universiteit twente · contents contents i 1 introduction 1 1.1 motivation . . . . . . . . . . . ....

STEMMUS:Simultaneous Transfer of Energy,

Mass and Momentum inUnsaturated Soil

Yijian Zeng & Zhongbo Su

ITC-WRS Report

ITC, P.O. Box 217, 7500 AE Enschede, The Netherlands

ISBN: 978–90–6164–351–7

© Yijian Zeng & Zhongbo Su, Enschede, The NetherlandsAll rights reserved. No part of this publication may be reproduced without theprior written permission of the author.

Contents

Contents i

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 How to use this manual . . . . . . . . . . . . . . . . . . . . . . 31.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Brief Review of Previous Work 72.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 STEMMUS: Governing Equations and Constitutive Equations 133.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . 193.3 Simplification of Governing Equations . . . . . . . . . . . . . 263.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 STEMMUS: Finite Element Solution of Governing Equations 354.1 Galerkin’s Method of Weighted Residuals . . . . . . . . . . . 364.2 Integrals’ Calculation . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Finite Difference Time-Stepping Scheme . . . . . . . . . . . . 404.4 MATLAB Implementation . . . . . . . . . . . . . . . . . . . . . 444.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 STEMMUS: Structures, Subroutines and Input Data 495.1 Structure of STEMMUS: Overview . . . . . . . . . . . . . . . . 505.2 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . 535.3 List of STEMMUS Variables . . . . . . . . . . . . . . . . . . . . 545.4 How to Run STEMMUS . . . . . . . . . . . . . . . . . . . . . . . 575.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

i

1Introduction

1

1. Introduction

1.1 Motivation

STEMMUS (Simultaneous Transfer of Energy, Mass and Momentum inUnsaturated Soil) is a program for simulating coupled liquid water, watervapor, dry air and heat transfer in unsaturated soil. The program numer-ically solves the Richards equation with modification made by Milly [1].The main driving factors are discussed as below:

A traditional conceptual model for water flow in unsaturated soil neg-lects the flow-of-gas phase, as can be seen in the statement: "Unsaturatedflow . . . is nothing but a special case of simultaneous flow of two immis-cible fluids, where the nonwetting fluid is assumed to be stagnant" [2].According to this statement, air always remains at atmospheric pressureand is free to escape from or enter into the vadose zone. This assumptionis widely used in the traditional coupled moisture and heat flow model,based on the theory by Philip and de Vries [3] (hereafter PdV model,referring to this traditional model), even when water vapor movement isconsidered. However, the flow in porous media is actually a two-phaseflow problem [4], in need of complicated mathematical analysis of the re-sponse of soil to atmospheric forcing, caused by nonlinearities, moistureretention hysteresis, soil heterogeneity, as well as by multiple length andtime scales [1]. The usual approach to solving the numerical model ofwater flow in soil involves the above cited assumption, which is oftencalled the Richards approximation [5]. Although successful applicationof this assumption has been demonstrated in many cases, there aresituations where the air phase can significantly retard or speed up infilt-ration [6, 7]. To properly describe these gas influenced infiltrations, atwo-phase model must be used.

To improve the model representation of multiphase systems, therehave been continuous efforts to build comprehensive multiphase mod-els. At present, the modern two-phase heat and mass flow models havegenerally overcome most of the above mentioned difficulties [8, 9], andthe weaknesses of the single-phase approach have been systematicallydiscussed in these studies. Some of the existing comprehensive mul-tiphase models, e.g. CODE_BRIGHT [8] and TOUGH2 [9], have been mainlyapplied to geothermal problems [10], without paying attention to thesoil-atmosphere interacting mechanisms mentioned in section 1.1. Suchcomprehensive multiphase models (e.g. fluid flow in the deformableporous media) do not make including a two-phase heat and mass flowmodel into the land surface model straightforward. It is also not easyto understand why a two-phase flow mechanism is better than the tra-ditional PdV model in the context of soil-atmosphere interaction. ThePdV model has been widely used in many unsaturated flow simulators[11, 12, 13, 14], and subsequently adopted as sub-model in hydrologicalintegration models [15, 16] and land surface models [17, 18, 19, 20].Therefore, in aiming to understand the need to include a two-phaseflow mechanism in the PdV model, it is necessary to know how the air-flow influences the performance of the PdV model when simulating soilmoisture and soil temperature.

2

1.2. Structures

On the other hand, the isothermal two-phase fluid flow in unsaturatedsoil, excluding heat flow and transport (e.g. considering air-water flowonly), has seen an increase in development because of its indispens-able role in simulating, modeling and studying waste storage, geologicalstorage, underground natural resource recovery and environmental re-mediation etc. [21, 22]. However, most development has been focusingon multiscale problems or numerical formulations and solutions [23, 24],and very little attention has been paid to investigating how heat flow canaffect the isothermal two-phase flow (air-water flow only) in unsaturatedsoil.

Therefore, to understand how airflow interact with coupled moistureand heat flow in soil, it’d better to develop a two-phase mass and heattransfer model (e.g. the STEMMUS Model) to investigate the interactionsbetween them. This technical manual introduces the main physics behindthe STEMMUS model, the numerical method the model uses, the programstructure and the input data (e.g. boundary & initial condition) forrunning the model.

1.2 Structures

In the following chapter, a brief review on previous works is discussed.Chapter 3 shows the governing equations for STEMMUS, and the physicsbehind them. How the constitutive equations were chosen to have aclosure for the model is briefly discussed. Chapter 4 introduces the finiteelement method used for solving the model. The structure of the modelis presented in Chapter 5. How individual subroutine is designed toserve the main program and how to generate the boundary and initialcondition for the model were also introduced.

1.3 How to use this manual

This technical manual is a documentation helping users to understandhow STEMMUS is build and how to make it running. The application ofthe model is limited to very dry soil (e.g. in a desert environment) for thecurrent version. The test of the model in other cases has not been madeavailable at this stage. In future, the updated STEMMUS will be releasedregularly. Whenever you find there were any typos or errors or mistakes,please help us to improve the manual and the STEMMUS by dropping anyline to [email protected] (/[email protected]) or [email protected].

1.4 Bibliography

[1] P. C. D. Milly. Moisture and heat transport in hysteretic, inhomo-geneous porous media: A matric head-based formulation and anumerical model. Water Resources Research, 18(3), 1982.

3

1. Introduction

[2] J. Bear. Dynamics of fluid in porous media. Dover, New York, 1972.

[3] J. R. Philip and Vries D. De Vries. Moisture movement in porousmaterials under temperature gradient. Trans. Am. Geophys. Union,38(2):222–232, 1957.

[4] B. A. Schrefler and X. Y. Zhan. A fully coupled model for water-flow and air-flow in deformable porous-media. Water Resour Res,29(1):155–167, 1993.

[5] M. A. Celia and P. Binning. A mass conservative numerical-solutionfor 2-phase flow in porous-media with application to unsaturatedflow. Water Resour Res, 28(10):2819–2828, 1992.

[6] J. Touma and M. Vauclin. Experimental and numerical analysis oftwo-phase infiltration in a partially saturated soil. Transport PorousMed, 1(1):22–25, 1986.

[7] L. Prunty and J. Bell. Soil water hysteresis at low potential. Pedo-sphere, 17(4):436–444, 2007.

[8] S. Olivella, A. Gens, J. Carrera, and E. E. Alonso. Numerical formula-tion for a simulator (code-bright) for the coupled analysis of salinemedia. Engineering Computations, 13(7):87–&, 1996.

[9] L. Prunty. The tough codes - a family of simulation tools for mul-tiphase flow and transport processes in permeable media. VadoseZone Journal, 3(3):738–746, 2004.

[10] C. L. Zhang, K. P. Kröhn, and T. Rothfuchs. Unsaturated Soils: Nu-merical and Theoretical Approaches, volume 94, pages 341–357.Springer, 2005.

[11] M.J. Fayer. Unsat-h version 3.0:unsaturated soil water and heatflow model-theory, user manual, and examples. Technical ReportPNNL-13249, Pacific Northwest National Laboratory, 2000.

[12] G. N. Flerchinger. The simultaneous heat and water (shaw) model:techincal documentation. Technical Report NWRC 2000-09, NWRC,USDA ARS, 2000.

[13] H. Saito, J. Simunek, and B. P. Mohanty. Numerical analysis ofcoupled water, vapor, and heat transport in the vadose zone. VadoseZone Journal, 5(2):784–800, 2006.

[14] J. Simunek and M. T. van Genuchten. Modeling nonequilibriumflow and transport processes using hydrus1d. Vadose Zone Journal,7(2):782–797, 2008.

[15] J. Simunek Twarakavi, N. K. C. and S. Seo. Evaluating interactionsbetween groundwater and vadose zone using the hydrus1d-basedflow package for modflow. Vadose Zone Journal, 7(2):757–768, 2008.

[16] P. Groenendijk R. F. A. Hendriks van Dam, J. C. and J. G. Kroes.Advances of modeling water flow in variably saturated soils withswap. Vadose Zone Journal, 7(2):640–653, 2008.

4

1.4. Bibliography

[17] R. Avissar and R. A. Pielke. A parameterization of heterogeneousland surfaces for atmospheric numerical-models and its impact onregional meteorology. Monthly Weather Review, 117(10):2113–2136,1989.

[18] R. Avissar. Conceptual aspects of a statistical-dynamic approach torepresent landscape subgrid-scale heterogeneities in atmosphericmodels. Journal of Geophysical Research-Atmospheres, 97(D3):2729–2742, 1992.

[19] I. Braud, A. C. Dantasantonino, M. Vauclin, J. L. Thony, and P. Ruelle.A simple soil-plant-atmosphere transfer model (sispat) developmentand field verification. Journal of Hydrology, 166(3-4):213–250, 1995.

[20] R. S. Jassal, M. D. Novak, and T. A. Black. Effect of surface layerthickness on simultaneous transport of heat and water in a bare soiland its implications for land surface schemes. Atmosphere-Ocean,41(4):259–272, 2003.

[21] Y. Wu and P. A. Forsyth. On the selection of primary variablesin numerical formulation for modeling multiphase flow in porousmedia. Journal of Contaminant Hydrology, 48(3-4):277–304, 2001.

[22] M. A. Celia and J. M. Nordbotten. Practical modeling approaches forgeological storage of carbon dioxide. Ground Water, 47(5):627–638,2009.

[23] H. Rajaram Celia, M. A. and L. A. Ferrand. A multiscale computationalmodel for multiphase flow in porous-media. Advances in WaterResources, 16(1):81–92, 1993.

[24] B. A. Ashitiani and D. R. Ardekani. Comparison of numerical for-mulations for two-phase flow in porous media. Geotechnical andGeological Engineering, 28(4):373–389, 2010.

5

2Brief Review of Previous Work

7

2. Brief Review of Previous Work

Under the Richards approximation (e.g. no gas phase transport inthe soil), using practical or phenomenological approaches, commonempirical constitutive relations (e.g. the laws of Darcy, Fick, Henry,Dalton and Kelvin) are directly incorporated in formulating the modelfor heat and mass transfer in a partially saturated porous medium, oftencalled diffusion-based model with roots in Philip and de Vries [1]. Thediffusion-based model has been adopted and extended by numerousauthors [2, 3, 4, 5, 6]. Yet, unsatisfactory discrepancy is often foundbetween theory prediction and field data. For example, Cahill and Par-lange [7] demonstrated the significant underestimation of the magnitudeas well as the incorrect direction predicted by the diffusion-based theoryfor vapor flux in their field experiments. Heitman et al. [8] revealedthe noticeable differences between measured and calculated patternsof heat and moisture redistribution when the boundary temperaturegradient was instantly reversed. As a result, both Cahill and Heitmansuggested that to further develop the theory for better description offield conditions additional mechanisms need to be taken into considera-tion. Although no additional mechanism is pointed out specifically, thegas phase flow, which involves dry air and vapor flow, is an importantmechanism that needs to be taken into account.

The gas phase flow in unsaturated soil has been studied for more thana century, since Buckingham [9] described the movement of air in soilin response to atmospheric pressure. However, not much attention waspaid to this until the importance of gas flow in various engineering fieldsbecame apparent. Especially in environmental engineering, where gasflow is the major mechanism for assessing contaminant mass depletiondue to volatilization and removing the volatile organic chemicals fromthe vadose zone by vapor extraction.

From a theoretical point of view, the gas phase flow problem hasbeen analyzed using numerical or analytical approaches. Although manynumerical simulators were developed to simulate gas flow problems incomplex conditions [10, 11, 12], the capacity of analytical solutions toverify numerical codes led to the development of analytical solutions forsoil gas flow in the vadose zone in the past two decades. For example,Massmann [13] and McWhorter [14] presented analytical solutions tosolve one-dimensional radial gas flow with simple initial and boundaryconditions; Shan [15, 16] developed analytical solutions for transient gasflow caused by barometric pumping in both one and two dimensions.These studies provided a good knowledge base on the induced gasflow field in soil, needed to successfully design a cleanup system forcontaminated vadose zones.

At the same time, simultaneous flow of water and air through unsat-urated porous media was intensively studied due to its vast applicationsin petroleum reservoir engineering, which also led to the developmentof either numerical or quasi-analytical solutions [17, 18, 19]. However,for the analysis of simultaneous flow of heat and mass through unsatur-ated porous media, a two phase flow model should also take the energybalance equation into account.

