Computers and Geotechnics 55 (2014) 494–505

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Computers and Geotechnics

journal homepage: www.elsevier .com/ locate/compgeo

Finite volume coupling strategies for the solutionof a Biot consolidation model

0266-352X/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compgeo.2013.09.014

⇑ Corresponding author.E-mail addresses: roza_asadi@mehr.sharif.ir (R. Asadi), ataie@sharif.edu (B.

Ataie-Ashtiani), craig.simmons@flinders.edu.au (C.T. Simmons).

Roza Asadi a,⇑, Behzad Ataie-Ashtiani a,b, Craig T. Simmons b,c

a Department of Civil Engineering, Sharif University of Technology, PO Box 11155-9313, Tehran, Iranb National Centre for Groundwater Research and Training, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australiac School of the Environment, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 April 2013Received in revised form 2 August 2013Accepted 21 September 2013

Keywords:Finite volume methodBiot consolidation modelFully coupledIteratively coupledExplicitly coupledLoosely coupled

In this paper a finite volume (FV) numerical method is implemented to solve a Biot consolidation modelwith discontinuous coefficients. Our studies show that the FV scheme leads to a locally mass conservativeapproach which removes pressure oscillations especially along the interface between materials with dif-ferent properties and yields higher accuracy for the flow and mechanics parameters. Then this numericaldiscretization is utilized to investigate different sequential strategies with various degrees of couplingincluding: iteratively, explicitly and loosely coupled methods. A comprehensive study is performed onthe stability, accuracy and rate of convergence of all of these sequential methods. In the iterative andexplicit solutions four splits of drained, undrained, fixed-stress and fixed-strain are studied. In looselycoupled methods three techniques of the local error method, the pore pressure method, and constant stepsize are considered and results are compared with other types of coupling methods. It is shown that thefixed-stress method is the best operator split in comparison with other sequential methods because of itsunconditional stability, accuracy and the rate of convergence. Among loosely coupled schemes, the porepressure and local error methods which are, respectively, based on variation of pressure and displace-ment, show consistency with the physics of the problem. In these methods with low number of totalmechanical iterations, errors within acceptance range can be achieved. As in the pore pressure methodmechanics time step increases more uniformly, this method would be less costly in comparison withthe local error method. These results are likely to be useful in decision making regarding choice of solu-tion schemes. Moreover, the stability of the FV method in multilayered media is verified using a numer-ical example.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The coupled process of fluid flow and mechanics in geotechnicswas first introduced by Terzaghi in 1924 as a consolidation phe-nomenon. This theory described one dimensional consolidationanalytically and has since been widely used in practice to calculateground settlements [1]. Subsequently in 1941 Biot generalized Ter-zaghi’s theory to three-dimensional porous media based on a linearstress–strain constitutive relationship and a linear form of Darcy’slaw [2]. The fluid flow-stress analysis in porous media is of increas-ing importance today in a diverse range of engineering fields in-cluded reservoir engineering, biomechanics, and environmentalengineering [3].

As the solutions of Biot system in closed forms are only avail-able in special cases, numerical methods are commonly used for

solving the respective initial-boundary value problem. However,numerical approximations based on different forms of the govern-ing equations and numerical methods can lead to significantly dif-ferent results for the cases of non-linear flow equations in porousmedia [4,5]. In spite of extensive research that has been carriedout for the numerical solution of the Biot equations, there still existchallenging issues which are as follow. First is the instabilities thatoccur because of sharp transient gradients. One of the numericalmethods that suffers from this kind of instability is the standard fi-nite element method (FEM). The FEM is widely used in solvingporoelasticity systems, especially in the cases when dealing withcomplex geometry or adaptive grids [1,6–8]. Although the standardFEM provides accurate results for the problems with smooth solu-tions, when strong pressure gradients appear, these methods maynot be stable in the sense that strong nonphysical oscillations oc-cur in the approximation of the pressure field [9]. To avoid thesedifficulties, a staggered finite difference discretization for the poro-elasticity equations in a single layer was examined by Gaspar et al.[10]. The approach from [10] was further developed by Naumovich

