Estimation of Relative Permeability from a Dynamic Boiling Experiment

ESTIMATION OF RELATIVE PERMEABILITY FROM A DYNAMIC BOILING EXPERIMENT

A REPORT SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

Marilou Tanchuling Guerrero June 1998

Abstract Thermal and multiphase flow properties of a Berea sandstone core were estimated by inverse calculation using temperature, pressure, steam saturation, and heat flux data. The development of the two-phase flow region was strongly related to the temperature conditions in the core since heat was the only driving force in the experiment. As a result, the heat input as well as the thermal properties of the sandstone and insulation materials were a major consideration in understanding the system behavior. The high sensitivity of the insulation materials to the observation data and their strong correlation with the other parameters of interest made it difficult to obtain accurate estimates. Although the linear relative permeability model gave the best fit in the two cases presented in the paper, all models yielded similar matches. This indicated that the data did not contain sufficient information to distinguish the different models, giving a non-unique solution. Nonetheless, almost all models gave a consistent estimate for Sgr, which was around 0.10.2. In addition, the choice for the capillary pressure model depended on the condition in the core, whether there was single-phase steam or two phases present. Since the distribution of steam depended heavily on the capillary pressure, the relative permeability would have been more accurately estimated had the capillary pressure been known. The comprehensive analysis of all available data from a transient non-isothermal two-phase flow experiment provided an insight into relation of processes and correlation of parameters.

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Acknowledgments I sincerely would like to thank Dr. Cengiz Satik for his patience and understanding while this project was underway. His insights about the experiment itself and possible solutions to the inverse problem helped me approach the problem from the proper perspective. I am also deeply grateful to Dr. Stefan Finsterle, whose guidance, support, and ITOUGH2 expertise were truly valuable. His insightful suggestions helped me understand the difficulties I encountered in the estimation process. I also would like to thank Prof. Roland Horne for his guidance, encouragement, and sense of humor. His easy-going nature made my life as a graduate student more bearable and truly enjoyable.

To my all friends, especially Apu Kumar, thank you for the fond memories. When the going got tough, you were there to lighten the burden.

To my family, thank you for your love, prayers and support. To Jason, you served as my inspiration to reach for the stars.

This project was funded by the U.S. Department of Energy under grant number DEFG07-95ID13370.

Malou T. Guerrero June 5, 1998

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Table of Contents Abstract ...............................................................................................................................ii Acknowledgments ..............................................................................................................iii Table of Contents ...............................................................................................................iv List of Figures.....................................................................................................................vi List of Tables....................................................................................................................... x Introduction ......................................................................................................................... 1 Review of Related Literature............................................................................................... 3 2.1 Relative Permeability and Capillary Pressure............................................................ 3 2.2 Heat and Mass Transfer ............................................................................................. 5 2.3 Previous Work ........................................................................................................... 6 Relative Permeability and Capillary Pressure Models ........................................................ 8 3.1 Linear Model.............................................................................................................. 8 3.4 Corey Model............................................................................................................... 9 3.5 Leverett Model ......................................................................................................... 10 3.2 Brooks-Corey Model................................................................................................ 11 3.3 van Genuchten Model .............................................................................................. 12 Experimental Apparatus and Procedure ............................................................................ 14 Numerical Simulation ....................................................................................................... 18 5.1 Model ....................................................................................................................... 18 5.2 Forward Calculation................................................................................................. 20 5.2.1 Input Data .................................................................................................................................... 20 5.2.2 Sensitivity Analysis...................................................................................................................... 21 5.2.3 Initial Guesses.............................................................................................................................. 58

5.3 Parameter Estimation ............................................................................................... 58 5.3.1 Inverse Modeling ......................................................................................................................... 58 5.3.2 Results and Discussion................................................................................................................. 60

Conclusion......................................................................................................................... 96 References ......................................................................................................................... 97 Appendix A ..................................................................................................................... 100 Appendix A.1.1 TOUGH2 input file. ................................................................................................. 100

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A.1.2 ITOUGH2 input file.................................................................................................................. 113 A.2 Linear model calibration input files ............................................................................................. 118 A.2.1 TOUGH2 input file. .................................................................................................................. 118 A.2.2 ITOUGH2 input file.................................................................................................................. 130