8

2.1. Bibliography

In nuclear or geothermal engineering, where an assessment of coupledliquid water, vapor, dry air and heat transport is required, the two-phaseheat and mass flow model has been widely researched [20, 21, 22, 23,24, 25]. Application of the two-phase heat and mass flow model is notlimited to this. It is also important in the drying technology, where aprecise thermo-hydro-mechanical model is highly recommended [26] forobtaining dried products of good quality, as well as being importantin the storage of liquefied natural gas [27], where a comprehensiveunderstanding of the processes of freezing and thawing is necessaryfor safety assessment, and in the CO2 sequestration system in a faultenvironment [28], where assessment of fault instability due to the impactof CO2 injection is a typical two-phase heat and mass flow problem, andso on. Although all these studies analyze the heat and mass transferwith a two-phase flow approach, which may fill the gap between theoryand measurement pointed out by Cahill and Parlange [7] and Heitmanet al. [8], further consideration of the coupled liquid, vapor, dry air andheat transport mechanism is needed.

From a mechanistic point of view, primary mechanisms of two phaseheat and mass flow in unsaturated porous media include convection(movement with the bulk fluid) and hydrodynamic dispersion (mech-anical dispersion and molecular diffusion) [29]. Mechanical dispersion,resulting from variations in fluid velocity at the micro pore scale, is theproduct of dispersion and convection. In most cases, the mechanicaldispersion of gas phase is neglected due to its very small velocity com-pared to the dominant molecular diffusion [30]. This assumption is alsoapplied in two-phase heat and mass flow models mentioned in the aboveparagraph. However, it has been argued that standard phenomenologicalapproaches to modelling two-phase flow based on empirical constitutiverelations (simplification of transport mechanisms) are not founded on anentirely sound physical basis [31]. Based on a series of work by Hassaniz-adeh and Gray [32, 33, 34], Niessner and Hassanizadeh [35, 36] presenteda non-equilibrium two-phase heat and mass flow model including theinterfacial area as state variable. The reliability of their physically-basedmodel was made convincing by capturing additional physical processes(hysteresis) compared to the standard model. However, the focus oftheir research is on microscopic scale problems, which is not suitable forsolving the field-scale problem yet.

2.1 Bibliography


[2] P. C. D. Milly and P. S. Eagleson. The coupled transport of waterand heat in a vertical soil column under atmospheric excitation. PhDthesis, 1980.

9



[4] I. N. Nassar and R. Horton. Heat, water, and solution transferin unsaturated porous media: I–theory development and transportcoefficient evaluation. Transport in Porous Media, 27(1):17–38, 1997.


[6] M. Sakai, N. Toride, and J. Simunek. Water and vapor movementwith condensation and evaporation in a sandy column. Soil ScienceSociety of America Journal, 73(3):707–717, 2009.

[7] A. T. Cahill, M. B. Parlange, A. Prosperetti, and S. Whitaker. Convect-ively enhanced water vapor movement at the earth’s surface. WaterResour. Res.,(submitted), 1998.

[8] J. L. Heitman, R. Horton, T. Ren, I. N. Nassar, and D. D. Davis. A testof coupled soil heat and water transfer prediction under transientboundary temperatures. Soil Science Society of America Journal,72(5):1197–1207, 2008.

[9] E. Buckingham. Contributions to our knowledge of aeration of soils.Technical report, 1904.

[10] C. A. Mendoza and E. O. Frind. Advective-dispersive transport ofdense organic vapors in the unsaturated zone .1. model develop-ment. Water Resour Res, 26(3):379–387, 1990.

[11] R. W. Falta, K. Pruess, I. Javandel, and P. A. Witherspoon. Numericalmodeling of steam injection for the removal of nonaqueous phaseliquids from the subsurface .1. numerical formulation. Water ResourRes, 28(2):433–449, 1992.

[12] L. Prunty. The tough codes - a family of simulation tools for mul-tiphase flow and transport processes in permeable media. VadoseZone Journal, 3(3):738–746, 2004.

[13] J. W. Massmann. Applying groundwater-flow models in vapor ex-traction system-design. J Environ Eng-ASCE, 115(1):129–149, 1989.

[14] D. B. McWhorter. Unsteady radial flow of gas in the vadose zone. J.Contam. Hydrol., 5:297–314, 1990.

[15] C. Shan. Analytical solutions for determining vertical air permeabil-ity in unsaturated soils. Water Resour Res, 31(9):2193–2200, 1995.

[16] C. Shan. An analytical solution for the capture zone of two arbitrarilylocated wells. J Hydrol, 222(1-4):123–128, 1999.

[17] J. Y. Parlange, R.D. Braddock, and R.W. Simpson. Optimizationprinciple for air and water movement. Soil Sci, 133(1):4–9, 1982.

[18] H.J. Morel-Seytoux and J. A. Billica. A tow-phase numerical modelfor prediction of infiltration: Application to a semi-infinte column.Water Resour. Res, 21(4):607–615, 1985.

10

2.1. Bibliography

[19] G. J. Moridis and D. L. Reddell. Secondary water recovery by airinjection .1. the concept and the mathematical and numerical-model.Water Resour Res, 27(9):2337–2352, 1991.

[20] E. E. Alonso, A. Gens, and A. Josa. A constitutive model for partiallysaturated soils. Geotechnique, 40(3):405–430, 1990.

[21] D. Gawin, P. Baggio, and B. A. Schrefler. Coupled heat, water andgas-flow in deformable porous-media. Int J Numer Meth Fluid, 20(8-9):969–987, 1995.

[22] H. R. Thomas, H. T. Yang, Y. He, and A. D. Jefferson. Solving coupledthermo-hydro-mechanical problems in unsaturated soil using asubstructuring-frontal technique. Communications in NumericalMethods in Engineering, 14(8):783–792, 1998.

[23] B. A. Schrefler and R. Scotta. A fully coupled dynamic model fortwo-phase fluid flow in deformable porous media. Comput MethAppl Mech Eng, 190(24-25):3223–3246, 2001.

[24] B. A. Schrefler. Multiphase flow in deforming porous material. Int JNumer Meth Eng, 60(1):27–50, 2004.

[25] H. R. Thomas, P. J. Cleall, D. Dixon, and H. P. Mitchell. The coupledthermal-hydraulic-mechanical behaviour of a large-scale in situ heat-ing experiment. Geotechnique, 59(4):401–413, 2009.

[26] S. J. Kowalski. R&d in thermo-hydro-mechanical aspect of drying.Drying Technology, 26(3):258–259, 2008.

[27] K. M. Neaupane, T. Yamabe, and R. Yoshinaka. Simulation of a fullycoupled thermo-hydro-mechanical system in freezing and thawingrock. Int J Rock Mech Min Sci, 36(5):563–580, 1999.

[28] Q. Li, Z. S. Wu, Y. L. Bai, X. C. Yin, and X. C. Li. Thermo-hydro-mechanical modeling of co2 sequestration system around faultenvironment. Pure Appl Geophys, 163(11-12):2585–2593, 2006.


[30] B. R. Scanlon. Uncertainties in estimating water fluxes and residencetimes using environmental tracers in an arid unsaturated zone.Water Resources Research, 36(2):395–409, 2000.

[31] J. Niessner and S. M. Hassanizadeh. Non-equilibrium interphaseheat and mass transfer during two-phase flow in porous media-theoretical considerations and modeling. Adv Water Resour,32(12):1756–1766, 2009.

[32] M. Hassanizadeh and W. G. Gray. General conservation equationsfor multi-phase systems: 1. averaging procedure. Adv Water Resour,2:131–144, 1979.

[33] M. Hassanizadeh and W. G. Gray. General conservation equationsfor multi-phase systems: 2. mass, momenta, energy, and entropyequations. Adv Water Resour, 2:191–203, 1979.

11


[34] M. Hassanizadeh and W. G. Gray. General conservation equationsfor multi-phase systems: 3. constitutive theory for porous mediaflow. Adv Water Resour, 3(1):25–40, 1980.

[35] J. Niessner and S. M. Hassanizadeh. A model for two-phase flow inporous media including fluid-fluid interfacial area. Water ResourRes, 44(8), 2008.

[36] J. Niessner and S. M. Hassanizadeh. Modeling kinetic interphasemass transfer for two-phase flow in porous media including fluid-fluid interfacial area. Transport Porous Med, 80(2):329–344, 2009.

12

3STEMMUS: Governing Equationsand Constitutive Equations

13

3. STEMMUS: Governing Equations and Constitutive Equations

3.1 Governing Equations

In this section, two main groups of equations (conservation equationsand constitutive equations) have been used to describe the two-phaseheat and mass flow model. The Milly’s equations [1], considering thepredominantly vertical interactive process between atmosphere and soil,have been introduced first to describe the traditional coupled heat andmass flow model scheme. Then, the two-phase heat and flow modelhas been developed on this basis. Next, considering dry air as a singlephase and the main component of the gaseous phase in soil, the balanceequation of dry air was introduced, based on Thomas’ work [2]. Henry’slaw has been used to express the equilibrium of dissolved air in liquid.In addition, the thermal equilibrium assumption between phases hasbeen adopted and the equation for energy balance established takinginto account the internal energy in each phase (liquid, vapor, and dryair).

3.1.1 Liquid Transfer

Soil water is present in a liquid and a gaseous phase, and following Milly[1], the total moisture balance is expressed as

∂∂t(ρLθL + ρVθV ) = −

∂∂z(qL + qV ) (3.1)

where ρL (kg m−3) represents the density of liquid water; ρV (kg m−3)the density of water vapor; θL the volumetric water content; θV thevolumetric water vapor content; z (m) the vertical space coordinate,positive upwards; qL and qV the liquid mass flux and the vapor mass flux(kg m−2 s−1). The liquid flux is expressed by a general form of Darcy’sflow

qL = −ρLK( ∂hwγw + z)∂z

(3.2)

where hw (Pa) is the pore-water pressure; γw (kg m−2 s−2) the specificweight of water; and K (m s−1) the unsaturated hydraulic conductivity.According to Groenevelt and Kay [3], the effect of the heat of wetting onthe pressure field and the resulting flow is taken into account by Milly [1],which leads to an additional liquid flow term in equation (3.2) resultingin

qL = −ρLK( ∂hwγw + z)∂z

− ρLDTa∂T∂z

(3.3)

where DTa (m2 s−1 ◦C−1) is the transport coefficient for adsorbed liquidflow due to temperature gradient; and T ( ◦C−1) the temperature. Accord-ing to the definition of capillary potential, hw could be expressed as the

14

3.1. Governing Equations

difference between the pore-air pressure and the pore-water pressure[4, 5, 2]

h = hw − Pgγw

(3.4)

where h (m) is the capillary pressure head; and Pg (Pa) the mixed pore-air pressure (e.g. including vapor and dry air pressure). Substitutingequation (3.4) into equation (3.3) yields

qL = −ρLK∂(h+ Pg

γw + z)∂z

− ρLDTa∂T∂z

(3.5)

Equation (3.5) can be rewritten [6] as

qL = −ρL[K∂(h+ Pg

γw )

∂z︸︷︷︸qLh+qLa

+DTa∂T∂z︸︷︷︸

qLT

+K] (3.6)

in which qLh and qLT are the isothermal and thermal liquid flux (kgm−2 s−1); qLa (= ρL Kγw

∂Pg∂z = KLa

∂Pg∂z ) the advective liquid flux due to

soil air pressure gradient (kg m−2 s−1); and KLa (s) the advective liquidtransport coefficient.

3.1.2 Vapor Transfer

The vapor flux is expressed by a generalized form of Fick’s law

qV = −DV∂ρV∂z

(3.7)

where DV (m2 s−1) is the molecular diffusivity of water vapor in soil.When dry air is considered, the vapor flow is assumed to be induced inthree ways: firstly diffusive transfer, driven by a vapor pressure gradientequation (3.7); secondly advective transfer, as part of the bulk flows of air(ρV

qaaρda ); and thirdly dispersive transfer, due to longitudinal dispersivity

(−DVg ∂ρV∂z ). Accordingly, equation (3.7) can be rewritten as

qV = −DV∂ρV∂z︸︷︷︸

Diffusion

+ ρVqaaρda︸︷︷︸

Advection

−θVDVg∂ρV∂z︸︷︷︸

Dispersion

(3.8)

where qaa (= −ρdaKg ∂Pg∂z ) is the advective dry air flux (kg m−2 s−1); ρda(kg m−3) the dry air density; DVg(m2 s−1) the gas-phase longitudinaldispersion coefficient; and Kg the gas conductivity (m s−1).

Considering vapor density to be a function of matric potential andtemperature, the vapor flux can be divided into isothermal and thermal

15


components. According to the chain rule for partial derivatives, the vaporflux in equation (3.8) could be rewritten using the three state variables as

qV = qVh + qVT + qVa= −[

(DV + θVDVg

) ∂ρV∂h

∂h∂z

(3.9)

+(DV + θVDVg

) ∂ρV∂T

∂T∂z+ ρVKg

∂Pg∂z]

where qVh, qVT , and qVa is the isothermal vapor flux, the thermal vaporflux, and the advective vapor flux (kg m−2 s−1).

Combining the governing equations for liquid water (equation 3.6)and vapor flow (equation 3.9) leads to the governing differential equationfor moisture transfer:

∂∂t(ρLθL + ρVθV ) = −

∂∂z(qL + qV )

= − ∂∂z(qLh + qLT + qLa)−

∂∂z(qVh + qVT + qVa)

= ρL∂∂z[K(

∂h∂z+ 1)+DTa

∂T∂z+ Kγw∂Pg∂z] (3.10)

+ ∂∂z[DVh

∂h∂z+DVT

∂T∂z+DVa

∂Pg∂z]

where, DVh (= (DV + θVDVg)ρV∂h) is the isothermal vapor diffusioncoefficient (kg m−2 s−1); DVT (= (DV + θVDVg)ρV∂T ) the thermal vapordiffusion coefficient (kg m−1 s−1 ◦C−1); and DVa (= ρVKg) the advectivevapor transfer coefficient (s).

