a continuum-discrete model using darcy's law: formulation and verification

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2014)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2319

A continuum-discrete model using Darcy’s law:formulation and verification

Majid Goodarzi*,†, Chung Yee Kwok and Leslie George Tham

Department of Civil Engineering, University of Hong Kong, Hong Kong

SUMMARY

This paper presents a numerical scheme for fluid-particle coupled discrete element method (DEM), which isbased on poro-elasticity. The motion of the particles is resolved by means of DEM. While within the prop-osition of Darcian regime, the fluid is assumed as a continuum phase on a Eulerian mesh, and the continuityequation on the fluid mesh for a compressible fluid is solved using the FEM. Analytical solutions of tradi-tional soil mechanics examples, such as the isotropic compression and one-dimensional upward seepageflow, were used to validate the proposed algorithm quantitatively. The numerical results showed very goodagreement with the analytical solutions, which show the correctness of this algorithm. Sensitivity studies onthe effect of some influential factors of the coupling scheme such as pore fluid bulk modulus, volumetricstrain calculation, and fluid mesh size were performed to display the accuracy, efficiency, and robustnessof the numerical algorithm. It is revealed that the pore fluid bulk modulus is a critical parameter that canaffect the accuracy of the results. Because of the iterative coupling scheme of these algorithms, high value offluid bulk modulus can result in instability and consequently reduction in the maximum possible time-step. Fur-thermore, the increase of the fluid mesh size reduces the accuracy of the calculated pore pressure. This studyenhances our current understanding of the capacity of fluid-particle coupled DEM to simulate the mechanicalbehavior of saturated granular materials. Copyright © 2014 John Wiley & Sons, Ltd.

Received 24 June 2013; Revised 15 March 2014; Accepted 3 July 2014

KEY WORDS: discrete element method; FEM; fluid-coupled simulation; Darcy’s law

1. INTRODUCTION

Discrete element method (DEM) proposed by Cundall and Struck [1] has been widely used to studygranular materials in a variety of engineering applications. Outstanding successes achieved by usingthis approach have made it one of the major research tools in computational mechanics. Thesimplicity of the motion equations and the input data along with its capability of simulating complexbehaviors of particulate systems is considered as the main advantages of DEM over other methods.Most published DEM-related research in geomechanics has concerned the simulations of drygranular material and outstanding achievements obtained through micro-mechanical studies. Despiteits popular use in simulating dry granular materials, it is much awaited to extend the conventionalDEM to study fluid-particle interactions. Important applications of fluid-particle DEM in engineeringfields include saturated soil mechanics, sand production problems in petroleum engineering,fluidized bed-related operations in chemical engineering, and sedimentation and segregation ofparticles in mineral processing. Three approaches have been proposed for incorporating fluid effectinto DEM simulations, which are the constant volume method, sub-particle-scale method, andcoarse-grid method [2].

*Correspondence to: M. Goodarzi, Department of Civil Engineering, Haking Wang Building University of Hong Kong,Pokfulam Road, Hong Kong.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

M. GOODARZI, C. Y. KWOK AND L. G. THAM

Constant volume approach was widely adopted for simulating undrained biaxial and triaxial tests onfully saturated soil samples based on the assumption of incompressibility of fluid and soil grains. Thereis no computational effort for the fluid phase, and the undrained effect is caused by keeping the volumeof the sample constant during shearing. The procedure is simple, and when the top and bottom platensare being moved, the lateral boundaries will be moved in such a way that the total volume of thesample is kept constant (Figure 1(a)). The pore pressure of the sample can be obtained from thedifference between the current confining stress and the initial confining stress. This approach islimited to the simulation of undrained shear tests, and it has been used by researchers to simulateundrained monotonic and cyclic loading behavior of fully saturated soil [3–6].

Sub-particle-scale method can be considered as the most precise way for fluid-coupled simulation ofparticulate media. In this approach, the fluid flow is explicitly considered inside pores, and all particlesare assumed as the boundary for fluid flow through pores. The fluid flow can be simulated with variousmethods such as solving Navier–Stokes equations on a Eulerian mesh with sub-particle resolution(Figure 1(b)) [7], Lattice–Boltzmann [8, 9], and smooth particle hydrodynamic [10]. As thesemethods attempt to simulate fluid particles system as close as possible to the real conditions, theycan be used for highly fundamental studies; however, the computational demand is extremely high,which would limit the size of simulations.