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List of Figures Figure 2.1 Relative permeability as a function of liquid saturation. ................................... 4 Figure 2.2 Relative permeability obtained by Ambusso (1996).......................................... 7 Figure 2.3 Relative permeability obtained by Satik (1998)................................................. 7 Figure 3.1 Linear relative permeability. .............................................................................. 9 Figure 3.2 Corey relative permeability.............................................................................. 10 Figure 3.3 Brooks-Corey relative permeability. ................................................................ 12 Figure 3.4 van Genuchten relative permeability. .............................................................. 13 Figure 4.1. Schematic diagram of the experimental apparatus (after Satik, 1997b). ........ 14 Figure 4.2 Experimental temperature data. ....................................................................... 16 Figure 4.3 Experimental pressure data. ............................................................................. 16 Figure 4.4 Experimental steam saturation data. ................................................................ 17 Figure 4.5 Experimental heat flux data. ............................................................................ 17 Figure 5.1. Schematic diagram of the 4×51 TOUGH2 model. Ring 4 is not shown since it represents ambient conditions. .......................................................................................... 19 Figure 5.2. ITOUGH2 input used to generate the elements and connections. .................. 20 Figure 5.3 Base scenario: αs=4.930 W/m2, αb=0.150 W/m2, αi=0.115 W/m2, αh=2.885 W/m2 and αe=0.577 W/m2. ............................................................................................... 26 Figure 5.4 αs=4.300 W/m2, αb=0.150 W/m2, αi=0.115 W/m2, αh=2.885 W/m2 and αe=0.577 W/m2.................................................................................................................. 27 Figure 5.5 αs=4.930 W/m2, αb=0.090 W/m2, αi=0.115 W/m2, αh=2.885 W/m2 and αe=0.577 W/m2.................................................................................................................. 28 Figure 5.6 αs=4.930 W/m2, αb=0.150 W/m2, αi=0.175 W/m2, αh=2.885 W/m2 and αe=0.577 W/m2.................................................................................................................. 29 Figure 5.7 Base case linear model: Slr=0.20, Sgr=0.20, Sls=0.80, Sgs=0.80, Pcmax=105 Pa 30 Figure 5.8 Linear model: Slr=0.02, Sgr=0.10, Sls=0.80, Sgs=0.80, Pcmax=105 Pa................ 31 Figure 5.9 Linear model: Slr=0.02, Sgr=0.20, Sls=0.80, Sgs=0.70, Pcmax=105 Pa................ 32 vi

Figure 5.10 Linear model: Slr=0.02, Sgr=0.20, Sls=0.80, Sgs=0.80, Pcmax=104 Pa.............. 33 Figure 5.11 Base case: Corey relative permeability and linear capillary model: Slr=0.20, Sgr=0.20, Pcmax=105 Pa...................................................................................................... 34 Figure 5.13 Corey relative permeability and linear capillary model: Slr=0.20, Sgr=0.10, Pcmax=105 Pa. ..................................................................................................................... 36 Figure 5.14 Corey relative permeability and linear capillary model: Slr=0.20, Sgr=0.20, Pcmax=104 Pa. ..................................................................................................................... 37 Figure 5.15 Base case: Corey relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Po=105. Slr (CP)=0.20.................................................................. 38 Figure 5.16 Corey relative permeability and Leverett capillary pressure: Slr(RP)=0.10, Sgr=0.20, Po=105. Slr (CP)=0.20. ...................................................................................... 39 Figure 5.17 Corey relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.10, Po=105. Slr (CP)=0.20. ...................................................................................... 40 Figure 5.18 Corey relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Po=104, Slr (CP)=0.20. ...................................................................................... 41 Figure 5.19 Corey relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Po=105, Slr (CP)=0.10. ...................................................................................... 42 Figure 5.20 Base case: Linear relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Sls=0.80, Sgs=0.80, Po=105, Slr (CP)=0.20................................... 43 Figure 5.21 Linear relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.10, Sls=0.80, Sgs=0.80, Po=105, Slr (CP)=0.20 ........................................................ 44 Figure 5.22 Linear relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Sls=0.80, Sgs=0.70, Po=105, Slr (CP)=0.2 .......................................................... 45 Figure 5.23 Linear relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Sls=0.80, Sgs=0.80, Po=104, Slr (CP)=0.20 ........................................................ 46 Figure 5.24 Linear relative permeability and Leverett capillary pressure: Slr(RP)=0.20, Sgr=0.20, Sls=0.80, Sgs=0.80, Po=105, Slr (CP)=0.10 ........................................................ 47 Figure 5.25 Base case: Brooks-Corey model: Slr=0.20, Sgr=0.20, λ=2.0, pe=2000 Pa ..... 48 Figure 5.26 Brooks-Corey model: Slr=0.10, Sgr=0.20, λ=2.0, pe=2000 Pa....................... 49 Figure 5.27 Brooks-Corey model: Slr=0.20, Sgr=0.10, λ=2.0, pe=2000 Pa....................... 50

