modeling of soil deformation and water flow in a swelling soil

Ž .Geoderma 92 1999 217–238

Modeling of soil deformation and water flow in aswelling soil

D.J. Kim a,), R. Angulo Jaramillo b, M. Vauclin b, J. Feyen c,S.I. Choi d

a Department of Earth and EnÕironmental Sciences, Faculty of Science, Korea UniÕersity,Anam Dong 5-1, Sungbuk Ku, Seoul, South Korea

b Laboratoire d’Etude des Transferts en Hydrologie et EnÕironment CNRS URA 1512, INP6. UJF,BP53, 38041 Grenoble Cedex 9, France

c Institute for Land and Water Management KUL, Vital Decosterstraat 102, LouÕain, Belgiumd Department of EnÕironmental Engineering, Faculty of Engineering, Kwangwoon UniÕersity,

Seoul, South Korea

Received 15 July 1998; accepted 26 April 1999

Abstract

Soil deformation and unsaturated transient water flow in swelling soil on a laboratory scale ispredicted using a one-dimensional numerical model. The model is based on a soil water flow

Ž .equation and extended to soil deformation using Lagrangian description LD . The specificfeatures of the model are inclusion of an overburden component in the total potential of the flow

Ž .equation, introduction of a shrinkage–swelling characteristic SSC known as a third soil hydraulicfunction, and two-dimensional analysis of soil deformation using a geometry factor. This paperdescribes an evaluation of the model, which was previously verified with a case of shrinkage ofmarine clay soil, with a data set from an infiltration experiment performed under swelling

Ž .conditions. Soil hydraulic properties such as moisture retention characteristic MRC , hydraulicŽ Ž ..conductivity function K h and SSC were derived from simultaneous measurement of volumetric

Ž .moisture content VMC , bulk density and pressure heads in a framework of the EulerianŽ . Ž .description ED . Two different K h relationships, with respect to moving solid particles, were

obtained from both the LD and ED. Simulation was performed using two different hydraulicŽ .conductivities, i.e., Eulerian descriptive hydraulic conductivity EDHC and Lagrangian descrip-

Ž .tive hydraulic conductivity LDHC . Model performance was evaluated using four differentstatistical parameters representing the goodness of model predictions on the transient variations ofsoil moisture contents. Results of the evaluation reveal that the model predicts reasonably well themoisture content changes, although the use of LDHC gives slightly better results than the EDHC.

) Corresponding author. Fax: q82-2-927-6180; E-mail: [email protected]

0016-7061r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0016-7061 99 00033-6

( )D.J. Kim et al.rGeoderma 92 1999 217–238218

Overprediction of moisture contents by the simulation model especially near the bottom of thesample is attributed to the incomplete dissipation of swelling pressure during expansion of the soilmaterial. As a result, the soil shows anisotropic swelling although it was forced to deform in theone-dimensional vertical direction by experimental design. Unlike other one-dimensional consoli-dation models, the model can be used for two-dimensional analysis of soil deformation in porousmedia with either self-weight or external load application, at either shrinking or swelling phaseand under both saturated and unsaturated flow conditions. q 1999 Elsevier Science B.V. All rightsreserved.

Keywords: swelling; overburden; geometry factor; moisture content; hydraulic conductivity

1. Introduction

Strictly speaking, the problem of soil deformation and water flow can beconsidered under various conditions according to the way it is imposed. Theconditions include the deformation phase of a porous medium, its dimensions,degree of saturation, zone of flow domain and imposition of internal or externalload. Agricultural practices often encounter problems related to two-dimensionalanalysis for shrinking and swelling of soil materials in an unsaturated zone withapplication of internal load or self-weight. Soil mechanical problems, however,are mostly one-dimensional deformation or consolidation under the shrinkingphase in a saturated zone with external load.

The first type of problem is usually solved using the governing equation thatrelies on the Richards’ equation extended to two-dimensional deformation.