Nomenclature

b body force (M L�2 T�2)c consolidation coefficient (L2 T�1)cbr solid grain compressibility (M�1 L T2)cM vertical uniaxial compressibility (M�1 L T2)er relative error of local error methoderg goal local errorf flow source or sink (T�1)g gravitational acceleration (L T�2)h size of control volume (L)�K hydraulic conductivity tensor (L T�1)Ku undrained bulk modulus (M L�1 T�2)L length of domain (L)m fluid contentM Biot modulus (M L�1 T�2)N number of nodesp fluid pore pressure (M L�1 T�2)PL overburden (M L�1 T�2)t time (T)T time interval (T)u displacement (L)v Darcy’s velocity (L T�1)a Biot coefficientb fluid compressibility (M�1 L T2)

C frontier of the domain of coupled partial differentialequations

e strain2 acceptable tolerance for the L2-norm of two iterative

solutions in one time steph time weighted factorj harmonic averaging of �K=qg over the interval (xi-1, xi)

(M�1 L3 T)k Lame constant (M L�1 T�2)l shear modulus (M L�1 T�2)m harmonic averaging of kþ 2l over the interval (xi�0.5, -

xi+0.5) (M L�1 T�2)n interface (L)q fluid density (M L�3)r effective stress (M L�1 T�2)rt total mean stress (M L�1 T�2)s time step size (T)/ porosityv spatial weighted factor for location of interface�xp spatial grids for pressure�xu spatial grids for displacementxT time grids

R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505 495

et al. [11] to the case of multilayered deformable porous media,based on finite volume discretization. The order of convergenceof this scheme is analyzed by Ewing et al. [12].

One of the numerical methods that is widely used in computa-tional fluid dynamics problems is finite volume method (FVM). Themain advantage of the FVM is that, it produces accurate discretiza-tion for systems of partial differential equations with discontinu-ous coefficients. The corresponding schemes yield local massconservation and lead to higher accuracy for stresses and fluxesat the interfaces [13]. Therefore, it is worthwhile to examineFVM for addressing the instability problem in numerical solutionof poroelasticity systems.

A second challenge is related to the coupling strategies. Thereare four types of strategies for solving the coupled flow andmechanics problem: fully coupled, iteratively coupled, explicitlycoupled and loosely coupled. In the fully coupled method, the gov-erning equations of flow and mechanics are solved simultaneouslyat each time step [14–16] but in the iterative approach, by parti-tioning the coupled problem, one of the flow or mechanical sub-problem is solved first and then the other is solved using the inter-mediate solution information. This procedure iterates at each timestep until the solutions converge to the fully coupled approach[17–20]. Explicitly coupled is a kind of iterative scheme where onlyone iteration is considered [21]. In loosely coupled, the mechanicsequation is not solved in each time step [22] and after multipleflow steps are taken, the solution of the mechanical sub-problemis updated. In this method the number of flow steps depends onthe change of the pore-pressure or displacement and the specifiedallowable error [22].

The fully coupled approach is unconditionally stable but it leadsto a large algebraic system that requires huge computationalefforts for large problems. This linear system may be severely ill-conditioned [23]. To resolve these problems, three sequentialmethods including, iteratively, explicitly and loosely coupledmethods are employed. In addition, sequential methods have theadvantage of using existing flow and geomechanics simulators.

Various sequential methods are different in the stability, accuracyand efficiency behaviors [24]. The loosely coupled approach is aninexpensive method compared with other types of coupling butthe iterative and then the explicit methods exhibit higher accuracythan this approach [25]. Due to the need for balance between accu-racy and efficiency, comprehensive investigations of these strate-gies are performed in this study.

In the iterative methods, based on the partitioning strategy, dif-ferent operator splits are considered. In the splits where themechanical equation is solved first, the drained and undrainedsplits are applied. In a drained split, it is assumed that there isno pressure change in mechanical equation and this yields condi-tional stability. On the other hand, in an undrained split, fluid massremains constant during the mechanical step. The undrained splitis unconditionally stable except for an incompressible system inwhich this operator split is not convergent [17]. In other iterativeschemes, which have been applied in reservoir engineering, theflow problem is solved first [25,26]. Fixed-strain and fixed-stressmethods are two operator splits of this type of sequential schemes.In a fixed-strain split the rate of total volumetric strain is consid-ered constant during the flow calculation while in a fixed-stresssplit the rate of total volumetric stress is a constant parameter.In a fixed-strain split conditional stability occurs but a fixed-stresssplit provides unconditional stability even if the system is incom-pressible [18]. Examples of models based on the iterative coupledapproaches are given by Jha and Juanes [19] who have investigatedsequential schemes by employing a FEM for the mechanical prob-lem and a mixed FEM for the flow problem. Also Kim et al. ana-lyzed stability and convergence of iterative methods based on aFEM and a FVM for the mechanical and the flow problems, respec-tively [24,17,18].