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Figure 5.28 Brooks-Corey model: Slr=0.20, Sgr=0.20, λ=1.5, pe=2000 Pa....................... 51 Figure 5.29 Brooks-Corey model: Slr=0.20, Sgr=0.20, λ=2.0, pe=1000 Pa....................... 52 Figure 5.30 Base case van Genucthen model: Slr=0.2, Sgr=0.2, n=2.0, 1/α=500 ............ 53 Figure 5.31 van Genucthen: Slr=0.1, Sgr=0.2, n=2.0, 1/α=500........................................ 54 Figure 5.32 van Genucthen: Slr=0.2, Sgr=0.1, n=2.0, 1/α=500........................................ 55 Figure 5.33 van Genucthen: Slr=0.2, Sgr=0.2, n=2.5, 1/α=500........................................ 56 Figure 5.34 van Genucthen: Slr=0.2, Sgr=0.2, n=2.0, 1/α=1000...................................... 57 Figure 5.35 Inverse modeling flow chart (after Finsterle et al., 1998).............................. 60 Figure 5.36 Measured and calculated temperature after single-phase period calibration. 64 Figure 5.37 Measured and calculated heat flux after single-phase period calibration. ..... 64 Figure 5.38 Measured and calculated temperature............................................................ 69 Figure 5.39 Measured and calculated pressure................................................................. 70 Figure 5.40 Measured and calculated steam saturation.................................................... 70 Figure 5.41 Measured and calculated heat flux................................................................. 71 Figure 5.42 Temperature with respect to time................................................................... 71 Figure 5.43 Pressure with respect to time. ........................................................................ 72 Figure 5.44 Steam saturation with respect to time. ........................................................... 73 Figure 5.45 Heat flux with respect to time........................................................................ 74 Figure 5.46 Temperature with respect to distance from the heater. .................................. 74 Figure 5.47 Pressure with respect to distance from the heater. ......................................... 75 Figure 5.48 Steam saturation with respect to distance from the heater............................. 75 Figure 5.49 Heat flux with respect to distance from the heater......................................... 76 Figure 5.50 Inverse modeling relative permeability results compared with Ambusso’s results (1996)..................................................................................................................... 77 Figure 5.51 Leverett capillary pressure. ............................................................................ 77 Figure 5.52. Relative permeability estimates compared with Ambusso’s (1996) results. 78 Figure 5.53. Relative permeability estimates compared with Satik’s (1996) results. ....... 78 Figure 5.54 No Sensor 1 data: Measured and calculated temperature after calibration of model under single-phase conditions. ............................................................................... 79

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Figure 5.55 No Sensor 1 data: measured and calculated heat flux after calibration of model under single-phase conditions. ............................................................................... 80 Figure 5.56 No Sensor 1 data: Measured and calculated temperature. ............................. 84 Figure 5.57 No Sensor 1 data: Measured and calculated pressure. ................................... 84 Figure 5.58 No Sensor 1 data: Measured and calculated steam saturation. ...................... 85 Figure 5.59 No Sensor 1 data: Measured and calculated heat flux. .................................. 85 Figure 5.60 No Sensor 1 data: Temperature with respect to time. ................................... 88 Figure 5.61 No Sensor 1 data: Pressure with respect to time............................................ 89 Figure 5.62 No Sensor 1 data: Steam saturation with respect to time............................... 90 Figure 5.63 No Sensor 1 data: Heat flux with respect to time. ......................................... 91 Figure 5.64 No Sensor 1 data: Temperature with respect to distance from heater............ 92 Figure 5.65 No Sensor 1 data: Pressure with respect to distance from heater. ................. 92 Figure 5.66 No Sensor 1 data: Steam saturation with respect to distance from heater. .... 93 Figure 5.67 No Sensor 1 data: Heat flux with respect to distance from heater. ................ 93 Figure 5.68 No Sensor 1 data: Relative permeability estimate compared with Ambusso’s results (1996)..................................................................................................................... 94 Figure 5.69 No Sensor 1 data: Linear capillary pressure................................................... 94 Figure 5.70 No Sensor 1 data: Relative permeability estimates compared with Ambusso’s results. ............................................................................................................................... 95 Figure 5.71 No Sensor 1 data: Relative permeability estimates compared with Satik’s results. ............................................................................................................................... 95

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List of Tables Table 4.1. Physical properties of the cores, epoxy, core insulation, and heater insulation. .................................................................................................................................... 15 Table 5.1. Observation used for model calibration. .......................................................... 61 Table 5.2 Parameter estimates after single-phase calibration period. ............................... 62 Table 5.3 Variance-covariance matrix (diagonal and lower triangle) and correlation matrix (upper triangle) after single-phase period calibration. .................................... 63 Table 5.4 Statistical measures and parameter sensitivity after single-phase period calibration. .................................................................................................................. 63 Table 5.5 Total sensitivity of observation, standard deviation of residuals, and contribution to the objective function (COF) after single-phase period calibration... 64 Table 5.6 Parameter estimates for linear relative permeability and Leverett capillary pressure case. .............................................................................................................. 67 Table 5.7 Variance-covariance and correlation matrices. ................................................. 67 Table 5.8 Statistical measures and parameter sensitivity. ................................................. 68 Table 5.9 Total sensitivity of observation, standard deviation of residuals, and contribution to the objective function (COF). ............................................................ 69 Table 5.10 No Sensor 1 data: Parameter estimates after the single-phase calibration period. ......................................................................................................................... 80 Table 5.11 No Sensor 1 data: Variance-covariance and correlation matrices after singlephase period calibration.............................................................................................. 81 Table 5.12 No Sensor 1 data: Statistical measures and parameter sensitivity after singlephase period calibration.............................................................................................. 81 Table 5.13 No Sensor 1 data: Total sensitivity of observation, standard deviation of residuals, and contribution to the objective function (COF) after single-phase period calibration. .................................................................................................................. 82 Table 5.14 No Sensor 1 data: Parameter estimates. .......................................................... 86

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Table 5.15 No Sensor 1 data: Variance-covariance and correlation matrices................... 86 Table 5.17 No Sensor 1 data: Total sensitivity of observation, standard deviation of residuals, and contribution to the objective function (COF). ..................................... 87 Table 5.18 No Sensor 1 data: Total sensitivity of observation, standard deviation of residuals, and contribution to the objective function (COF). ..................................... 87

