (a)-(f). The combination of each pair of three state variables are plotted against time and the projection on 2D plane consisting of every two variables are also shown after each 3D plot.
(a)
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(b)
(c)
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(d)
(e)
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(f) Figure 28 (a) 3D contour plot of Saturation-Interfacial area-Time step (b) a projection of 3D contour plot of SaturationInterfacial Area-Time step on Saturation-Interfacial area 2D plane (c) 3D contour plot of Capillary pressure-Interfacial AreaTime step (d) a projection of 3D contour plot of Capillary pressure-Interfacial area-Time on 2D Capillary pressure-Interfacial area plane (e) 3D contour plot of Capillary pressure-Saturation-Interfacial area with contour color bar representing Time step (d) a projection of 3D contour plot of Capillary pressure-Saturation-Interfacial area on 2D Capillary pressure-Saturation plane with contour color bar marking the Time step
From Figure 28 (a) and (b), there is a clear demonstration of the temporal evolution of wetting phase saturation-interfacial area within time domain. Checking through the color bar representing the time step, it is easy to find that there is less variation of saturation-interfacial area relationship after 1*105 time steps. Beforehand, there is a different saturation-interfacial area plot within each time step. The similar time-dependent phenomenon can also be found for the temporal evolution of capillary pressureinterfacial area relationship from Figure 28 (c) and (d). For Figure 28 (a)-(d), the curves for drainage is located on the upper side while the curves for imbibition is located downside. The saturation-interfacial area relationship follows different paths for hydraulic loading history. The saturation-interfacial area curve for drainage is above imbibition. This also replicates the physical phenomenon that meniscus area is pulled larger for drying than for wetting process in which the curvature of the meniscus alleviated by fluid inertial. It is also an explanation that there is no relationship between dynamic contact angle of meniscus and static hysteresis of SWRC, because the dynamic contact angle contributes to the capillary pressure difference between dynamic and static, but the data points of static hysteresis SWRC is taken on equilibrium state. However, including dynamic contact angle as one of the reason of static hysteresis scanning curves in SWRC is still recorded in textbook of unsaturated soil mechanics and review papers on non-uniqueness of SWRC (D. G. Fredlund & Rahardjo, 1993; Lu & Likos, 2004; Malaya & Sreedeep, 2011). Other reason for this difference could be that within a fast drainage process, more wetting phase discontinuity is generated (V. Joekar-Niasar & S. M. Hassanizadeh, 2012). However, compared to dynamic pore network model (V. Joekar-Niasar & S. M. Hassanizadeh, 2012), our setup has the limitation in beads package having large pore channels and inevitable fluid compressibility of SCLBM. As a result, there are only a little wetting phase trapping in a transient process, and finally
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they disappear after equilibrium achieved. Thus the different interfacial area for hydraulic history is mainly due to the meniscus dynamics only for this simulation, while the amount of phase trapping does not contribute to interfacial area increase during the drainage obviously. The different saturationinterfacial area curves for drainage and imbibition were also confirmed by S Galindo-Torres et al. (2013) using a 3D SCLBM in a 5*5*5 particle package. The Figure 28 (c) shows the variation of the 3D capillary pressure-saturation-interfacial area with time forwarding to the next equilibrium state for each density difference boundary condition. The temporal evolution of imbibition is on the left side while the other temporal evolution of drainage is on the right side. As can be seen from the colorbar of temporal evolution in Figure 28 (c) and (d), it is obvious that the equilibrium can only be achieved when there is no variation of specific interfacial area. Most of the dynamic variation of capillary pressure and interfacial area are exactly located within 0 - 1*105 time steps. Only within this time scale there is significantly saturation varying with time, indicating that there is real flux seepage through this assumed Darcy scale REV. Once the equilibrium achieved, the static SWRC can be plotted but there is no flow at all. When the dynamic process is ongoing from one state to next state, the capillary pressure-saturation-interfacial area is continuously changing with time steps. There is no any uniqueness of SWRC and the 3D constitutive surface for the evolution of the state. A more clear 2D projection of this 3D surface on capillary pressure-saturation plane is plotted in Figure 28 (d) where the color bar mark the time variation. Based on Figure 28 (d), one physical fact cannot be doubted is that when two-phase flow is displacing each other in a porous media, there is no exact steady state. The transition from one state to the next cannot follow the capillary pressure-saturation curve only on the static state. Also, the larger the density boundary (pressure boundary of LBM) applied, the larger the dynamic capillary pressure response and larger interfacial area variation appear during a transient process. These findings might lead to a further question on how the effective hydraulic conductivity of each phase can be evaluated based on a pure static SWRC using any empirical and statistical HCF models. According to the nonlinear diffusion theory, the temporal evolution of the volumetric water content should be equal to the divergence of the flux in 3D space. If the flux for each dimension cannot be approximated using the hydraulic conductivity predicted by static SWRC, the purpose on using such theory somehow becomes unreasonable. More interestingly, so far the HCF models are all correlated to only the static drainage SWRC under phase fraction network assumptions. Under the transient process, there is no guarantee for this continuous phase flowing through the fraction they occupied. Dynamic SWRC is an indication of the dynamic response of two phase displacement in porous media. The application of dynamic SWRC on the estimation of seepage flux so far has not been investigated yet. Each dynamic SWRC only exists on each time step and finally return to static SWRC. Incorporating dynamic SWRC into Richard’s equation, therefore, seems inapplicable. For further investigating the dynamic multiphase flow, flow governing equation for calculating two phase seepage flux should be further studied using SCLBM. From Figure 28, there are some anomalies of the numerical result. First, the capillary pressure turns to negative to become positive water pressure for imbibition curve. This is due to the large pore channels in LBM domain. Due to highly computational expenditure, the domain is controlled to a 500*500 lattice square cell. To ensure the resolution of fluid in void space, the constriction size of pore channels at least has to be several lattice unit length. Thus, the pore size is too large compared to real soil particle package. To completely saturate the loose packing beads, a positive pressure of wetting phase has to be injected into the domain from the bottom. The each saturation of imbibition curves is so achieved by only the positive pressure of wetting phase. The second problem is the reduction of capillary pressure for low saturation on primary drainage curve. When the virtual soil are packed more loosely, under large density difference boundary condition, the connection between bottom boundary and bottom layer particles will be lost. Losing fluid-particle contact represents totally drain out of wetting phase in virtual soil, so the capillary pressure return back to a smaller value or even to zero, because the entire pore structure is almost fully occupied by non-
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wetting phase. This explains why there is a variation between static SWRC and quasi-steady steady SWRC at low saturation range on drainage curves in Figure 28 (f). The loss of fluid-particle is also impacted by dynamic effect. After equilibrium, there is no such anomalies. It is expected that this numerical artefact can be avoided in a quasi-2D or 3D D3Q19 SCLBM. Here, the limitation of 2D packing is confirmed. For exact 2D setup, bottom layer of soil package has to be very close to the bottom boundary to provide a good wetting phase fluid-solid contact. Third, this numerical experiment have not considered the dimensionless capillary number to quantify the difference between dynamic SWRC and static SWRC. Thus the time step replaces it to indirectly quantify the dynamic variation. Author originally stressed on the dynamic effect in SWRC instead of HCF. To characterise the SWRC and Pc-S-Anw surface, there is only need of density field upscaling and phase pixels upscaling. Therefore, the velocity field upscaling code has not been developed. This information will be further worked out in the future plan. The last, the physical variables in this result does not match any physical experimental setup yet. It is hard to draw conclusion on how long the dynamic effect maintain. The qualitative analysis is more convincible than the quantitative analysis based on this numerical experiment.
5. Summary of preliminary progress and the entire Research Schedule The preliminary work on investigation of dynamic multiphase flow in granular porous media can be summarised into following points:
Four types of soil including two medium sands, one fine loamy sand and one silty silica powder are prepared for one-step/multistep inflow/outflow test. Zero dimensional REV scale standard experiment on Soil Water Retention Curve have a scale effect. Soil Water Retention Curve does not cease on zero pressure of wetting phase while there might be more information between postive pressure and volumetric water content. Saturated hydraulic conductivity tests have only been conducted for certain porosities and the prediction of KC equation on different porosities needs further validation using constant head test. Relative hydraulic conductivity test of unsaturated are estimated using available HCF models and it needs large unsaturated soil traxial to verify their prediction. One dimensional IPM test are on the final stage of preparation. Suction logging, volumetric water content logging and accumulative flux logging system are prepared already. The logging system need to be testified several times with purely water height variation before loading the soil into column. Two dimensional transparent pore network model is still on design. Potential options for generating transparent pore network model will be decided in future. Numerical simulation of two phase displacement in virtual soil using SCLBM clearly demonstrates the paradox that there is no flux of each phase fluid if there is static SWRC, and under flowing condition there is no static SWRC. SCLBM result does not only unveil the physical dynamic condition in two phase flow but also shows the capability of candidate on understanding and utilising multiphase flow LBM package in free Library Mechsys. In the future, the model development of multiphase flow in porous media using SCLBM will be the main objective of another candidate investigating dynamic effect in HCF. Candidate will merely focus on experimentally validating and calibrationg SCLBM using the 2D transparent pore network experiment. The findings so far only reveals the dynamic response of two phase fluid seepage but for validating advanced theories and mathematically capturing dynamics with hysteresis, more
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experimental data need to be generated to help us a better understanding of something having never been discovered. A flow chart of experimental methods on the investigation of dynamic two-phase flow in porous media is shown in Figure 29. There are three experimental stages: the standard test on REV scale of soil bulk, the 1D instantaneous profile method test in a soil column with automatically logging state variables, and 2D transparent pore network model for both visualization of phenomenon and calibration of SCLBM. Finally, the project schedule and progress are given in Figure 30. Literature Review on dynamic effect in multiphase flow in porous media Research Question and innovation identification
Improving available advanced theories No Dynamic capillary pressure relaxation theory and phase redistribution relaxation theory Standard SWRC and HCF experiment
Theory validated by experimental result
1D IPM soil column experiment Scale effect and spatial averaging for suction and saturation
Advanced thermodynamic basis theory accounting interfacial area gradient for Helmholtz free energy
2D or quasi-2D ShanChen Lattice Boltzmann Simulation experimental validation
2D Transparent soil or pore network model
Yes
Engineering Application and simulation
REV scale and upscaling microscale state variables
Figure 29 A flow chart of experimental methods on studying dynamic two-phase flow in granular porous media
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Research Schedule Gant Chart
Figure 30 Research schedule Gant Chart
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Appendix 1: Full Version of Literature Review A1.1 Overview of unsaturated soil mechanics Unsaturated soil mechanics, compared to saturated soil mechanics, additionally include air phase and the interface between soil moisture and air. The framework is still same with conventional soil mechanics referring to fundamental properties and classification, fluid seepage, stress state variables, earth pressure theory, strength and failure theory, stress-deformation constitutive relationship, soil consolidation theory and hydro-mechanical coupling, each of which is assisted with corresponding experiment to determine coefficients. Here the entire framework will not be given in details. Instead, the contribution of multiphase physics into each part will be stated as a summary of unsaturated soil mechanics.
