Transp Porous Med (2014) 103:295–314 DOI 10.1007/s11242-014-0302-y
Flow Partitioning in Fully Saturated Soil Aggregates Xiaofan Yang · Marshall C. Richmond · Timothy D. Scheibe · William A. Perkins · Haluk Resat
Received: 19 January 2013 / Accepted: 8 March 2014 / Published online: 30 March 2014 © Springer Science+Business Media Dordrecht 2014
Abstract Microbes play an important role in facilitating organic matter decomposition in soils, which is a major component of the global carbon cycle. Microbial dynamics are intimately coupled to environmental transport processes, which control access to labile organic matter and other nutrients that are needed for the growth and maintenance of microorganisms. Transport of soluble nutrients in the soil system is arguably most strongly impacted by preferential flow pathways in the soil. Since the physical structure of soils can be characterized as being formed from constituent micro-aggregates which contain internal porosity, one pressing question is the partitioning of the flow among the “inter-aggregate” and “intraaggregate” pores and how this may impact overall solute transport within heterogeneous soil structures. The answer to this question is particularly important in evaluating assumptions to be used in developing upscaled simulations based on highly resolved mechanistic models. In our synthetic model of soils, firstly we statistically generated a number of micro-aggregates containing internal pores. Then we constructed a group of diverse multi-aggregate structures with different packing ratios by stacking those micro-aggregates and varying the size and shape of inter-aggregate pore spacing between them. We then performed pore-scale flow simulations using computational fluid dynamics methods to determine the flow patterns in these aggregate-of-aggregates structures and computed the partitioning of the flow through intraand inter-aggregate pores as a function of the spacing between the aggregates. The results of these numerical experiments demonstrate that soluble nutrients are largely transported via flows through inter-aggregate pores. Although this result is consistent with intuition, we
X. Yang · M. C. Richmond · T. D. Scheibe · W. A. Perkins Hydrology Group, Pacific Northwest National Laboratory, PO Box 999, MS K9-36, Richland, WA 99352, USA H. Resat (B) Computational Biology and Bioinformatics Group, Pacific Northwest National Laboratory, PO Box 999, MS J4-33, Richland, WA 99352, USA e-mail:
[email protected] H. Resat School of Chemical Engineering and Bioengineering, Washington State University, Pullman, WA 99164, USA
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have also been able to quantify the relative flow capacity of the two domains under various conditions. For example, in our simulations, the flow capacity through the aggregates (intraaggregate flow) was less than 2 % of the total flow when the spacing between the aggregates was larger than 18 µm. Inter-aggregate pores continued to be the dominant flow pathways even at much smaller spacing; intra-aggregate flow was less than 10 % of the total flow when the inter- and intra-aggregate pore sizes were comparable. Although the results may not be exactly the same as those obtained from actual soil systems, such studies are making it possible to identify which model upscaling assumptions are realistic and what computational methods are required for detailed numerical investigation of hydrodynamics and microbial carbon cycling dynamics in soil systems. Keywords Soil aggregate · Porous media · Pore-scale modelling · Water saturation · Flow partitioning · Hydraulic conductivity · Preferential flow pathways · Nutrient transport · Computational fluid dynamics
1 Introduction Organic matter transformation by microorganisms in terrestrial ecosystems is one of the fundamental components of the global carbon cycle, and it is regulated by a multitude of environmental factors. One such factor is the distribution of soluble nutrients in the soil, which is largely controlled by the material transport by water flow through pores in the soil. Recent advances in ecological studies, combined with computational advances, have begun to allow the construction of highly complex predictive mathematical models to simulate the fluxes of carbon and other organic materials in terrestrial ecosystems. These mechanistic models may play a critical role in predicting how manipulations of soil systems can affect carbon cycling processes, the role of microbes in organic matter decomposition, and the consequences of climate changes for ecosystems. These integrated models must take into account the physical, chemical, and biological conditions as well as the interplay between them (Prosser et al. 2007; Konopka 2009; Sinsabaugh and Shah 2012). As the physical host to such studies, a better understanding of the dynamic properties of soil is crucial for the success of these research efforts. Soils are highly complex systems in terms of their composition, dynamics, and heterogeneity (Or et al. 2007; Young et al. 2008; Feeney et al. 2006), and the spatial structure of the soil is non-homogeneous (Nunan et al. 2006, 2002); roughly half of the space in the top soil consists of water- and air-filled pores whose location and size distribution can vary according to environmental conditions and the soil material. Complexity of the soil organic matter can be traced to the micro-scales (Lehmann et al. 2008), and arguably the most fundamental building block of the soil system is the soil aggregate. Here we refer to an “aggregate” as a soil particle on the order of 50–250 µm in diameter (often referred to as a “micro-aggregate” in the soil science literature), comprising several smaller mineral particles bound together by organic matter and other cohesive forces and thereby containing internal micro-porosity (typically up to 10 µm intra-aggregate pore apertures). There can be significant variability of microbial activity and function between individual aggregates (Bailey et al. 2012), and the microbial community compositions and activity of the aggregates are dependent on their local environmental conditions within a given soil type. In this study we consider the soil aggregate to be the primary habitat structure and use it as the fundamental unit to construct physical models of soil by forming “granules/lumps” containing aggregates-of-aggregates. Soil ecosystems can then be conceptualized as a hierarchy of aggregates of different sizes that contain spatially varying microbial communities.