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Chapter 1 Introduction

Relative permeability is one of the parameters required in describing multiphase flow in porous media since it governs movement of one phase or component with respect to another. It has been studied widely for oil applications (e.g. waterflooding and gas injection), but less so for geothermal applications, which is the main concern of this study. So far, relative permeability relations for steam and liquid water have been based on theoretical methods using field data (e.g. Grant, 1977; Horne and Ramey, 1978), and laboratory experiments (e.g. Ambusso, 1996; Satik, 1998). Although relative permeability is best determined through laboratory experiments, it is difficult to measure due to capillary forces that introduce nonlinear effects on the pressure and saturation distribution at the core exit (Ambusso, 1996). Also, relative permeability is difficult to measure directly for steam-water flows due to experimental constraints (Satik, 1997b). Thus, this study describes a different approach to estimating the relative permeability by matching data from a transient boiling experiment performed on a Berea sandstone to results obtained from numerical simulation. This method provides a way to examine the validity of the relative permeability measurements taken from previous experiments, as well as to estimate capillary pressure.

Although the focus of this research project is relative permeability, capillary pressure has also been studied. These two parameters cannot be separated from each other since they are both important in the behavior of multiphase flow in porous media. However, like relative permeability, capillary pressure in steam-water systems is unknown. This makes the estimation process more difficult because the higher number of unknowns in the

problem. In addition, other unknown parameters, such as heat input and heat losses, affect the successful estimation of the relative permeability. To decrease the correlation between the thermal properties and two-phase parameters of interest, the inversion was performed in two steps. Several relative permeability and capillary pressure models were used to solve the inverse problem. The function that best matches the observed data without over-parameterization can be considered the likely scenario.

This report will begin by discussing the concepts of relative permeability, capillary pressure, and heat and mass transfer. Results of recent relative permeability experiments will also be discussed. Following this, a brief summary of the relative permeability and capillary pressure models used in the study will be given. Then a description of the experimental apparatus and procedures will be given, followed by a discussion of the steps involved in the numerical simulation. This report will end with an analysis of results and conclusion of the study.

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Chapter 2 Review of Related Literature In this chapter, relative permeability and capillary pressure will be defined in order to understand their role in mass flow calculations. Since the system being considered is a boiling system, a discussion of heat transfer will also be included. Finally, two recent relative permeability experiments will be reviewed to provide a reference for comparison for the simulation results.

2.1 Relative Permeability and Capillary Pressure When two fluids are flowing simultaneously in a porous medium, each fluid has its own effective permeability, kβ (Dake, 1978). The effective permeability is related to the absolute permeability, k, a property that is independent of the properties of the singlephase fluid flowing through the porous medium, by the following equation: k β = k rβ k

β = l (liquid), g (gas) ………………………………………… (2.1)

where krβ is the relative permeability of the phase, β. The effective permeabilities are normally expressed as a function of the saturation of each fluid, and their sum is usually less than the absolute permeability. For single-component two-phase systems (e.g. liquid and gas phases in water), the liquid would not flow (kl=0) when the liquid saturation, Sl, is equal to the residual liquid saturation, Slr. Also, when Sl=1, or when the rock is fully saturated with water, kl=k. Similarly, the gas would not flow (kg=0) when the gas saturation, Sg=1-Sl, is equal to the residual gas saturation, Sgr. Also, kg=k when Sg=1. In terms of relative permeability, krl=0 when Sl=Slr, krl=1 when Sl=1, kgr=0 when Sg=Sgr, and kgr=1 when Sl=1. A simple illustration of the relative permeability as a function of 3

liquid saturation is shown in Figure 2.1. Apart from saturation, other factors can influence relative permeability or phase distribution in a porous medium. They include matrix (or pore) structure, history, interfacial tension, wettability, viscosity ratio, and density ratio (Dullien, 1992; Kaviany, 1995). 1 0.9 k rl 0.8 krg

0.7

kr

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5 Sl

Slr

0.6

0.7

0.8

0.9

1

Sg r

Figure 2.1 Relative permeability as a function of liquid saturation.

Incorporating relative permeability in the generalized Darcy’s law (Dullien, 1992),

(

 kk rβ  ∇pβ − ρβ g  µβ 

(δQ / δA)n = 

) ………………………………………………… (2.2)

where Q is the volumetric flow rate, A is the normal cross sectional area of the porous medium, n is the unit normal vector of surface area A, µβ is the viscosity of β, ∇pβ is the pressure drop in β across two points in the porous medium, ρβ is the density of β, and g is the acceleration due to gravity. The pressures pβ in the two phases are related to each other by the capillary pressure, pc. Hence, if gas is the non-wetting phase and liquid is the wetting phase, the two pressure gradients are related to each other by the gradient of the capillary pressure (Leverett, 1941):

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∇pc = ∇pg − ∇pl ……………………………………………………………………. (2.3) The general expression for calculating the capillary pressure at any given point on an interface between liquid and gas is given by the Young-Laplace equation  1 1 …………………………………………………...……… (2.4) pc = pg − pl = σ  +  r1 r2  where σ is the interfacial tension between the two liquid and gas phases, and r1 and r2 are the principal radii of curvature at any point on the interface where the pressures in the liquid and gas are pl and pg, respectively. Some of the more commonly used relative permeability and capillary pressure models will be discussed in the next chapter.