A1.1.1 Soil Suction and Soil Water Retention Curve This total suction is defined as the free energy state of soil water. A thermodynamic-based expression of total suction could be given by Kelvin’s equation, which gives a relation between this free energy of the soil water and the partial pressure of the pore water vapour (D. G. Fredlund & Rahardjo, 1993). According to the explanation, from D. G. Fredlund and Rahardjo (1993), the total suction consists of matrix suction and osmotic suction. The matrix suction is further decomposed into suction induced by the capillary effect and short range adsorption effects including electrical and van der Waals force field (Lu & Likos, 2004). Lu and Likos (2004) stated that short range adsorption effects are most significant for fine soil having large surface area and most related to low water content condition. In this study, the non-cohesive granular soil is selected. Hence, the capillary effect, shown in Figure 28 (a), will be taken as the only one contributes to matrix suction. Also, to reduce experimental complexity, a solute-free deionized water will be used to neglect osmotic suction. The equation for soil suction is
t
2 cos RT ln( RH ) m o (ua uw ) s CRT (1 B2C 2 B3C 3 ) vw r
(A1)
where ψt = total soil suction (kPa), R = universal gas constant [8.31432J/(mol K)], T = absolute temperature (T = 273.16⁰+t⁰), vw = partial molar volume of liquid water (1.8 cm3/mol at 20⁰C), RH = relative humidity [RH = uv/uv0, uv = partial pressure of vapour (kPa), uv0 =saturation pressure of water vapour over a flat surface of pure water at the same temperature (kPa)], ψt = matrix suction (kPa), ψt = osmotic suction (kPa), ua = air pressure (kPa), uw = water pressure (kPa), π = osmotic suction (kPa), σs = surface tension of soil water (N/m), ϴ = contact angle of water-air interface, r = pore radius (m), C = molar concentration of the pore solute solution (mol/m3), Bi= viral coefficients (Shaw, 1992).
(b)
(a)
Figure 31 (a) capillary suction; (b) ink-bottle effect (Jotisankasa, 2005)
To reduce experimental complexity, the soil suction used in this study is just the matrix suction that is dominated by capillary effect
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t
2 cos RT ln( RH ) m (ua uw ) s vw r
(A2)
where the notation is same as above. In unsaturated soil mechanics, the conceptual model of the pore matrix is usually considered as a bundle of capillary tubes of tube radius statistically distributed in a soil representative elementary volume (REV). The soil suction estimated from this REV scale is an upscale matrix suction by averaging. The concept of Soil Water Retention Curve (SWRC) is the relationship between this macro scale matrix suction and saturation of soil water. After measuring the moisture content at different equilibrium capillary pressure, an S-shape curve can be plotted along the collection of these data points. An example is shown in Figure 29.
Figure 32 A sketch of Soil Water Retention Curve with first scanning curves inside for a uniform graded sandy soil
In the domain of static SWRC (Figure 2), there are mainly two boundary curves (primary drainage and primary imbibition curves) which constrains all possible SWRCs under varying history of hydraulic loading. Due to the ink-bottle effect inside pore matrix (Figure 1(b)), according to equation (2), capillary water prefer to pass through smaller pores rather than large pore, into which requires more water pressure to penetrate. Every SWRCs, except primary drainage curves, own this nature, so the hysteresis scanning loops exist between two boundaries SWRCs. Near to the high suction range, there is a small amount of irreducible film water because of short-range adsorption effect. In this condition, soil water exists in the form of molecule film strongly attracted by mineral particle surface charge and van de Waal force. Gravity or high gas pressure cannot drain them out unless the high-temperature oven is used. The water content (or saturation) in this suction range is residual moisture content (or residual saturation). Air entry value, also named as a bubbling point, is a suction point where saturation commences decreasing. This concept is usually used when the capillary flow is modelled using drainage SWRC. Based on the conceptual model of pore matrix, this point can be identified as a suction threshold where meniscus initiates in the largest pores. In fact, it is almost impossible to represent natural soil SWRC by drainage curve due to the previously mentioned gas trapping mechanism, whereas numerical modellers still prefer to drainage curve. Soil Water Retention Function (SWRF) is a continue function fitted into experimental data. This function allows continuously extracting relation between water content and soil suction for numerical modelling. As for non-deformable soil, four SWRFs are usually used in geotechnical engineering and hydrogeology (D. G. Fredlund & Xing, 1994; Jacob, 1972; Lu & Likos, 2004). They are summarised in Table 1. Fitting parameters impacts on the shape and air entry value of SWRF. The effect of adjusting fitting parameters can be checked in Lu and Likos (2004).
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Table 15 Soil Water Retention Functions
Model authors
Model functions
1 (A3) 1 G nG
Se
WR Gardner (1958a)
AEV
1 Se BC nBC BC AEV
Brooks (1964)
AEV
1 Se nVG 1 VG
Van Genuchten (1980)
Notations where r S Sr is Se s r 1 Sr the effective saturation or content (dimensionless);
(A4)
s ,r ( Sr ) are saturated water content and residual water content (or residual saturation); (S ) [ s ,r ] is water content (or saturation);
mVG
(A5)
i , ni , mi (i G, BC ,VG, FX )
are
fitting parameters for each model;
ln(1 ) 1 r r is the soil suction Se 1 6 mFX 10 nFX for irreducible water ) ln(1 ) ln e ( content r FX
D. G. Fredlund and Xing (1994)
(dimensionless)
(A6) Soil water retention function initially appeared as an empirical function fitted into data. The only physical interpretation was given in D. G. Fredlund and Xing (1994). In this work, D. G. Fredlund and Xing (1994) derived each previous SWRF by assuming different pore size distribution functions (PSD). This connection is based on both Young-Laplace equation and statistical distribution of pore matrix. The derivation of SWRF from PSD is
( R)
R
Rmin
f (r )dr
max
f(
2 s cos
)d
max
f(
2 s cos 2 s cos ) d ( ) 2
(A7)
where ϴ is volumetric water content (dimensionless); f(r) is the function of pore size distribution; R is the pore radius (m); Rmin is the smallest pore radius; ψ is the soil suction (kPa); ψmax is the maximum soil suction (kPa); σs = surface tension of soil water (N/m); ϑ is dummy variable of soil suction (kPa) (D. G. Fredlund & Xing, 1994). This derivation opens a gate for studying SWRC of deformable soil, where the variation of PSD can be measured using Mercury Intrusive Porosimetry (MIP) (Hu et al., 2013). For non-deformable soil, SWRC can be fitted by these four models. Leong and Rahardjo (1997) review these four equations against a large amount of soil water characteristic data from previous publications, and eventually suggested that D. G. Fredlund and Xing (1994) model provides the best fit among all, but Van Genuchten (1980) model and D. G. Fredlund and Xing (1994) model without suction correction factor have better fitting performance for sandy soil. This recommendation was later agreed by Zapata,
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Houston, Houston, and Walsh (2000) that both of these two models provide similar fitting curves except high suction range. Also, they did not recommend using correction factor in D. G. Fredlund and Xing (1994) model because it lacks physical support for absolute zero water content at the suction of 106 kPa. The uniqueness of Soil Water Retention Curve is influenced by several factors. J. Zhou and Yu (2005) reviewed three effects impacting the uniqueness of SWRC, which are the initial void ratio, initial water content, stress state and high suction values. Malaya and Sreedeep (2011) reviewed another impact in suction measurement methodology, stress history, additives, aging and measuring range of suction. Regarding these two review studies, the effects impacting the uniqueness of SWRC are mainly governed by two characteristics. The first one is geometry characteristic of soil (e.g. soil structure, pore structure) influenced by sample fabrication, stress history and wetting-drying history. The other one is chemical characteristic, which is affected by soil mineralogy, solute concentration and wetting phase temperature variation (H. H. Liu & Dane, 1993; Romero, Gens, & Lloret, 2001). Jotisankasa (2005) concluded that SWRC is firstly dominated by soil matrix at the low suction range and the other characteristic have more influence on SWRC at high range. Therefore, to research on one factor impacting the non-uniqueness of SWRC, other conditions have to be maintained through the entire experimental operation. In accordance of a review from Malaya and Sreedeep (2011), following several impact factors are worth to be further studied:
The SWRC variation of deformable soil because of hydro-mechanical loading; Hysteresis of SWRC additionally involving stiffness variation of deformable soil caused by hysteresis, densification of collapsing soil induced by hysteresis; The time-dependent change of SWRC because of wetting phase reconfiguration at transient state.