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These aggregates have different sizes, and the porosity of the soil and the relative positioning of the aggregates will impact how nutrients penetrate within and exchange among aggregates within the soil. Such variations in local environmental conditions are expected to lead to a widely varying spatio-temporal distribution in the dynamics of the terrestrial organic matter transformation (Or et al. 2007). In terms of mathematical modeling, this translates to have a heterogeneous system in which participating objects have different classes of dynamical properties (Young and Crawford 2004). Ideally, mathematical models for soil ecosystems need to incorporate the possible effects of its non-homogeneous three-dimensional structure on system dynamics and hydrodynamic processes into a unified model to improve our understanding of the coupling between structural heterogeneity and environmental conditions, such as the wetting cycles, seasonal changes, and global warming. One key aspect of microbe-facilitated reaction-kinetics is the coupling between the microbial kinetics and porescale hydrodynamics, because of inter-aggregate exchanges that occur through transport of microbial cells, enzymes, and hydrolysis products by pore fluid flow. This can significantly affect patterns of bacterial diversity and survival, and hence, the role of biomass in organic matter transformation in terrestrial systems (Or et al. 2007; Porporato et al. 2003; Schlegel and Jannasch 2006; Young et al. 2008). Spatial patterns of solute transport are well known to be heterogeneous in soil; there will be preferential flow pathways because fluid flow occurs along the path of least resistance. As the dominant routes of transportation, these preferential flow pathways are likely to exert primary control over the distribution of the soluble factors. However, in part because of the challenge of modeling three-dimensional systems with multi-scale heterogeneity, preferential flow in soils is poorly understood (Sammartino et al. 2012). Because of the pore size differences, the role of the advective solute transport is expected to be different within and between aggregates. Within aggregates, which have small pores with limited connectivity, diffusion may be the primary transport mechanism, whereas inter-aggregate pore spaces that are relatively large may be advection-dominated. Quantification of the relative magnitudes of advection with inter- and intra-aggregate pore spaces under various conditions is an important step toward defining appropriate assumptions that will facilitate upscaling of these highly resolved mechanistic simulations to environmentally relevant scales. In particular, we wish to identify the conditions under which we can neglect advection and consider only diffusion in intra-aggregate pores, which greatly simplifies the simulation of microbial processes within an individual aggregate and facilitates upscaling to aggregate-of-aggregates. In this study, we investigated fully saturated soil systems and did not consider the situation in which the soil is partially saturated. Wetting and drying cycles (dynamics of soil moisture) are very important to ecosystem function, and the characteristic states of soils may give rise to a broad range of pore sizes which allow the coexistence of water and air phases (Young and Crawford 2004; Or et al. 2007). Saturation level changes may also give rise to soil shrinkage and swelling thus causing possible rearrangement of pore size distributions. Such complex effects were omitted to keep the simulations feasible. In this work, we quantify the partitioning of fully saturated flow between the “interaggregate” and “intra-aggregate” pores, and determine under which conditions the interaggregate regions can be designated as the dominant flow pathways in the soil system. Under such conditions, advective flow will be expected to dominate the material flow through inter-aggregate pores while the transport of soluble substrates within aggregates would mainly occur through diffusion and not by advective flow (Kausch and Pallud 2013). To this end, we have utilized computational fluid dynamics (CFD) simulations to determine the flow patterns in synthetically generated multi-aggregate structures whose internal structures were based on the statistical distribution of pore sizes that were reported in the literature