Ž .Examples of such cases are documented by the work of Bronswijk 1988 andŽ .Kim et al. 1992a , where a geometry factor responsible for the deformation

geometry under either shrinking or swelling condition as well as shrinkage–Ž .swelling characteristic SSC was introduced. The latter type, however, is

handled by a one-dimensional consolidation theory that introduces the soilengineering properties most likely under saturated condition. Gibson et al.Ž .1967 reported a detailed work of the theory of one-dimensional consolidationof saturated clays. They proposed an improved equation of the classical consoli-

Ž .dation theory Terzaghi, 1925 by delimiting the conditions imposed on thestrains and variation of soil compressibility and permeability during consolida-tion, based on the LD. In fact, it has been proven that the governing equationaccounting for water flow and solid particle movement, derived using the

Ž .approach of Philip 1969 based on the overburden potential, is identical to theŽ .equation proposed by Gibson et al. 1967 since both of them are described in

the framework of Lagrangian formulation. More recent work of this type wasŽ .presented by Narasimhan and Witherspoon 1977 attempting to model one-di-

mensional vertical deformation of variably saturated heterogeneous porousŽ . Ž .media. De Smedt et al. 1992 presented a simulation model CONSOL for

one-dimensional consolidation phenomena based on the consolidation function

( )D.J. Kim et al.rGeoderma 92 1999 217–238 219

Ž .as well as moisture retention characteristic MRC and SSC of the soil material.Ž .The work of De Smedt et al. 1992 can be considered a hybrid of those types.

With respect to the formulation of the governing equation for soil deforma-tion and water flow, there exist two different approaches: Eulerian descriptionŽ . Ž .ED and Lagrangian description LD , depending upon the manner referring toan element in motion in a fluid system. The former focuses attention on fixedpoints of the element, whose dimensions do not change with time. The spatialcoordinates are independent variables and flow properties such as velocities andpressure heads become dependent variables. In the latter approach, the focus ison the history of individual particles in the element rather than the space. Thespatial coordinates of a moving particle are dependent variables as a function oftime. These coordinates are termed fluid, material or Lagrangian coordinates.Although both viewpoints are supposed to give the same result, the latter is

Ž .preferable when dealing with material front movement Bear, 1972 since in thelatter method, the coordinates of moving particles are represented as functions oftime and easily obtained.

Ž .Since the one-dimensional consolidation theory of Terzaghi 1925 , therehave been a number of attempts to model soil deformation and water flow, yetno practical model has been proposed that considers simultaneously soil defor-mation under saturated andror unsaturated conditions, shrinkage androrswelling, and self-weight andror external load with two-dimensional analysis ofsoil deformation in the framework of LD. A recent one-dimensional soildeformation study on the swelling case was performed by Angulo Jaramillo et

Ž .al. 1990a, b . Based on the measurement of water and solid particle movementin the framework of ED, they proposed a new method to determine the hydraulicconductivity of deformable porous media using both the approaches of ED andLD.

A numerical model, based on the governing equation developed by PhilipŽ .1968 and accounting for one-dimensional soil deformation and associated

Ž .water flow, was developed by Kim et al. 1992a . The model was evaluated withexperimental data obtained from laboratory soil columns under shrinking condi-tions. For two-dimensional analysis of soil deformation, a geometry factorŽ .Bronswijk, 1990; Kim et al., 1992b can be introduced to obtain vertical andhorizontal components of shrinkage and swelling from the SSC of a soil. Based

Ž .on the geometry factor, Kim et al. 1992c showed that the model could beapplied to the two-dimensional soil deformation system subject to a shrinkingcondition. The model retains additional merits compared to other one-dimen-sional consolidation models, such as simultaneous prediction of soil deformationand water flow under saturatedrunsaturated conditions and internalrexternalload application with two-dimensional analysis. However, it still lacks in theapplicability to a swelling soil. The primary target of this paper is, therefore, tostudy the potential use of the model to the swelling condition based on the

Ž .experimental data Angulo Jaramillo et al., 1990b . This study also highlights


implementation of a numerical model based on the LD with experimental dataobtained from the ED and the results of a simulation using the hydraulicconductivities obtained from both the ED and LD.

2. Theory

2.1. Eulerian approach

The process of transient fluid flow in a deformable porous medium can bedescribed in either ED or LD. The ED is based on a definite volume, fixed inspace, called ‘control volume’. The amount and identity of matter in the controlvolume may change with time, but the shape and position of this volume remainfixed.

Ž . Ž .Nakano et al. 1986 and Angulo Jaramillo 1989 used the ED for estimationof soil hydraulic conductivity in a deformable porous medium. The movementof fluid and solid particles was treated separately on the condition that the flowsystem was one-dimensional in the vertical direction, the fluid was homoge-neous and incompressible, and there was no restriction of air flow in the porousmedium. Given these assumptions, the continuity equation for each phase of theporous medium becomes:

Eu rEtsyEq rEz 1aŽ .w wr o

Eu rEtsyEq rEz 1bŽ .s s r o

Ž . w 3 y3 xwhere u and u are the volumetric moisture content VMC L L andw sŽ . w 3 y3 xvolumetric solid particles content VSPC L L , and q and q thewro sro

w y1 xvolumetric fluxes L T of water and solid particles relative to the referenceŽ . wpoint vertical axis z and positive downward . The pore water velocity V Lwro

y1 xT is associated with two components:

V sV qV 2aŽ .wr o wr s s r o

q sq qq u ru 2bŽ . Ž .wr o wr s s r o w s

where V and q are the relative pore water velocity and volumetric fluxwrs wrs

with respect to the moving solid particles. Each specific flux can be given byDarcy’s law on the assumption that transport of solid particles is described by a

Ž .phenomenological transport law Yong, 1973 :

q syK EC rEz syD Eu rEz 3aŽ . Ž . Ž .wr o wr o w wr o w

q syK EC rEz syD Eu rEz 3bŽ . Ž . Ž .wr s wr s w wr s w

q syK EC rEz syD Eu rEz 3cŽ . Ž . Ž .s r o s r o s s r o s


with

D sK EC rEu ,Ž .wr o wr o w w

D sK EC rEu and D sK EC rEu 3dŽ . Ž . Ž .wr s wr s w w s r o s r o s s

Ž .where K is the Eulerian descriptive hydraulic conductivity EDHC withwrsw y1 xrespect to the moving particles L T , and C , C are the matric potentials forw s

w xwater and solid phase l , and D , D the apparent capillary diffusivity ofwro srow 2 y1 xpore water and solid phase L T . Here, the potentials are defined as energy

w xper unit weight of water, equivalent to the hydraulic head L .The matric potentials C and C are related by:w s

EC rEzsyx EC rEz 4Ž . Ž .s w

with

0FxF1

where x is an empirical parameter equal to 1 for a saturated porous medium. Itis noted that C and C are analogous to the effective mechanical stress and thes w

Ž . Ž . Ž .pore water pressure, respectively. Substituting Eqs. 3a , 3b and 3c into Eq.Ž . Ž .2b , and using Eq. 4 gives:

K sK yK x u ru 5Ž . Ž .wr o wr s s r o w s

The ratio of the diffusivity D and D results in:sro wro

D rD sy K rK x Eu rEu 6Ž . Ž .Ž . Ž .s r o wr o s r o wr s w s

Ž . Ž .Combination of Eqs. 5 and 6 gives the following relationship:

K sK 1y u ru D rD Eu rEu 7Ž . Ž . Ž .Ž .wr s wr o w s s r o wr o s w

Ž . Ž . Ž . Ž .Introducing Eqs. 3a and 3c into Eq. 1a and 1b results in:

Eu rEtsErEz D Eu rEz 8aŽ . Ž .w wr o w

Eu rEtsErEz D Eu rEz 8bŽ . Ž .s s r o s

that, associated with the initial conditions and the specific limits of the consid-ered problem, describe simultaneously the movement of particles and that of

Ž .water with respect to the observation, which is fixed. Eq. 7 expresses theEDHC in terms of u , u , D , D and K , which can be determinedw s wro sro wro

experimentally.

2.2. Lagrangian approach

In the LD, the analysis is carried out with respect to a specified mass of fluid.This mass always remains unchanged, although the particle may change inshape, position or other properties as it moves. One way to approach the LD is


Ž .to introduce a material coordinate Philip, 1968; Smiles and Rosenthal, 1968 ,m, defined by:

y1dmrd zs 1qe 9Ž . Ž .Ž .where e is the void ratio y . Using the continuity equation given in the form

Ž . Ž .of Eq. 1a and 1b , we have:

EqrEzsyEu rEt 10Ž .w

Expressing u as a function of moisture ratio and void ratio:w

u sqr 1qe 11Ž . Ž .w

Ž . Ž . Ž . Ž .where q is the moisture ratio y . Substituting Eqs. 9 and 11 into 10 , andassuming that the variation of void ratio is negligible for a relatively small time,it follows that:

EqrEmsyEqrEt 12Ž .