The first research question draws more attention from geotechnical researchers because it highly concerns the determination of stress state variables, stress-strain behaviour and hardening behaviour of unsaturated plastic soil. Many studies were given to study SWRC variation under changing void ratio, soil plasticity (Arroyo, Rojas, de la Luz Pérez-Rea, Horta, & Arroyo, 2015; Domenico Gallipoli, 2012; D Gallipoli, Wheeler, & Karstunen, 2003; Hu et al., 2013; Marinho, 2005; Pham & Fredlund, 2008; A.N. Zhou, Sheng, & Carter, 2012). All of them have one thing in common. It is to construct a threedimensional space of suction-saturation-void ratio (or specific volume). In this way, the usual SWRC measured in conventional axis translation is just a projection of the state surface of the suctionsaturation-void ratio in the two-dimensional suction-saturation plane. With the addition of void ratio axis, the variation of SWRC caused by both mechanical and hydraulic loading can be tracked from the Soil Water Retention Surface (SWRS) rather than a single curve having a blurry specification of soil deformation history. D Gallipoli et al. (2003) embedded specific volume change into air entry value of Van Genuchten (1980) model to capture this behaviour. The other method is to derive SWRS according to PSD variation (Hu et al., 2013). Comparing both of them, it seems that the SWRS derived from PSD is more physically reasonable because PSD can be directly measured from the deformed specimens. Simply using varying air entry value to account soil deformation is not able to reveal PSD variation. However, corresponding experimental effort simultaneously increases with the theory expansion (a lot of MIP tests instead of standard ATT), and, therefore, limits the application of new research findings. The second research question is concerned by both geotechnical researchers and hydrogeologists because hysteresis behaviour does not only change the stress state variables of unsaturated soil but also the specific capillary storage. D. G. Fredlund, Sheng, and Zhao (2011) studied the shift between drying curve and wetting curve and finally indicated that a median curve should be fitted between two curves for engineering application. They clearly confirmed the shortage of this method but also stated the variability of in situ SWRC, so they encouraged further research on this topic. As for the hysteresis SWRC model of non-deformable soil, Pham, Fredlund, and Barbour (2005) reviewed many physical-
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based models and empirical models. Through comparing these models with 34 soil datasets, they concluded that Mualem (1974) appears to be the most accurate model and simplest model for scanning loops prediction in engineering practice (Pham et al., 2005). A novel model for SWRC hysteresis was recently developed by Pedroso and Williams (2010) in the University of Queensland. This model can be calibrated using genetic algorithms (Pedroso & Williams, 2011) and has been successfully applied into a numerical simulation of unsaturated soil stress-strain with SWRC hysteresis (Pedroso, 2014). Nevertheless, all of these hysteresis models, in fact, neglected the volume change effect, which might be more significant than hysteresis effect. Also, these hysteresis models were usually developed for solely modelling multiphase flow in an assumption that rigid soil matrix does not affect capillary storage. To solve unsaturated soil stress-strain problem in the geotechnical engineering domain, soil deformation and hysteresis simultaneously happens. This is less considered into the numerical simulation because the 3D hysteresis SWRS and corresponding experimental procedure are still on the stage of development. Three available hysteresis models accounting void ratio variation are given in Domenico Gallipoli (2012), Hu et al. (2013) and Tsiampousi et al. (2013). In their works, hysteresis curves are constructed between primary drying Soil Water Retention Surface (SWRS) and primary wetting SWRS. Comparing Hu et al. (2013) with Domenico Gallipoli (2012) and Tsiampousi et al. (2013), the prediction of SWRS from PSD variation has more physical meaning because it does not only capture the air-entry value changing with specific volume but also the SWRC gradient changing with the void ratio. The last research question is one of the most interesting one, which is more explored by soil scientists, petroleum engineers, and hydrogeologists but less concerned in geotechnical engineering. Time dependence of SWRC can also be defined as the SWRC for transient multiphase flow or SWRC for dynamic (unsteady state) multiphase flow. The SWRC is defined as a constitutive relationship that intrinsically exists in the unsaturated soil. If there is a non-negligible discrepancy between dynamic SWRC and static SWRC, the theory of entire unsaturated soil mechanics might still miss this important piece. Also, there should be no doubt that this effect will be coupled with soil deformation and hysteresis as well. This is the research question in this thesis. The literature review will be given in details from capillary theory to the state of the art of dynamic SWRC in the last three sections of this chapter. Despite these three popular research questions, there are still other studies working on simplifying the procedure of SWRC determination. Due to the nature of SWRC being strongly related to void structure, SWRC can be estimated from Grain Size Distribution (GSD) with appending parameters, such as soil density, fine particle content, organic content and atterberg limits. According to the literature review from M. D. Fredlund et al. (2002), there are two types of Pedotransfer Functions to transfer GSD to SWRC. The first one is purely regression approach, seeking the correlation between SWRF fitting parameters and grain size index or grain size fraction, combined with density or plastic index (Lee & Ro, 2014; Perera et al., 2005; Rawls & Brakensiek, 1985; Scheinost, Sinowski, & Auerswald, 1997). The other one is physico-empirical model approach, for which estimation of SWRC is given based on particle geometry assumption, pore size distribution and capillary equation (Arya & Paris, 1981; Basile & D'Urso, 1997; M. D. Fredlund et al., 2002; Scheuermann & Bieberstein, 2007). The SWRC estimated by these methods actually cannot be perfectly fitted into experimental data, but physico-empirical models show a better prediction for uniform grade granular soil (M. D. Fredlund et al., 2002). The SWRF is originally designed for multiphase flow simulation. To model porous media flow, Darcy Law (Darcy, 1856) is often introduced. In next section, the governing equation derivation of multiphase flow in porous media will be briefly depicted with the Hydraulic Conductivity Function (HCF) of the unsaturated soil and association between SWRC and HCF.
A1.1.2 Steady state and unsteady state multiphase flow theory The theory of dynamic multiphase flow in porous media was invented by L. A. Richards (1931). L. A. Richards (1931) constructed this model in three steps. First, an explanation of negative water pressure
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or suction idea was invented to determine the state variable for the mathematical model. This state variable is
h hc z (A8) where h = the total water head (m); hc = the soil suction or negative water pressure (m); z = the gravitational potential in elevation (m). Second, it convinces readers that capillary flow is a flow predominated by viscous (laminar flow). Hence, Poiseuille Law can be applied to the single capillary channel, and the empirical Darcy Law can be used to describe capillary flow motion in an REV. The multiphase Darcy Law is
q ks kr h (A9) where q = the volume of water passing through unit area in unit time (m/s), ks = the saturated hydraulic conductivity of REV (m/s); kr =kunsat/ks the relative hydraulic conductivity (dimensionless); h = the total water head (m). Last step, the two-phase Darcican form seepage equation, usually named as DarcyBuckingham Law, is inserted into continuity equation (mass conservation) to represent the flux. Finally, the L. A. Richards (1931)’ model is constructed to a non-linear partial differential equation that is
(ks kr h)
nS t
(A10)
where n = porosity of soil REV (dimensionless); S = saturation of soil REV (dimensionless); t = time (s); other parameters are same with notation above; density is neglected because of incompressible fluid assumption. This equation is the entire formularization of groundwater flow equation. By ignoring different terms in equation (A10), different ground-water PDEs can be derived. The details could be sourced from Jacob (1972). For steady state and unsteady state multiphase flow in rigid porous media, Equation (10) can be written in following forms
hc hc (hc z ) k k ( ) k k ( ) k k ( ) 0 (A11) s r s r s r x x y y z z
h h (h z ) h ks kr ( ) c ks kr ( ) c ks kr ( ) c C (hc ) c (A12) x x y y z z t where C(hc) is the specific capillary moisture capacity (kPa-1), other notations are same as above. Equation (A11) is for steady state flow in the unsaturated soil while Equation (A12) is for the transient condition. L. A. Richards (1931) model can be rewritten in different forms including the soil suction head (hc) based, volumetric water content (ϴ) based and mixing based forms (Lu & Likos, 2004). All of them can be transformed into the form of diffusion theory partial differential equation. They are individually listed in Table 2. Table 16 Richards' model forms on different state variable basis (Celia et al., 1990)
Forms hc base
PDE
h h (h z ) h ks kr ( ) c ks kr ( ) c ks kr ( ) c C (hc ) c x x y y z z t (A12)
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ϴ base
k ( ) D( ) D( ) D( ) x x y y z z z t where D( )
Mixing
(A13)
k ; k ks kr ( ); C C hc
hc hc (hc z ) k k ( ) k k ( ) ks kr ( ) s r s r x x y y z z t
(A14)
To solve Richards’ partial differential equation, two functions have to be offered to account the variation of hydraulic conductivity by saturation and specific capillary moisture capacity changing with soil suction. This requirement promoted the development of Soil Water Retention Function (SWRF) and Hydraulic Conductivity Function (HCF) until recent years. Since the development of estimation of HCF from SWRF (Brooks, 1964; D. Fredlund et al., 1994; Mualem, 1978; Van Genuchten, 1980), currently, SWRC is the only function provide coefficients in equation (A12). A summary of popular HCFs frequently used in numerical solving equation (A12) is given in Table 17. Table 17 Hydraulic Conductivity Functions
Model authors Model equations x E. C. Childs and Collisdx George (1950) 2 r
( x) (A15) x s ( x)2 dx kr exp(G hc ) (A16) kr
s
Notations Se based statistical model; D. Fredlund et al. (1994) rewritten it into continuum form;
r
WR Gardner (1958b)
Brooks (1964)
1 23n kr BC BC hc (A17)
hc hc AEV
Increase the ease of analytical solution but loss fitting performance (hc based empirical model); hc based empirical model;
hc hc AEV
Brooks (1964)
kr Se3 2/ nbc (A16)
Se based empirical model;
Mualem (1976)
w d w well-developed SWRF 0 hc inserted into equation; kr Se 0.5 (A18) s d w 0 hc kr Se0.5 [1 (1 Se1/ mVG )mVG ] (A19) mVG =1-1/nVG; Se based
2
Statistical model; Require
Van Genuchten (1980)
D. Fredlund et al. (1994)
( y) ( ) ' ( y )dy y2 kr (A20) ( y) ' s y 2 ( y)dy r
r
aev
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model; VG SWRF inserted into Equation (16); Insert D. G. Fredlund and Xing (1994) SWRF (Equation (6)) into Equation (19); Suction based HCF;
According to the Table 3, available HCF can be divided into two groups: suction (or suction head) based HCF and effective saturation (or effective volumetric water content) based HCF. Equation (16) is usually used to derive the analytical solution of L. A. Richards (1931) model in one dimension (Zhan & Ng, 2004) or two and three dimensions (Tracy, 2006). Nevertheless, this selection is just for simplifying the analytical derivation. Due to the hysteresis nature of SWRC, suction-based HCFs also have hysteresis behaviour (D. G. Fredlund & Rahardjo, 1993). The unique relation between relative hydraulic conductivity and effective saturation is assumed because the hydraulic paths are determined by fluid fractions filled into pores. D. G. Fredlund and Rahardjo (1993) also demonstrated a series of experimental data that wetting and drying HCFs collapse into a unique curve. Therefore, Se based HCF is always considered for numerical solving Richard’s equation. Leij et al. (1997) investigated prediction performance of a large amount of HCFs against 346 Se(h)-K(h) and 557 Se(h)-K(Se) data sets, and recommended using equation (19) for HCF and equation (5) for SWRF. However, the little hysteresis of Se based HCF was questioned by E. Childs (1969) and Lu and Likos (2004) that different hydraulic loading path may not guarantee same hydraulic path in one saturation. Also, those HCFs seem only satisfied with sandy soil that has negligible soil deformation during the drying-wetting process. Because Equation (18) is determined from PSD (Mualem, 1976), soil deformation induced by purely hydraulic loading might result into non-uniqueness of HCF. Therefore, HCF is also encountering the same problem with SWCF for deformable soil. The second semianalytical method to solve the transient process of water invading into the unsaturated soil is a one-dimensional Green-Ampt model. Green and Ampt (1911) proposed a transient infiltration model by assuming that there is a sharp wetting front clearly separating zone of saturation and dry zone. From Figure 3, this sharp wetting front is used to replace the water distribution along the vertical axis (the Soil Water Retention Curve). The water content of saturated zone is assumed to be effective porosity (ϴe). An initial water content (ϴi) is given to the dry zone. The wetting front infiltration rate, therefore, can be derived using Darcy Law as
q
h hs dQ (e i )dz k (1 0 ) (A21) dt dt z
where q = infiltration rate (m/s); t = time (s); k = effective hydraulic conductivity (m/s); h0 = water head of ponding water above top soil surface (m); hs = capillary suction head at wetting front (m), z = depth of wetting front (m). Solving this ordinary differential equation gives cumulative infiltration displacement as
Q Q (h0 hs )(e i ) ln 1 kt (A22) (h0 hs )(e i ) where Q = cumulative infiltration displacement (m); other notation are same as above. Equation (A21) can be solved by interation if ponding depth (h0), effective hydraulic conductivity (k) and initial water content of dry zone (ϴi) are known.