In the explicit approach, there is no iteration in each time stepto ensure convergence of the solution and there is only single passbetween mechanics and flow equations. As iterative methods, onecan use all operator splits described above: the drained, undrained,fixed-strain and fixed-stress splits [24].

Fig. 1. Staggered grid in one dimension [27].

496 R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505

The other approach is the loosely coupled approach, in which asimulator solves the flow equation at each time step and performscalculations of the mechanics equation on selected time steps. Thefrequency of mechanics updating depends on the pressure or dis-placement changes during flow time steps. Three techniques ofthe local error method, the pore pressure method, and constantstep size can be considered in this category [22].

In this paper two objectives are investigated: the first objective isto study the stability and the computational performance of FVM inmodeling of mechanical and flow equations in multilayered media.The second objective is focused on the assessment of different sequen-tial approaches with fully-coupled method to conduct a comprehen-sive study of performance of these methods. In this paper theadvantages and disadvantages of all methods are analyzed quantita-tively and systematically. A similar study has not been done previ-ously. Linear elasticity and single-phase unidirectional flow isconsidered in order to isolate coupling effects and study the advanta-ges and disadvantages of the various coupling strategies themselves.

The remainder of the paper is organized as follows. Biot poro-elasticity system and the respective FV discretization of the prob-lem are presented in Section 2. In Section 3, the couplingstrategies that are used in this paper are illustrated. Numerical re-sults and comparisons are presented in Section 4, and some con-clusions are drawn at the end.

2. Formulation

2.1. Governing equations

The interaction between a granular material and the pore fluidis governed by equilibrium equation coupled to a mass balanceequation of pore fluid, based on Terzaghi’s effective stress principlethat links the grain forces to the fluid pore pressure [14]. The mod-el can be written as the following system of partial differentialequations [1]:

�r � rþ arp ¼ b ð1Þ

@

@t1M

pþ ar � u� �

þr � v ¼ f ð2Þ

where r ¼ lðruþ ðruÞTÞ þ kr � uI is a second order symmetricstress tensor, M = [/b + (a � /)cbr]�1 is the Biot modulus andv ¼ � K

�

qgrp is Darcy’s velocity, k and l are the lame constants, ais the Biot coefficient, b is the body force, u is the medium displace-ment, p is the fluid pore pressure, cbr is the solid grain compressibil-ity, / is the medium porosity, b is the fluid compressibility, K

�is the

hydraulic conductivity tensor, qg is the fluid specific weight, t istime and f is a flow source or sink.

Eqs. (1) and (2) form a coupled partial differential system definedon a domain X bounded by the frontier C with u and p as unknowns.This system can be solved with appropriate boundary conditions:

u ¼ uD on Cu

r:n ¼ SD on Cr

p ¼ pD on Cp

v:n ¼ q on Cq

8>>><>>>:and initial conditions:

u ¼ u0

p ¼ p0

(at t ¼ 0 and on X

uD, SD, pD, q are the prescribed displacement, traction, pore pressureand flux respectively on the corresponding boundaries, namely Cu,Cr, Cp, Cq and n is the unit normal vector of the boundary [14]. Alsoin Eqs. (3) and (4) Cu [ Cr = Cp [ Cq = C and Cu \ Cr = Cp \ Cq = £.

To complete the model in the case of the multi-layered medium,the discontinuity of coefficients across the interface n, should be con-sidered. For this purpose, the continuity conditions on the interfacebetween different layers are applied to the system. By assuming a per-fect contact, the interface conditions will be as follows [11]:

½u� ¼ 0; ½p� ¼ 0 ð5Þ

which mean continuity of the displacement and the fluid pressureacross the interface. Also, for the continuity of the stress and fluidflux across the interface, we have [11]

½r� ¼ 0; ½v� ¼ 0 ð6Þ

In the Eqs. (5) and (6) for an arbitrary parameter r,[r] = rn+0 � rn�0.