Darcy’s law is given in the following form:

qsyK =F 13Ž .wr s

w xwhere F is total potential l , i.e., sum of capillary and gravitational compo-Ž .nents. According to Philip 1969 , for shrinking and swelling soils, an additional

component, namely overburden potential, is to be included in the total potential:

FscqzqV 14Ž .w xwhere V is the overburden potential l defined as the work performed per unit

weight of water added or extracted. This potential arising from the localmovement of solid particles upon addition or extraction of any quantity of waterto the porous medium is given by:

Vs derdq p z 15aŽ . Ž . Ž .

with

p z sp 0 y g d z 15bŽ . Ž . Ž .Hand

gs qr qr r 1qe 15cŽ . Ž . Ž .w s

Ž . w x Ž .where p z is the total vertical stress l at the depth of z, p 0 the external loadw x w y3 xl on the soil surface, g the apparent wet specific density m L of soil, and

w y3 xr , r the specific densities m L of water and particle. Assuming that rw s w


Ž . Ž .equals to 1 and substituting the flux q defined through Eqs. 13 to 15a, b, c ,Ž .the following general form Philip, 1969 is obtained:

2E K dC d e Eq E dewr syp 0 y K y1qgŽ . wr s2 ½½ 5Em 1qe dq dq Em Em dq

2 md e 1 dqy g 1qe dm yEqrEts0 16Ž . Ž .H2 5dq 1qe dm0

Ž .Kim et al. 1992a, c proposed the following simplified form in case of noexternal load:

E K dC Eq Eqwr sqSF qSF y s0 17aŽ .1 2ž /Em 1qe dm Em Et

with

deSF s 1qe y qqr rr 17bŽ . Ž . Ž .1 s w dq

and2 md e

SF s qqr rr dm 17cŽ . Ž .H2 s w2dq 0

Ž .where the term SF represents the components of gravitational 1qe and1Ž .overburden potential qqr rr derdq and SF a component of overburdens w 2

potential arising from the second-order derivatives of the SSC. It was provedfrom the numerical and experimental evidence that the external load applied onthe top boundary does not influence water movement inside the flow boundarysince the vertical stress applied on the top affects the soil matrix of each innernode equally during numerical solution. Rather, existence of the external load is

Ž .to be treated as a boundary value Dirichlet Condition . For details, refer to theŽ .work of Kim et al. 1992a .

Neglecting the contribution from gravitational and overburden potential andŽ .introducing capillary diffusivity, Eq. 17a reduces to:

Eq E dCs K 18aŽ .mž /Et Em dm

with

K sK r 1qe 18bŽ . Ž .m wr s

or

Eq E dqs D 18cŽ .mž /Et Em dm


with

D sK dC rdq 18dŽ . Ž .m m w

where K and D are the hydraulic conductivity and the capillary diffusivitym m

with respect to the moving solid particles in material coordinates. From therelationship between the void ratio and the VSPC:

u s1r 1qe 19Ž . Ž .s

Ž . Ž . Ž .Substitution of Eqs. 11 and 19 into Eq. 18d gives the following relationshipbetween the D and the K , i.e., Lagrangian descriptive hydraulic conductiv-m wrs

Ž .ity LDHC :

dC u duw w s2D s K u 1y . 20Ž .m wr s sž / ž /du u duw s w

Ž .Eq. 17a is a q-based equation that introduces a material coordinate, i.e.,Lagrangian system, and includes an overburden potential in the Darcy’s flow.This overburden potential is responsible for the movement of solid particles as aresult of water flow through the soil matrix under self-weight or external load.The advantage of this equation is that a more detailed analysis of soil deforma-tion can easily be made using the geometry factor. This factor enables one topredict two-dimensional soil deformation from the empirical relationship:

rsDV D H1y s 1y 21Ž .ž / ž /V H

where r is the geometry factor, and DV and D H the changes of volumetrics

shrinkage andror swelling of soil matrix, and V and H are the initial volumeŽ .and height. The soil deformation varies from unidimensional shrinkage r s1s

Ž . Ž .to equidimensional shrinkage r s3 Kim et al., 1992b .s

3. Materials and methods

3.1. Experimental

Two different sets of vertical infiltration experiments were performed byŽ . Ž .Angulo Jaramillo 1989 on a mixture of Bentonite 20% mass percentage and

Ž .loam, compacted in a glass cell 6 cm in diameter and height resting on aŽ .metallic support Fig. 1 . In this study, the first set of experiments was used for

calibration of the model parameters and the second set for evaluation of themodel. Described below is the first set of experiments, since both experimentswere conducted under identical conditions. The only difference between them isthe initial condition of the soil sample. The infiltration was realized with a

Žporous plate placed on the top of the soil sample in a relatively dry state suction


Fig. 1. Experimental set-up for monitoring swelling of a soil mixture during infiltration.