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Figure 33 Sketch of 1-D vertical Green-Ampt infiltration model (Kale & Sahoo, 2011)
Philip (1957) stated that Green-Ampt model is an exact solution of Richard’s model if the diffusivity (D(ϴ)) is considered as a Dirac-Delta function with a non-zero water content in the saturated zone. Hence, the Green-Ampt model is just a simplified tools for approximating wetting front depth. Kale and Sahoo (2011) reviewed Green-Ampt model with modified versions and concluded that the prediction of this model is sensitive to the effective hydraulic conductivity. The main advantage of this model is less requirement of input parameters, compared to Richard’s model in need of SWCC and HCF. For Richard’s model, SWRC and HCF have to be experimentally measured in the laboratory for a long time due to equilibrium condition achievement. In contrast, the Green-Ampt model only needs effective hydraulic conductivity, ponding head, porosity and dry zone initial water content, each of which can be easily acquired using standard soil mechanic tests in a short period. However, ignoring water content distribution in unsaturated soil constrains the accuracy of this model. Furthermore, due to the inhomogeneous nature soil, the determination of this averaged effective hydraulic conductivity is also a challenging task (Kale & Sahoo, 2011). As a result, Richards’ model is a better method for predicting the unsaturated flow, whereas Green-Ampt model can be used for some engineering application under certain conditions. The third method for simulation of multiphase flow in porous media is from petroleum engineering as Buckley and Leverett (1942) analysis. This method is almost identical to Richard’s model derivation and based on Darcy-Buckingham seepage motion equation as well. Therefore, this method is just another rewritten form of Richards’ model. The model performance is also dependent on SWRC and HCF. As above review of macroscale theory for multiphase flow in porous media, it is not sophisticated to find that the solution of each theory is coefficient determined, and experimental determination of SWRC and HCF is the core of entire macroscale theory of multiphase flow in porous media.
A1.1.3 Mechanical behaviour of unsaturated soil Mechanical behaviour of a material usually refers to the stress-strain behaviour and strength determination for the purpose of earth structure design. The unsaturated soil is just a special material that soil suction need to be additionally included, compared to saturated soil where positive water pressure is the only neutral stress changing soil skeleton stress. Many textbooks already give the stress state variable for unsaturated soil, which is
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' ( ua ) (ua uw ) (A23) where σ’ = effective stress of unsaturated soil (kPa), σ = total normal stress (kPa), ua = pore air pressure (kPa), uw = pore water pressure (kPa), χ = a parameter depends on soil saturation and is between 0 and 1 (Bishop, 1960). In most of the case, pore air pressure is assumed at atmosphere pressure as a zero reference pressure so the pore water pressure can be defined as a negative pore water pressure that is the soil suction. Equation (23) has been experimentally validated by D. G. Fredlund and Morgenstern (1977) and also derived in the textbook of Lu and Likos (2004) with spherical packing assumption. Although there are some other forms of unsaturated soil effective stress additionally considering solute suction (osmotic suction) (Aitchison, 1965; B. Richards, 1966), bishop effective stress form is still the most common equation used in geotechnical engineering research. Based on Equation (23), the classical Mohr-Coulomb (MC) criterion is further developed for unsaturated soil as
c ' ' tan ' c ' c '' ( ua ) tan ' c ' ( ua ) tan ' (ua uw ) tan ' (A24) where τ = the shear stress (kPa), c’ = cohesion at zero suction (kPa), φ’ = the friction angle for zero suction, other notation is same with Equation (23) (Lu & Likos, 2006). Equation (24) can be seen in two different ways. First, the suction effect enhances the total cohesion between soil particles. This cohesion can be separated into original cohesion (c’) and external cohesion (c’’) induced by soil suction that is sensitive to the environmental condition. The second way is to treat suction as another stress variable varying the shear failure criterion line. Therefore, the original 2-D MC can be extended to a 3D MC criterion for unsaturated soil (see Figure 4), also called Modified Mohr-Coulomb criterion. When suction is zero, it is the original 2-D MC for saturated soil. With the suction increase, the criterion envelope line increase. Another suction-shear stress plane is generated perpendicular to the original MC plane. The envelope instead of a line eventually becomes a surface. Also, the suction friction angle is later found in nonlinear (D. G. Fredlund & Rahardjo, 1993). Nowadays, many studies suggest this angle is determined by both original friction angle and effective saturation (Khalili, Geiser, & Blight, 2004; Lu, Godt, & Wu, 2010; Lu & Likos, 2006). On the other hand, due to the addition of stress term containing both suction and saturation, the Ranking theory of earth pressure is also extended for unsaturated soil. in chapter 11 of D. G. Fredlund and Rahardjo (1993), the at rest, active and passive earth pressure were derived by using Bishop effective stress instead of Terzaghi’ effective stress and corresponding K0, Ka and Kp were all given. The common unsaturated behaviour captured in extended theory is that suction increases earth tensile strength. Suction induced soil matrix internal stress enhances the soil cohesion, subsequently decrease the active earth pressure and increase passive earth pressure. Also high suction and unevenly distributed soil moisture lead to tensile stress concentration on low moisture content region of the top soil further inducing soil cracking as a preferential path for surface recharge. Suction profile along the vertical axial of earth dominates the earth pressure of unsaturated soil. Therefore, the earth pressure is highly dependent on hydraulic boundary condition variation. Same issue happens to the determination of the bear capacity of shallow foundation design and slope stability safety analysis as well. There is no apparent different with conventional knowledge of these engineering design calculation but adding the Bishop effective stress and taking consideration of suction profile variation. If the SWRF is not written into the new derived theory to be looked terrifying, each equation is still same as origin soil mechanics but with a new term counting the internal stress resulting from capillary pressure.
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Figure 34 3-D Mohr-Coulomb failure envelope surface for unsaturated soils (D. G. Fredlund & Rahardjo, 1993)
The Bishop’s effective stress parameter (χ) governs the nonlinear 3-D MC envelope surface. Its determination is still being debated in academic research. According to the textbook of Lu and Likos (2004), it can be written in either suction based function and saturation based function. A list of Bishop parameter equations is summarised in Table 4. As for the saturation based bishop’s parameter, SWRF has to be introduced to calculate the effective saturation from various soil suctions. In fact, even Khalili and Khabbaz (1998) claims that there is a unique relationship between suction and bishop’s parameter, the relationship given in their work is still another expression of Brooks (1964)’ SWRF with a fixed power parameter at -0.55. Hence, the SWRC is also the most important constitutive relationship for the determination of unsaturated soil effective stress. Table 18 Bishop effective stress parameter equations
Author Equation Alonso, Gens, and Josa (A25) (1990) Öberg and Sällfors S (A26) (1997) Khalili and Khabbaz u u r w (1998) a ua uw uaev uaev ua uw uaev 1 (A27) S. Vanapalli and Fredlund (2000) (A28)
s
Lu et al. (2010)
r s r
S Sr Se (A29) 1 Sr
Notation α is a content; S is saturation; Suction basis; have been an air entry value; r is a fitting parameter, usually equal to -0.55 ϴ is effective volumetric water content; κ is a fitting parameter; Derived from Suction stress concept (Lu & Likos, 2006); Interfacial energy considered in Lu et al. (2010) but finally neglected for mechanics
The effective stress concept is originally introduced in saturated soil mechanics by Terzaghi et al. (1943) to study saturated soil consolidation. Unsaturated soil researchers therefore further extended it to serve unsaturated soil deformation. Nevertheless, Matyas and Radhakrishna (1968) found that χ is not an intrinsic material parameter but dependent on loading path. Using bishop’s effective stress form cannot express the wetting induced soil collapse and swell. Bishop’s parameter can go into negative value
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when there is no deformation or swelling by decreasing suction. Matyas and Radhakrishna (1968) eventually recommended that applied stress and suction should be two independent stress variables. Based on this point of view, the unsaturated soil deformation constitutive relationship is using two independent stress rather than one effective stress (Alonso et al., 1990; D. G. Fredlund & Rahardjo, 1993). The 2-D unsaturated soil effective stress-strain space is expanded to a stress-suction-strain (p-sv) 3-D constitutive surface (Figure 5).
Figure 35 An example of 3-D unsaturated soil constitutive surface for studying soil deformation (Zhang & Li, 2010)
Barcelona Basic Model (BBM) first developed by Alonso et al. (1990) provides a good basis for studying unsaturated soil deformation. Modern constitutive models are variants of BBM and their cores are all based on BBM as well (Zhang & Li, 2010). In BBM, the unsaturated elastoplasticity deformation is considered in two stress paths: Loading Collapse (LC) and Suction Increase (SI) (Alonso et al., 1990). On the loading collapse stress path, unsaturated soil deformation consists of loading induced plastic deformation, suction induced elastic deformation and loading induced elastic deformation. Due to the deformation coefficients and reference stress state depending on suction, also, an experimental equation is used to account the soil stiffness varying with suction. On Suction Increase stress path, a yield suction is defined as threshold suction, over which suction induced plastic deformation initiates. These two stress paths enclose an elastic surface (A-B-C in Figure 5). Surpassing this elastic surface, the plastic deformation occurs. BBM formulation and modelling parameters determination in details can be sourced from (Alonso et al., 1990; Jotisankasa, 2005; Zhang & Li, 2010). Here, interpretation of BBM is not going to be expanded because this thesis does not stress on unsaturated soil deformation. However, no matter of using the effective stress concept or separately using two independent stress variables, a well-developed SWRF is also necessary to account the suction induced elastoplasticity deformation. Also, to provide a correct suction prediction for effective stress or BBM, an SWRF of deformable soil should be taken into calculation. Ignoring SWRF variation under both deformation and hysteresis could be a reason of model inaccuracy. D'Onza et al. (2011) provided a comprehensive study on the benchmark of unsaturated soil constitutive model. In their work, seven modern constitutive models with different SWRFs are validated against the blind test of same sets of experimental results from unsaturated soil direct shear, oedometer and triaxial test. The prediction of each model shows significant discrepancy with blind test data. D'Onza et al. (2011) concluded that model prediction is sensitive to the selection of SWRF and experimental calibration procedure, and the model’s prediction governed by SWRF shows fewer differences to experimental data than prediction governed by mechanical model. Therefore, the selection of an accurate and well-calibrated SWRF is the most important step in modelling unsaturated soil deformation and hydromechanical coupling.