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(Peth et al. 2008; Sleutel et al. 2008; Strong et al. 2004). In this study, we have concentrated on computing the saturated flow patterns in multi-aggregate structures to identify the dominant mechanisms of nutrient transportation in soil. This is important because nutrient availability through redistribution impacts microbial growth and survival kinetics in terrestrial systems. Stacking the aggregates with variable inter-aggregate distances also allowed us to compute the partitioning of the flow between intra- and inter-aggregate pores as a function of the spacing between the aggregates. The simulated results have shown that the soluble factors are mainly transported via flows through inter-aggregate pores and that the flow capacity through the aggregates was only a small fraction of the total flow. The numerical techniques used in the current studies are explained in Sect. 2, which includes the construction of the micro-aggregate pore structures and the CFD methods. In Sect. 3, the results for the computed flow fields in different aggregates and the calculated hydraulic conductivities are presented and discussed. The conclusions are given in Sect. 4. 2 Mathematical Models and Methods Primary elements of the numerical methods used for these studies were (1) the construction of synthetic micro-aggregate pore structure as the fundamental structural element of soil using statistical method, (2) formation of multi-aggregate structures by combining individual aggregate structures, (3) specification of initial and boundary conditions on the multi-aggregate structures, (4) application of CFD methods to simulate low-Reynolds number Stokes flow within the multi-aggregate structures, and (5) analysis of model outputs to determine the allocation of flow between inter- and intra-aggregate pore spaces. 2.1 Computational Fluid Dynamics Simulations CFD has long been an important tool for the analysis of fluid flows in a wide variety of applications (e.g., Lee et al. 2013; Xia and Sun 2002). CFD methods have recently been applied to the simulation of fluid flow through porous media at the pore scale, in which the pore geometry is explicitly specified and the Navier–Stokes equations are solved within the fluid domain on a highly discretized computational mesh (e.g., Molins et al. 2012; Logtenberg and Dixon 1998; Dixon and Nijemeisland 2001; Morais et al. 2009; Nijemeisland and Dixon 2004; Zaretskiy et al. 2010; Eppinger et al. 2011; Cardenas 2008). CFD is a computationally efficient approach for saturated (single-phase fluid) simulations, and the accuracy of computed velocities using CFD methods in micro-channels has been validated through the application of magnetic resonance velocimetry (Elkins and Alley 2007) in medical applications (Rochefort et al. 2007; Yiallourou et al. 2011). We have recently compared pore-scale flow predictions obtained using CFD simulations to the experimental data and shown that agreement was excellent (Yang et al. 2013). Because of its good performance in related studies, we chose the CFD method to perform the simulations in the present study. However, we note that other computational methods for simulating fluid flow including smoothed particle hydrodynamics (Tartakovsky et al. 2008, 2009), Lattice-Boltzmann (e.g., Kang et al. 2010; Sukop et al. 2008), and pore network models (e.g., Celia et al. 2000; Blunt 2001) could also be suitable for pore-scale flow simulations. 2.1.1 Numerical Methods The CFD calculations of fully saturated pore-scale laminar fluid flow through multi-aggregate structures were performed using the transient energy transport hydrodynamics simulator
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(TETHYS) code (Richmond et al. 2013; Yang et al. 2013). TETHYS, developed at Pacific Northwest National Laboratory, is written in C++ and uses the finite volume method (Ferziger and Peric 2001) to solve the governing mass, momentum, and scalar conservation equations on general unstructured meshes. In the current study, computational meshes imported into TETHYS for simulation were Cartesian orthogonal meshes. The cubic cells defining the aggregate structures were used directly as mesh elements in the TETHYS simulations. Each mesh element corresponds either to a void space (a pore occupied by water) or to a solid volume inaccessible to the fluid. TETHYS utilizes parallel high-performance computing clusters [parallel implementation uses MPI (MPI-Forum 2009) and the Global Array toolkit (Nieplocha et al. 1996, 2006)] to quantify the complex 3-D flow patterns in heterogeneous systems. TETHYS solves the three-dimensional Navier–Stokes equations describing incompressible fluid flow at the pore scale. The standard conservation equations of mass and momentum for an incompressible fluid with constant dynamic viscosity are: Mass Conservation equation : ∇ · v = 0,