Coarse-grid method is formulated to overcome the high computational efforts of the sub-particle-scale approach by using larger cells that encompass several particles in one cell. In fact, the fluidflow through individual pores is not considered directly. The velocities and pressures of severalpores are averaged and assigned to a fluid cell that encircles those pores (Figure 1(c)). This method wasfirst proposed by Tsuji et al. [11] to simulate fluidized bed. The averaged form of Navier–Stokesequations proposed by Anderson and Jackson [12] was solved on a Eulerian mesh, and the effect ofparticle movement on those fluid cells is taken into account by calculating the change of the porosity.Kafui et al. [13] followed this idea in fluidized bed simulations, and Shimizu [14] added the heattransfer equation and improved it to thermal-fluid scheme. Zeghal and El Shamy [15, 16] implementedthis approach for geomechanical applications. They studied the liquefaction process of cemented anduncemented soil deposits. In addition, some geotechnical aspects of simulation were investigated intheir research. For example, because the permeability is not an input data for averaged Navier–Stokesequations, the numerical permeability obtained by this method was compared with the availablelaboratory data. For the specific sands that they considered, the errors were as much as 10–20%. Thiserror is predictable as porosity is the only soil parameter in the averaged form of the Navier–Stokesequation, and both pressure and fluid velocity are unknowns. This may cause some errors becauseexperimental investigations have revealed that permeability is not only dependent on porosity but alsoon particle size distribution [17]. They also looked at the possibility of turbulent flow during dynamicexcitement in their models, and it was revealed that the Reynolds’s numbers were significantly lowerthan the Darcy’s limit, which shows that the assumption of Darcy’s regime in geotechnical engineeringapplication is not far from the reality.

Figure 1. Different algorithms for adding fluid into DEM simulation: (a) constant volume, (b) sub-particleresolution method, and (c) coarse-grid method.

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2014)DOI: 10.1002/nag

A CONTINUUM-DISCRETE MODEL USING DARCY’S LAW

To simplify the calculation of fluid flow, some researchers accept the validity of Darcy’s law andignore the momentum equation. Hakuno and Tarumi [18] developed a method to study liquefactionbased on detecting all the pores and connecting them by some pipes. The fluid flow between eachpore and its adjacent pores was based on Darcy’s law; this idea later is used by Shimizu [19].Chareyre et al. [20] and Catalano et al. [21] followed the same methodology and developed ahydromechanical pore-scale model, which is quantitatively in agreement with classical problemssuch as the oedometer test. In their method, a fluid mesh is generated by connecting the pores usingthe Delaunay triangulation. The fluid flow equation is solved on the mesh in pore-scale using finitevolume method. Jensen and Preece [22] avoided pore pressure calculation in pore scale and usedlarger triangular cells at steady state condition to find the forces acting on grains around an oil welland simulated sand production process. This method was a one-way coupling scheme that can onlyconsider the effect of fluid on particles. Nakasa et al. [23] proposed an algorithm as a combinationof these two ideas [11, 18]. They implemented square-cell elements that could have around 15particles, and the flow fluid between these cells was based on Darcy’s law. The pore pressuregeneration due to external forces in each cell corresponded to particle movement in the neighboringcells. Finite difference scheme was applied to solve the flow equation. Shafipour and Soroush[5] compared the results of this algorithm with a constant volume method for a biaxial sheartest on a soil sample. Although this coupling scheme has been recognized as a nonphysicalidea in the study of O’Sullivan [2], they obtained satisfying compatibility between the twoapproaches. Koyama et al. [24] used discontinuous deformation analysis with a simple coupledalgorithm similar to the study of Jensen and Preece [22], and they solved the continuityequation with FEM for steady state condition, but their method does not take into account theeffect of particle movement on fluid.

Following the idea of previous researches [5, 23], in this paper, a new description for iterative fluid-coupled DEM using Darcy’s law is presented based on the poro-elasticity theory. The continuityequation for unsteady state flow of a compressible fluid is discretized using FEM. Qualitativevalidations in the previous studies have not revealed the accuracy and efficiency or any possibledrawbacks in their formulations. Unlike the previous studies, the results of this continuum-discretealgorithm are compared with well-known analytical solutions in geomechanics, isotropiccompression, and 1D upward seepage flow, to evaluate its accuracy and to validate the couplingscheme. Through these quantitative validations, different aspects of this algorithm are investigated,and sensitivity analyses are performed to reveal the effect of fluid bulk modulus, fluid mesh size,and volumetric strain calculation method. The results are used to make useful suggestions forefficient fluid-coupled DEM simulations.