.head ranging from y250 to y350 cm and linked to a teflon support and amariotte device to constantly control the hydraulic head. Variation of the sampleheight, initially 3 cm, was automatically measured using a comparator. Two

Ž241 137 .radioactive sources Am and Cs emitting co-linear gamma rays, mountedon a mobile platform moving in 0.5-cm interval, were used to simultaneously

Ž .measure the spatial and temporal 3-h interval evolution of VMC, u , and dryw

bulk density, r , of the sample. Beer’s law was used to obtain those properties,d

describing the gamma ray attenuation in the soil material as a function of theVMC and dry bulk density. The VSPC, u , was obtained from:s

u z ,t sr z ,t rr . 22Ž . Ž . Ž .s d s

The SSC was constructed from the relationship between u and u . Onew s

micro-tensiometer cup was installed at the middle of the soil sample to measureŽ .changes of the pressure heads matric suction heads . Correction of the mea-

sured pressure heads was made by taking into account an additional pressurearising from the swelling of the soil sample as the following:

C sC yh 23aŽ .c m c


with

h sgrr j D z 23bŽ .c w

w xwhere c , c are the corrected and measured pressure heads l , h the swellingc m c

pressure per unit weight of water, j the slope of the SSC, and D z the distanceof the tensiometer cup from the soil surface. The gamma ray counting periodwas 180 s. The infiltration test was conducted for 550 h. Data acquisition wasunder computer control.

3.2. Determination of hydraulic conductiÕities

Ž .The EDHC, K , given by Eq. 7 was obtained from the K , D andwrs wro wroŽ .D . The apparent hydraulic conductivity K given by Eq. 3d was obtainedsro wro

from the MRC and the apparent capillary diffusivity D . D and D werewro wro sro( ) ( )obtained from the data of u z,t and u z,t . The LDHC was obtained from Eq.w s

Ž . Ž .20 with known D , which can be calculated for the initial condition q z,0 smŽ .q , and the Dirichlet boundary condition, q 0 ,t sq , imposed on one end of0 1

y1r2 Ž .the soil. Introducing the Boltzmann variable, lsmt , Eq. 18c can betransformed into an ordinary differential equation for which the integrationbetween q and q results in:0 1

1 dl q1D sy l dq . 24Ž .Hm 2 dq q0

3.3. Model description

The model for soil deformation and water flow in shrinking and swelling soilin this study is based on the one-dimensional model presented by Kim et al.Ž . Ž .1992a . According to Haines 1923 , three distinct shrinkage andror swelling

Ž . Ž .phases exist for a deformable soil: 1 normal shrinkage where DqsDe, 2Ž .residual shrinkage where Dq-De, and 3 zero shrinkage where e remains the

Ž .same but q still decreases. From the fact that the term SF Eq. 17c is zero2

during the normal shrinkage and swelling phase since d2erdq 2 becomesŽnegligible compared to SF during residual and zero shrinkage phases Kim et1

.al., 1992c , the governing equation used in the model has the following formŽ .reduced from Eq. 17a :

E K dC ECwr sqSF sC 25Ž .1ž /Em 1qe dm Et

where C is the differential moisture capacity equal to dqrdC . The solution ofŽ .Eq. 25 is obtained with a finite difference scheme by discretizing c in space

m and time t. Due to the strong non-linearity of the equation, an implicit finite


difference approximation relative to time with explicit linearization of C, K wrs

and SF is adopted. To solve the linear equations, either the Neumann condition1

or the Dirichlet condition can be applied to both the top and bottom boundaries.In order to enhance the accuracy of the solution, the Newton–Raphson iteration

Ž .technique Carnahan et al., 1969 was used. The input requirements of the modelare the soil hydraulic functions of SSC, MRC and hydraulic conductivity as wellas a geometry factor. In addition, information on initial conditions as well asboundary conditions are required. The boundary conditions used in this studyare the Dirichlet condition at the top and the Neumann condition at bottom,which were determined from the experimental design. The initial soil moistureprofile data were segmented into six compartments, each having an equalthickness of 0.5 cm.

4. Calibration of model parameters

4.1. Shrinkage-swelling characteristics

The parameters of the soil hydraulic functions were obtained from the first setof infiltration experiments, and an identical soil material was used for the secondset. Since the experimental data were obtained from the framework of the ED,and the numerical model was developed based on the LD, the data of soilhydraulic functions need to be transformed from ED to LD. Fig. 2 shows twotypes of SSCs. The first type represents the relationship between VMC and

Ž .volumetric solid particle content VSPC . The second is the transformed SSC

Fig. 2. Measured and transformed swelling characteristic data. Empty circles indicate the adjustedswelling characteristic in the domain of moisture and void ratio.