2.2 Heat and Mass Transfer For two phases of a single component in porous media, the conservation of energy can be simplified as

M 1 = (1 − φ ) ρ R C R T + φ

∑ S β ρ β uβ …………………………………………….… (2.5)

β =l , g

where φR is the rock porosity, ρR is the rock grain density, CR is the specific heat of the rock, T is the temperature, Sβ is the saturation of phase β (liquid or gas), and uβ is the specific internal energy of β. The first term in Equation 2.1 is the energy of the rock, and the second term is the summation of the energy of the two phases present. The phase transition, pressure-work, and viscous dissipation terms are neglected since they are small in liquid-dominated systems in the absence of an adsorbed phase (Garg and Pritchett, 1977). For a multiphase single component system, the conservation of mass is written as

M2 =φ

∑ S β ρβ X β

…………………………………………………………….… (2.6)

β =l ,g

Pruess and Narasimhan (1985) expressed the energy-balance and mass-balance equations in integral form. For an arbitrary flow domain, Vn (κ ) d (κ ) (κ ) M dv = F .nΓ + q dv ∫ ∫ ∫ dt Vn Vn Γn

κ=1:heat and κ=2:water …………….. (2.7)

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The first term on the right-hand side of Equation 2.7 is the flux term, where the heat flux is

F 1 = − K∇T −

∑ hβ Fβ …………………………………………………………….. (2.8)

β = l ,g

and the mass flux is

F2 =

∑ Fβ

…………………………………………………………………………. (2.9)

β = l ,g

The parameter, K, is the heat conductivity of the rock-fluid mixture, and hβ is the specific enthalpy of water in β. Equation 2.8 includes conductive and convective heat terms only, neglecting dispersion. The mass flux in each phase is F

β

=

k rβ

∑ − k µβ β =l,g

ρ β X β ∇Pβ − ρ β g ) − δ βg Dva ρ β ∇X β ….……………………. (2.10)

The last term in Equation 2.10 represents a binary diffusive flux and contributes only to gas phase flow (Pruess, 1987).

2.3 Previous Work A wide range of relative permeability relations has been reported. However, only a few studies on single-component two-phase systems, particularly steam and liquid water systems, have been done. In the past, attempts to measure the relative permeability of single-component two-phase flow directly produced questionable results (Sanchez et al, 1980; Verma, 1986; Clossman and Vinegar) since measurement of the liquid saturation was inaccurate and the assignment of pressure gradient may have been inappropriate (Ambusso, 1996). Recently, significant improvements in measuring water and steam saturation and collecting data from steam-water flow experiments were achieved. Results of the steady-state experiment conducted by Ambusso (1996) on a Berea sandstone indicated a linear relationship for steam-water relative permeability (Fig. 2.2). In attempting to repeat these results, Satik (1998) improved the design of the experimental apparatus. A successful experiment was conducted and steam-water relative permeability was obtained. These recent results suggest a more curvilinear relationship (Fig. 2.3) that

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is different from the results obtained by Ambusso (1996) (linear relationship). Satik (1998) suggested that the difference in the relative permeability obtained from the two experiments may have been due to permeability effects. Ambusso used a Berea sandstone with k=600 md, whereas Satik used another Berea sandstone with k=1200 md. Sanchez (1987) also reported a relative permeability relation similar to Satik’s results. Nevertheless, variations in results must be expected since the experimental methods and cores used differ. 1 0.9 0.8 0.7

kr

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sl

Figure 2.2 Relative permeability obtained by Ambusso (1996). 1 0.9 0.8 0.7

kr

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Sl

Figure 2.3 Relative permeability obtained by Satik (1998).

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0.9

1

Chapter 3 Relative Permeability and Capillary Pressure Models Several semi-empirical relative permeability and capillary pressure relationships have been proposed by different authors. However, only four relative permeability and four capillary pressure models are included in this study. These are the linear, Corey, and Leverett models, and coupled relative permeability and capillary pressure functions, the Brooks-Corey and van Genuchten models. The selection of the models was based on the experimental results obtained by Ambusso (1996) and Satik (1998).

3.1 Linear Model The linear functions comprise the simplest relative permeability (Fig. 3.1) and capillary pressure model. They are given by krl increases linearly from 0 to 1 in the range S lr ≤ S l ≤ S ls …………………………. (3.1) krg increases linearly from 0 to 1 in the range S gr ≤ S g ≤ S gs …….………………….. (3.2) Pcmax increases linearly from S l = 0 to S l = 1 ………….……….……………………. (3.3) where krl is the liquid relative permeability, krg is the gas relative permeability, Sl is the liquid saturation, Slr is the residual liquid saturation or the liquid saturation at krl=0, Sls is the liquid saturation at krl=1, Sgr is the residual gas saturation or the gas saturation at krg=0, Sgs is the gas saturation at krg=1, and Pcmax is the maximum capillary pressure.