2. FORMULATION OF THE CONTINUUM-DISCRETE MODEL

2.1. Theory

A realistic model to simulate saturated soil should consider both the solid and fluid phase and theirinteractions. If the equations of fluid and solid are solved simultaneously, it is called fully coupledalgorithm. Contrarily, when the soil and flow equations are solved separately for each time-step andthe information is exchanged between them, it is called partially coupled approach. Such methodsare further divided into two categories, explicit and iterative, which depend on whether theequations are performed once per each time-step or several times until the convergence of soil orflow unknowns, respectively [25]. Partially coupled approaches are less precise but more flexible;they can implement new developments in both solid and fluid simulations.

The different nature between the DEM and the FEM leads to the partially coupled scheme. In thisstudy, a continuum-discrete model is assumed for the saturated soil. The soil grains are representedby circular disks, and the fluid flow is considered as a continuum system on a Eulerian grid. In eachcell, variables such as fluid velocity, pressure, and density are locally averaged quantities. The flowbetween cells is considered as laminar so that the use of Darcy’s law is valid. The procedure of porepressure generation and dissipation can be summarized as follows.



Particles are displaced by forces originated from external loading or fluid pressure. This movementleads to pore volume change in fluid cells, which will be followed by excess pore pressure generation.The fluid flows because of the differential pore pressure between cells, and the pore pressure will beredistributed in the domain. In reality, these steps are a simultaneous procedure that can besummarized in a simple equation based on the poro-elastic theory as follows [26]:

ΔVf

V¼ n

ΔVp

Vpþ ΔPKef f

� �(1)

where ΔP is the change in pore pressure, n is the porosity, ΔVf is the change of the volume of fluid dueto flow in or out, ΔVp is the change in void volume, Vp is the void volume, V is the total volume of thesolid–fluid system, and Keff is the effective bulk modulus of the mixture. It should be noted that forincompressible solid particles, the volume change of the system is equal to the volume change ofthe pores, so the volumetric strain (εv) is

εv ¼ nΔVp=Vp (2)

Moreover, with this assumption, the effective bulk modulus of the mixture will change to effectivebulk modulus of pore fluid, which can be obtained as following:

1Kef f

¼ ns

Kfþ n 1� sð Þ

Ka(3)

where s is the degree of saturation and Kf and Ka are the fluid and air bulk modules, respectively.Although the overall behavior of soil may not be elastic, it can be assumed that the small

movement in each time-step is an elastic deformation. In the continuum-discrete model, thiscoupled procedure in each time-step will be divided into two steps, the effect of solid on fluid andthe effect of fluid on solid. This algorithm is similar to partially coupled approach in which solidand fluid equations are solved separately.

2.2. Solid phase

Considering partially coupled system for each time-step, the particles move in one time-step (dynamicsimulation), or they move until they reach equilibrium (quasi-static simulation), while the amount offluid is constant in each fluid cell. This can be considered as the pore pressure change because ofvolumetric strain in undrained condition, which implies that ΔVf= 0; therefore, the change in porepressure, using Eq. (1), will be

ΔP ¼ Kef f

nεv (4)

The movement of particles is treated by DEM. The calculations are simple in which the Newton’ssecond law of motion is considered for each particle for both translational (v•p ) and rotational (ω• p )accelerations as in the following text:

mpv•p ¼

Xc

f Nc þ f f þ f g (5)

Ipω•p ¼ rp

Xc

f Sc (6)

where f Nc and f Sc are the normal and shear contact forces acting on a particle by other particles or thewall, and rp is the particle radius, ff is the particle–fluid interaction force acting on the particle,which includes both buoyancy force and drag force in the current case, and fg is the gravitational force.

Contact forces are generated between two particles because of their relative movement. Differentcontact models have been proposed for the relationship between these shear and normal relative



movements and the corresponding forces. In this study, linear elastic contact model is implemented.These relationships can be summarized as follows:

df n ¼ kndun þ cndvn (7)

df s ¼ ksdus þ csdvs (8)

in which fn is the normal contact force, un and vn are the relative displacement and velocity of twoparticles along the line that connect their centers, fs is the shear contact force, us and vs are therelative displacement and velocity of two particles perpendicular to the line that connect theircenters, kn and ks are the normal and shear contact stiffness, and cn and cs are the normal and shearviscous damping coefficients. More details about this method can be found in the study ofO’Sullivan [2].

To calculate volumetric strain for fluid cells, two methods can be implemented. One way is to useEq. (2) and calculate the exact change in pore volume inside each cell. This can be carried out bycalculating the exact volume of the particles inside each fluid cell, which might be very time-consuming for a big model [27]. It is also possible to use an approximation method to obtain strainrate or strain tensor in a discrete system as described in the following text [28].