The terms within the square brackets in equation (3.6) represent the

liquid flux. The term (Pgγw ) is the mixed soil air pressure expressed as

the height of a water column. The terms within the square bracketsin equation (3.8) represent the water vapor flux, with the first termrepresenting the diffusive flux (Fick’s law), the second representing theadvective flux (Darcy’s law) and the third the dispersive flux (Fick’s law).

Equation (3.3) shows clearly that only thermal (e.g. explicitly throughthe temperature dependence of K) and isothermal liquid advection andwater vapor diffusion are considered in the traditional coupled heat andmass transport model (PdV model). However, equation (3.6) shows thatdry air is considered to be a single phase. Thus not only diffusion, butalso advection and dispersion become included in the water vapor trans-port mechanism. As for the liquid transport, the mechanism remainsthe same, but with atmospheric pressure acting as one of driving forcegradients.

3.1.3 Dry Air Transfer

Dry air transport in unsaturated soil is driven by two main gradients, thedry air concentration or density gradient and the air pressure gradient.The first one diffuses dry air in soil pores, while the second one causes

16

3.1. Governing Equations

advective flux of dry air. At the same time, the dispersive transfer of dryair should also be considered. In addition, to maintain mechanical andchemical equilibrium, a certain amount of dry air will dissolve into liquidaccording to Henry’s law. Considering the above four effects, the balanceequation for dry air may be presented [2] as

∂∂t[ερda(Sa +HcSL)] = −

∂qa∂z

(3.11)

and the dry air flow qa (kg m−2 s−1) is given by

qa = −DV∂ρda∂z

− ρdaKg∂Pg∂z+Hcρda

qLρL− θaDVg

∂ρda∂z

(3.12)

where Hc (= 0.02 for air at 1 atm and 25 ◦c−1) Henry’s constant; Sa(= 1 − SL) the degree of air saturation of soil; SL (= θL/ε) the degreeof saturation of soil; and ε the porosity. In the right hand side (RHS)of equation (3.12), the first term depicts diffusive flux (Fick’s law), thesecond term advective flux (Darcy’s law), the third dispersive flux (Fick’slaw), and the fourth advective flux due to dissolved air (Henry’s law).Considering dry air density is a function of matric potential, temperatureand air pressure, equation (3.12) could be rewritten with three statevariables. Combining equation (3.12) with equation (3.11), the governingequation for dry air can be expressed as

∂∂t[ερda(Sa +HcSL)]

= − ∂∂(z)

(qah + qaT + qaa)

= ∂∂z[Kah

∂h∂z+KaT

∂T∂z+Kaa

∂Pg∂z]

+Vah∂h∂z+ VaT

∂T∂z+ Vaa

∂Pg∂z+Hcρda

∂K∂z

(3.13)

where qah, qaT , and qaa are the isothermal air flux, the thermal airflux, and the advective flux (kg m−2 s−1). Accordingly, the transportcoefficients for air flux are

Kah = (DV + θaDVg)∂ρda∂h

+HcρdaK

KaT = (DV + θaDVg)∂ρda∂T

+HcρdaDTa (3.14)

Kaa = (DV + θaDVg)∂ρda∂Pg

+ ρda(Kg +HcKγw)

where Vah, VaT and Vaa are introduced in section 3.3.2.

3.1.4 Energy Transfer

In the vadose zone, the mechanisms for energy transport include conduc-tion and convection. The conductive heat transfer contains contributions

17


from liquids, solids and gas. Conduction is the main mechanism forheat transfer in soil and contributes to the energy conservation by solids,liquids and air. Advective heat in soil is conveyed by liquid flux, vaporflux, and dry air flux. On the other hand, heat storage in soil includesthe bulk volumetric heat content, the latent heat of vaporization anda source term associated with the exothermic process of wetting of aporous medium (integral heat of wetting) [7]. Accordingly, followingthe general approach by de Vries [7], the energy balance equation inunsaturated soil may be written as four parts

• Solid:

∂[ρsθscs(T − Tr )]∂t

= ∂∂z(λsθs

∂T∂z) (3.15)

• Liquid:

∂[ρLθLcL(T − Tr )]∂t

= ∂∂z(λLθL

∂T∂z)− ∂∂z[qLcL(T −Tr )] (3.16)

• Air and Vapor:

∂∂t[(ρdaca + ρVcV )θg(T − Tr )+ ρVL0θg] (3.17)

= ∂∂z(λgθg

∂T∂z)− ∂

∂z[qV (cv(T − Tr )+ L0)+ qaca(T − Tr )]

• Heat of Wetting:

Hw = −ρLW∂θL∂t

(3.18)

where λs , λL and λg represent the thermal conductivities of solids, liquidsand pore gas (= λa + λV , including dry air and water vapor) respectively(W m−1 ◦c−1); θs the volumetric content of solids in the soil; θg thevolumetric content of gas (= θV = θa) in the soil; cs , cL, ca and cVspecific heat of solids, liquids, air and vapor, respectively (J kg−1 ◦c−1);Tr ( ◦c−1) the reference temperature; ρs the density of solids in the soil(kg m−3); L0 the latent heat of vaporization of water at temperatureTr (J kg−1); and W the differential heat of wetting (the amount of heatreleased when a small amount of free water is added to the soil matrix)(J kg−1). The latent heat of vaporization varies with T according toL(T) = L0−(cL−cV )(T−Tr ) ≈ 2.501×106−2369.2×T [8]. In accordancewith equation (3.16) to equation (3.18), the conservation equation forenergy transfer in the soil is given as

∂∂t[(ρsθscs + ρLθLcL + ρdaθgca + ρVθgcV )(T − Tr )+ ρVL0θg]− ρLW

∂θL∂t

= ∂∂z(λeff

∂T∂z)− ∂

∂z[qLcL(T − Tr )+ qV (cv(T − Tr )+ L0)+ qaca(T − Tr )]

(3.19)

18

3.2. Constitutive Equations

where λeff is the effective thermal conductivity (W m−1 K−1), combiningthe thermal conductivity of solid particles, liquid, vapor and dry air inthe absence of flow. The parameters in the first term in the left hand side(LHS) of equation (3.19) and can be determined by de Vries’ [7] scheme.Equation (3.10), (3.13) and (3.19) are solved jointly with the specifiedboundary and initial condition of the solution domain to obtain spatialand temporal variations of the three prime variables h, T and Pg . Ifthe advective flux conveyed by the dry air flux (qaca(T − Tr )) and thebulk volumetric heat content of dry air (ρdaθaca) were to be neglected,equation (3.19) would result in the heat balance equation of Milly [1].

3.2 Constitutive Equations

The constitutive equations link the independent state variables (un-knowns) and the dependent variables. Each governing equation is solvedfor a single unknown (a state variable), for example, equation (3.10) formatric potential, equation (3.13) for atmospheric pressure and equa-tion (3.19) for temperature. The closure of the model developed aboverequires all dependent variables to be computable from the set of un-knowns. The governing equations are finally written in terms of theunknowns when the constitutive equations given below are substituted.

3.2.1 Unsaturated Hydraulic Conductivity

The pore-size distribution model of Mualem [9] was used to predictthe isothermal unsaturated hydraulic conductivity from the saturatedhydraulic conductivity [10]:

K = KsKr = KsSle[1(1− S1me )m]2 (3.20)

where Ks is the saturated hydraulic conductivity (m s−1); Kr , the relativehydraulic conductivity (m s−1); Se (= θL−θres

θsat−θres ) the effective saturation(unitless) (In STEMMUS, the Se is approximated as SL); θres , the residualwater contents; and l and m empirical parameters (l = 0.5). The para-meter m is a measure of the pore-size distribution and can be expressedasm = 1− 1

n , which in turn can be determined by fitting van Genuchten’sanalytical model [10]

θ(h) =

θres +θsat−θres[1+|αh|n]m , h < 0

θsat , h ≥ 0(3.21)

where α (m−1) is related to the inverse air-entry pressure. According toequation (3.20), the unsaturated hydraulic conductivity is a function ofθL, which subsequently is a function of h (e.g. from equation (3.21)). Theh is actually temperature dependence, hTemCorr = he−Cψ(T−Tr ), where∂h∂T = Cψh and Cψ is assumed to be a constant (= 0.0068 ◦c−1)[11].Such temperature dependence is implied by the temperature dependence

19


of surface tension and viscous flow [12]. Therefore, the temperatureapparently has an effect on the unsaturated hydraulic conductivity, dueto the temperature dependence of h, which is given by Milly [1] as

K(θ, T) = KsKr (θ)KT (T) (3.22)

and KT is given as

KT =µw(Tr )µw(T)

(3.23)

where µw is the viscosity of water. The dynamic viscosity of water isgiven [13] as

µw = µw0exp[

µ1

R(T + 133.3)

](3.24)

where µw0 = 2.4152× 10−2 the water viscosity at reference temperature(Pa s), µ1 = 4.7428 (kJ mol−1), R = 8.314472 (J mol−1 ◦c−1), and T isin ◦c.

3.2.2 Gas Conductivity [14]

In unsaturated soil, the pore space generally is occupied by gas andliquid. Under the ideal assumption that there is no interaction betweenthe fluids (which is actually not the case in the reality), Darcy’s low isapplied to determine the gas conductivity. When the pore space is filledgradually by a single fluid, the permeability with respect to the other fluidwill decreased accordingly, because of the cross-sectional area availablefor the flow of that fluid (other than the fluid occupying the pore space)is less. According to Darcy’s law, the gas conductivity can be expressedas

Kg =Krg(Sa)KsµwρLgµg

(3.25)

where µg is gas viscosity (kg m−1 s−1), and the air viscosity, µa (=1.846× 10−5 kg m−1 s−1); Krg the relative gas conductivity, which is afunction of effective gas saturation and is defined by Van Genuchten-Mualem model as

Krg =(1− S0.5

a

){1−

[1− (1− Sa)1/m

]m}2

(3.26)

3.2.3 Gas-Phase Density

Gas phase density includes water vapor density and dry air density. Thewater vapor density is given by Philip and de Vries [6] as

ρV = ρSVHr , Hr = exp(hgRVT

) (3.27)

20


where ρSV is the density of saturated water vapor (exp(31.3716 −6014.79/T − 7.92495 × 10−3T)10−3

T ) (kg m−3); Hr the relative humid-ity; RV the specific gas constant for vapor (= 461.5 J kg−1 K−1); g thegravitational acceleration (m s−2); and T in K. The gradient of the watervapor density with respect to z can be expressed as

∂ρV∂z

= ρSV∂Hr∂T

∣∣∣∣h+ ρSV

∂Hr∂h

∣∣∣∣T+Hr

∂ρSV∂T

∂T∂z

(3.28)

Assuming that pore-air and pore-vapor could be considered to beideal gas, air and vapor density can be expressed as

ρda =PdaRdaT

, and, ρV =PVRVT

(3.29)

where Rda is the specific gas constant for dry air (= 287.1 J kg−1 K−1);Pda (Pa) and PV (Pa) are the dry air pressure and vapor pressure; andT is in K. According to Dalton’s law of partial pressure, the mixed soilair pressure should be equal to the sum of the dry air pressure and thevapor pressure: Pg = Pda + PV . Therefore, the dry air density could begiven as

ρda =PgRdaT

− ρVRVRda

(3.30)

Differentiating equation (3.30) with respect to time and space yields

∂ρda∂t=Xaa

∂Pg∂t+XaT

∂T∂t+Xah

∂h∂t

∂ρda∂z

=Xaa∂Pg∂z+XaT

∂T∂z+Xah

∂h∂z

(3.31)

where

Xaa=1

RdaT

XaT=−[ PgRdaT 2

+ RVRda

(Hr∂ρSV∂T

+ ρSV∂Hr∂T

)](3.32)

Xah=−∂ρV∂h

3.2.4 Vapor Diffusivity

The vapor diffusion described by Fick’s law in the atmosphere has beenmodified so as to apply in porous media by Rollins [15] as

qV = −Datmντθa∇ρV (3.33)

where Datm is the molecular diffusivity of water vapor in air (m2 s−1); τa tortuosity factor allowing for the extra path length; θa the volumetricair content of the medium (θa = θV = θg = 1 − θL); and ν the ’mass-flow factor’ introduced to allow for the mass flow of vapor arising from

21


the difference in boundary conditions governing the air and the vaporcomponents of the diffusion system. Equation (3.33) has been describedas the ’simple theory’ of vapor transfer by Philip and De Vries [6] andhas been proven not enough to explain the vapor transfer in soil [6].In equation (3.33), the vapor diffusivity in soil can be expressed as:DV_Sim_h = Datmντθa, which is called as isothermal vapor diffusivityfor the ’simple theory’.

With the thermodynamic relationship between the vapor density andthe relative humidity in soil, which is a function of temperature andcapillary pressure [16], the thermal vapor diffusivity for the ’simpletheory’ can be expressed as

DV_Sim_T = DatmντθaHrβ (3.34)

where β is for ∂ρ0∂T , and DV_Sim_T , the thermal vapor diffusivity for the

’simple theory’. In a single air-filled pore, τ equals to 1 and θa is con-sidered as a unity as well, presuming similarity of temperature andvapor fields in the pore. Therefore, the vapor transfer due to the airtemperature gradient ((∇T)a) in the pore can be expressed as

qV = −DatmνHrβ(∇T)a (3.35)

On the other hand, Philip and De Vries [6] assumed that the vapor cantransfer through ’liquid island’ (i.e. the liquid capillary connecting soilparticles) by condensing on the cold side of the island and evaporatingon the warm side. With such assumption, the cross section available forvapor transfer is equal to that occupied by air and liquid. Now, assumingthat the mean flux density in the connecting liquid island is equal to thatin the air-filled pores, we can get

−(θa + θL)DatmνHrβ(∇T)a = −DV_PdV_T∇T (3.36)

where DV_PdV_T represents the thermal vapor diffusivity following Philipand De Vries’ assumption on ’liquid island’, which employs the conceptof air temperature gradient in a single pore and increases cross-sectionfor vapor transfer. The ratio of the thermal vapor diffusivity proposedby Philip and De Vries to that of ’simple theory’ is expressed as

η = θa + θLτθa

(∇T)a∇T (3.37)

where η is the ratio and called as the enhancement factor for thermalvapor transfer in soil.