(1)
Momentum Conservation equation : ρ∂v/∂t + ρv · ∇v = −∇ p + μ∇ 2 v,
(2)
where v is the local velocity vector, t is time, p is pressure, ρ is the fluid bulk density, and μ is the dynamic viscosity. Equations (1) and (2) are solved under fixed boundary conditions over time until a steady solution is reached. These equations are discretized using standard finitevolume techniques (Ferziger and Peric 2001) to obtain a system of algebraic equations, which are then solved using advanced linear algebra methods. In TETHYS, the momentum equations are solved using a 2nd-order central difference scheme on the orthogonal mesh. The iterative SIMPLE algorithm (Patankar 1980) is used to couple the velocity and pressure fields. In the predictor step the estimated pressure field is used to solve the momentum equation, resulting in an intermediate velocity field. This velocity field will in general not satisfy the continuity (mass conservation) equation. In the next step of the algorithm, a pressure correction equation is solved. Then the pressure field is updated followed by the correction of the velocity field. This process is repeated until the solution is converged. Iterative methods provided in PETSc (Balay et al. 2012) are used to solve the system of linear algebraic equations corresponding to the discretized governing equations and the SIMPLE algorithm. We have successfully performed extensive testing of the TETHYS code (Richmond et al. 2013) and demonstrated its ability to reproduce spatial velocity patterns for a variety of benchmark cases, in comparison to other CFD codes, and against experimental data using magnetic resonance velocimetry in porous media (Yang et al. 2013). Although we have simulated fully saturated conditions, it is important to note that CFD techniques to simulate unsaturated flow in pore-scale domains are available (Raeini et al. 2012). The capability of simulating unsaturated flow will be included in the future versions of TETHYS. 2.1.2 Simulation Setup and Boundary Conditions For the CFD simulations, the laminar flow was assumed to be isothermal and physical properties of the incompressible fluid were that of water. System dimensions are listed in Table 1. At the inlet of the system (bottom layer in Fig. 1 below), a specified mass-flux boundary condition (fixed incoming velocity) was used. The boundary condition at the outlet was imposed as having a specified pressure at the top layer of the grid structure. No-slip conditions were imposed on the lateral boundaries (by default in TETHYS for the solid cells that are not shown in Fig. 1). Thus, the flow is conducted through the aggregates in the vertical
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300 Table 1 Model configuration parameters (for a single unit) and boundary conditions
X. Yang et al. Parameter
Symbol (unit)
Value 105
Unit length
L (µm)
Unit width
W (µm)
105
Unit height
H (µm)
117
Aggregate porosity
ε
0.2, 0.3
Reynolds number
Re
0.001
Fluid density
ρ (kg/m3 )
997.561
Fluid dynamic viscosity
μ (Pa s)
8.887 × 10−4
Fig. 1 A synthetic representative micro-aggregate unit. It consists of 35 × 35 × 35 grid elements, which had a 3 µm edge length. Two additional grid layers were added on both ends along the z-axis to better disperse the in- and out-flows. So the overall simulation volume for the single aggregate was 105 × 105 × 117 µm3
z-direction from bottom to top as denoted in Fig. 1 (because of the small system size, gravity effects are neglected in the current simulations). The grain-scale Reynolds number can be calculated as Re = vρΔl/μ where Δl is the size of the grain (cell). Velocity was specified such that the Reynolds number was 0.001, i.e., it was much smaller than unity and the flow was in the laminar flow (Stokes flow) regime. 2.2 Hydraulic Conductivity Calculation Darcy’s Law specifies a linear relationship between the specific flux (mass flux per unit area) and the hydraulic head gradient as follows: q = Q/A = K dh/dz,
(3)
where q is the specific flux, Q is the total fluid flux, A is the cross-sectional area of the domain perpendicular to the flow, and K is the hydraulic conductivity. The hydraulic head h