Strain rate and strain are continuum concepts that can be directly calculated in continuum methodsusing displacement function. On the other hand, in a discrete system, the particle displacements andvelocities are not continuum values. To obtain the strain-rate tensor for a particulate system, aselected area, such as inside a fluid cell, is chosen, and a continuum function is assumed for thevelocity of each point in that area. If U′ is considered as the velocity function of the area, with NP

particles inside it, which can relate velocity to position for each point, its variation form can bewritten as

dU′i ¼ U′

i; j dxi ¼ α′ijdxj (9)

where α′ij is called velocity-gradient tensor or strain-rate tensor. The volumetric strain rate and volumetricstrain can be obtained by Eq. (10) provided that a suitable strain-rate tensor can be derived.

Ui ¼ U′idt so εv ¼ Ui;i ¼ U′

i;idt ¼ α′11 þ α′22� �

dt (10)

The components of strain-rate tensor can be computed through the following procedure.For NP particles in the desired region, the mean velocity and position are

U′i ¼

XNP

U′ pð Þi

NPand xi ¼

XNP

x pð Þi

NP(11)

where U′ pð Þi and x pð Þ

i are the translational velocities and centroid positions of the NP particles,respectively, and the measured relative velocities for each particle is

eU′ pð Þi ¼ U′ pð Þ

i � Ui′

(12)

For an assumed α′ij, the relative velocities at the centroid location of the particles can be predicted as

eu′ pð Þi ¼ α′ijex pð Þ

j (13)

ex pð Þi ¼ x pð Þ

i � xi (14)

where eu′ pð Þi is the predicted relative velocity and ex pð Þ

i is the relative position vector.



The least-squares method is implemented to find a strain-rate tensor that can minimize the sum ofthe errors between predicted and measured values for the NP points (Eqs. (15) and (16)).

z ¼XNP

eu′ pð Þi � eU′ pð Þ

i

�� 2 (15)

∂z∂α′ij

¼ 0 (16)

By substituting Eqs. (13) and (15) into Eq. (16) and differentiating, the following set of linear equationswill be obtained.

XNP

ex pð Þ1 ex pð Þ

1

XNP


1XNP


2

XNP


2

26643775 α′i1

α′i2

( )¼

XNP

eU′ pð Þi ex pð Þ

1

XNP

eU′ pð Þi ex pð Þ

2

8>>>><>>>>:

9>>>>=>>>>; (17)

The four components of the strain-rate tensor will be obtained by solving this set of equations.

2.3. Fluid phase

In the second step, the solid phase is fixed, and the fluid is exchanged between cells because ofdifferential pressure. This causes redistribution of pore pressure in the whole system under constantvolumetric strain (εv = nΔVp/Vp = 0). Hence, from Eq. (1), we have

n

Kef fΔP ¼ ΔVf

V(18)

Using this equation and assuming that the fluid exchange between cells is under Darcy’s regime, thecontinuity equation for a slightly compressible fluid can be written as

n

Kef f

∂P∂t

¼ ∂∂x

kxγw

∂P∂x

� �þ ∂∂y

kyγw

∂P∂y

� �(19)

where kx and ky are the hydraulic conductivity in x and y directions, respectively, and γw is the uniteweight of water. This equation can be solved using different numerical scheme but usually in theprevious relevant studies, finite difference method (FDM) is implemented. Here, FEM is adopted forthis purpose because of the following reasons: in the FDM, the boundary conditions are applied onelements that can be internal elements inside or virtual elements outside the physical boundaries.This may introduce some errors especially in the case of small-scale DEM simulations. In FEM, theboundary condition is imposed on element edges that can be better matched with the physicalboundaries of the model. In addition, in the case of high gradient of pore pressure, the accuracy ofFEM can be increased using the same mesh size with higher-order elements.