According to the equations and concept described above, the iso-thermal vapor diffusivity is given as the same as the isothermal vapordiffusivity for the ’simple theory’

DIso_V = DV∂ρV∂h

= Datmντθa∂ρV∂h

(3.38)

where ν is set as 1, τ = θ2/3a , and Datm = 0.229

(1+ T

273

)1.75(m2 s−1).

22


While the thermal vapor diffusivity is given by considering the en-hancement factor as

DNoniso_V = DV∂ρV∂T

= Datm

Enhancement Factor︷︸︸︷(θa + f(θa)θL)︸︷︷︸Cross Section(ηcross)

(∇T)a∇T

∂ρV∂T

= Datmη∂ρV∂T

(3.39)

where f(θa) is a factor introduced to account for decreasing cross-section due to increasing moisture content and increasing degree ofliquid continuity, and is given by De Vries [7] as

f(θa) =

1 , θL ≤ θkθaε−θk , θk < θL

(3.40)

Substitute θL = ε− θa into equation (3.39), the cross-section for vaportransfer will be equal to θa + θa

ε−θk (ε − θa), when θk < θL. On theother hand, when the moisture content decreases till θk(θk = θL), thecross sectional area equals to θa + θL = θa + θk. At this every momentwhen θk merely equal to θL but less than θL, we can approximate thatθa + θa

ε−θk (ε − θa) ≈ θa + θk ≈ ε. It is easy to know that θa(ε − θa) =θk(ε− θk), which means θa ≈ θk. This indicates that the cross sectionalarea reach maximum when θa ≈ θL ≈ θk = 0.5ε. It means that atmoderate moisture content, continuity of both liquid and vapor phasesreaches maximum, togehter with islands of both phases.

In the enhancement factor, (∇T)a∇T (ηT )is regarded as the local temperat-ure gradient effect due to a higher average pore-air temperature gradientcompared to the average temperature gradient of the bulk medium, andis given by Philip and De Vries [6] as

ηT =(∇T)a∇T = (∇T)a

5∑i=1

(∇T)iθi

−1

(3.41)

where (∇T)i is the thermal gradient in the ith constituent, and θi thevolumetric fraction of the ith constituent. Based on De Vries’ appraoch[17], the ratio of the average temperature gradient in the ith constituentto the average temperature gradient of the bulk medium can be expressedby a conceptual model

ki =23

[1+

(λiλ1− 1

)gi]−1

+ 13

[1+

(λiλ1− 1

)(1− 2gi)

]−1

(3.42)

where ki is the ratio; λi the thermal conductivity of the ith constituent(J cm−1 S−1 ◦c−1); and gi the ’shape factor’ of the ith constituent (seeTable 3.1). For the solid particles, constant values for gi as given inTable (3.1) are assumed. For liquid water, no value is needed since its

23


coefficient is zero. The value of g2 is considered a function of moisturecontent as follows [18]

g2 =

0.013+(

0.022θwilting(pF=4.2) +

0.298ε

)θL , θL < θwilting

0.035+ 0.298ε θL , θL > θwilting

(3.43)

By applying the definition of the ki [17], equation (3.41) can be trans-formed as

ηT = k2

5∑i=1

kiθi

(3.44)

The value of ηT is valid for θL down to θk. For θL = 0, ηT may becalculated using air as the continuous phase, and may be interpolatedfor θL between 0 and θk [11].

Table 3.1 Properties of Soil Constituents [17]

Ci λiConstituent i (J cm−3 ◦c−1) (J cm−1 S−1 ◦c−1) giLiquid Water 1 1.0 5.73× 10−3 . . .Air 2 1.25× 10−3 2.5× 10−4 . . .

+LDa(∂ρV/∂T)|hQuartz 3 2.66 8.8× 10−2 0.125Other Materials 4 2.66 2.9× 10−2 0.125Other Materials 5 1.3 2.5× 10−3 0.5

3.2.5 Gas Dispersivity

The gas phase longitudinal dispersivity, DVg , is given by Bear [19] as

DVg = αL_iqi, i = gas/liquid (3.45)

where qi is the pore fluid flux in phase i, and αL_i the longitudinaldispersivity in phase i. αL_i has been evaluated by various authors fordifferent levels of soil saturation. Laboratory studies have shown thatαL_i increases when the soil volumetric water content decreases. In thisstudy, as done by Grifoll et al. [20], a correlation made from simulationresults [21] and experimental data obtained by Haga et al. [22] is used:

αL_i = αL_Sat

[13.6− 16× θg

ε+ 3.4× (θg

ε)5]

(3.46)

As Grifoll et al. [20] pointed out, the lack of dispersivity values led to,the saturation dispersivity, αL_Sat , used in the above correlation to beset at 0.078m, the Figure reported in the field experiments by Biggar andNielsen [23] and shown to be a reasonable value in previous modelingstudies [24].

24


3.2.6 Thermal Property [17]

3.2.6.1 Heat Capacity

The volumetric heat capacity of a soil is a weighted average of thecapacities of its components

C =5∑i=1

Ciθi (3.47)

where θi and Ci are the volumetric fraction and the volumetric heatcapacity of the ith soil constituent (J cm−3 ◦c−1). The five componentsare (1) water, (2) air, (3) quartz particles, (4) other minerals, and (5)organic matter (see Table (3.1)).

3.2.6.2 Thermal Properties

The effective thermal conductivity of a moist soil is given by

λeff = 5∑i=1

kiθiλi

5∑i=1

kiθi

−1

(3.48)

the unit of which is (J cm−1 S−1 ◦c−1) (see Table (3.1)).

3.2.6.3 Differential Heat of Wetting

The differential heat of wetting, W (J Kg−1), is the amount of heatreleased when a small amount of free water is added to the soil matrixand is original expressed by Edlefsen and Anderson [16] as

W = −ρL(ψ− TψT) = −0.01g(h+ Tah) = −0.01gh(1+ Ta) (3.49)

whereψ J kg−1 = 0.01gh (cm) at T = 293K with the value of a = 0.0068(K−1) [25]. Thus, Prunty [26] expressed the differential heat of wetting as

W = −0.2932h (3.50)

3.2.6.4 Transport Coefficient for Adsorbed Liquid Flow

The transport coefficient for adsorbed liquid flow due to temperaturegradient is expressed by Groenevelt and Kay [3] as

DTa =HwεbτµwT

(1.5548× 10−15) (3.51)

where Hw is the integral heat of wetting (J m−2); b = 4× 10−8 (m); T isin ◦c.

25


3.3 Simplification of Governing Equations

The governing differential equations are derived to be in a simple formcontaining neatly the three state variables and their coefficients. Thisallows the conversion of the governing equations to the non-linear ordin-ary differential equations whose unknowns are the values of the primevariables at a finite number of nodes by using Galerkin’s method ofweighted residuals, which will be explained in Chapter 4.

3.3.1 Soil Moisture Equation

The left hand side (LHS) of equation (3.10) can be expanded, consideringthe dependence of moisture content on matric potential and temperatureas follows

∂∂t(θL +

ρVρLθV )

=[∂θL∂t+ ρVρL∂(ε− θL)∂t

+ θV∂∂t

(ρVρL

)]=[(

1− ρVρL

)∂θL∂t+ θVρL∂ρV∂t

]

=[(

1− ρVρL

)(∂θL∂h

∂h∂t+ ∂θL∂T

∂T∂t

)+ θVρL

(∂ρV∂h

∂h∂t+ ∂ρV∂T

∂T∂t

)]

= Chh∂h∂t+ ChT

∂T∂t

(3.52)

where ρL associated with θL has been eliminated by dividing ρL on theboth sides of equation (3.10), the coefficients in above equation can beexpressed accordingly as:

Chh =(

1− ρVρL

)∂θL∂h

+ θVρL∂ρV∂h

ChT =(

1− ρVρL

)∂θL∂T

+ θVρL∂ρV∂T

(3.53)

In the right hand side (RHS) of equation (3.10), the term associatedwith DVa should be expanded as

∂(DVa

∂Pg∂z

)∂z

= DVa∂(∂Pg∂z

)∂z

+ ∂DVa∂z

∂Pg∂z

= DVa∂(∂Pg∂z

)∂z

+Kg∂Pg∂z∂ρV∂z

= DVa∂(∂Pg∂z

)∂z

+Kg∂Pg∂z

(∂ρV∂h

∂h∂z+ ∂ρV∂T

∂T∂z

)(3.54)

26

3.3. Simplification of Governing Equations

Accordingly, the last bracket in the RHS of equation (3.10) can be rewrit-ten as

1ρL∂∂z

[DVh

∂h∂z+DVT

∂T∂z+DVa

∂Pg∂z

](3.55)

= 1ρL∂∂z

[DVh

∂h∂z+DVT

∂T∂z

]+ 1ρL

(DVa

∂∂z

(∂Pg∂z

)+Kg

∂Pg∂z∂ρV∂z

)

We can rewrite the RHS of equation (3.10) as

RHS = ∂∂z

[Khh

∂h∂z+KhT

∂T∂z+Kha

∂Pg∂z

]+ ∂KLh∂z

+VVh∂h∂z+VVT

∂T∂z

(3.56)

where

Khh = K +DVhρL

VVh = −1ρLVa∂ρV∂h

KhT = DTa +DVTρL

VVT = −1ρLVa∂ρV∂T

(3.57)

Kha =Kγw+ DVaρL

Va = −Kg∂Pg∂z

Combining equation (3.52), (3.53), (3.56) and (3.57) yield the moistureequation in the form as below:

Chh∂h∂t+ ChT

∂T∂t

(3.58)

= ∂∂z

[Khh

∂h∂z+KhT

∂T∂z+Kha

∂Pg∂z

]+ ∂KLh∂z

+ VVh∂h∂z+ VVT

∂T∂z

3.3.2 Dry Air Equation

The same procedure applied to equation (3.13), and recall that Sa = 1−SLand SL = θL/ε) we can have

∂∂t[ερda(Sa +HcSL)] =

∂∂t[ερda(1− SL +HcSL)]

= ε∂ρda∂t

+ (Hc − 1)ρda∂θL∂t+ (Hc − 1)θL

∂ρda∂t

= [ε+ (Hc − 1)θL](Xah

∂h∂t+XaT

∂T∂t+Xaa

∂Pg∂t

)

+(Hc − 1)ρda(∂θL∂h


∂T∂t

)= Cah

∂h∂t+ CaT

∂T∂t+ Caa

∂Pg∂t

(3.59)

27


where

Cah = [ε+ (Hc − 1)θL]Xah + (Hc − 1)ρda∣∣∣const

∂θL∂h

CaT = [ε+ (Hc − 1)θL]XaT + (Hc − 1)ρda∣∣∣const

∂θL∂T

(3.60)

Caa = [ε+ (Hc − 1)θL]Xaa + (Hc − 1)ρda∣∣∣const

∂θL∂Pg

The RHS of equation (3.13) can be rewritten as below

∂∂z[Kah

∂h∂z+HcρdaK +KaT

∂T∂z+Kaa

∂Pg∂z]

= ∂∂z

[DV∂ρda∂z

+ ρdaKg∂Pg∂z−Hcρda

qLρL+DVg

∂ρda∂z

]

= ∂∂z

[(DV + θaDVg)

(Xah

∂h∂z+XaT

∂T∂z+Xaa

∂Pg∂z

)]

+ ∂∂z

[ρdaKg

∂Pg∂z−Hcρda

qLρL

]

= ∂∂z

[(DV + θaDVg)

(Xah

∂h∂z+XaT

∂T∂z+Xaa

∂Pg∂z

)+ ρdaKg

∂Pg∂z

]

+Kg∂Pg∂z∂ρda∂z

−Hcρda∂∂z

(qLρL

)−Hc

qLρL∂ρda∂z

= ∂∂z

[(DV + θaDVg)

(Xah

∂h∂z+XaT

∂T∂z+Xaa

∂Pg∂z

)+ ρdaKg

∂Pg∂z

]

+[Kg∂Pg∂z−Hc

qLρL

](Xah

∂h∂z+XaT

∂T∂z+Xaa

∂Pg∂z

)

+Hcρda∂∂z

[K∂∂z

(h+ Pg

γw+ z

)+DTa

∂T∂z

](3.61)

The RHS of equation (3.13) can be rewritten as

RHS = ∂∂z

[Kah

∂h∂z+KaT

∂T∂z+Kaa

∂Pg∂z

]

+Vah∂h∂z+ VaT

∂T∂z+ Vaa

∂Pg∂z+Hcρda

∂K∂z

(3.62)

28


where

Kah = (DV + θaDVg)Xah +HcρdaKKaT = (DV + θaDVg)XaT +HcρdaDTaKaa = (DV + θaDVg)Xaa + ρda(Kg +Hc

Kγw)

Vah =[Kg∂Pg∂z−Hc

qLρL

]Xah (3.63)

VaT =[Kg∂Pg∂z−Hc

qLρL

]XaT

Vaa =[Kg∂Pg∂z−Hc

qLρL

]Xaa

Combining equation (3.59) and equation (3.62), the dry air mass equationcan be represented as below:

Cah∂h∂t+ CaT

∂T∂t+ Caa

∂Pg∂t

= ∂∂z

[Kah

∂h∂z+KaT

∂T∂z+Kaa

∂Pg∂z

]

+Vah∂h∂z+ VaT

∂T∂z+ Vaa

∂Pg∂z+Hcρda

∂K∂z

(3.64)

3.3.3 Energy Equation

Equation (3.19) can be rewritten in a brief form as:

∂Cunsat(T − Tr )∂t

+ L0∂ρVθg∂t

− ρLW∂θL∂t

= ∂∂z

(λeff

∂T∂z

)(3.65)

− ∂∂z

[qLcL(T − Tr )+ qV (cV (T − Tr )+ L0)+ qaca(T − Tr )

]The LHS of equation (3.65) can be expressed as:

∂∂t[Cunsat(T − Tr )]+ L0

∂ρVθg∂t

− ρLW∂θL∂t

= ∂∂t[Cunsat(T − Tr )]+ L0

∂ρV (ε− θL)∂t

− ρLW∂θL∂t

(3.66)

= ∂∂t[Cunsat(T − Tr )]+ L0(ε− θL)

∂ρV∂t− (L0ρV + ρLW)

∂θL∂t

29


Where ∂∂tCunsat(T − Tr ) can be expanded as:

∂∂t[Cunsat(T − Tr )]

= Cunsat∂(T − Tr )

∂t+ (T − Tr )

∂Cunsat∂t

= Cunsat∂T∂t+ (T − Tr ) (3.67)[

ρLcL∂θL∂t+ (ε− θL)ca

∂ρda∂t

− ρdaca∂θL∂t+ (ε− θL)cV

∂ρV∂t− ρVcV

∂θL∂t

]After substituting equation (3.67) into equation (3.66) and rearrangement,we have the LHS of equation (3.65) as

LHS = [(T − Tr )(ρLcL − ρdaca − ρVcV )− (L0ρV + ρLW)]∂θL∂t

+(T − Tr )(ε− θL)ca∂ρda∂t

+ [(T − Tr )(ε− θL)cV + L0(ε− θL)]∂ρV∂t+ Cunsat

∂T∂t

= [(T − Tr )(ρLcL − ρdaca − ρVcV )− (L0ρV + ρLW)](∂θL∂h


∂T∂t

)+ (T − Tr )(ε− θL)ca

(Xah

∂h∂t+XaT

∂T∂t+Xaa

∂Pg∂t

)

[(T − Tr )(ε− θL)cV + L0(ε− θL)][∂ρV∂h

∂h∂t+ ∂ρV∂T

∂T∂t

]+ Cunsat

∂T∂t

= CTh∂h∂t+ CTT

∂T∂t+ CTa

∂Pg∂t

(3.68)

where

CTh = [(T − Tr )(ρLcL − ρdaca − ρVcV )− (L0ρV + ρLW)]∂θL∂h

+(T − Tr )(ε− θL)caXah + [(T − Tr )(ε− θL)cV + L0(ε− θL)]∂ρV∂h

CTT = [(T − Tr )(ρLcL − ρdaca − ρVcV )− (L0ρV + ρLW)]∂θL∂T

+(T − Tr )(ε− θL)caXaT+ [(T − Tr )(ε− θL)cV + L0(ε− θL)]

∂ρV∂T

+ CunsatCTa = (T − Tr )(ε− θL)caXaa (3.69)

Now, using relation of cV (T −Tr )+L0 = cL(T −Tr )+L and qm = qL+qV ,applying the same procedure above to the RHS of equation yields (3.65)

∂∂z

(λeff

∂T∂z

)− ∂∂z

[qLcL(T − Tr )+ qV (cV (T − Tr )+ L0)+ qaca(T − Tr )

](3.70)

= ∂∂z

[λeff

∂T∂z− (qLcL + qaca)(T − Tr )− qV (cL(T − Tr )+ L)

]30


Substituting expressions of qL, qV and qa into equation (3.70), we havethe RHS as

RHS = ∂∂z

[λeff

∂T∂z− (qmcL + qaca)(T − Tr )− qVL

]−(qmcL + qaca)

∂T∂z− qV

∂L∂T∂T∂z

= ∂∂z

(λeff

∂T∂z

)+cL(T − Tr )[

∂∂z

(Khh

∂h∂z+KhT

∂T∂z+Kha

∂Pg∂z

)

+Vvh∂h∂z+ VvT

∂T∂z+ ∂K∂z]

+ca(T − Tr )[∂∂z

(Kah

∂h∂z+KaT

∂T∂z+Kaa

∂Pg∂z

)

+Vah∂h∂z+ VaT

∂T∂z+ Vaa

∂Pg∂z+Hcρda

∂K∂z]

+L ∂∂z

[(DV + θVDVg)

(∂ρV∂h

∂h∂z+ ∂ρV∂T

∂T∂z

)+ ρVKg

∂Pg∂z

]

+[(DV + θVDVg

) ∂ρV∂z

+ ρVKg∂Pg∂z

]cL∂T∂z

+[(DV + θaDVg

) ∂ρda∂z

+ ρdaKg∂Pg∂z

]ca∂T∂z

(3.71)

+(cL + caHc

ρdaρL

)[K∂∂z

(h+ Pg

γw

)+DTa

∂T∂z+K

]∂T∂z

= ∂∂z

(KTh

∂h∂z+KTT

∂T∂z+KTa

∂Pg∂z

)+ VTh

∂h∂z+ VTT

∂h∂z+ VTa

∂h∂z+ CTg

where

KTh = cL(T − Tr )Khh + ca(T − Tr )Kah + L(DV + θVDVg)∂ρV∂h

KTT = λeff + cL(T − Tr )KhT + ca(T − Tr )KaT + L(DV + θVDVg)∂ρV∂T

KTa = cL(T − Tr )Kha + ca(T − Tr )Kaa + LρVKgVTh = cL(T − Tr )VVh + ca(T − Tr )Vah+(DV + θVDVg)

cLρL∂ρV∂h

+ ca(DV + θaDVg)Xah + (cL +HcρdacaρL)K

VTT = cL(T − Tr )VVT + ca(T − Tr )VaT+(DV + θVDVg)

cLρL∂ρV∂T

+ ca(DV + θaDVg)XaT + (cL +HcρdacaρL)DTa

VTa = cLKgρVρL+ ca(T − Tr )Vaa

+ca[(DV + θaDVg)Xaa + ρdaKg

]+ (cL +Hcρda

caρL)Kγw

(3.72)

CTg = [cL(T − Tr )+ ca(T − Tr )Hcρda]∂K∂z+ (cL +Hcρda

caρL)K

31


Combing equation (3.68) and equation (3.71), the energy equation canbe rewritten as

CTh∂h∂t+ CTT

∂T∂t+ CTa

∂Pg∂t

(3.73)

= ∂∂z

(KTh

∂h∂z+KTT

∂T∂z+KTa

∂Pg∂z

)+ VTh

∂h∂z+ VTT

∂h∂z+ VTa

∂h∂z+CTg

3.4 Bibliography


[2] H. R. Thomas and M. R. Sansom. Fully coupled analysis of heat,moisture, and air transfer in unsaturated soil. Journal of EngineeringMechanics, 121(3):392–405, 1995.

[3] P. H. Groenevelt and B. D. Kay. On the interaction of water and heattransport in frozen and unfrozen soils: Ii. the liquid phase. Soil Sci.Soc. Am. Proc, 38:400–404, 1974.

[4] W. G. Gray and S. M. Hassanizadeh. Unsaturated flow theory includ-ing interfacial phenomena. Water Resources Research, 27(8):1855–1863, 1991.

[5] D. G. Fredlund and H. Rahardjo. Soil Mechanics for Unsaturated Soils.John Wiley and Sons, New York, 1993.


[7] D.A. De Vries. Simultaneous transfer of heat and moisture in porousmedia. Trans. Am. Geophys. Union, 39(5):909–916, 1958.


[9] Y. Mualem. A new model for predicting the hydraulic conductivityof unsaturated porous media. Water Resour Res, 12(3):513–522,1976.

[10] M. T. van Genuchten. A closed-form equation for predicting thehydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J,44(5):892–898, 1980.

[11] P. C. D. Milly. Simulation analysis of thermal effects on evaporationfrom soil. Water Resources Research, 20(8), 1984.

[12] E.E. Miller and R.D. Miller. Physical theory for capillary flow phe-nomena. J. Appl. Phys., 27:324–332, 1956.

[13] R. L. Fogel’son and E. R. Likhachev. Temperature dependence ofdiffusion coefficient. Phys Met Metalloved, 90(1):58–61, 2000.

32

3.4. Bibliography

[14] B. R. Scanlon. Uncertainties in estimating water fluxes and residencetimes using environmental tracers in an arid unsaturated zone.Water Resources Research, 36(2):395–409, 2000.

[15] Ralph L. Rollins. Movement of soil moisture under a thermal gradi-ent. 1954.

[16] N. E. Edlefsen and A. B. C Anderson. Ther thermodynamics of soilmoisture. Hilgardia, 16:231–299, 1943.

[17] D.A. De Vries. Thermal Properties of soils. North-Holland, Amster-dam, 1966.

[18] B. A. Kimball, R. D. Jackson, F. S. Reginato, F. S. Nakayama, and S. B.Idso. Comparison of field-measured and calculated soil-heat fluxes.Soil Science Society of America Journal, 40(1), 1976.


[20] J. Grifoll, J.M. Gast, and Y. Cohen. Non-isothermal soil water trans-port and evaporation. Advances in Water Resources, 28:1254–1266,2005.

[21] M. Sahimi, A.A. Heiba, H.T. Davis, and L.E. Scriven. Dispersion inflow through porous media. ii. two-phase flow. Chem. Eng. Sci.,41(8):2123–2136, 1986.

[22] D. Haga, Y. Niibori, and T. Chida. Hydrodynamic dispersion andmass transfer in unsaturated flow. Water Resources Research,35(4):1065–1078, 1999.

[23] J.W. Biggar and D.R. Nielsen. Spatial variability of the leachingcharacteristics of a field soil. Water Resour Res, 1(1):78–84, 1976.

[24] Y. Cohen and P. A. Ryan. Chemical transport in the top soil zone. therole of moisture and temperature gradients. Journal of HazardousMaterials, 22(3):283–304, 1989.

[25] P. C. D. Milly and P. S. Eagleson. Parameterization of Moisture andHeat Fluxes Across the Land Surface for Use in Atmospheric GeneralCirculation Models. Dept. of Civil Engineering, School of Engineering,Massachusetts Institute of Technology, 1982.

[26] L. Prunty. Soil water heat of transport. Journal of Hydrologic Engin-eering, 7(6):435–440, 2002.

33

4STEMMUS: Finite Element Solutionof Governing Equations

35

4. STEMMUS: Finite Element Solution of Governing Equations

4.1 Galerkin’s Method of Weighted Residuals

The governing differential equations are converted to non-linear ordin-ary differential equations whose unknowns are the values of the primevariables at a finite number of nodes by using Galerkin’s method ofweighted residuals. Then, a finite-difference time-stepping scheme isapplied to evaluate the time derivatives, which is solved by a successivelinearization iterative scheme. To describe the spatial discretization andtime stepping of the governing equations, an example of the derivationis presented below, using the moisture equation. For the dry air equationand the energy equation, the procedure is similar. The standard piece-wise linear basis functions for approximation of the prime variables areexpressed as

h(z, t) = h1φ1 + h2φ2 =2∑j=1

hj(t)φj(z)

T (z, t) = T1φ1 + T2φ2 =2∑j=1

Tj(t)φj(z) (4.1)

Pg(z, t) = Pg1φ1 + Pg2φ2 =2∑j=1

Pgj(t)φj(z)

where j is the node index,and φj(z) the usual shape function definedelement by element (φ1 = ze+1−z∆ , φ2 = z−ze∆ ). If the approximations givenby equation (4.1) are substituted into the equations (3.10), (3.13) and(3.19), residuals are obtained for each governing differential equation,which are then minimized using Galerkin’s method. Introducing the newnotation for the coefficients in the moisture mass conservation equation,equation (3.10) becomes

Mmoisture(h, T) (4.2)

= c1∂h∂t+ c2

∂T∂t− ∂∂z

(c3∂h∂z+ c4

∂T∂z+ c5

∂Pg∂z+ c6

)− c7

∂h∂z− c8

∂T∂z

where c1 to c8 are defined implicitly by equation (3.10) and (4.2). Follow-ing Galerkin’s method of weighted residuals [1] for each element, theresiduals obtained by substituting h, T and Pg into equation (4.2) arerequired to be orthogonal to the set of trial functions:

∫z

[c1∂h∂t+ c2

∂T∂t− ∂∂z

(c3∂h∂z+ c4

∂T∂z+ c5

∂Pg∂z+ c6

)− c7

∂h∂z− c8

∂T∂z

]φidz = 0 (4.3)

where z is the solution domain, and i = 1,2. We apply integration bypart (

∫udv = uv −

∫vdu) to the third, fourth and fifth term, which may

36

4.2. Integrals’ Calculation

be recognized as the flux divergence.