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is the sum of the pressure head ψ, which is the height of an equivalent water column above some datum (ψ = p/ψg where p is pressure, ψ is fluid density, and g is the gravitational acceleration constant), and the elevation head z, which is simply the elevation relative to the datum. Since flow is in the z direction in our system and taking the elevation datum as corresponding to the bottom plane of the simulation domain (z = 0), one obtains: Q/A = −
K dp . ρg dz
(4)
For the multi-aggregate system, we specify the total flux Q as a boundary condition on the bottom plane. We approximate the pressure gradient d p/dz by calculating the average pressure p¯ on the inlet and outlet faces (not including the added layers), and dividing the difference p¯ (average pressure at the top minus average pressure at the bottom) by the thickness of the multi-aggregate system. Then we can rearrange the equation to solve for the hydraulic conductivity K as: K = −(Q/A)(ρgz/ p). ¯
(5)
All quantities on the right-hand side except p¯ are known or specified a priori. Thus, to compute the hydraulic conductivity, we use the above equation with the p¯ value determined from the simulations. 2.3 Porous Micro-Aggregate Structures In order to build a synthetic model of soils, we have recently developed a “worming algorithm” to construct random pore structures internal to an individual soil aggregate (Resat et al. 2013, submitted). Briefly, using the statistical estimates of the pore sizes and the distances between interstitial sites in soil aggregates that are becoming available (Peth et al. 2008; Sleutel et al. 2008; Strong et al. 2004) as reference, our algorithm creates a connected random pore structure by traversing through the aggregate volume defined on a rectangular grid in a wormlike manner. These pore spaces within aggregates are not created by macro-invertebrate burrowing; they are much smaller than worm burrows in soil, which are well-known to create macropore structures at larger scales, and the reference to “worming” in this context is only by analogy. The traversing direction during pore creation is determined by assigning probabilities to (a) the continuation of movement in the current direction, and (b) turning to change the movement direction (Resat et al. 2013, submitted). The relative ratio of these two probabilities determines whether the constructed pore distribution will be mostly 1-D rod-like pores (the case when the probability of continuing in the same direction to the probability of turning ratio is large, which enhances movement in a single direction) or a collection of 3-D spherical lobes (the case when the aforementioned ratio is small, which enhances random movement around a spot). The algorithm starts by defining the aggregate as a fully solid unit, picks a starting point in the aggregate and an initial traveling direction randomly, and then uses the described worming motion to create open pore spaces for the grid units that the worm passes through. The pore creation process is ended when the desired mean porosity level, which we define as the percentage of the total aggregate volume that is open pore space, is achieved. Pores constructed using this algorithm are all interconnected, i.e., there are not any totally enclosed pores that cannot drain. In this study we have constructed cubic porous aggregates with an edge length of ∼ 105 µm (Fig. 1). Our algorithm used a grid spacing of 3 µm, i.e., the aggregate was defined on a 35 × 35 × 35 rectangular grid. Aggregate structures with the mean internal porosity of 20–30 % were considered in this study. A similar aggregate
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structure was used in our earlier study to investigate bacterial carbon degradation dynamics within individual aggregates (Resat et al. 2012). Internal structures of the aggregates statistically constructed by our worming algorithm can be contrasted to the available experimental information. Peth et al. have investigated aggregates from conventionally tilled (CT) and grassland (GL) sites (Peth et al. 2008). They have used voxel resolutions of 3.214 µm (CT) and 5.403 µm (GL), and analyzed cube-shaped samples with edge lengths 1.286 mm (CT) and 2.161 mm (GL). The observed statistical distribution of pore sizes had a peak at ∼18 µm [cf., Fig. 3b in (Peth et al. 2008)]. Because of the utilized resolution and sample volume, pore sizes observed in this study mostly corresponds to inter-aggregate pores. Sleutel et al. used much smaller voxel sizes (as small as 0.84 µm) in their studies (Sleutel et al. 2008). They classified the pores into 1–5 and 5–20 µm sized pores, and also discussed that microbial activities differ between fine (