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1 0.9 0.8 0.7

kr

0.6 0.5 0.4

k rl

0.3

k rg

0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sl

Figure 3.1 Linear relative permeability.

3.4 Corey Model The Corey relative permeability functions (Fig. 3.2), obtained in 1954, are given by

k rl = S e 4 ……………………….……………………………………………………. (3.4) k rg = (1 − S e ) 2 (1 − S e 2 ) ………….………………………………………………….. (3.5) where S e = ( S l − S lr) / (1 − S lr − S gr ) …….…………………………………………………. (3.6) These relationships were determined from experiments with a variety of porous rocks that had a pore size distribution index of around 2, which is a typical value for soil materials and porous rocks (Corey, 1994).

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1 0.9 0.8 0.7

kr

0.6 0.5 0.4

k rl

0.3

k rg

0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sl

Figure 3.2 Corey relative permeability.

3.5 Leverett Model Leverett (1941) defined a reduced capillary pressure function, which was eventually named Leverett J function by Rose and Bruce (1949) and used for correlating capillary pressure data. Expressing the Leverett J function in terms of capillary pressure p c = − Po σ (T ) J …………………………………………………………………… (3.7) where J = 1.417 (1 − S * ) − 2 .120 (1 − S * ) 2 + 1.263 (1 − S * ) 3 ……………………………….. (3.8)

S * = (S l − S lr ) / (1 − S lr ) …………………………………………………………….. (3.9) σ is the surface tension of water and Po is a scale factor. One limitation of the function is that it cannot account for the individual differences between the pore structures of various materials since the scale factor, Po=√k/φ, is inadequate (Dullien, 1992).

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3.2 Brooks-Corey Model The Brooks-Corey functions (Fig. 3.3), obtained in 1964, are given by k rl = S ek( 2 − 3λ ) / λ .........................................……………………………………………. (3.10) krg = (1 − S ek )2 (1 − S ek

( 2 −3λ )/ λ

) ...........…........…………………………………….…… (3.11)

where S ek = ( S l − S lr ) / (1 − S lr − S gr ) ......…........…………………………………………. (3.12) and λ is the pore size distribution index. These relationships are simplifications of the generalized Kozeny-Carman equations and were verified experimentally by Brooks and Corey (1964), and Laliberte et al. (1966). Brooks and Corey found that for typical porous media, λ is 2. Soils with well-developed structures have λ values less than 2, and sands have λ values greater than 2 (Corey, 1994).

The Brooks-Corey capillary pressure function is given by

pc = − pe [ε / (1 − S lr

[

)]

−1/ λ

− (pe / λ ) ε / (1 − S lr

)

(1− λ )/ λ

](S − S l

lr

− ε)

pc = − pe ( S ec ) −1/ λ

for S l < ( S wr + ε ) (3.13) for S l ≥ ( S wr + ε ) (3.14)

where S ec = ( S l − S lr ) / (1 − S lr ) ...................….....………………………………………. (3.15)

and pe is the gas entry pressure. Equations 3.10 and 3.11 are modified Brooks-Corey equations. To prevent the capillary pressure from increasing to infinity as the effective saturation approaches zero, a linear function is used for Sl> heat CONDUCTIVITY >>> MATERIAL: BEREA >>>> VALUE >>>> RANGE: 3.0 6.0 >>>> DEVIATION: 0.2 >>>> STEP :0.05 MATERIAL: HEATB >>>> VALUE >>>> RANGE: 0.001 2.0 >>>> DEVIATION: 0.05 MATERIAL: INSUL >>>> VALUE >>>> RANGE: 0.01 2.0 >>>> DEVIATION: 0.05 SOURCE: HTR_1 +4 >>>> ANNOTATION : H2 >>>> PARAMETER No.: 2 >>>> DEVIATION : 0.1 >>>> VALUE >>>> STEP : 0.1 SOURCE: HTR_1 +4 >>>> ANNOTATION : H3 >>>> PARAMETER No.: 3 >>>> DEVIATION : 0.1 >>>> VALUE >>>> STEP : 0.1 SOURCE: HTR_1 +4 >>>> ANNOTATION : H4 >>>> PARAMETER No.: 4 >>>> DEVIATION : 0.1 >>>> VALUE >>>> STEP : 0.1 SOURCE: HTR_1 +4 >>>> ANNOTATION : H5 >>>> PARAMETER No.: 5 >>>> DEVIATION : 0.1 >>>> VALUE >>>> STEP : 0.1 SOURCE: HTR_1 +4 >>>> ANNOTATION : H6 >>>> PARAMETER No.: 6 >>>> DEVIATION : 0.1 >>>> VALUE >>>> STEP : 0.1 > TEMPERATURE >>> ELEMENT: B9__2 >>>> ANNOTATION: T1 >>>> COLUMNS: 1 3 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 degree C ELEMENT: B5__2 >>>> ANNOTATION: T2 >>>> COLUMNS: 1 4 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 deg C ELEMENT: B3__2 >>>> ANNOTATION: T3 >>>> COLUMNS: 1 5 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 deg C ELEMENT: AZ__2 >>>> ANNOTATION: T4 >>>> COLUMNS: 1 6 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 deg C ELEMENT: AV__2 >>>> ANNOTATION: T5 >>>> COLUMNS: 1 7 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 deg C ELEMENT: AT__2 >>>> ANNOTATION: T6 >>>> COLUMNS: 1 8 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 deg C ELEMENT: AQ__2 >>>> ANNOTATION: T7 >>>> COLUMNS: 1 9 >>>> PICK : 10 >>>> DATA in DAYS on FILE: temp.dat >>>> DEVIATION: 1.0 deg C ELEMENT: AM__2