∫z

(c1∂h∂t+ c2

∂T∂t

)φidz (4.4)

+∫z

(c3∂h∂z+ c4

∂T∂z+ c5

∂Pg∂z+ c6

)φ′idz +

∫zc7hφ′idz +

∫zc8Tφ′idz

=[((

c3∂h∂z+ c4

∂T∂z+ c5

∂Pg∂z+ c6

)+ c7h+ c8T

)φi

]z2

z1

= [−Qmφi]z2z1

where z1 and z2 are two points in one element and subscripted accordingto a local numbering system. According to the definition of c3, c4, c5, c6,c7 and c8, Qm is implicitly seen as the sum of liquid and vapor mass flux.Now, substituting from equation (4.1) into equation (4.4) yields

2∑j=1

h′j

∫c1φjφidz +

2∑j=1

T ′j

∫c2φjφidz +

2∑j=1

hj∫c3φ′jφ

′idz

+2∑j=1

Tj∫c4φ′jφ

′idz +

2∑j=1

Pgj∫c5φ′jφ

′idz +

∫c6φ′idz

+2∑j=1

hj∫c7φjφ′idz +

2∑j=1

Tj∫c8φjφ′idz = [−Qmφi]

z2z1

(4.5)

4.2 Integrals’ Calculation

In equation (4.5), the primed state variables are time derivatives. Equation(4.5) can be interpreted as approximate mass conservation relations fornode 1 and 2. Their sum is a mass conservation condition for the element.The ’c’ coefficients are assumed to vary linearly within the element. Theirvalues at the nodes could be given by

cm1 = cm(h1, T1)cm2 = cm(h2, T2) (4.6)

where m = 1,2, . . . ,8. If c is discontinuous at a node, it is evaluated asthe limit of c that is approached from inside the element. Given thisform for the variation of the coefficient, we may proceed to evaluatethe integrals in equation (4.5), using the definitions from equation (4.1).Taking the first item in equation (4.5) as an example, we have∫

c1φjφidz =∫c11φ1φjφidz +

∫c12φ2φjφidz (4.7)

where in general, we can get∫φkφjφidz for equation (4.7). Recall that,

φ1 equals to ze+1−z∆ and φ2 equals to z−ze∆ .

37


When i = j = k:∫φkφjφidz =

∫ ze+1

zeφ3

1dz

=∫ ze+1

ze

(ze+1 − z∆

)3

dz =[−∆

4

(ze+1 − z∆

)4]ze+1

ze= ∆

4∫φkφjφidz =

∫ ze+1

zeφ3

2dz (4.8)

=∫ ze+1

ze

(z − ze∆

)3

dz =[∆

4

(z − ze∆

)4]ze+1

ze= ∆

4

when otherwise:∫φkφjφidz =

∫ ze+1

zeφ2

1φ2dz

=∫ ze+1

ze

(ze+1 − z∆

)2 (z − ze∆)dz

= −∆3

∫ ze+1

ze

z − ze∆ d(ze+1 − z∆

)3

(4.9)

= −∆3z − ze∆

(ze+1 − z∆

)3∣∣∣ze+1

ze+ ∆

3

∫ ze+1

ze

(ze+1 − z∆

)3

dz − ze∆

= 13

∫ ze+1

ze

(ze+1 − z∆

)3

dz = 13

[−∆

4

(ze+1 − z∆

)4]ze+1

ze= ∆

12

Above result is the same for the integrals containing c2, which couldbe defined as the first type of integrals. As for the second type ofintegrals related to c3, c4 and c5, we have∫

c3φ′jφ′idz =

∫c31φ1φ′jφ

′idz +

∫c32φ2φ′jφ

′idz (4.10)

The general form for the above integrals is∫φkφ′jφ

′idz. Accordingly,

When i = j:∫φkφ′jφ

′idz =

∫ ze+1

zeφ1φ′1φ

′1dz (4.11)

=∫ ze+1

ze

(ze+1 − z∆

)(− 1∆

)2

dz =(− 1∆

)2[−∆

2

(ze+1 − z∆

)2]ze+1

ze= 1

2∆when i ≠ j:∫

φkφ′jφ′idz =

∫ ze+1

zeφ2φ′1φ

′1dz (4.12)

=∫ ze+1

ze

(z − ze∆

)(− 1∆

)2

dz =(− 1∆

)2[∆

2

(z − ze∆

)2]ze+1

ze= − 1

2∆38

4.2. Integrals’ Calculation

The same method applied to the integrals associated with c6:∫c6φ′idz = c61

∫φ1φ′idz + c62

∫φ2φ′idz (4.13)

where in general, we can get∫φkφ′idz.

When i = 1:∫φkφ′idz =

∫ ze+1

zeφ1φ′1dz (4.14)

=∫ ze+1

ze

(ze+1 − z∆

)(− 1∆

)dz =

(− 1∆

)[−∆

2

(ze+1 − z∆

)2]ze+1

ze= −1

2

when i = 2:∫φkφ′idz =

∫ ze+1

zeφ1φ′1dz (4.15)

=∫ ze+1

ze

(z − ze∆

)(1∆)dz =

(1∆)[∆

2

(z − ze∆

)2]ze+1

ze= 1

2

For the integrals associated with c7 and c8, we have∫c7φjφ′idz = c71

∫φ1φjφ′idz + c72

∫φ2φjφ′idz (4.16)

where in general, we can get∫φkφjφ′idz. The integrals associated with

c7 and c8 are more complicated than the integrals mentioned above.

When k = j and i = 1:∫φkφjφ′idz

=∫ ze+1

ze

(ze+1 − z∆

)2 (− 1∆

)dz = − 1∆

[−∆

3

(ze+1 − z∆

)3]ze+1

ze= −1

3∫φkφjφ′idz (4.17)

=∫ ze+1

ze

(z − ze∆

)2 (− 1∆

)dz = − 1∆

[∆3

(z − ze∆

)3]ze+1

ze= −1

3

when k ≠ j and i = 1:∫φkφjφ′idz

=∫ ze+1

ze

(ze+1 − z∆

)(z − ze∆

)(− 1∆

)dz

= − 1∆∫ ze+1

ze

∆2

(ze+1 − z∆

)d(z − ze∆

)2

(4.18)

= − 1∆ ∆2(ze+1 − z∆

)(z − ze∆

)2∣∣∣ze+1

ze+ 1∆

∫ ze+1

ze

∆2

(z − ze∆

)2

d(ze+1 − z∆

)= − 1∆2

∫ ze+1

ze

∆2

(z − ze∆

)2

dz = − 1∆2

∆2

[∆3

(z − ze∆

)3]ze+1

ze= −1

6

39


when k = j and i = 2:

∫φkφjφ′idz

=∫ ze+1

ze

(ze+1 − z∆

)2 ( 1∆)dz = 1∆

[−∆

3

(ze+1 − z∆

)3]ze+1

ze= 1

3∫φkφjφ′idz (4.19)

=∫ ze+1

ze

(z − ze∆

)2 ( 1∆)dz = 1∆

[∆3

(z − ze∆

)3]ze+1

ze= 1

3

when k ≠ j and i = 2:

∫φkφjφ′idz

=∫ ze+1

ze

(ze+1 − z∆

)(z − ze∆

)(1∆)dz

= 1∆∫ ze+1

ze

∆2

(ze+1 − z∆

)d(z − ze∆

)2

(4.20)

= 1∆ ∆2(ze+1 − z∆

)(z − ze∆

)2∣∣∣ze+1

ze− 1∆

∫ ze+1

ze

∆2

(z − ze∆

)2

d(ze+1 − z∆

)= 1∆2

∫ ze+1

ze

∆2

(z − ze∆

)2

dz = 1∆2

∆2

[∆3

(z − ze∆

)3]ze+1

ze= 1

6

We may evaluate the right-hand-side term of equation (4.5) as:

[−Qmφi]ze+1ze =

Qm

∣∣∣ze, i = 1

−Qm∣∣∣ze+1, i = 2

(4.21)

4.3 Finite Difference Time-Stepping Scheme

Having evaluated the coefficients in equation (4.5), we now rewrite thatequation system in matrix form:

[A11 A12

A21 A22

][h′1h′2

]+[B11 B12

B21 B22

][T ′1T ′2

]+[C11 C12

C21 C22

][h1

h2

](4.22)

+[D11 D12

D21 D22

][T1

T2

]+[E11 E12

E21 E22

][Pg1

Pg2

]+[F1

F2

]=[ Qm∣∣∣z1

−Qm∣∣∣z2

]

40

4.3. Finite Difference Time-Stepping Scheme

where

Aij = c1

∫φjφidz, Bij = c2

∫φjφidz

Cij = c3

∫φ′jφ

′idz + c7

∫φjφ′idz

Dij = c4

∫φ′jφ

′idz + c8

∫φjφ′idz (4.23)

Eij = c5

∫φ′jφ

′idz, Fij = c6

∫φ′jdz

Now, considering that the boundaries separating individual elements arepoints at which discontinuities may occur, coupling will be accomplishedthrough the matching of boundary conditions at the discontinuities. Thecoupling conditions give us

h12 = h2

1

T 12 = T 2

1

P1g2 = P2

g1 (4.24)

Qm∣∣∣z1

2= Qm

∣∣∣z2

1

Expanding equation (4.22) from one element to two elements, introducingthe global index for the coordinate z and for the state variables (i.e.superscript takes on values of 1 and 2), and remembering the elementindices increase in the positive upward z direction, we get

A111 A1

12 0

A121 A1

22 +A211 A2

12

0 A221 A2

22

h′1h′2h′3

+B1

11 B112 0

B121 B1

22 + B211 B2

12

0 B221 B2

22

T ′1T ′2T ′3

+

C111 C1

12 0C1

21 C122 + C2

11 C212

0 C221 C2

22

h1

h2

h3

+D

111 D1

12 0D1

21 D122 +D2

11 D212

0 D221 D2

22

T1

T2

T3

+

E111 E1

12 0E1

21 E122 + E2

11 E212

0 E221 E2

22

Pg1

Pg2

Pg3

+ F1

1

F21 + F1

2

F22

=Qm

∣∣∣z1

0

−Qm∣∣∣z3

(4.25)

From equation (4.25) we know that the extension of this procedure willcover N nodes (N-1 elements) strightforward. The final pair of matrixequation for moisture balance equation is

Akh′ + BkT ′ + Ckh+DkT + Ekpg + Fk = Qk (4.26)

Where k = 1, 2 and 3. To obtain the temporal solution to above equation,a fully implicit, backward difference scheme is used. This means that allterms other than the time derivative are evaluated at the end of the timestep

Akhk − hk−1

∆t +BkTk − T k−1

∆t +Ckhk+DkT k+EkPgk+Fk = Qmk (4.27)

41


where ∆t is the time increment. An implicit integration scheme is usuallymuch more stable than an explicit one. Considering hk to be unknownyields

(Ak∆t + Ck

)hk = A

k

∆t hk−1−(Bk∆t +Dk

)T k+ B

k

∆t T k−1−EkPgk−Fk+Qmk

(4.28)

where k is a time index and ∆t is the length of the time step. Thecoefficient matrices in equation (4.27) are to be evaluated at the new timelevel with an iterative scheme, which updates the coefficient matricesat each iteration until desired convergence criteria are achieved. TheIterative solution strategy is implemented in STEMMUS [2]:

1. Extrapolate the solutions for the last two time steps (k − 2 andk−1) forward to obtain an estimate of hk, T k and Pgk (If this is thefirst time stop, only the k− 1 ’solution’ is available - it is the initialcondition. In this case, we can assume hk and T k are given by theinitial conditions for a first guess.);

2. Use the estimated states to evaluate all components (except hk) inequation (4.27). Solve the resulting tridiagonal matrix equation forhk;

3. Use the latest estimates of hk, T k and Pgk to evaluate and solve for

Pgk in air equation similar to equation (4.27);

4. Use the latest estimates of hk, T k and Pgk to evaluate and solve for

T k in energy equation similar to equation (4.27);

5. Repeat steps 2 and 4 until some convergence criterion is met.

With this algorithm, we never need to solve a matrix equation anymore complex than one in which the matrix is tridiagonal. A very fastprocedure exists for the solution of such equations (e.g. Thomas al-gorithm). Note that it is possible to incorporate any of the boundaryconditions, including the non-linear ones, into this iterative procedure,using the methods shown below:

In a problem where flux boundary conditions are prescribed, therelevant Q′s are substituted into the right-hand-side vector, and thefour unknown states are found by integrating the system. On the otherhand, if one or more of the states is fixed at a node (first-type boundarycondition), the equations containing the unknown flux through thatnode is eliminated from the system of equations, keeping the solutionuniquely determined. After solution, the neglected equations may beused to determine boundary fluxes [2].