In this study a one-dimensional (1-D) Biot model with discon-tinuous coefficients is considered, so the Eqs. (1) and (2) are rewrit-ten in the following way:

� @r@xþ a

@p@x¼ bðx; tÞ; x 2 ð0; LÞ; t 2 ð0; T� ð7Þ

@

@t1M

pþ a@u@x

� �þ @v@x¼ f ðx; tÞ; x 2 ð0; LÞ; t 2 ð0; T� ð8Þ

2.2. Numerical discretization

The FVM is developed for both of flow and mechanics Eqs. (7)and (8) in the following sections.

2.2.1. Staggered grids and grid notationsThe interval (0, L) is subdivided into N > 1 equally spaced subin-

tervals of size h = 2L/(2N � 1). To eliminate numerical instability,which often come from the discretization of the model on the col-located grids, the staggered grids was applied in [10,18,19]. Twodifferent spatial grids of x

�p and x

�u are used to discretize the pres-

sure and the displacement equations, respectively (so-called stag-gered grids). Also a grid in time t with a step-size s is employed.This discretization is as follows:

xp¼fxi : xi¼ ih; i¼0; . .. ;N�1g; xp¼fxi 2xP ; i¼1; . .. ;N�1gxu¼fxi�0:5 : xi�0:5¼xi�0:5h; i¼1; . .. ;Ng; xu¼fxi�0:5 2xu; i¼1; . .. ;N�1gxT ¼ftn : tn¼ns; n¼1;2; . .. ;jg ð9Þ

The sketch of the grids isdepicted in Fig. 1. According to these grids,the position of the interface n could be shown with the equationn ¼ xiint�0:5 þ vh in which 0 < iint < N is an integer and 06 v < 1 [11].

R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505 497

The notations for discrete parameters, defined on xP �xT andxu �xT , are introduced as follows [11,12]:

u :¼ un :¼ uni :¼ uðxi�0:5; tnÞ

p :¼ pn :¼ pni :¼ pðxi; tnÞ

ph :¼ hpnþ1 þ ð1� hÞpn

uh :¼ hunþ1 þ ð1� hÞun ð10Þ

pn and un are pressure and displacement at time tn. pni and un

i denotepressure and displacement at node i and i � 0.5, respectively, whileph and uh stand for pressure and displacement at time tn + hs

2.2.2. Integral form of the governing equationsBy employing FVM for discretization of the differential prob-

lems (7) and (8), the mechanical equation is integrated over eachinterval (xi�1, xi) and the flow equation on (xi�0.5, xi+0.5). In orderto approximate the stress r(x) and flux v(x) in the formulae (7)and (8) in the necessary grid points, one integrates the equationr ¼ ðkþ 2lÞ @u

@x over the interval (xi�0.5, xi+0.5), and the equationv ¼ � K

qg@p@x over the interval (xi�1, xi). Also weighted discretization

in time with the weight parameter h is applied. Using non-indexnotations, the coupled differential equations for the discreteapproximate of u ¼ un

i at point (xi�0.5, tn) e xu �xT and p ¼ pni at

grid point (xi, tn) e xp �xT can be represented in the followingway [11,12]:

� muh�x

� �xþaph

�x ¼ bh; x¼ xi�0:5 2xu n x0:5f g ði¼2; . . . :;N�1Þ; t2xT

ð11Þah

pþaux

� �t� jph

�x

� �x ¼ f h; x¼ xi 2xp nfxN�1g i¼1; . . . ;N�2ð Þ; t2xT

ð12Þ

where uh�x and ph

�x are the backward spatial derivative of displacementand pressure at time tn + hs. Other parameters are defined asfollows:

px :¼ px;i ¼ pðxiþ1Þ�pðxiÞð Þ=h; p�x :¼ p�x;i ¼ pðxiÞ�pðxi�1Þð Þ=h ð13Þux :¼ux;i ¼ uðxiþ0:5Þ�uðxi�0:5Þð Þ=h; u�x :¼u�x;i ¼ uðxi�0:5Þ�uðxi�1:5Þð Þ=h ð14Þpt :¼ pn

t :¼ ptðxi;tnÞ¼ ðpnþ1i �pn

i Þ=s; xi 2xp ð15Þut :¼un

t :¼utðxi�0:5;tnÞ¼ ðunþ1i �un

i Þ=s; xi�0:5 2xu ð16Þf h :¼ f h

i :¼ hf nþ1i þð1�hÞf n

i ; bh:¼ bh

i :¼ hbnþ1i þð1�hÞbn

i ð17Þ

also coefficients a, j and m are calculated according to the followingformulae [11,12]:

ai ¼Z xiþ0:5

xi�0:5

1M

dx ð18Þ

mi ¼1h

Z xiþ0:5

xi�0:5

dxkþ 2l

!�1

ð19Þ

ji ¼1h

Z xi

xi�1

dxK=qg

!�1

ð20Þ

and the right hand side f and b are defined as [11,12]:

fiðtÞ ¼1h

Z xiþ0:5

xi�0:5

f ðx; tÞdx

!; biðtÞ ¼

1h

Z xi

xi�1

bðx; tÞdx

!ð21Þ

This system can be solved with appropriate boundary condi-tions that mentioned in Eq. (3). These formulations which arebased on finite volume method for a uniform structured mesh,could be developed by approximating pressure field with piece-wise constant shape functions and the displacement interpolationfunctions with the C0 – continuous isoparametric functions in the

weak form of the governing equations, which has done by Kimet al. [17]. But in this way of modeling, the mesh should conformto the boundaries and an extra node must be placed at the inter-face, but as shown for the FVM, the interface could be simulatedindependently of the mesh structure [28].

3. Coupling strategies

In this section different techniques of coupling are discussed.These methods include: fully, iteratively, explicitly and looselycoupled approaches which are all different in the degree of cou-pling. Also, different operator splits for the sequential solutionmethods are reviewed.

3.1. Fully coupled approach

The best feature of the fully coupled approach is its stability. Migaet al. [29] show that the fully coupled method for the coupled flowand mechanics problem is unconditionally stable for h P 0:5. The re-sults of this method have been considered as a benchmark to com-pare other competitive sequential methods with this approach.

3.2. Iteratively coupled approach

In contrast to the fully coupled approach, in the iteratively cou-pled approach, information is exchanged between the mechanicsand flow equations sequentially until the solution converges with-in an acceptable tolerance at each time step [24]. Here four sequen-tial methods including: drained, undrained, fixed-stress and fixed-strain splits are considered.

3.2.1. Drained splitIn the drained split, by keeping the system at constant pressure,

the mechanical step is solved and then, the fluid flow problem issolved with a frozen displacement field. The diagram of Fig. 2 illus-trates the solution procedure by this method. The discretized equa-tions for flow and mechanics are written as [17]:

� mi

h unþ1;kþ1iþ1 � unþ1;kþ1

i

� �þ 1� hð Þ un

iþ1 � uni

� �h

0@

�mi�1

h unþ1;kþ1i � unþ1;kþ1

i�1

� �þ ð1� hÞ un

i � uni�1

� �h

1A

þ ah pnþ1;ki � pnþ1;k

i�1

� �þ a 1� hð Þ pn

i � pni�1

� �¼ hbh

i ð22Þ

aipnþ1;kþ1

i � pni

sþ a

unþ1;kþ1iþ1 � un

iþ1

s� a

unþ1;kþ1i � un

i

s

þ �jiþ1

h pnþ1;kþ1iþ1 � pnþ1;kþ1

i

� �þ ð1� hÞ pn

iþ1 � pni

� �h

0@

1A

þ ji

h pnþ1;kþ1i � pnþ1;kþ1

i�1

� �þ ð1� hÞ pn

i � pni�1

� �h

0@

1A

¼ hf hi ð23Þ

the pressure term pn+1,k in Eq. (22) is obtained from the previous iter-ation (kth) step. The other variables in Eqs. (22) and (23) are un+1,k+1

and pn+1,k+1, which are unknown at the present (k + 1)th step.Stability analysis in [17] indicates that, for the backward Euler

scheme h = 1, the drained split is conditionally stable and the sta-bility condition is:

a2Mkþ 2l

6 1 ð24Þ

1A

Fig. 2. Flowchart of the drained and undrained splits.

498 R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505

The parameter a2Mkþ2l which is called the coupling strength, is de-

fined by the ratio of the bulk stiffness of the fluid and solid skeleton[17]. Eq. (24) indicates that for the backward Euler time discretiza-tion, the size of the time step does not affect the stability conditionof the drained operator split and the stability of the solution fol-lows under the proper conditions of material properties [17].