Implementing weighted residual method by multiplying a trial function at both sides and integratingover the whole domain yields

∫Ω

wn

Kef f

∂P∂tdΩ ¼ ∫

Ω

w∂∂x

kxγw

∂P∂x

� �þ ∂∂y

kyγw

∂P∂y

� �� dΩ (20)

Applying Gauss–Green theorem and eliminating the boundary integral for no-flow boundary,Eq. (20) will change to



∫Ω

wn

Kef f

∂P∂tdΩ ¼ ∫

Ω

∂w∂x

∂∂x

kxγw

∂P∂x

� �� dΩþ ∫

Ω

∂w∂y

∂∂y

kyγw

∂P∂y

� �� dΩ (21)

The FDM is used to discretize derivatives with respect to time, and it is assumed that porosity,permeability, and compressibility will not change significantly in each time-step; therefore, usingGalerkin method (w= [Np] and w,i = [Bp]), we have

∫Ω

ni

Kief f

NTPNPdΩ

" #1Δt

Piþ1N � Pi

N

� � ¼ ∫Ω

kixγw

∂NTP

∂x∂NP

∂xþ kiyγw

∂NTP

∂y∂NP

∂y

!dΩ

" #Piþ1N (22)

PTN ¼ p1 p2 :::: pN �½ (23)

NTP ¼ N1 N2 :::: NN �½ (24)

where N is the number of nodes for each element, PN is the nodal pore pressure, NP is the shapefunction, Δt is the time-step, and i+ 1 and i are the next and current steps, respectively.

The simple one-step solution of fluid pressure helps to avoid the excessive computational effort ofthe iterative solution of the Navier–Stokes momentum equations. The total pore pressure at the end ofeach step is the sum of the induced excess pressure due to particle movement and the residual porepressure after dissipation (Eq. (25)).

PFinaltþΔt ¼ PtþΔt þ ΔPεv (25)

This equation is essential to generate coupled behavior for the algorithm such that solid movementwill induce pore pressure. The fluid cells are assumed to be the FEM elements, and for each FEM node,the volumetric strains of the surrounding elements can be averaged and assigned to that node as thepore pressure is a nodal value. To complete the coupling procedure, the solid particles should alsoreceive the effects of the fluid phase. The fluid forces exerted on particles can be divided into twoparts, hydrostatic and hydrodynamic forces. Buoyancy force is a typical hydrostatic force. It is dueto the pressure gradient around particles, and it can be directly obtained by calculating the pressuregradient around each fluid cell. Hydrodynamic forces consist of drag force, virtual mass force, andlift force among which the virtual mass and lift forces can be neglected for laminar flows [15]. Thedrag force generated due to the difference between the particle velocity and the fluid velocity isapplied parallel to the flow direction. The drag force can be estimated using semi-empiricalequations. The Ergun’s equation is one of the most well-known equations in this field, which can beused to estimate the drag force on particles from laminar to turbulent flow. Thus, the total forceacting on each particle in a fluid cell is

f i ¼ � β1� n

vf � vp� �� ∇P

� �Vp (26)

where ∇P is the hydrostatic pressure gradient around the cell, vf is the pore fluid velocity insidethe cell, vp is the translational velocity of the particle, Vp is the particles volume, and β is theempirical coefficient that can be determined through the Ergun’s equation. For porosity smallerthan 0.8 [2],

β ¼ μ 1� nð Þd2n

150 1� nð Þ þ 1:75vp � vf� �

ρf ndμ

" #(27)

where d is the mean particle diameter, μ is the fluid viscosity, and ρf is the fluid density. It isnoted that the velocity vf in this equation is the true velocity of the fluid inside pores, which isdifferent form superficial fluid velocity (v*), obtained on the fluid mesh nodes. The vf has been



mistakenly used in some studies including the study of Shafipour and Soroush [5]. The relationshipbetween the superficial fluid velocity and the true pore fluid velocity is

v� ¼ n:vf (28)

The permeability is one of the key input data in this model, and it can be evaluated using differentempirical equations. Among all of them, the well-known empirical Kozeny–Carman equation(Eq. (29)) shows the best correlation with the experimental results [17].

k ¼ d2n3ρf g

150μ 1� nð Þ2 (29)

The overall procedure of this continuum-discrete model can be summarized in Figure 2.

3. NUMERICAL RESULTS

3.1. Isotropic compression

Isotropic compression is the most common step of starting a triaxial test during which the soil sample iscompressed by the same load in all directions. In the case of an undrained test, the result of this step canbe analyzed to evaluate the quality of saturation. Theoretically, when the fluid and soil grains are fullyincompressible and the sample is fully saturated, there will be no volumetric change, and the wholeload will be sustained by fluid. On the contrary, in real condition, both fluid and soil grains arecompressible, so there should be some volumetric strain to mobilize pore pressure and effectivestress. For an elastic saturated soil sample under isotropic compression, the ratio between thedeveloped pore pressure due to an applied load is called B-value, which has the following closed-form solution (Figure 3):

B ¼ ΔuΔσ

¼ 11þ nKs=Kef f

(30)

where Δu is the change in pore pressure, Δσ is the change in total stress, and Keff and Ks are the bulkmodules of pore fluid (Eq. (3)) and soil skeleton, respectively.