Most of investigators have found that the form of the storage matricesgenerated by the Galerkin method often lead to numerical difficulties. Acommonly accepted means of overcoming these problems is to diagon-alize, or lump the storage matrix. Consider the first matrix of equation

42

4.3. Finite Difference Time-Stepping Scheme

(4.22), we obtain

2∑j=1

h′j

∫c1φjφidz =

A11 A12

A21 A22

h′1h′2

=

c11φ1φ1φ1 + c12φ2φ1φ1 c11φ1φ2φ1 + c12φ2φ2φ1

c11φ1φ1φ2 + c12φ2φ1φ2 c11φ1φ2φ2 + c12φ2φ2φ2

h′1h′2

= ∆

14c11 + 1

12c121

12c11 + 112c12

112c11 + 1

12c121

12c11 + 14c12

h′1h′2

= ∆

13c11 + 1

6c12 0

0 16c11 + 1

3c12

h′1h′2

(4.29)

In the above storage matrix, the modification is made with the c11 = c12,which leads to the final form of the storage matrix shown as below

2∑j=1

h′j

∫c1φjφidz =

[A11 A12

A21 A22

][h′1h′2

](4.30)

= ∆1

3c11 + 16c12 0

0 16c11 + 1

3c12

h′1h′2

= ∆1

2c11 0

0 12c12

h′1h′2

The same procedure applied to the terms related to Cij and Dij , we

have2∑j=1

hj∫c3φ′jφ

′idz +

2∑j=1

hj∫c7φjφ′idz

=

(c31φ1 + c32φ2)φ′1φ′1 (c31φ1 + c32φ2)φ′2φ

′1

(c31φ1 + c32φ2)φ′1φ′2 (c31φ1 + c32φ2)φ′2φ

′2

h′1h′2

+

(c71φ1 + c72φ2)φ1φ′1 (c71φ1 + c72φ2)φ2φ′1(c71φ1 + c72φ2)φ1φ′2 (c71φ1 + c72φ2)φ2φ′2

h′1h′2

=

(c31 + c32) 12∆ −(c31 + c32) 1

2∆−(c31 + c32) 1

2∆ (c31 + c32) 12∆

h′1h′2

+

(−1

3c71 − 16c72

) (−1

6c71 − 13c72

)(

13c71 + 1

6c72

) (16c71 + 1

3c72

)h′1h′2

(4.31)

=

(c31 + c32) 12∆ −

(13c71 + 1

6c72

)−(c31 + c32) 1

2∆ −(

16c71 + 1

3c72

)−(c31 + c32) 1

2∆ +(

13c71 + 1

6c72

)(c31 + c32) 1

2∆ +(

16c71 + 1

3c72

)h′1h′2

Applying the same procedures to the terms associated with Eij and

Fij and combining the lumped storage assumptions for equation (4.27),

43


we obtain

∆[12c11 0

0 12c12

] 1∆t(hk1 − hk−1

1

)1∆t(hk2 − hk−1

2

)+∆[12c21 0

0 12c22

] 1∆t(T k1 − T k−1

1

)1∆t(T k2 − T k−1

2

)+

12∆ (c31 + c32)−

(13c71 + 1

6c72

)− 1

2∆ (c31 + c32)−(

16c71 + 1

3c72

)− 1

2∆ (c31 + c32)+(

13c71 + 1

6c72

)1

2∆ (c31 + c32)+(

16c71 + 1

3c72

)hK1hK2

+

12∆ (c41 + c42)−

(13c81 + 1

6c82

)− 1

2∆ (c41 + c42)−(

16c81 + 1

3c82

)− 1

2∆ (c41 + c42)+(

13c81 + 1

6c82

)1

2∆ (c41 + c42)+(

16c81 + 1

3c82

)TK1TK2

+

12∆ (c51 + c52) − 1

2∆ (c51 + c52)

− 12∆ (c51 + c52) 1

2∆ (c51 + c52)

PKg1

PKg2

+

−12 (c61 + c62)

12 (c61 + c62)

= Qm

∣∣∣z1

−Qm∣∣∣z2

(4.32)

4.4 MATLAB Implementation

Through the discretization scheme introduced in above sections, wenow can translate these equations into MATLAB. Be aware of that, theexamples shown above is only for soil moisture balance equation, thesimilar procedure can be applied for dry air equation and energy equation.In STEMMUS, the subroutines for implementing the task explained aboveinclude: “hPARM.m”, “h_MAT.m” and “h_EQ.m” (see Chapter 5)

In “hPARM.m”, the parameters related to matrix coefficients (e.g. c1-c8) are evaluated. The code is shown in figure 4.1. The subroutine inSTEMMUS is coded as a function. It is not necessary to code in thisway (i.e. to build a subroutine as a function instead of a ’script’ file).However, one advantage to build subroutine as a function is that it iseasy to check the code with Matlab, which automatically detects errors,warnings and so on by using different font color. One disadvantage ofbuilding subroutines as functions is that the variables claimed before in“Constants.m” have to be claimed again in the subroutine, as figure 4.1shows.

As described above, the first part of “hPARM.m” includes the nameof ‘function’, which is the same as the file name, followed by the globalvariable claiming. The first “for” loop is for calculating the derivative ofsoil moisture content with respect to matric potential (DTheta_LLh, ∂θ∂h )

and temperature (DTheta_LLT, ∂θ∂T ). The second “for” loop is to calculatethe matrix coefficients: Chh, ChT, Khh, KhT, Kha, Vvh, VvT and Chg,expressions of which can be found in equation (3.53) and equation (3.57).

After the evaluation of the matrix coefficients, “h_MAT.m” assemblesthe coefficient matrices of the Galerkin expressions for the conservationequations. It uses a functional coefficient scheme to evaluate the matrixelements (see figure 4.2).

44

4.4. MATLAB Implementation

With all the coefficient matrics of the Galerkin expressions available,now we can proceed to perform the finite difference of the time derivat-ives of the matrix equation as equation (4.32) shows. The subroutine of“h_EQ.m” is employed for this purpose (see figure 4.3).

The same implementation procedures are applied to solve dry airequation and energy equation (see Chapter 5). After these three sub-routines, we can get a trigonal matrix for each balance equation. To solvethese trigonal matrics, the Thomas algorithm is employed.

Figure 4.1 Matlabe code of hPARM.m in STEMMUS

45


4.5 Bibliography

[1] G.F. Pinder and W. G. Gray. Finite Element simulation in surface andsubsurface hydrology. Academic, New York, 1977.

[2] P. C. D. Milly and P. S. Eagleson. The coupled transport of water andheat in a vertical soil column under atmospheric excitation. PhD thesis,1980.

46

4.5. Bibliography

Figure 4.2 Matlabe code of h_MAT.m in STEMMUS

47


Figure 4.3 Matlabe code of h_EQ.m in STEMMUS

48

5STEMMUS: Structures, Subroutinesand Input Data

49

5. STEMMUS: Structures, Subroutines and Input Data

This chapter provides documentation of the computer code for solv-ing STEMMUS theoretical equations outlined in Chapter 3 & 4. Theframework of STEMMUS will be present with a flowchart, which is definedby the main program of the code. Accordingly, each subroutine willbe introduced with its specific role in the algorithm. The execution ofSTEMMUS requires the determination of input data including the systemparameters, initial conditions and boundary conditions. The format ofsuch input data can be easily perceived in the specific subroutines whichwill be indicated in this section. An extensive list of definitions of theMATLAB variables used in STEMMUS is also given.

5.1 Structure of STEMMUS: Overview

Figure 5.1 illustrates the main program of STEMMUS. After initial inputand computations, STEMMUS code starts to loop through the subroutinesfollowing the flowchart. Inside a given simulation period, which includesmany time steps, a time step loop is executed as many times as necessaryto reach the end of the simulation period, given the constraints on timestep size. For a given time step, the balance equations for soil moisture,temperature and dry air are solved alternatively to obtain successiveestimates of three independent state variables: h, T , and Pg .

In a time step, a specified maximum iteration number is given toconsider “accuracy” and numerical “convergence”. It is also given tobalance the simulation “accuracy” and the computation cost. Duringa simulation, the maximum iteration number is rarely reached unlessthe change of state variables (e.g h, T , or Pg) exceeds the predefinedconstrains. In that case, the time step will be adjusted (e.g usually bereduced with a given factor), and the current time step will be repeated,with a set of new updated state variables. The maximum changes ofstate variables in one time step are usually predefined with a value,within which the STEMMUS estimates of h, T , and Pg in one time stepare accepted for producing “converged” successive estimates.

The subroutines can be divided into four groups playing differentroles in the main program: Initialization Group, Parameterization Group,Processing Group and Post-Process Group. One caution here is that somesubroutines can be involved in more than one single group and will becalled during looping by different groups.

[The Initialization Group]:

MainLoop.m: The main routine of STEMMUS;

Constants.m: All constants needed for STEMMUS;

Dtrmn_Z.m:Determination of nodal space ∆z;

StartInit.m: Initialization for a specific simulation period;

SOIL2.m:Calculation of soil moisture θL;

50

5.1. Structure of STEMMUS: Overview

Figure 5.1 Flowchart of STEMMUS Main Program

Forcing_PARM.m: Generating forcing data.

[The Parameterization Group]:

CondL_h.m: Calculation of K considering temperature dependence;

CondL_T.m: Calculation of KLT (Disabled, reasons to be discussed);

Density_V.m: Calculation of vapor density ρV ;

CondL_Tdisp.m: Calculation of transport coefficient for adsorbedliquid flow due to temperature gradient DTa;

Latent.m: Calculation of latent heat L;

Density_DA.m: Calculation of dry air density ρda;

CondT_coeff.m & EfeCapCond.m: Calculation of thermal propertiesλeff , C , and η;

Condg_k_g.m: Calculation of gas conductivity kg ;

51


CondV_DE.m: Calculation of vapor diffusivity DV ;

CondV_DVg.m: Calculation of vapor dispersivity DVg , and qaa;

Evap_Cal.m: Calculation of evaporation.

[The Processing Group]:

SOIL1.m: Updating the wetting history;

h_sub.m: Solve liquid equation for a new estimate of hh, which meansthe matric potential at the end of time step, while h means thematric potential at the end of last time step;

hPARM.m:Calculation of coefficient matrices for liquid equation;

h_MAT.m:Assembles the global coefficient matrices of the Galerkinexpressions for liquid equation;

h_EQ.m:Performs the finite difference of the time derivatives in thematrix equation ;

h_BC.m:Determines the boundary condition for solving liquidequation;

h_Solve.m:Using Thomas Algorithm to solve the matrix equation;

h_Bndry_Flux.m:Calculation of liquid flux on the boundary node;

Air_sub.m: Solve dry air equation for a new estimate of Pgg , whichmeans the mixed soil air pressure at the end of time step, while Pgmeans the mixed soil air pressure at the end of last time step;

AirPARM.m:Calculation of coefficient matrices for dry air equation;

Air_MAT.m:Assembles the global coefficient matrices of the Galerkinexpressions for dry air equation;

Air_EQ.m:Performs the finite difference of the time derivatives in thematrix equation ;

Air_BC.m:Determines the boundary condition for solving dry airequation;

Air_Solve.m:Using Thomas Algorithm to solve the matrix equation;

Enrgy_sub.m: Solve energy equation for a new estimate of TT , whichmeans the soil temperature at the end of time step, while T meansthe soil temperature at the end of last time step;

52

5.2. Initial and Boundary Conditions

EnrgyPARM.m:Calculation of coefficient matrices for energy equation;

Enrgy_MAT.m:Assembles the global coefficient matrices of the Galerkinexpressions for energy equation;

Enrgy_EQ.m:Performs the finite difference of the time derivatives in thematrix equation ;

Enrgy_BC.m:Determines the boundary condition for solving energyequation;

Enrgy_Solve.m:Using Thomas Algorithm to solve the matrix equation;

Enrgy_Bndry_Flux.m:Calculation of energy flux on the boundary node.

[The Post-processing Group]:

TimestepCHK.m: Assessing the change in boundary conditions afterone time step;

CnvrgnCHK.m: Checking if the convergence can be reached withoutadjusting time step;

PlotResults.m: After the convergence and the end of simulation periodis reached, plotting the results.

5.2 Initial and Boundary Conditions

Initial conditions include soil moisture content (θL), soil temperature (T ),atmospheric pressure (Pg) (there is no different symbol for this variable,with respect to mixed soil air pressure. It is assumed as the main forcingfor soil air flow), air temperature (Ta), relative humidity (Hra), wind speed(U ), precipitation (P ), and soil surface temperature (Ts ).

The initial soil moisture content profile and soil temperature pro-file are built by linearly extrapolating or interpolating from the pointmeasurement at different depths. The time step in STEMMUS is adjustedautomatically during computing (1 to 1800 s). Accordingly, the timeinterval of the meteorological inputs need to be adjusted to match eachnew time step. However, in the field measurement, the time interval forrecording data is fixed with half of an hour. To match this requirementof flexible time step, the Fourier transform method need to be appliedfor producing the successive forcing data, through approximating thefrequency domain representation of the meteorological forcing data.After the transformation, the parameters for producing meteorologicalforcing data should be put in the subroutine of “Constants.m”.

Considering the aerodynamic resistance and soil surface resistanceto water vapor transfer from soil to atmosphere, the evaporation is

53


expressed as [1]

E = ρVS − ρVara + rs

(5.1)

where ρVS (kg m−3) is the water vapor density at the soil surface; ρVa(kg m−3) the atmospheric vapor density; rs (s m−1) the soil surface res-istance to water vapor flow; and ra (s m−1) the aerodynamic resistance.Equation (5.1) forms the surface boundary condition for moisture trans-port. Without taking ponding and surface runoff into consideration, soilsurface is open to the atmosphere and the measured atmospheric pres-sure is adopted as the surface boundary condition for dry air transportin the soil. The measured soil surface temperature is set as the boundarycondition for heat transport. The surface energy balance equation alsocan be used for this purpose, which can be implemented in STEMMUS byintroducing a specific subroutine for that.

Following van de Griend and Owe [2], the aerodynamic resistance (ra)and soil surface resistance (rs ) was expressed as

ra =1k2U

[ln(Zm − d− Zom

Zom

)−ψsm

][ln(Zm − d− Zoh

Zoh

)−ψsh

]rs = rslea(θmin−θsur ) (5.2)

where k is the von Karman constant (= 0.41); U (m s−1) the measuredwind speed at certain height; Zm (m) the height of wind-speed meas-urement; d (m) the zero plane displacement (= 0 for bare soil); ZOm(= 0.001 m) the surface roughness length for momentum flux; ψsm theatmospheric stability correction factor for momentum flux; ZOh (= 0.001m) the surface roughness length for heat flux; ψsh the atmosphericstability correction factor for heat flux; rsl (= 10 s m−1) the resistance tomolecular diffusion across the water surface itself; a (= 35.63) the fittedparameter; θmin (= 0.15 m−3 m3) the empirical minimum value abovewhich the soil is able to deliver vapor at a potential rate; and θsur thesoil water content in the topsoil layer.