Ž . Ž .using Eqs. 11 and 19 in the expression of void ratio and moisture ratio. In thefirst type, the sum of VMC and VSPC exceeds 1, which is physically infeasible.This is attributed to the independent measurement of VMC and VSPC in thedual gamma ray technique, in which otherwise, the sum obtained would have

Ž .been unity Angulo Jaramillo, 1989 . The transformed SSC was then adjustedfor the model simulation so that it follows the saturation line as the void ratiocannot practically be less than the moisture ratio under any conditions.

4.2. Moisture retention characteristic

For deformable porous media, two different types of expression for moisturecontent exist since the volume of soil changes according to the variation in

Ž .moisture content Kim et al., 1992d . One is an actual volumetric moistureŽ .content AVMC taking into account changes of the soil volume as the soil

deforms upon drying and wetting. This expression gives the MRC in theframework of ED, an approach in which focus is fixed on the relative amount ofsolid particles for a given volume, i.e., equivalent to, ‘‘What’s inside a control

Ž .volume?’’ The other is a fictitious volumetric moisture content FVMC , inwhich moisture content is calculated with reference to an initial soil volume.The FVMC is equivalent to the expression of a moisture ratio which is alsoobtained with reference to the fixed amount of soil particles. These expressionsrefer to the LD since evolution of the fixed volume of solid particles is focusedon, i.e., equivalent to, ‘‘How do the given particles move along the pathway?’’The experimentally determined MRC relating the matric suction heads to theAVMC and the transformed MRC expressing the suction heads in terms of

Ž . Ž .either FVMC or moisture ratio are shown in Fig. 3 a and b . Conversion of theAVMC to either FVMC or moisture ratio is given by:

q su r 1yu 26aŽ . Ž .i wi wi

qsu 1qq 26bŽ . Ž .wa i

u sqr 1qq 26cŽ . Ž .wf i

where u , q are the initial AVMC and moisture ratio, and u , u the AVMCwi i wa wf

and the FVMC. The transformed MRC data expressed in terms of either FVMCŽ .or moisture ratio are fitted to the van Genuchten model van Genuchten, 1980

with five parameters:MN

Ses1r 1q aC 27aŽ . Ž .w

with

Ses QyQ r Q yQ 27bŽ . Ž . Ž .r s r

Ž .where a , N and M 1y1rN are the parameters that determine the shape ofthe MRC, c the matric suction head, and Q can either be the FVMC orw


Ž . Ž .Fig. 3. a Measured MRC data with AVMC in ED and with FVMC in LD and b transformedand fitted MRC with moisture ratio in LD.

moisture ratio, and subscripts r and s refer to residual and saturated moisturecontent, respectively. The parameters Q , a and N, Q fixed as zero, weres r

Ž .obtained using the RETC code van Genuchten et al., 1991 , which utilizes theŽ .algorithm of Marquardt’s maximum neighborhood method Marquardt, 1963 .

4.3. Hydraulic conductiÕity function

The unsaturated hydraulic conductivities, EDHC and LDHC, obtained usingŽ . Ž . Ž .Eqs. 7 and 20 are shown in Fig. 4 a . The LDHC is slightly lower than the

EDHC due to neglecting the contribution of the overburden potential in Eq.Ž .18a . Transformation of the experimental data expressed in terms of AVMC to

Ž .the data in matric suction head was made using the MRC presented in Fig. 3 a .Ž Ž ..The transformed data points Fig. 4 b are fitted with the combined form of the


Ž . Ž .Fig. 4. a Measured hydraulic conductivity data using both the ED and LD and b transformedand fitted hydraulic conductivity data.

Ž .moisture retention model van Genuchten, 1980 and the Mualem model ofŽ .hydraulic conductivity Mualem, 1976 :

2ML 1r MK Se sK Se 1y 1ySe 28Ž . Ž . Ž .s

w y1 xwhere K is the saturated hydraulic conductivity L T and L, M thes

parameters determining the shape of the curve.

4.4. Geometry factor

Calibration of the geometry factor was performed by comparing measure-ments on vertical soil deformation with simulation results obtained for various


Fig. 5. Comparison of simulation results with observations on the sample height using variousŽ . Ž .geometry factors for a EDHC and b LDHC.

values of the geometry factor using previously described soil hydraulic functionsas model input. Fig. 5 shows the comparison between measured and simulatedresults of the soil surface movements with time. The simulation using EDHCŽ Ž ..Fig. 5 a slightly overpredicts the soil surface changes when compared to the

Ž Ž ..LDHC Fig. 5 b . This is due to the higher hydraulic conductivity in the EDthan in the LD. In either case, the simulations approximate the measurementswhen the geometry factor ranges from 1.5 to 2.0. Since this soil parameter isunique for a given soil system and independent of the approach used inmathematical description, the low limit of the range was selected by taking intoaccount that the soil was allowed to deform only in the vertical direction byexperimental design.