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A1.2 Review of unsaturated soil experiment In this section, experiments for soil suction measurement, water content measurement and relative hydraulic conductivity experiments are summarised from previous studies. A review and discussion are also given to identifying the advantage and shortage of each measuring technique. Based on this review, the experimental method of this study is finally determined.
A1.2.1 Experimental Techniques for Soil Water Retention Curve To determine a Soil Water Retention Curve (SWRC), there are two variables that have to be measured. The first one is water content or degree of saturation. The other one is the soil suction. Traditionally, the gravimetric water content can be easily measured using oven drying method (ASTM D2216-10, 2010). To further determine volumetric water content and degree saturation, the soil packing condition, including specific unit weight (ASTM D5550-06, 2006) and density (ASTM D7263-09, 2009), needs to be measured in order to transfer gravimetric water content into volumetric content. However, as for deformable soil, dry density (also the porosity) will not keep at a constant, so shrinkage behaviour has to be measured to determine the volumetric water content (ASTM D4943-08, 2008). However, this is only necessary for the deformable soil like clay, silt or mixture of clay and silt but not needed for the coarse soil. Other methods also involve directly measuring specimen using a scale (Method C of ASTM D6836-02 (2003)), measuring water expelled from specimen by a burette (Method A and B of (ASTM D6836-02, 2003)), and water content sensors like Time Domain Reflectometer (TDR) (ASTM D7664-10, 2010). For laboratory scale suction measurement, the methods can be divided into two streams: suction control and suction measurement. The mainstream one is a suction control test, one of which is Axis Translation Technique (ATT). During the suction control process, the water content is measured for each static capillary pressure under an equilibrium condition. According to ASTM D6836-02 (2003), there are two standard axis translation techniques: Buchner funnel (also named Hanging Column) method (Haines, 1930), Pressure Chamber Method (L. A. Richards, 1931). The only difference is their suction control approach. As for the Hanging Column method, soil specimen is located on a saturated High Air Entry (HAE) value ceramic plate embedded into a hanging column. The top of a soil specimen is directly contacted with the ambient atmosphere. A measuring burette is attached to the bottom output. Ensuring the good moisture contact between specimen and HAE ceramic disk, matrix suction can be applied through changing the burette elevation in which there is the water table. By applying different water elevation, the matrix suction head can be vertically measured. The water expelled from soil specimen is measured from the incremental volume in this burette. A sketch of Hanging Column method is shown in Figure 6(a). As for the other axis translation, suction is controlled by increasing air pressure. Soil specimen is placed on the HAEV disk located in the bottom of a gas-liquid proof chamber. The bottom disk is attached to a water burette for measuring the volume of expelled water. The high air pressure is injected from the top of the cell. After each air pressure increase, there will be a static capillary condition achieved until there is no further water expelled from the specimen. Water content decrease can be either measured from the burette or measured the entire weight of pressure chamber by a bench scale. An example of Pressure Chamber method is depicted in Figure 6 (b).
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(b)
(a)
Figure 36 Standard Axis Translation Techniques: (a) A sketch of Hanging Column Method; (b) the sketch of Pressure Chamber Method (S. K. Vanapalli et al., 2008)
Although Axis Translation Technique has been confirmed as a standard test for determination of SWRC of unsaturated soil, it has some limitations. For Buchner funnel, the most important issue is the evaporation from the top of the specimen and short range of measurable suction. Evaporation can be reduced by decreasing the ambient temperature. Nonetheless, the surface tension variation induced by temperature change has to be calibrated back to normal surface tension under laboratory temperature. Also, it is questionable that there might be different water distribution in the specimen under different surface tension induced by the temperature difference. Due to the limitation of laboratory vertical space, the standard Buchner funnel can only measure suction from 0 - 20 or 30 kPa (0-2 or 3m) (S. K. Vanapalli et al., 2008). Hence, this method is usually for sandy soil having maximum suction head less than 30 kPa. Recently, this method is further developed in need of less vertical space using vacuum control system (ASTM D6836-02, 2003; Romano, Hopmans, & Dane, 2002). This new method increases the upper limit to about 40 kPa (S. K. Vanapalli et al., 2008). However, this range is still insufficient for fine soil, so pressure chamber is an irreplaceable equipment in SWRC determination of fine soil in the laboratory. In accordance of ASTM D6836-02 (2003), pressure chamber method covers the suction range from 0 to 1500 kPa and the recommended suction loading sequence is 10, 50, 100, 300, 500, 1000 and 1500 kPa. Also, from the previous researchers’ experience, it is hard to collect enough data points to plot an SWRC of coarse soil using pressure chamber, because pore water will be easily drained out by adjusting the low resolution of air pressure on the control panel. This will subsequently result in none identification of coarse soil AEV. The pressure chamber is more suitable for fine soil within a suction range 10 - 1500 kPa. S. K. Vanapalli et al. (2008) reviewed the ATT limitations, especially stressing on pressure chamber method. The main problem lead to a discrepancy between suction of pressure chamber and field is that the water pressure in the field is negative, but it is positive in the chamber. The high air pressure causes air diffusion into the soil water. Moreover, because of low air permeability in the initial step of air pressure loading, soil specimen will be compressed. The theory for pressure chamber is only valid for soils with completely interconnected pore-air void and for soil grains that are incompressible and only when air-water interphase is continued (S. K. Vanapalli et al., 2008). Also, with more air diffused into soil water, there will be more compressed air bubbles expanded during and after passing through HAE disk. This is because the low permeability of HAV disk leads to a large pressure drop between pore water pressure in the chamber bottom and positive hydrostatic pressure in outlet burette.
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Therefore, air bubble flushing is very necessary under the bottom of HAEV disk (S. K. Vanapalli et al., 2008). The vital shortage of ATT is that the low permeability of HAEV disk also induced a low speed of equilibrium achievement. After one step suction increase, the pressure difference between air pressure and back pressure (hydrostatic pressure in burette) is not the matrix suction under dynamic capillary flow, because the back pressure will be much smaller than real pore pressure above the HAEV disk. Hence, even ATT is accepted as the standard test; it is unable to determine the dynamic suction or dynamic negative water pressure. Except the mainstream test-Axis Translation Technique, there are also some other test recorded into Lu et al. (2010) and ASTM D6836-02 (2003), such as Chilled-Mirror Hygrometer, Contact and nonContact filter paper method, Centrifuge Method and Tensiometer. Chilled-Mirror Hygrometer is a total suction control method. It utilises humidity control technique. Through controlling the ambient temperature to dewpoint temperature constantly by chilled-mirror sensing technology, the total suction can be determined by Kelvin’s equation (equation (2)). This method is usually used for measuring a high suction range of fine soil (1000-450000 kPa) (Lu & Likos, 2006). Gubiani, Reichert, Campbell, Reinert, and Gelain (2013) compared this method with other methods to conclude that the lowest limit of Dewpoint meter should be 7000 kPa instead of 1000 kPa. Contact and non-Contact filter paper methods are an indirect suction measuring method. It measures the unsaturated soil suction from water content transfer from the sample to itself in an enclosed space. It cannot directly measure the soil suction so a calibration between suction and moisture absorbed into filter papers should be accomplished before experiment implementation. Contact filter paper has intimate contact with ambient soil to give the association between absorbed moisture and matrix suction. The non-contact filter paper is placed into the ambient environment of soil, so it can indirectly measure the total suction. Details of using this method can be sourced from Lu and Likos (2004). The advantage of contact filter paper method is that it covers the entire suction range from 0 to 106 kPa. However, this method is quite time-consuming because it usually takes 7 to 10 days for reaching an equilibration (Lu & Likos, 2004). Centrifuge method is the last method mentioned in ASTM D6836-02 (2003). It is a suction control technique. Various suction can be generated with different angular velocity. It is used for the suction range 0 to 120 kPa. The advantage of this method is less consumption, but the relatively high cost constrain its popularity. A tensiometer is a very powerful suction measuring technique originally developed by Willard Gardner et al. (1922). It is a water-filled shaft with a High Air Entry Value (HAEV) tip at one end, and the other end is connected to a pressure sensor. A micro-sized tensiometer (UMS T5 tensiometer® Umweltanalytische Mess-Systeme GmbH) and diagram are shown in Figure 7(a) and (b). After inserting tensiometer into soil body, the water in the shaft will be absorbed into the soil by matrix suction. As a result, this matrix suction can be transferred to a negative pressure vacuuming the water in the shaft. This vacuum pressure finally is measured by the pressure sensor at the other end. This method only measures matrix suction because HAE tip is not solute impermeable. The accuracy of tensiometer measurement is not only dependent on the equipment itself but also users’ careful installation. The key to ensuring precise suction reading is to ensure a good contact between unsaturated soil and ceramic tip (Lu & Likos, 2006). The measuring range of conventional tensiometer is usually 0 to 100 kPa. Recently, high suction tensiometer measuring range goes up to 1500 kPa. Toll, Lourenço, and Mendes (2013) reviewed the characteristics of recently developed high suction tensiometers and find that the pressure transducer range (highest is 15 MPa) is much higher than AEV tip (highest is 1.5 MPa). The suction range is constrained by the highest Air Entry Value (AEV) of this tip. Compared it with all other methods, Tensiometer has following advantages:
Fast and ease of installation (ensure tip contact condition) Application in both Laboratory condition and situ field condition
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Fast responding time (less than 1min for low suction capacity micro-tensiometer) Long-term measurement (ensure no drain out of water in shaft )
(b)
(a)
Figure 37 An example of tensiometer (a) UMS T5 Micro-Tensiometer (b) A sketch of T5 structure (UMS, 2009)
According to author’s using experience, tensiometer should be carefully prepared before taking a measurement. Water in the shaft should be totally bubble free above highest suction that will be encountered in selected sample. If a bubble with similar size of the shaft is generated under high suction, the hydraulic path in the shaft will be significantly reduced, subsequently leading to slow extremely responding speed or even incorrect reading. Because the fast responding speed of T5 micro tensiometer, it is very useful for studying dynamic capillary or dynamic soil suction in the unsteady multiphase flow. In this study, the T5 sensor is selected as a suction transducer for continuum suction measurement.