114

>>>> ANNOTATION: T8 >>>> COLUMNS: 1 10 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AH__2 >>>> ANNOTATION: T9 >>>> COLUMNS: 1 11 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AB__2 >>>> ANNOTATION: T10 >>>> COLUMNS: 1 12 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: A4__2 >>>> ANNOTATION: T11 >>>> COLUMNS: 1 13 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C >>> ANNOTATION: HF1 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 3 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: B5__2 B5__3 >>>> ANNOTATION: HF2 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 4 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: B3__2 B3__3 >>>> ANNOTATION: HF3 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 5 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AZ__2 AZ__3 >>>> ANNOTATION: HF4 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 6 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2

115

CONNECTION: AV__2 AV__3 >>>> ANNOTATION: HF5 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 7 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AT__2 AT__3 >>>> ANNOTATION: HF6 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 8 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AQ__2 AQ__3 >>>> ANNOTATION: HF7 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 9 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AM__2 AM__3 >>>> ANNOTATION: HF8 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 10 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AH__2 AH__3 >>>> ANNOTATION: HF9 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 11 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AB__2 AB__3 >>>> ANNOTATION: HF10 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 12 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: A4__2 A4__3 >>>> ANNOTATION: HF11 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 13 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 >> max. no. of ITERATIONS: 50 >>> IQIT=3: 50 >>> CONSECUTIVE: 100 >>> LEVENBERG: 0.01 >>> STEP : 0.5 JACOBIAN >>> FORWARD: 9 >>> PERTURB: 0.01 OUTPUT >>> PLOTFILE: COLUMNS >>> DAYS >>> generate file with CHARACTERISTIC curves >> MATERIAL: BEREA >>>> LOGARITHM >>>> INDEX: 3 >>>> DEVIATION: 0.2 MATERIAL: BEREA >>>> PARAMETER No.: 1 >>>> ANNOTATION : Slr >>>> standard DEVIATION : 0.05 >>>> max STEP : 0.05 >>>> PERTURB : -0.001 >>>> estimate VALUE >>>> RANGE : 0.00 0.50 MATERIAL: BEREA >>>> PARAMETER No.: 2 >>>> ANNOTATION : Sgr >>>> standard DEVIATION : 0.05 >>>> max STEP : 0.05 >>>> PERTURB : -0.001 >>>> estimate VALUE >>>> RANGE : 0.00 0.50 MATERIAL: BEREA >>>> PARAMETER No.: 3 >>>> ANNOTATION : Sls >>>> standard DEVIATION : 0.05 >>>> max STEP : 0.05 >>>> PERTURB : -0.001 >>>> estimate VALUE >>>> RANGE : 0.51 1.00 MATERIAL: BEREA >>>> PARAMETER No.: 4 >>>> ANNOTATION : Sgs >>>> standard DEVIATION : 0.05 >>>> max STEP : 0.02 >>>> estimate VALUE >>>> RANGE : 0.51 1.00 >>>> PERTURB : -0.001 MATERIAL: BEREA >>>> PARAMETER No.: 1 >>>> ANNOTATION : Pc max >>>> standard DEVIATION : 0.50 >>>> max STEP: 0.2 >>>> estimate LOGARITHM >>> ANNOTATION : >>>> PARAMETER No.: >>>> DEVIATION : >>>> VALUE >>>> STEP : SOURCE: HTR1_ +4 >>>> ANNOTATION : >>>> PARAMETER No.: >>>> DEVIATION : >>>> VALUE >>>> STEP : TEMPERATURE >>> ELEMENT: B3__2 >>>> ANNOTATION: T1 >>>> COLUMNS: 1 2 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: B5__2 >>>> ANNOTATION: T2 >>>> COLUMNS: 1 4 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: B3__2 >>>> ANNOTATION: T3 >>>> COLUMNS: 1 5 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AZ__2 >>>> ANNOTATION: T4 >>>> COLUMNS: 1 6 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C
>>> ANNOTATION: T5 >>>> COLUMNS: 1 7 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AT__2 >>>> ANNOTATION: T6 >>>> COLUMNS: 1 8 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AQ__2 >>>> ANNOTATION: T7 >>>> COLUMNS: 1 9 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AM__2 >>>> ANNOTATION: T8 >>>> COLUMNS: 1 10 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AH__2 >>>> ANNOTATION: T9 >>>> COLUMNS: 1 11 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: AB__2 >>>> ANNOTATION: T10 >>>> COLUMNS: 1 12 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C ELEMENT: A4__2 >>>> ANNOTATION: T11 >>>> COLUMNS: 1 13 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1.0 deg C >>> ANNOTATION: P1 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 2 >>>> PICK : 10 >>>> DATA in DAYS on FILE: pres.dat >>>> DEVIATION: 1000.0 Pa
>>> ANNOTATION: P2 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 3 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: B3__1 >>>> ANNOTATION: P3 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 4 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: AZ__1 >>>> ANNOTATION: P4 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 5 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: AV__1 >>>> ANNOTATION: P5 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 6 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: AT__1 >>>> ANNOTATION: P6 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 7 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: AQ__1 >>>> ANNOTATION: P7 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 8 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: AM__1 >>>> ANNOTATION: P8 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 9 >>>> PICK : 10 >>>> DATA in DAYS on FILE: >>>> DEVIATION: 1000.0 Pa ELEMENT: AH__1 >>>> ANNOTATION: P9 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 10 >>>> PICK : 10