To investigate the accuracy of pore pressure generation due to volumetric strain, a simple saturatedsoil sample that can replicate elastic behavior is simulated and isotropically compressed in anundrained condition. Figure 4 shows the sample for this example, and the parameters adopted forthe simulation are summarized in Table I. First, the sample is isotropically compressed, while the

Figure 2. The procedure of one time-step of this continuum-discrete model.


Figure 3. Schematic model of isotropic compression.

Figure 4. Continuum-discrete model of the isotropic compression example.

Table I. The details of the isotropic compression example.

Parameters Values

Number of particles 400 disc shapedParticles diameter (d) 0.001mParticles density (ρs) 2650 kg/m3

Contact stiffness (kn) 105N/mFluid density (ρf) 1000 kg/m3

Initial porosity 0.2146Fluid viscosity (μ) 1.004 × 10�3 N/s/m2

Effective pore fluid bulk modulus (Keff) 106 and 105 PaGravity (g) 9.80665m/s2

Permeability (k) 0.001m/s


fluid can drain from all boundaries to obtain the bulk modulus of the soil skeleton under drainedcondition. Because of the 2-KPa change in total stress (Δσ) in all boundaries, the volumetric strainof the soil sample is 0.0376, which is the ratio of the change in the volume of the sample to itsinitial volume, so the bulk modulus of the soil skeleton would be Ks =Δσ/εv= 53KPa.

Then, the fluid boundary condition is changed to undrained, and the same simulation is carried outfor two different values of the fluid bulk modulus. The comparison between analytical and numericalresults of B-values is presented at Table II. It can be seen that the numerical results agree well with theanalytical results for the two different pressures.


Table II. Comparison between the numerical and analytical B-values.

Effective pore fluid bulk modulus (Keff) 1000KPa 100KPa

Analytical B-value 0.9887 0.9008Numerical B-value 0.9804 0.8976


The effect of a small amount of air bubble on fluid bulk modulus and consequently on the porepressure is not infinitesimal. For instance, when the susceptibility to liquefaction is investigatedthrough cyclic undrained test, the degree of saturation can change the required number of cycles forliquefaction noticeably [29, 30]. Constant volume method is based on fully saturated sample andincompressible fluid, which implies that B-value should be one. This simple algorithm can beimplemented to simulate undrained cyclic triaxial test with different B-values.

3.2. One-dimensional upward seepage flow

One-dimensional upward seepage flow through a column of soil is a classical problem in soilmechanics, which can be described using Terzaghi’s theory of consolidation provided that thethree following assumptions are made: flow is a laminar and one dimensional, fluid isincompressible, and the permeability is constant. The pressure at upper boundary is set to zero,and a constant fluid velocity is applied at the bottom; moreover, the initial excess pore waterpressure in the column is zero (Figure 5(a)). The following partial DE with the mentionedboundaries and initial conditions can be analytically solved to obtain the pressure at differenttimes and depths [31]:

∂p∂t

¼ Cv∂2p∂z2

; Cv ¼ k

mvγw(31)

where mv is the coefficient of volumetric compressibility, Cv is the coefficient of consolidation, andγw is the unite weight of water. The closed-form solution for the excess pore pressure is [32]:

Figure 5. (a) Schematic model and (b) continuum-discrete model of upward seepage flow.



p z; tð Þ ¼ Hγwu�

k

z

H�X1n¼0

8 �1ð Þnπ2 2nþ 1ð Þ2 sin

π 2nþ 1ð Þz2H

e�Tv 2nþ1ð Þ2π2=4 !

(32)

where z is the distance from the top of the soil layer; H is the soil column height, which is the lengthof the longest possible drainage path; and Tv is called time factor, and for single drainage in this case,it can be defined as

Tv ¼ Cvt=H2 (33)

The uplift of the ground surface can be derived by integrating the strain due to the change ineffective stress.

St ¼ �∫H

0

εdz ¼ �∫H

0

mvpdz ¼ u�H2

2Cv1�

X1n¼0

32 �1ð Þnπ3 2nþ 1ð Þ3 e

�Tv 2nþ1ð Þ2π2=4 !

(34)

The analytical solution of upward seepage flow was used by Suzuki et al. [32] and Chen et al. [27]to verify their DEM-finite volume method codes using Navier–Stokes equations because this examplecan cover different aspects of coupling process.