A1.2.2 Unsaturated soil hydraulic conductivity experiments Unsaturated soil hydraulic conductivity (kunsat) is the hydraulic conductivity of unsaturated soil under various suctions or water content. It can be calculated by saturated hydraulic conductivity (ks) timed by relative hydraulic conductivity (kr). In the section of steady and unsteady multiphase flow in porous media, the prediction of kr from Soil Water Retention Function (SWRF) was already introduced. However, this approach is just an approximation of kunsat with an aim to reduce experimental complexity and time consumption. The most reliable result, in fact, is still based on direct experimental measurement. Here, the experiments for measuring kunsat will be summarised and discussed. Based on the theory of multiphase flow in porous media, there are two types of experiment: steady state measurement and transient measurement. In steady state experiment, conventional Axis Translation Technique (ATT) is often introduced whereas there are some shortcomings. Centrifuge Method is another one included in steady state method. The determination of kunsat using ATT is quite straightforward. Through applying suction by varying air chamber pressure and measuring outflow from ATT chamber, the kunsat can be simply calculated by Darcy Law and then plotted against the corresponding average suction of soil specimen. ASTM D7664-10 (2010) offers two subcategories of ATT: rigid wall (odometer) type ATT ( Figure 6(b)) or flexible wall (soil triaxial cell) type ATT (Figure 7 (a)). The former is more suitable for coarse grain soil and latter is more suitable for fine grain soil. However, as the discussion of ATT shortages in previous content in terms of air diffusion, air compressing specimen and impedance of HAEV disk with bubble expanding after pressure drop, ATT can only be used for an approximation of the Hydraulic Conductivity Function (HCF) of unsaturated soil under steady state (ASTM D7664-10, 2010). The testing procedure and calculation will be given in Section of Standard unsaturated soil test in the Chapter of Methodology in details. In general, according to the method B of ASTM D7664-10 (2010), the air-water seepage in unsaturated soil is
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controlled by applying air pressure on specimen top and water pressure (back pressure) at the bottom under the HAEV disk; the tensiometer is recommended to measure each depth of specimen, but TDR with previous tested SWRC could be a replacement. The volumetric water content change in a certain time interval can be integrated to give the water flux in this period. The hydraulic head gradient can be given between each depth in the soil where suction and water content are measured. With both calculated water flux, cross-section Area of the chamber and hydraulic head, the kunsat is simply calculated by Darcy Law (Details sourced from (ASTM D7664-10, 2010)).
(b)
(a)
Figure 38 kunsat measurement by ATT (a) flexible wall ATT (ASTM D7664-10, 2010) (b) Constant head method by ATT(L. A. Richards, 1931)
In Lu and Likos (2004) and Masrouri, Bicalho, and Kawai (2009), another steady state methods based on ATT are also introduced. One is constant head method additionally added a constant head on the top of unsaturated soil specimens during suction control by ATT (L. A. Richards, 1931). A sketch is shown in Figure 7(b). In this method, suction is maintained during a two-phase flow seepage. The kunsat is calculated by Darcy Law under various suction maintained by ATT. Water content can be measured by the destructive method (Oven drying method) or nondestructive method (TDR or other moisture problems). For the destructive method, more identical specimens have to be prefabricated. In this way, the HCF can be finally plotted with either suction basis or water content basis. The other one is the constant flow rate method (Olsen, Willden, Kiusalaas, Nelson, & Poeter, 1994). The core of experimental setup is same with constant head method (suction controlled by ATT) but it requires a highly complex constant flow control system to generate constant flux in unsaturated soil triaxial cell, whose details are given in Olsen et al. (1994). After measurement of suction, water content and controlled flux, the kunsat is also calculated by Darcy Law. The other steady state method, in exception of ATT, is centrifuge method. It uses a centrifugal force as driving force instead of gravity (ASTM D7664-10, 2010). The water is allowed to infiltrate into unsaturated soil specimens and finally reach the outflow reservoir. Higher horizontal field force can be produced by increasing angular velocity in centrifugal operation. In such high horizontal field force, vertical gravity is neglected. Instead, the horizontal force is the only force generating water pressure in the system. The pressure gradient could be determined by angular velocity and seepage radius. The infiltration rate could be measured from outflow reservoir with a corresponding time interval. The kunsat is eventually calculated by Darcy Law. Nevertheless, due to the high cost of the geotechnical centrifuge, it is not often available in most of geotechnical laboratory. Masrouri et al. (2009) recommended that centrifuge method is only suitable for non-deformable soil specimens with a pore structure insensitive to the state of stress because of high net normal stress applied by a centrifugal operation.
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In unsteady state experiment, there are mainly three methods: multistep in-outflow method (WR Gardner, 1956), Infiltration Method (Bruce & Klute, 1956), Instantaneous Profile method (IPM) (S. J. Richards & Weeks, 1953). The first transient experiment still relies on ATT showed in Figure 6(b) and Figure 8. By changing the state of suction within a small interval but large enough for flux measurement, the volume change of pore water expelled for applied suction increment can be recorded for later calculating unsaturated soil water diffusivity using an analytical solution of 1-D Richards’ equation in hc based diffusion form. Finally, with previously determined capillary specific storage (SWRC slope), the sunset can be calculated by notation of Equation (13). However, this experiment also has the same limitations for SWRC and steady state HCF using ATT. Lu and Likos (2004) summarised the six assumptions of multistep outflow method: (1) Each suction increase interval must be small enough, so kunsat can be assumed as a constant in this interval (which requires very careful suction control); (2) The relation between soil suction and water content is linear (but it is not only nonlinear but also hysteresis); (3) HAEV disk does not cause any hydraulic resistivity (but it is a large impedance especially for sandy soil); (4) Flow is just one dimension; (5) Gravity effect can be ignored; (6) The testing specimen is homogeneous and non-deformable (which is only available for sandy soil). Therefore, it is not hard to find that transient flow experiment based on ATT is just a method for approximation. Masrouri et al. (2009) comment on this method is that it owns simplicity and is good at mass control, but there is few reliable result for comparison with other methods. ASTM D7664-10 (2010) also records this method as one transient method in the standard but clearly mentioned its limitation as well. The second method (Bruce & Klute, 1956) indirectly measures unsaturated soil hydraulic conductivity by measuring the unsaturated soil water diffusivity. Though a process of water is invading into the unsaturated soil, the moisture content along invading profile, invading distance and the corresponding time interval can be recorded. The Boltzmann variable (λ=xt1/2), which is a function of both invading displacement (x) and duration (t), can be plotted against corresponding water content as a moisture content function of Boltzmann variable. The diffusivity can be calculated from Boltzmann transformation of 1-D Richards’ model (ϴ based diffusion equation), which needs integration of the function of diffusivity against Boltzmann variable. With the addition of previous measured SWRC, the kunsat can be finally calculated by notation of Equation (13). The last but the best method is the Instantaneous Profile Method (IMP), which is the one of transient methods recommended in ASTM D7664-10 (2010). This experiment is to replicate a natural vertical soil column in the laboratory. By spatially inserting suction sensors and moisture sensors along soil column, the suction and water content can be continuously measured under different wetting and drying paths. This method allows flexible application of boundary condition. ASTM D7664-10 (2010) offers four types of hydraulic loading paths: top infiltration, bottom imbibition, top drainage and evaporation. This experiment can produce data for a dynamic effect of both suction and water content. The water flux can be easily calculated by integration of moisture content variation along investigated time interval. Corresponding suction head to vertical depth can also be given to determine the suction gradient. Darcy Law is finally used to calculate kusat. Moreover, it is also a good setup for studying the dynamic effect of SWRC. Therefore, in this study, Instantaneous Profile Method is chosen as the basic framework of macroscale experiment. Due to surface tension depending on temperature, evaporation test will not be considered in this study. Other options will be considered into the experimental design. Nevertheless, this method lacks stress state measurement and volumetric measurement as well (Masrouri et al., 2009). Hence, the only sandy soil is selected to conduct the soil column test in this
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study. This is also the reason this study merely focuses on granular porous media rather than wellgraded natural soil where there is a certain amount of plastic fines causing significant deformation. The details of MIP used in this study is given in Chapter of Methodology.
A1.3 Dynamic water retention behaviour in porous media In this section, the dynamic effects in water retention behaviour will be initiated from the genesis of experimental findings. Then a review on paradoxes of conventional theory will be summarised regarding previous studies. To identify the problem physically, in the third part, a microscale dynamic capillary physic will be reviewed from the state of the art of interfacial physics. Subsequently, the only improved theory capturing dynamic effect is summarised against conventional one. The state of the art of studies based or not based on this theory is all review in the last part.
A1.3.1 Original findings of dynamic effects in Soil Water Retention Curve For soil science or geotechnical engineering, capillary flow was not originally studied from microscale view of physics rather a macroscale constitutive relationship-Soil Water Retention Curve (SWRC). This somehow constrains our understanding of two-phase flow into a unit cubic-Representative Elementary Volume (REV). To calculate the two-phase flow in soil domain made up of soil REVs, a constitutive relationship, for instance, an intrinsic mechanical-geometry mathematical relationship like stress-strain elastic system, has to be assumed in this REV. SWRC is just another constitutive relationship assumed by original creators to account the intrinsic behaviour of multiphase flow in porous media. But whether SWRC is intrinsic or not has been being continually debated by experimental studies (Goel & O'Carroll, 2011; S. M. Hassanizadeh et al., 2002; Hui et al., 2010; O'Carroll et al., 2010; O'Carroll et al., 2005; Sakaki et al., 2010; Schultze et al., 1997; Wildenschild et al., 2001). Since early studies constrained by experimental technique, ATT is the standard experimental method for determination of SWRC. With the development of tensiometer and water moisture sensors, Topp et al. (1967) studied the SWRC under unsteady state flow condition using tensiometer for suction measurement and gamma-ray system for moisture measurement and compared it with static SWRC. As tensiometer can provide an instant response of soil suction, the SWRC measured under transient flow condition shows a significant discrepancy with static SWRC. For same water content, the dynamic suction is higher than static suction around 25% of static suction in a fine sand package (Topp et al., 1967). However, in early studies, the study of dynamic effect merely focus on drainage cycle. This finding strongly questioned the validity of Richard’s model that can be used to simulate dynamic multiphase flow in porous media whereas it using a constitutive relationship having a dynamic effect. This issue is not only discovered by experimental studies on SWRC but some studies on the experimental validation of Richard’s Model. S. M. Hassanizadeh et al. (2002) reviewed some early studies of Richard’s equation validation to find that the diffusivity depends on the speed of wetting front but is not a material intrinsic. Some literature reviewed from early study clearly draw the conclusion that diffusion equation cannot be verified for moisture transport, and the application of Darcy’s Law is questionable (D. Nielsen et al., 1962). Also the relationship between diffusivity and moisture content losses uniqueness for transient flow condition (S. M. Hassanizadeh et al., 2002). The wetting front speed dependence of diffusivity raised the concern that both SWRC and HCF or each of them has a dynamic effect. The dynamic effect in drainage SWRC founded by Topp et al. (1967) was also experimentally measured from latter studies (Elzeftawy & Mansell, 1975; Smiles et al., 1971; Stauffer, 1978; Vachaud et al., 1972; Wana-Etyem, 1982). All of these works used a similar setup but in exception of Wana-Etyem (1982) most of them solely focused on drainage curve. Despite dynamic effect in SWRC, Stauffer (1978) also checked the HCF and found it also owns dynamic behaviour. Smiles et al. (1971) confirmed that dynamic drainage curves have higher suction than static drainage curve, but they concluded that imbibition curve had no observable dynamic effects. Nevertheless, this was later challenged by Wana-
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Etyem (1982) that dynamic effects happen for both drainage and imbibition SWRC. S. M. Hassanizadeh et al. (2002) summarised their findings into following points: (5) (6) (7) (8)
The dynamic effect is not significant in fine-textured soil; The higher rate of water content variation, the larger the dynamic effect; The dynamic effect is more significant in coarse textured sand; The dynamic effect in primary drainage curves is more significant than the dynamic effect in main drainage curves. In the original findings of dynamic effect in SWRC, experimental results were the only evidence for demonstration. Because of lacking of understanding of multiphase thermodynamic, there was no theoretical mathematical formulation to capture such behaviour, and most of the works were purely experimental exploration. Therefore, to theoretical derive a mathematical equation accounting this dynamic effect, the conventional theory has to be critically reviewed to identify the shortage or missing part so the Richards’ model can be further improved for a true simulation of dynamic multiphase flow in unsaturated soil.