pres.dat

pres.dat

pres.dat

pres.dat

pres.dat

pres.dat

pres.dat

133

>>>> DATA in DAYS on FILE: pres.dat >>>> DEVIATION: 1000.0 Pa ELEMENT: AB__1 >>>> ANNOTATION: P10 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 11 >>>> PICK : 10 >>>> DATA in DAYS on FILE: pres.dat >>>> DEVIATION: 1000.0 Pa ELEMENT: A4__1 >>>> ANNOTATION: P11 >>>> SHIFT: 1.0065E5 >>>> COLUMNS: 1 12 >>>> PICK : 10 >>>> DATA in DAYS on FILE: pres.dat >>>> DEVIATION: 1000.0 Pa ELEMENT: B9__1 >>>> ANNOTATION: Sst1 >>>> SHIFT: -0.10 >>>> FACTOR: 1.1111 >>>> COLUMNS: 1 2 >>>> DATA in DAYS on FILE: sat.dat >>>> DEVIATION: 0.02 >>>> WINDOW: 5.00 8.00 [DAYS} ELEMENT: B7__1 >>>> ANNOTATION: Sst2 >>>> COLUMNS: 1 4 >>>> SHIFT: -0.075 >>>> FACTOR: 1.0811 >>>> DATA in DAYS on FILE: sat.dat >>>> DEVIATION: 0.01 >>>> WINDOW: 5.0 8.0 [DAYS} ELEMENT: B6__1 >>>> ANNOTATION: Sst3 >>>> COLUMNS: 1 5 >>>> SHIFT: -0.065 >>>> FACTOR: 1.0695 >>>> DATA in DAYS on FILE: sat.dat >>>> DEVIATION: 0.01 >>>> WINDOW: 5.0 8.0 [DAYS} ELEMENT: B5__1 >>>> ANNOTATION: Sst4 >>>> COLUMNS: 1 6 >>>> SHIFT: -0.07 >>>> FACTOR: 1.0753 >>>> DATA in DAYS on FILE: sat.dat >>>> DEVIATION: 0.01 >>>> WINDOW: 5.0 8.0 [DAYS} ELEMENT: B4__1 >>>> ANNOTATION: Sst5

134

>>>> COLUMNS: 1 7 >>>> SHIFT: -0.06 >>>> FACTOR: 1.0638 >>>> DATA in DAYS on FILE: sat.dat >>>> DEVIATION: 0.01 >>>> WINDOW: 5.0 8.0 [DAYS} ELEMENT: B3__1 >>>> ANNOTATION: Sst6 >>>> COLUMNS: 1 8 >>>> SHIFT: -0.06 >>>> DATA in DAYS on FILE: sat.dat >>>> DEVIATION: 0.01 >>>> WINDOW: 5.0 8.0 [DAYS} CONNECTION: B5__2 B5__3 >>>> ANNOTATION: HF2 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 4 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 20.0 W/m^2 CONNECTION: B3__2 B3__3 >>>> ANNOTATION: HF3 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 5 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AZ__2 AZ__3 >>>> ANNOTATION: HF4 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 6 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AV__2 AV__3 >>>> ANNOTATION: HF5 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 7 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AT__2 AT__3 >>>> ANNOTATION: HF6 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 8 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 CONNECTION: AQ__2 AQ__3 >>>> ANNOTATION: HF7 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 9

135

>>>

>>>

>>>

>>>

>>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 > ANNOTATION: HF8 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 10 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 > ANNOTATION: HF9 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 11 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 > ANNOTATION: HF10 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 12 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2 > ANNOTATION: HF11 >>>> FACTOR: -1.910E-3 [W/m^2 - W] >>>> COLUMNS: 1 13 >>>> PICK : 10 >>>> DATA in HOURS on FILE: hflux.dat >>>> DEVIATION: 10.0 W/m^2
>> max. no. of ITERATIONS: 20 >>> IQIT=3: 50 >>> CONSECUTIVE: 100 >>> INCOMPLETE: 20 >>> LEVENBERG: 0.1 >>> STEP : 1.0 JACOBIAN >>> FORWARD: 8 >>> PERTURB: 0.01 OUTPUT >>> PLOTFILE: COLUMNS >>> DAYS >>> generate file with CHARACTERISTIC curves