5. Evaluation of the model

5.1. Soil deformation

Simulation results of soil deformation and soil moisture content profiles wereobtained using different hydraulic conductivity data, i.e., EDHC and LDHC fora given set of initial and boundary conditions and other input soil hydraulicfunctions including the geometry factor. These results were compared with themeasurements obtained from the second set of infiltration experiments. Soildeformation as a result of water flow during swelling of the soil mixture wereobtained in terms of changes in the height of the soil sample as the glass cellrestricted the horizontal deformation. Fig. 6 shows the simulated and measuredresults of soil deformation in the vertical direction when the geometry factor was1.5. The simulation using EDHC seems to predict the measurement better thanthat obtained using LDHC. Both simulation cases predict well the verticaldeformation in the earlier stages but rather poorly in the later. The poorpredictions can be explained by the fact that the measurement of soil surfacechange is affected by the development of a dispersed soil layer at the top due tothe ponding condition, especially over a large time. This phenomenon, however,is not incorporated in the numerical model. Nevertheless, the model predictsreasonably well the soil deformation when taking into account the maximumrelative error of 25% for the last observation.

A notable feature of the soil deformation is that the soil given in theŽ .experimental design does not follow a unidimensional swelling r s1.0 al-s

though it was cast into such a condition. Supposing that a natural condition atwhich the soil can expand three-dimensionally is imposed, the soil would follow

Fig. 6. Prediction of vertical soil deformation using EDHC and LDHC.


Ž . Ž .an equidimensional swelling r s3.0 Bronswijk, 1990 . This means that anysŽ .deformable soil except for ripening soil under neither lateral restriction nor

vertical external load would show an equal expansion capacity in not only thevertical but also the horizontal direction. Given the experimental system used inthis study, upon infiltration, the soil would tend to expand equally in both thevertical and horizontal directions. However, due to lateral restriction, the soilcan expand only upward. This process will move from top to bottom. However,the propagation of this process will proceed less smoothly at the bottom than atthe top layer. The reason lies in the fact that the soil materials near the bottom

Ž .are confined three-dimensionally two lateral sides and bottom side . In addition,there exists overburden pressure arising from the weight of soil materials above.As a result, there will be development of additional swelling energy in the lowerregion of the soil due to incomplete swelling energy dissipation in the horizontaldirection. In fact, this swelling energy will be dissipated by upward movementof the soil material but much more slowly compared to the natural condition, inwhich there is no contribution of horizontal swelling to upward movement. Thespeed and degree of upward movement will largely depend on the energybalance between overburden load and additional swelling energy. These phe-nomena will be more pronounced near the bottom as the overburden loadincreases with depth. The overburden pressure taking place here has no mathe-matical connection with the one used in the description of water flow, sinceanalysis of soil deformation in the model is purely empirical-based and furtherexecuted after the computation of water flow. Full mathematical description ofwater flow and subsequent soil deformation will require a three-dimensionalapproach using a tensor to account for the energy balance in the mass flowdomain.

5.2. Water flow

Measured and simulated results of transient AVMCs are shown in Fig. 7. ForŽ Ž ..measured data Fig. 7 a , each data point corresponds to the middle of each soil

compartment and is extended to the top boundary as the sample swells.Ž . Ž .Simulation results of soil moisture content profile are shown in Fig. 7 b and c

for the use of EDHC and LDHC, respectively. The position of nodal points usedfor the numerical computation, corresponding to the middle of each compart-ment, is shown in the data points of day zero. The height of each nodal pointshifts upward as the sample swells. This is typical of the expression of aLagrangian system since the spatial coordinates are dependent variables. Use ofthe EDHC predicts higher moisture contents than the LDHC especially at thelower region of the sample. This is due to overestimation of the hydraulic

Ž .conductivity Fig. 4 resulting in the increase of downward water flow duringinfiltration. Although the LDHC predicts the moisture content profile better thanthe EDHC within the period of 8.33 days, it still overpredicts the moisture


Ž .Fig. 7. Measured soil moisture profile data of a AVMC, and computed moisture profile dataŽ . Ž .using b EDHC and c LDHC.

content especially near the bottom over large times. The overprediction is due tothe fact that the computation of water flow in the model is based on theone-dimensional vertical system that depends on only top and bottom boundaryconditions whereas the actual water flow in the soil sample is associated withthe additional potential component arising from the horizontal swelling energy.This additional component may be the main source for the relatively low valuesof measured moisture contents, thus, exhibiting differences in moisture contentsat the top and bottom of the soil due to the aforementioned swelling energydissipation. On the other hand, the measured moisture contents at the soilsurface are higher than those simulated. The main causes can be explained byboth the effects of dispersion of soil particles at the top layer and the lowestimate of saturated moisture content in the MRC function. As can be seen inFig. 3, the measured saturated moisture content is equal to 0.81, correspondingto the zero pressure head, but the saturated moisture content at the soil surface isobserved 0.89 at the end of the experiment.