A1.3.2 Paradoxes in Theory of Dynamic Multiphase flow in porous media As the summary of conventional unsaturated moisture diffusion theory in the second part of the first section, the theory was constructed in three steps: state variables, conservation of momentum and insertion of simplified momentum balance into mass balance. The reason this diffusion theory does not account the dynamic suction or water redistribution effect during a transient process is not due to utilization of conservation law but the poor expression of state variables and extending saturated diffusion to unsaturated diffusion by simply adding parameters (Gray & Hassanizadeh, 1991). Compared to the foundation of saturated porous media flow diffusion theory, for which the state variables (water pressure) and simplified version of momentum balance (Darcy’s Law) were solidly validated by experimental studies (Jacob, 1972), the unsaturated moisture diffusion does not have sufficient experimental studies to verify the reasonability of using static capillary pressure as matrix suction and two phase Darcy Law by addition of relative permeability. Therefore, the paradoxes just happen in these two points. As for the selection of state variables, the matrix suction is always assumed to be equal to macroscale averaged capillary pressure as given in Equation (1). Then this so-called negative water pressure (matrix suction) is empirically correlated with wetting phase saturation by the concept of SWRC based on some early experimental findings. Gray and Hassanizadeh (1991) criticized that the negative water pressure (the suction under zero reference atmosphere pressure) should not be simply determined by wetting phase saturation while it should be rigorously derived from Equation of State (EOS) with the addition of phase saturation. If this is separated, there is a paradox to define if a negative pressure is a function of saturation or a function of wetting phase density and temperature. The second paradox point Gray and Hassanizadeh (1991) argued is the range of negative water pressure. With zero reference pressure of the atmosphere, matrix suction should be bounded inside 0-1 atm. However, in the real measurement, matrix suction is usually above 100 kPa, which cannot be physically conceptualized. So they stated that this high suction is an energy that soil extract from a water molecule and simply assuming equation between high matrix suction and meniscus geometry cannot be proved. Gray and Hassanizadeh (1991) cited the conclusion of a study from Bolt and Miller (1958), that the pressure of most soil water in unsaturated soil remains positive pressure, and the large values of soil water tension (high suction) are actually an artificial variable to account the energy generated by attraction of soil particles. Hence, they concerned that pressure and adsorption effects should not be lumped into one negative pressure. The third paradox point exists in the relationship between matrix suction pressure and suction head. Gray and Hassanizadeh (1991) argued that the suction head cannot be simply determined from matrix
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suction divided by wetting phase density and gravitational accelerator. As the argument mentioned in last paradox, this energy generated by water-grain adsorption effect still depends on soil type and water molecule inter-attraction. Also, wetting phase density variation is not experimentally investigated. Gray and Hassanizadeh (1991) suspected using constant density into hydrostatic form to account matrix suction in elevation potential due to binding causing significant density variation. Hence, simply applying the hydrostatic form to represent elevation potential of soil suction has never been physically proved as well. The last paradox is on interface dynamics. Young-Laplace equation in fact only provides a relationship between meniscus geometry and surface tension that is a macroscale artificial tension variable accounting the combination of solid surface adsorption and molecule attraction effects for the single capillary tube. It can only be experimentally validated without external actions. When capillary water start to expel other non-wetting phase fluids or be expelled by such fluids, this equation has been experimentally verified to be failure (Baver et al., 2014; Calvo et al., 1991; De Gennes, 1988; Hoffman, 1975; Kim & Kim, 2012; Sheng & Zhou, 1992; Yang et al., 2011; Zhmud et al., 2000). A common physical behaviour from these physical studies indicates that dynamic capillary pressure is influenced by advancing velocity, which is usually lumped into the dimensionless Capillary number (Ca) as an additional term to original static capillary pressure. Gray and Hassanizadeh (1991) also proposed this paradox in their study but instead of looking into capillary physics they expected to add specific interfacial area (air-water interface per bulk volume) to quantify the dynamic interface variation because they expect a unique relationship between interfacial area and saturation. This is also derived in their improved multiphase flow theory that will be introduced in part after next. Before reviewing currently available macroscale theories of multiphase flow in porous media, in next part, the current physical research on dynamic capillary will be briefly reviewed to identify the state of the art of dynamic capillary theory in multiphase physics.
A1.3.3 Microscale dynamic capillary behaviour in interfacial physics The microscale mentioned here is not molecular scale in chemistry rather a pore scale among soil particles and can be simply conceptualised as a capillary tube. Although nature soil has complex and pore structure and statistically distributed paths, each single pore path can be simplified as a capillary tube to look into pore-size physics. Whether a theory is rigid should be clearly verified by the physics and experimental observation. Richard’s model using two-phase Darcy Law to represent immiscible phases displacement into a soil body can be simplified as a two-phase movement in a series of capillary tubes. In this case, the assumption, which viscosity force purely dominates fluid flow condition, and conductivity of each phase is due to phase fraction in tubes, overlooks the real flow phenomenon in even one single capillary tube. Compared with single phase flow in either porous media or single tube, two-phase flow in a pore channel has more driving forces dominated flow condition in the transient state. Unfortunately, some early studies just simply extend Poiseuille’s Law for studying two-phase displacement by adding capillary pressure in the form of Young-Laplace equation. The representative theory derivations in this way can be sourced from Bell and Cameron (1906), Lucas (1918) and Washburn (1921). Their eventually derived equation is
r cos l s t Dt (A30) 2 where l = advancing distance (m); σs = surface tension (N/m); r = capillary tube radius (m); ϴ = contact angle; μ = dynamic viscosity (kg/ms or Pa·s); D = diffusivity (m2/s); t = advancing period. The Equation (30) is the famous Lucas-Washburn (LW) equation for the mathematical formulation of dynamic
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capillary flow in begin of last century. There is an obvious overestimation of physics in this equation, which is the content angle is assumed to be constant during a moisture diffusion process. The Young’s equation is assumed to be held under transient capillary flow but is not. Hoffman (1975) stated that there are four types of forces control the capillary flow: viscous forces, inertial forces, interfacial forces at the liquid-gas interface and the liquid-gas-solid juncture which are a set of forces dominate flow depending on the system and flow rate. Consequently, an additional term (experimentally determined term K(Ca)x, K and x are empirical coefficients) added upon static capillary pressure is given to form the total capillary pressure or dynamic capillary pressure (Weitz et al., 1987). In order to continually use Poiseuille’s Law as a basis for multiphase flow in capillary tube, some studies focus on verification of dynamic capillary pressure or dynamic capillary contact angles against Ca, so that the physically validated dynamic term can be added into Poiseuille’s Law (Baver et al., 2014; Calvo et al., 1991; Hoffman, 1975, 1983; Sheng & Zhou, 1992). However, the coefficients of such an empirical relationship are more determined by experimental results for the single capillary tube, which varies from study to study. It somehow constrains upscaling to macroscale law for engineering application in natural porous media. Zhmud et al. (2000) reviewed and discussed incapability of Lucas-Washburn (LW) equation due to unphysical assumptions used for equation derivation. Starting from Newton dynamics equation assisted with capillary, gravity, viscous and turbulent drag terms, Zhmud et al. (2000) gave experimental, analytical and numerical analysis of dynamic capillary flow in single capillary tube and proved that LW equation fails for prediction under short timescales, small viscosity limit and turbulent drag induced meniscus damping oscillations. As for the damping effect, they concluded that it is required for long capillaries while it can be neglected for short capillaries like head zone is a few of radius. Zhmud et al. (2000) gave an explanation of dynamic capillary flow process: at the beginning of capillary flow, wetting phase absorbed into tube is dominated by capillary force which violates WS equation (x~t1/2) instead gives a x~t2 relation; after viscous drag balancing capillary force, flow reach to quasi-steady state obeying the LW equation; finally flow is eased by gravity. Kim and Kim (2012) reviewed recent physic studies on the dynamic capillary rise to find that the power number of advancing time gives different values for different packing beads. They suspected that this is because pores are not fully filled with wetting phase fluids. Also, not only for a short instant time-scale, but LW equation also fails to match experimental result for a large enough time (Kim & Kim, 2012). Despite the dynamic advancing interface theory and corresponding empirical function of Ca with endless fitting into various experimental results, another method to account dynamic capillary pressure is energy method. Yang et al. (2011) developed a modified LW equation by considering capillary driving force generated from an unleashed free energy. In this way, the capillary driving force can be divided into a static capillary force given by Young-Laplace equation and a dynamic free energy term, referring to the thickness of meniscus. Inserting suction terms into Posiesuille Law, Yang et al. (2011)’s model shows good agreement with their experimental results. Moreover, not only does dynamic energy term use for microscale capillary dynamics, but it also offers a chance for upscaling from the thermodynamic basis (S. M. Hassanizadeh & Gray, 1993a). Hence, the macroscale multiphase dynamics in porous media commenced having the theories built upon physics than experimental empirical findings. These advanced theories will be rather briefly introduced than showing each deriving step in following part.