Figure 5(b) shows the continuum-discrete model for the seepage example, and the details arepresented in Table III. To determine the coefficient of volumetric compressibility, the sample wasfirst generated in such way that particles have no overlap with each other, then it was allowed tosettle under gravity and buoyancy forces. Using the strain and the fully saturated density, thecoefficient of compressibility can be obtained as follows:

ε ¼ mvσ (35)

ρSaturated ¼ 1� nð Þρs þ nρf ; σ ¼ ρSaturated � ρf

gH=2 (36)

The amount of vertical strain due to the submerged settlement of the sample is 0.00245, and byusing Eq. (36), the averaged stress in the column is 200.14 Pa; therefore, from Eq. (35), the mv willbe 1.2235 × 10�5 Pa�1. A constant input fluid velocity equals to 0.0005m/s is applied at the bottomof the sample. Because the analytical solution is based on static condition, the damping should belarge enough to replicate quasi-static conditions, or for each time-step, the DEM code should iterateuntil the static equilibrium is satisfied for particles. Here, large contact damping was chosen by trialand error to avoid inertial effect. The uplift of the soil column and the pore fluid pressure atdifferent times are plotted in Figures 6 and 7, which show good agreements between analytical andnumerical results. The accuracy of these simulations revealed the capability of this algorithm toreplicate fluid-particle interactions.

Table III. The details of the upward seepage example.

Parameters Values

Number of particles 126 discParticles diameter (d) 0.001mParticles density (ρs) 2650 kg/m3

Contact stiffness (kn) 1 × 105N/mFluid density (ρf) 1000 kg/m3

Initial porosity 0.2099Fluid viscosity (μ) 1.004 × 10�3 N/s/m2

Effective pore fluid bulk modulus (Keff) 2 × 109 PaGravity (g) 9.8m/s2

Permeability (k) 9.65 × 10�4m/s


Figure 6. Comparison between analytical and numerical uplift of ground surface.

Figure 7. Comparison between analytical (points) and numerical (lines) pore pressure along the soil columnat different times.


3.3. Sensitivity analyses

Analytical solutions provide the opportunity to investigate the sensitivity of the results for differentparameters. The continuum-discrete method has some influential parameters, but among them, thefluid bulk modulus, the volumetric strain calculation, and the fluid cell size are of great importanceas they control the pore pressure. In addition, the validity of Darcy’s law should be monitored toavoid miscalculation.

3.3.1. Pore fluid bulk modulus. One of the inherent problems of partially coupled algorithms is theinstability problem caused by separate solutions of fluid and solid equations. It means that whenthe volumetric strain is progressing, the pore pressure is developing, while the fluid mesh is off, sothe pressure cannot be dissipated. Because the fluid is usually considered as a very stiff material ingeotechnical analyses, a small amount of volumetric strain or fluid injection may generate very largepore pressure, and it can cause instability. Thus, although both fluid and solid code calculations areapparently stable, the coupling process may not be stable. One way to solve this problem is to use asmall time-step, which however will significantly increase the computational time. Another possibleway is to avoid using the real value of fluid stiffness and consider a fluid that is stiffer than soilskeleton. To obtain a suitable value for fluid stiffness, sensitivity analyses can be performed.Figure 8 shows the sensitivity of the results of upward seepage example for some other differentvalues of fluid stiffness.

From this figure, it can be understood that increasing the fluid bulk modulus from 106 to2 × 109 will not improve the results, so even 106 is a suitable value for this example, whichcan satisfy the assumption of incompressibility of fluid. It is worthy to note that with thismodified fluid bulk modulus, the maximum time-step for stable calculation can be increasedmore than 10 times.


Figure 8. Effect of the effective fluid bulk modulus on the upward seepage.


3.3.2. Volumetric strain calculation. Two approaches have been described before to calculatevolumetric strain. One is based on the exact calculation of the change of porosity, and the other oneis based on the approximation of strain function. Both methods were implemented, and the resultswere compared in Figure 11. From this graph, it can be concluded that although the approximationmethod is less precise, the accuracy is good enough for common DEM simulation. Clearly, theperformance of numerical algorithms is very crucial, and the approximation method iscomputationally less expensive especially when large numbers of particles are simulated. Moreover,in some geotechnical applications, the effect of soil volumetric strain on fluid flow can be ignored,and only the seepage forces are considered on solid phase. This assumption is only applicable forspecial cases such as sand production to reduce the computation time. In these cases, both the solidand fluid phase may contribute to the pore pressure. This effect can be taken into account as thecoefficient that relates the change in porosity to the change in pore pressure. The seepage examplecan also be solved by applying this assumption, and for incompressible fluid, the coefficient ofcontinuity equation should be changed to the compressibility of the soil skeleton (1/mv). The upliftresult of this uncoupled simulation is also plotted in Figure 9.