5.3. Model eÕaluation

Since in both ED and LD, moisture content changes in any position and timeare directly related to changes in the density of soil materials, the moisturecontent data were used to evaluate the simulation model. A scatterplot con-


Fig. 8. Scatterplot of measured and simulated results of soil moisture content data for EDHC andLDHC.

structed for the computed and observed moisture content profile data is shown inFig. 8. In order to eliminate a possible confusion coming from the mixture ofdata obtained from temporal and spatial intervals, the measured and simulatedmoisture contents at different depths are compared for each monitoring time.

Table 1Results of statistical analysis on the comparison between measurements and simulations ofmoisture contentNote:

1r2n2RMSEs Siy Mi rn 100MŽ . Ž .Ý

is1

n n2 2

CDs Miy M Siy MŽ . Ž .Ý Ýis1 is1

n n n2 22EFs Miy M y Siy Mi Miy MŽ . Ž . Ž .Ý Ý Ý

is1 is1 is1

n n n

CRMs Miy Si MiÝ Ý Ýis1 is1 is1

where Si, S: Simulation data point and mean of simulation, and Mi, M: measurement data pointand mean of measurement.

RMSE CD EF CRM

LD 2.156 0.989 0.940 0.005ED 3.075 0.870 0.878 y0.044


Use of the EDHC overestimates the moisture contents whereas the LDHC showsa reasonable match with the measured data.

In order to quantify the appropriateness of model prediction, four differentstatistical criteria have been used. The results of statistical analysis on themeasurements versus simulations of moisture content changes are listed in Table

Ž . Ž .1. These are root mean square error RMSE , coefficient of determination CD ,Ž . Ž .modeling efficiency EF and coefficient of residual mass CRM used to

Ž .evaluate the quality of a simulation model Loague and Green, 1991 . TheRMSE value indicating the degree of deviation between measurements andsimulations, expressed as a percentage of the average of the measurements,shows 2% and 3% for each case of LDHC and EDHC. High values wereobtained for both CD and EF statistics. The CD describes the ratio between thescatter of measurements and the scatter of simulations. The EF indicates whetherthe simulations provide a better estimate of measurements than the averagevalue of the measurements. The CRM value indicates whether simulations tendto overestimate or to underestimate. A negative value indicates a tendency tooverpredict. This is the case for the simulation results derived using the EDHCand was clearly demonstrated in Fig. 8. From the results of statistical analysis, itappears that the model predicts the moisture content profile reasonably well.

6. Conclusions

Simultaneous prediction of soil deformation and unsaturated water flow in aswelling porous medium has been conducted using the numerical model basedon the q-type equation that is an extended version of the soil water flowequation. The numerical model described in the LD appears to be promising,although the simulation slightly overpredicts the moisture contents in the lowerregion of the soil sample and underpredicts at the soil surface especially overlarge times. Analysis of soil deformation reveals that anisotropic soil deforma-tion occurs with the geometry factor higher than 1.0, although the soil samplewas restricted to deform in the vertical direction only. Two different types of thesoil hydraulic conductivity do not seem to significantly affect the modelpredictability, although use of the EDHC gives slightly better results for soildeformation and worse for moisture content profile than the LDHC. Moreinvestigations will be required on swelling of a soil material in an unrestrictedway in order to verify the geometry factor in connection with the swellingenergy dissipation. In the present context of the developed numerical model, amechanistic description of soil deformation is not incorporated in the governingequation. Rather, the computation of soil deformation utilizes the empiricalparameter and is executed after each computational time step based on theresults of soil moisture content. Since the phenomena of soil deformation andwater flow occur in a three-dimensional mode, a mathematical description of the


governing equation in a two- or three-dimensional flow domain will be essentialto simulate the exact reality.

Acknowledgements

The author appreciates the financial support of Center for Mineral ResourcesŽ .Research CMR of Korea University to conduct this study.

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