A1.3.4 Advanced Theory of Dynamic Multiphase flows in porous media To the best knowledge of the author, from the 1970s to now, there are three theories invented to reveal the dynamic multiphase flow in porous media. They are the saturation rearrangement theory from G. Barenblatt (1971), dynamic capillary theory based on thermodynamic from either F.-M. Kalaydjian (1992) or S. M. Hassanizadeh and Gray (1993b). G. Barenblatt (1971) was the first one who clearly stated that with a fast nonmixing fluids displacement in porous media, a new saturation cannot be
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achieved immediately instead this process takes a finite relaxation period (τ). Based on the experimental phenomenon and this saturation relaxation assumption, the first dynamic theory is proposed as the difference between actual saturation (Sreal) and future saturation (Saim) is equal to a saturation relaxation term as
Saim Sreal B ( Sreal )
Sreal (A31) t
where τB is a redistribution time as a function of Sreal (s); t is time (s). So the capillary pressure-saturation relationship is a capillary pressure function of Sreal as
S Pnw Pw Pc (Sreal ) Pc Saim B (Sreal ) real (A32) t where Pc is dynamic capillary pressure (kPa), other notations are same with above. Because relative hydraulic conductivity is a function of saturation (Equation 16 or 18), therefore, the dynamic HCF can be given as
S kr ( Sreal ) kr Saim B (Sreal ) real (A33) t where kr is relative hydraulic conductivity (m/s). With these newly defined relations added into mass balance, and two phase Darcy’s Law, the first theory of dynamic multiphase flow in porous media was invented for petroleum engineering. In petroleum engineering, Richards’ model (Equation (A12)) is usually written in the form of Buckley and Leverett (1942) model to account the fraction of each phase in void space. This model has been solved by G. Barenblatt et al. (2003) which shows that the width of stabilised displacement front is not a linear relationship with the inverse of advancing velocity. Details of modified version of his original model and modelling result can be sourced from G. Barenblatt et al. (2003), and numerical solvers are available in the reference list of this work. Although the modelling results agree with experiments given in their series of studies, the phase distribution term does not have physical basis while just a term to account experimental findings. Also, in their studies, the two-phase Darcy Law is still assumed to be valid which is against physics of various flow regimes for dynamic capillary flow in porous media as reviewed in the previous part. The other two theories are all based on a thermodynamic basis. F.-M. Kalaydjian (1992) and S. M. Hassanizadeh and Gray (1993a) both started from microscale dynamic capillary flow and finally ended up with upscaling dynamic capillary pressure equation for macroscale porous media REV as
Pc dyn Pc stat
Sw (A34) t
where Pcdyn is dynamic capillary pressure (kPa), Pcstat is static capillary pressure (kPa), τ is a coefficient that may still have relation with saturation (Pa·s), which is same with unit of viscosity, Sw is wetting phase saturation and t is the advancing duration (s). The common of these two studies is that both of them introduced the second thermodynamic law (entropy inequality) as the basis to derive the macroscale dynamic capillary pressure, while the difference between these two theoretical works is their upscaling methods. S. M. Hassanizadeh and Gray (1993a) used the capillary tube as unit microscale system and upscale the units by volume averaging method, which is based on the concept of REV (M. Hassanizadeh & Gray, 1979). F. Kalaydjian (1987) used the weight function method. Both of two studies derived the mass balance, momentum balance and energy balance for not only each phase but also the interface. Through constructing the energy transfer between wetting phase (in most of the cases) and interface, the system is finally enclosed to provide a similar framework of conventional
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multiphase flow theory with some additional parameters accounting free energy released in a dynamic capillary flow phenomenon. For instance, the system of advanced multiphase flow theory simplified from S. M. Hassanizadeh and Gray (1993b) is summarised in Table 19. Table 19 A simplified system of advanced theory for dynamic multiphase flow in porous media (S. M. Hassanizadeh & Gray, 1993b)
Equations Forms Mass balance phase i nSi ( i qi ) 0 (A35) i=w,n (wetting and t nonwetting phase) Momentum balance Si 2 K i ii awn i Si (A36) of phase i (Darcian qi (Pi i g ) i awn Si like form) Dynamic capillary S Pc dyn Pc stat ( Pn Pw ) Pc stat w (A37) pressure t Capillary pressure P stat P stat (S , a , T ) (A38) c c w wn equation of state where ρi = density of phase i, n = porosity, Si = saturation of phase i, qi = volumetric flux in unit length (discharge velocity), Ki = intrinsic permeability of phase i, Pi = pressure of phase i, awn = specific interfacial area per REV, λii, Ωi = material coefficients (Sw·λwn=Sn· λnw), τ = dynamic coefficient, t = times, T = absolute temperature. However, the new parameters like specific interfacial area, two new material coefficients, so far cannot be measured using currently available experimental methods, in exception of the imaging technique for 2-D micro-sized physical model or microscale Computational Fluid Dynamics Simulation and Lattice Boltzmann Simulation. Although this theory has a rigid physical basis, its application requires improvement of measurement techniques and numerical experimental techniques to determine coefficients of this theory and check the performance of prediction. Comparing thermodynamically based theory to saturation redistribution phenomenological theory, the former owns rigorous physical origins. It does not only provide a macroscale vision looking into dynamic capillary effect but also gives a Darcian form flow motion equation with clear identification of the contribution of driving force released from interfacial energy transformation. In saturation redistribution theory, two-phase Darcy Law is still applicable with the requirement of HCF for both two phases in Buckley and Leverett (1942) form of Richards’ equation (Equation (12)). The contribution of dynamic capillary pressure lumped into HCF overlooks flows regimes occurred in one step imbibition or drainage process. Nevertheless, the dynamic SWRC and HCF can be easily determined by currently available experimental techniques. There is always a conflict of interest between rigorous physical methods and available measuring methods. In this study, the coefficients of both theories will be determined from experimental data, and both theories will be validated against experimental data generated from either 1-D soil column experiment or microscale LB simulation.
A1.3.5 State of art of Dynamic Multiphase flows in porous media The mainstream of recently experimental or numerical studies is on the validation of the thermodynamically based theory derived by S. M. Hassanizadeh and Gray (1993b). They can be separated into mainly three groups: (1) macroscale 1-D soil column experiments supported by pressure and moisture sensors with inverse analysis of Richards’ model; (2) micro-sized physical pore network model supported by image technique; (3) numerical experiments using Pore Network Model (PNM, solving Poiseuille form capillary flow Equation in artificial pore network), Direct Numerical Simulation (DNS, solving Navier-
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Stoke equation in artificial beads package) and Lattice Boltzmann Method (LBM, solving Discretised Boltzmann equation in virtual particles package). Despite the early studies on observation of dynamic effect in SWRC, one-dimensional soil column experiment is now being used to validate dynamic terms in Equation (33) or (34) or Richards’ model coupled with additional dynamic capillary pressure. Since the dynamic coefficient proposed in advanced theory, whether it is a constant or depends on other factors is being studied by Das and Mirzaei (2012) on saturation effect, Abidoye and Das (2014) on scale effect, Hanspal and Das (2012) on temperature impact, Goel and O'Carroll (2011) on fluid viscosity effects, Mirzaei and Das (2007) on micro-heterogeneity impact, O'Carroll et al. (2010) on wettability effects and Mirzaei and Das (2013) on hysteretic dynamic effect. Das and Mirzaei (2012) used a 1-D soil column setup to study the dynamic coefficient and found that dynamic coefficient is not a linear function of saturation but a nonlinear function in which dynamic coefficient can only be treated as a constant within high saturation 70%-100%. When saturation is lower than 60%, the dynamic coefficient increases nonlinearly with saturation (Das & Mirzaei, 2012). Abidoye and Das (2014) applied dimensional analysis to 9 variables (gravity g, isotropic intrinsic permeability K, bubbling pressure PAEV, the domain volume representing domain scale V, fluid density ρ, fluid viscosity μ, saturation S, porosity n, pore size distribution index λ), which are reported as important variables in determination of dynamic coefficient, to derive a nonlinear relationship between two dimensionless groups as b
VK 1.5 g b 1 0.25 a 3 a (A39) K PAEV n S where П1 is first dimensionless groups, П3 is the third dimensionless groups, a and b are fitting coefficient, as for experimental results from Das and Mirzaei (2012), a = 9e-14 and b = 1.31, other notations given in previous content. Prediction from Equation (39) shows good agreement with the experimental result not only from Das and Mirzaei (2012) but also Bottero (2009). Hence, this might be the first mathematical platform to quantify the dynamic coefficient impacted from other nine important variables especially including the domain scale effect. This is also quite worth to be checked against experimental results that will be produced in this study. Hanspal and Das (2012) carried out a numerical simulation of unsteady capillary flow in porous media between 20 and 80⁰C. Regarding their results, they concluded that dynamic coefficients were found to be nonlinear functions of temperature and saturation; and the dynamic coefficient increase with a temperature increase (Hanspal & Das, 2012). However, this study isolates non-isothermal condition so temperature effect will not be further expanded. Goel and O'Carroll (2011) experimentally studied the variation of dynamic coefficient impacted by the viscosity of non-wetting phase fluids using a 1-D sand column drainage test. In their study, three important points are mentioned: (1) there is a delay response of tensiometer due to permeability of ceramic cup, which implies using high conductivity tensiometer to reduce response postpone during dynamic experiment; (2) dynamic coefficient decreases for non-wetting phase fluids having small viscosity; (3) their work is the first experimental study used to validate previous numerical experiments for viscosity effect, and provided real data against some incorrect conclusions (inverse trend) from numerical studies. Mirzaei and Das (2007) conducted a numerical study on micro-scale heterogeneities influencing the dynamic multiphase flow in porous media. In this study, different distribution and intensity of microscale heterogeneities were generated in the solving domain to studying dynamic coefficient being changed by those two impact factors. This study demonstrates that dynamic coefficient is dependent on
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not only flow condition but also domain geometry. With the higher intensity of heterogeneity, the dynamic coefficient also increases. O'Carroll et al. (2010) implemented two-phase multistep outflow experiments with forward and inversed Richards’ model simulation to study the wettability variation on dynamic capillary effect in porous media. In addition to experimental exploration, O'Carroll et al. (2010) derived a microscale capillary advancing equation (Equation (A40)), which is in the same form of macroscale dynamic capillary pressure equation from S. M. Hassanizadeh et al. (2002), using Washburn (1921) equation coupled with current development of dynamic interfacial physics. 1
2 cos dl 2 8 L 2 P s (A40) dt r r r where l = advancing distance, t = advancing time, r = capillary tube radius, ζ = a coefficient of contact line friction, μ = fluid dynamic viscosity, L = total length of capillary tube, ΔP = two-phase pressure difference, σs = surface tension, ϴ = static contact angle. Comparing Equation (40) with Equation (34), it is obvious to see that if the macroscale dynamic capillary equation is applied to single capillary tube, the dynamic coefficient can be seen as
2 r
8 L (A41) r2
in which the calculation of the coefficient of contact friction (ζ) can be checked in details from O'Carroll et al. (2010), other notations are given in Equation (40). Through theoretical formulation by capillary tube conceptual model and experimental data processed by reversing model, O'Carroll et al. (2010) concluded that dynamic effect might be negligible for intermediate wetting conditions (60⁰