3.3.3. Fluid cell size. Almost all numerical methods show sensitivity to their spatial discretization,which is expected because of the assumptions and simplifications of different discretization. In thissection, the sensitivity of the simulation results with respect to the fluid cell size is investigated toshed light on the effect of fluid cell size.

The fluid cell illustrated in Figure 5 initially contains seven particles. The mesh size is increasedthree times and six times to encompass 21 and 42 particles. The changes of pore pressure along thesoil column at different time for these three models are compared in Figure 10. It is shown that theincrease of the fluid mesh size reduces the accuracy of the calculated pore pressure. This can bejustified with the fact that fluid mesh elements are four-nodes linear, and the pore pressuredistribution along the soli column, especially at the initial stage of the seepage example, isnonlinear, so the pore pressure cannot be precisely captured within these large linear elements. It

Figure 9. Uplift results obtained by different methods of volumetric strain calculation.


Figure 10. Pore pressure along the soil column for different fluid cell size.


can also be observed that because the pore pressure distribution and its induced strain at the steady statecondition are linear, the error becomes less at the later stage.

It should be noted that sensitivity analysis on fluid cell size is crucial for a sample withdifferent particle sizes as the strain approximation method needs enough particles in its desiredarea to give reasonable results [2]. It can be concluded that the optimum fluid cell size thatdepends on different parameters such as the mean particle size, the particle size distribution,and the fluid flow condition should be determined through sensitivity analyses for each specificproblem.

3.3.4. Validity of Darcy’s regime. Each numerical code has its own limitation, and this algorithm islimited to the Darcy’s regime. It is critical to test whether the flow condition can trespass theDarcy’s limit. For fluid flow through porous material, the particle Reynolds’s number (Rep) isdefined as:

ReP ¼ nρf d̄ vf � vp�� =μ (37)

This number can be used to estimate the condition of the flow. For values below one, the flowis creeping with no inertial influence, which is called Darcy’s regime. From one to 100, the flowcalled Forchheimer regime, the condition is laminar, and the inertial influence is enhancing byincreasing the Reynolds’s number [33]. The history of particle Reynolds’s number should bemonitored for all cells, especially for a system with unsteady flow. Figure 11 shows theReynolds’s number for bottom, middle, and top of the soil column for the upward seepageflow example. According to the histories, the flow has not exceeded the Darcy’s regime limit,so the simulation is valid.

Figure 11. Particle Reynolds’s number for different level in the soil sample.



4. CONCLUSIONS

In this paper, a coarse-grid model is presented to model fluid–soil interaction. The motion of particles issimulated using DEM, which allows a micro-mechanical study. The fluid is assumed as a continuumsystem on a Eulerian mesh, each cell of which contains several particles. The fixed coarse-grid iscomputationally more efficient than connecting all pores with pipes as used by other researchers. Therelationship between the soil movement and fluid flow is determined based on poro-elasticity theory.The flow between fluid cells is assumed to conform to Darcy’s regime, and this assumption is made toreduce the excessive computational effort, which is required for iterative solution of the Navier–Stokesequations. Two different examples with closed-form solutions were modeled to reveal the accuracy ofthis algorithm. The comparison between numerical and analytical results showed that this continuum-discrete model can accurately replicate saturated densely packed particles. The possibility of havingfluid bulk modulus as an input data can be used to simulate saturated soils with B-values less thanone, which is common in real experiments. In addition, it was revealed that to satisfy the assumptionof incompressible fluid, there is no need to use the real value of fluid bulk modulus, and as long as thefluid is stiffer than soil, the system can replicate incompressible fluid. The major limitation of thismethod is that the flow condition should remain laminar, which can be assured by controlling theparticle’s Reynolds number. This implies that this algorithm is generally suitable for geotechnicalengineering applications and not the problems involved with high-fluid flow velocity or fluid–solidsuspension. The study presented here illustrates the potential for DEM to be used to simulate themechanical behavior of granular materials in fluid in a simple and efficient way.

ACKNOWLEDGEMENTS

The support of the University of Hong Kong during this research is gratefully acknowledged. The authorswould also like to thank Dr. Feng Chen for his valuable discussion and guidance.

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a continuum-discrete model using darcy's law: formulation and verification

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