The above description of the boundary condition is far from complete.However, it shows the boundary condition used currently in STEMMUS forthe case in the Badain Jaran Desert [3]. A comprehensive boundary condi-tion description should consider water movement in soils occurs underboth saturated and unsaturated conditions, which will be discussed innext version of STEMMUS manual.

5.3 List of STEMMUS Variables

All variables used in STEMMUS have been commentated in the code. Thelist here is for the description of the most often used variables. Thecolumn of “Alias” indicates how the variables expressed in STEMMUScode.

54

5.3. List of STEMMUS Variables

Table 5.1: Most often used notations in STEMMUS

Variable Alias Description

θL Theta_L Volumetric moisture content at the start ofcurrent time step

Theta_LL Volumetric moisture content at the end ofcurrent time step

∂θL∂h DTheta_LLh Partial derivative of θL with respect to h∂θL∂T DTheta_LLT Partial derivative of θL with respect to T

θV Theta_g Volumetric air content

θsat Theta_s Saturated moisture content

θres Theta_r Residual moisture content

ε POR Porosity

Se Se Effective saturation, = θL−θresθsat−θres

≈ SL = θLε .

Sa Sa Degree of air saturation, = 1− SL = θgε .

ρL RHOL Density of liquid water

ρV RHOV Density of vapor∂ρV∂h DRHOVh Partial derivative of ρV with respect to h∂ρV∂T DRHOVT Partial derivative of ρV with respect to T

ρSV RHOV_s Density of saturated water vapor∂ρSV∂T DRHOV_sT Partial derivative of ρSV with respect to T

ρVS RHOV_sur Vapor density in the top soil layer

ρVa RHOV_A Vapor density in air at certain height

ρda RHOVDA Density of dry air

E Evap Evaporation rate

ra Resis_a Aerodynamic resistance for evaporation

rs Resis_s Surface resistance for evaporation

P Precip Precipitation rate

kah Kah Isothermal air transfer coefficient

kaT KaT Thermal air transfer coefficient

kaa Kaa Advective air transfer coefficient

Continued on next page

55


Table 5.1 – continued from previous page

Variables Alias Description

DTa D_Ta Transport coefficient for adsorbed liquidflow due to temperature gradient

DV D_V Vapor diffusivity in soil

Da D_a Vapor diffusivity in air

DVg D_Vg Longitudinal dispersion coefficient

τ f0 Tortuosity factor

γw Gamma_w Specific weight of water

µw MU_W Dynamic viscosity of water

µa MU_a Dynamic viscosity of air

h h Matric potential at the start of current timestep

hh Matric potential at the end of current timestep

hTemCorr hCor Temperature corrected matric potential

e−Cψ(T−Tr ) Cor Temperature correction factor for matricpotential

1∂T ∂hTemCorr

∣∣∣θL

DhU Rate of change of temperature correctedmatric potential with respect to temperat-ure, with no change of moisture content

= 1TT−T (hhTemCorr − hTemCorr ) =

hCorT = Cor hh−hTT−T + hh

∂Cor∂T

= Cor hh−hTT−T + hhCor (−Cψ)⇒ DhU = Cor(hh− h− hhCψ)T T Soil temperature at the start of current time

step

TT Soil temperature at the end of current timestep

Pg Pg Soil air pressure at the start of current timestep

Pgg Soil air pressure at the end of current timestep

56

5.4. How to Run STEMMUS

5.4 How to Run STEMMUS

To run the STEMMUS in MATLAB is straightforward in “MainLoop.m”.When you finished all numerical set-up and gave all input data to STEM-MUS, please open “MainLoop.m” in MATLAB and press ’F5’ to run themodel (or press the green arrow “run” button shown on the tool barin MATLAB). Typically, there are 5 steps to prepare before running themodel.

1. Setting time information and domain information;

2. Setting soil properties;

3. Setting observation information and initialization;

4. Setting meteorological forcing information;

5. Setting boundary conditions;

6. Running “MainLoop.m”;

5.4.1 Time and Domain Information

The time information setting includes the determination of maximumiteration number (i.e. ’NIT’) in one time step, duration of simulationperiod (i.e. ’DURTN’), initial time step (i.e. ’Delt_t’), maximum desirablechange of state variables (i.e. ’xERR’, ’hERR’, ’TERR’ and ’PERR’), totaldepth of the domain of interest (i.e. ’Tot_Depth’) AND the determinationof how to discrete the domain (e.g. equal spaces for elements or finnerdiscretization on the top layer) by the subroutine of “Dtrmn_Z.m”. Figure5.2 shows how the code for setting time and domain information lookslike.

For the maximum desirable change of state variables, PERR is not inuse yet by the subroutine of “TimestepCHK.m”. The different choicesof xERR and TERR sometime can give different outputs. The bigger themaximum change value, faster the STEMMUS running. However, this maycost the accuracy of the prediction by STEMMUS in regards to reality. It’dbe better to tune this parameter to optimize the balance between thecomputation cost and accuracy. Considering the example which will beshown latter in this chapter the STEMMUS running only takes about 30secs to finish, this indicates that the computation cost will not be a bigproblem for STEMMUS.

5.4.2 Soil Properties

The setting of soil properties includes saturated hydraulic conductivity(i.e. ’SaturatedK’), saturated moisture content (i.e. ’SaturatedMC’), re-sidual moisture content (i.e. ’ResidualMC’), soil porosity (i.e. ’porosity’),parameters of soil water retention curve (i.e. ’Coefficient_n’ & ’Coeffi-cient_Alpha’). In STEMMUS, the Van Genuchten-Mualem model is adopted(see section 3.2.1). In practically, residual moisture content is difficultto be detected in laboratory, and is usually determined by fitting the

57


Figure 5.2 Time and domain information block

experimental data to Van Genuchten model. Figure 5.3 shows how thecode for setting soil properties information looks like.

In STEMMUS, the heterogeneity of soil is considered by assigningvalues to ’IS’, which is defined as the index of soil type. The hysteresisis considered by detecting ’IH’ in every time step. The ’IH’ is defined asthe index of wetting history of soil which would be assumed as dry atthe first with the value of 1. Be aware of that ’IS’ and ’IH’ need to bedefined in “StartInit.m”. There are other two important parameters in“StartInit.m”: one is the ’XK’ that is associated with θk in equation (3.40)and is defined as 1/2ε; the other is the ’XWILT’ that is defined as wiltingpoint associated with a low matric potential of −1.5× 104 cm.

The treatment of hysteresis and soil heterogeneity follows Milly andEagleson’s work [4]. As for the soil wetting history, ’XWRE’ is a criticalparameter, which is calculated in subroutine of “SOIL2” and defined asthe value of the main wetting function evaluated at the effective reversalvalue of temperature-corrected matric potential ’h’.

5.4.3 Observation Information and Initialization

Observation information can be directly copied and pasted in the lastblock in “Constants.m” as figure 5.4 shows. ’Msr_Mois’, ’Msr_Temp’ and’Msr_Time’ represent respectively the measurement of moisture content,soil temperature and the measurement intervals (unit of seconds). Exceptfor the measurement data, the information about what the depth the

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measurement is taken need to be listed. Figure 5.5 shows how to inputthe information about the measurement depth and so on into STEMMUSfor generating initial soil moisture and soil temperature profile.

To generate the initial soil moisture and temperature profile, we needto input the observation positions/depths (i.e. ’InitND1’,. . . , ’InitND5’),measured soil moisture content and temperature at ’InitND?’ depth at thestart of simulation period (i.e. ’InitX1’,. . . , ’BtmX’, and ’InitT1’,. . . , ’BtmT’respectively). With these inputs, in “StartInit.m”, STEMMUS automaticallyinterpolate these information to generate initial profile information asfigure 5.6 shows. The measured soil matric potential can be input as wellwith minor changes in the code shown on figure 5.6.

5.4.4 Meteorological Forcing Information

The treatment of meteorological forcing data is a bit complicated com-pared to other inputs. The meteorological forcing data include mainlyair temperature (i.e. ’Ta’), air relative humidity (i.e. HRa), wind speed (i.e.’U’), surface temperature (i.e. ’Ts’), atmospheric pressure (i.e. ’TopPg’)and precipitation (i.e. Precip).

As stated above, the time step in STEMMUS is adjusted automaticallyduring computing (1 to 1800 s). Accordingly, the time interval of themeteorological inputs need to be adjusted to match the new time step(1 to 1800 s). However, the recording interval of meteorological forcingdata is usually fixed (e.g. half of an hour). To match this requirementof flexible time steps, the Fourier transformation is employed to cap-ture the variation of meteorological variables. This is done outside of

Figure 5.3 Soil properties information block

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Figure 5.4 Measurement information block

Figure 5.5 Observation information input

STEMMUS. One note here is appropriate, as long as you can express themeteorological data continuously by using a function of time, unit ofwhich is second, it does not matter what kind of method you use to dothat (e.g. methods other than Fourier transformation). In the currentversion of STEMMUS, the meteorological data is reproduced by Fouriertransformation. The related parameters for this are shown in figure 5.7.

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5.4.5 Boundary Conditions

The surface and bottom boundary conditions can be grouped accordingto different governing equations. For soil moisture balance equation,the surface boundary conditions (i.e. ’NBCh’) include specified matrichead (i.e. ’NBCh’=1), specified potential moisture flux (i.e. ’NBCh’=2) andspecified atmospheric forcing (i.e. ’NBCh’=2), while the bottom boundaryconditions (i.e. ’NBChB’) include specified matric head (i.e. ’NBChB’=1),specified moisture flux(i.e. ’NBChB’=2) and zero matric head gradient(or gravity drainage, i.e. ’NBChB’=3). For dry air equation (e.g. ’NBCP’and ’NBCPB’) and energy equation (e.g. ’NBCT’ and ’NBCTB’), we can findsimilar description of the surface and bottom boundary conditions. Allthe boundary conditions are implemented by subroutines with an affixof “_BC”, for example “h_BC.m”, “Enrgy_BC.m” or “Air_BC.m”. Figure 5.8shows how the code for setting boundary conditions looks like.

For the example shown in Zeng and coauthors’ work [3, 5], the surfaceboundary condition for soil moisture balance equation (’NBCh’) is set as3 for taking meteorological data into account, while the bottom condition(’NBChB’) is set as gravity drainage. For dry air equation, the NBCP is setas 3 to use the measured atmospheric pressure as the surface boundarycondition, and the NBCPB is set as 2 to use the specified soil air pressureat the bottom. For energy equation, the NBCT is set as 1 to adopt themeasure surface temperature as the driving force for heat transport, and

Figure 5.6 Automatic interpolation of observation information

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Figure 5.7 Fourier transformation parameters for meteorological forcing data

NBCTB is set as 1 to assume a zero temperature gradient at the bottom.There is a note for NBCh though, in the subroutine of “h_BC.m” the

precipitation is evaluated by using a fixed amount of rainfall, whichis evenly distributed during a certain period. For the case in Zeng etal.’s study, a fixed amount of rainfall (’Precip(KT)=0.35/3600’,cm s−1)

Figure 5.8 Boundary conditions setting parameters

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5.5. Bibliography

is used for the period spanning from 22nd hour to 27th hour (i.e. timereleased since the simulation started). This is in total about 1.2 cmprecipitation for the recorded event. However, in reality, the precipitationrecorded in the field is less and unevenly distributed during this spanof time. The adjustment of the precipitation (e.g. in a certain range)sometime is needed for getting behavioral performance (e.g. not only theprecipitation, but also other parameters like XERR, TERR and so on) ofthe STEMMUS model in representing reality. This is needed because theelimination of all the measurement error from device we used in the fieldis impossible if any. For example, the placement of the device is not at theexact depth as the STEMMUS describes numerically and the measurementerror of the device itself. Although the measurement error of the devicecan be determined in the laboratory, it can not be eliminated completely.We do can quantify these uncertainties by running the STEMMUS usingdifferent patterns of driving forces (e.g. by adopting data assimilationtechnique).

5.4.6 Running STEMMUS

After all these steps, we now can run STEMMUS by running the mainroutine, “MainLoop.m”, which links all subroutines in sequence to solvethe governing equations as figure 5.1 shows. There were several mannersto run the “MainLoop.m”:

1. Push the green “run” button after opening “MainLoop.m” in MAT-LAB;

2. Click “F5” after opening “MainLoop.m” in MATLAB;

3. Click “F9” without opening “MainLoop.m” in MATLAB, but withselecting “MainLoop.m” in the ’current folder’ navigator in MATLAB.

All are convenient normal commands used in MATLAB to run a ’*.m’script.

5.5 Bibliography

[1] P. J. Camillo and R. J. Gurney. Resistance parameter for bare-soilevaporation models. Soil Science SOSCAK, 141(2), 1986.

[2] Adriaan A. van de Griend and M. Owe. Bare soil surface resistanceto evaporation by vapor diffusion under semiarid conditions. WaterResources Research, 30(2):181–188, 1994.

[3] Y. Zeng, Z. Su, L. Wan, and J. Wen. A simulation analysis of advectiveeffect on evaporation using a two-phase heat and mass flow model.Water Resources Research, 47:doi:10.1029/2011WR010701, 2011.

[4] P. C. D. Milly and P. S. Eagleson. The coupled transport of water andheat in a vertical soil column under atmospheric excitation. PhD thesis,1980.

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[5] Y. Zeng, Zhongbo Su, Li Wan, and Jun Wen. Numerical analysis ofair-water-heat flow in the unsaturated soil is it necessary to considerair flow in land surface models. Journal of Geophysical Research-Atmospheres, 116:doi:10.1029/2011JD015835, 2011.

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universiteit twente · contents contents i 1 introduction 1 1.1 motivation . . . . . . . . . . . ....

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