3.2.2. Undrained splitIn the undrained split, by considering the initial solid phase at

constant fluid content, the pressure predictor for solving themechanical problem is evaluated, as follows [17]:

pnþ1;kþ1i ¼ pnþ1;k

i � aM@u@x

� �nþ1;kþ1

i

� @u@x

� �nþ1;k

i

!ð25Þ

Then Eq. (22) is replaced by:

� mi

h unþ1;kþ1iþ1 �unþ1;kþ1

i

� �þð1�hÞ un

iþ1�uni

� �h

�mi�1

h unþ1;kþ1i �unþ1;kþ1

i�1

� �þð1�hÞ un

i �uni�1

� �h

0@

� a2hhai

� �unþ1;kþ1

iþ1 �unþ1;kþ1i

h

!� a2hh

ai�1

� �unþ1;kþ1

i �unþ1;kþ1i�1

h

! !

þ a2hhai

� �unþ1;k

iþ1 �unþ1;ki

h

!� a2hh

ai�1

� �unþ1;k

i �unþ1;ki�1

h

! !þah pnþ1;k

i �pnþ1;ki�1

� �það1�hÞ pn

i �pni�1

� �¼hbh

i

ð26Þ

The flow equation of the undrained split is the same as Eq. (23).The flowchart of the undrained split is shown in Fig. 2. Kim et al. in[17] show that the undrained split is unconditionally stable for0:5 6 h 6 1.

3.2.3. Fixed-strain splitIn this approach, the mechanical contribution in the flow equa-

tion can be expressed by fixing the variation of the strain rate dur-ing solving the flow problem, which means [18]:

Den ¼ Den�1 ð27Þ

WhereDð�Þn ¼ ð�Þnþ1 � ð�Þn and e ¼ 12 ðruþ ðruÞTÞ is the linearized

strain under the assumption of infinitesimal transformation [18].Therefore after solving the flow problem for pressure at theðkþ 1Þth step, the mechanical problem will be solved. Full discret-ization of the fixed-strain split yields [24]:

aipnþ1;kþ1

i � pni

sþ a

unþ1;kiþ1 � un

iþ1

s� a

unþ1;ki � un

i

s

þ �jiþ1

h pnþ1;kþ1iþ1 � pnþ1;kþ1

i

� �þ ð1� hÞ pn

iþ1 � pni

� �h

0@

1A

þ ji

h pnþ1;kþ1i � pnþ1;kþ1

i�1

� �þ ð1� hÞ pn

i � pni�1

� �h

0@

1A

¼ hf hi ð28Þ

� mi

h unþ1;kþ1iþ1 �unþ1;kþ1

i

� �þð1�hÞ un

iþ1�uni

� �h

�mi�1

h unþ1;kþ1i �unþ1;kþ1

i�1

� �þð1�hÞ un

i �uni�1

� �h

0@

1A

þah pnþ1;kþ1i �pnþ1;kþ1

i�1

� �það1�hÞ pn

i �pni�1

� �¼hbh

i

ð29Þ

The flowchart of the fixed-strain split is shown in Fig. 3. Thecondition for linear stability is [18]:

For 0:5 6 h 6 1 :a2M

kþ 2l6 1 ð30Þ

For 0 < h < 0:5 :a2M

kþ 2l6 1 and

s 6kþ 2l

M� a2

� �h2

2ð1� 2hÞ kþ 2lð Þ �Kqg

� � ð31Þ

Eq. (31) indicates that for 0 < h < 0.5, in addition to the term forthe coupling strength, which is the same as Eq. (30), an additionalcondition for the time step size is required for numerical stability[18].

Fig. 3. Flowchart of the fixed-strain and the fixed-stress splits.

R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505 499

3.2.4. Fixed-stress splitIn this scheme, the geomechanical variable in flow equation is

the mean total stress in which its rate is kept constant duringthe pressure calculation (d _rt ¼ 0). So we have [18]:

Den ¼ akþ 2l

�Dpn � Dpn�1

�þ Den�1 ð32Þ

Similar to the fixed-strain method, the mechanical equation willbe solved by prescribing the pressure from the flow step. The dis-crete form of the flow equation of fixed-stress split is [18]:

ai þha2

kþ 2l

� �pnþ1;kþ1

i � pni

s� ha2

kþ 2lpnþ1;k

i � pni

s

þ aunþ1;k

iþ1 � uniþ1

s� a

unþ1;ki � un

i

s

þ �jiþ1

h pnþ1;kþ1iþ1 � pnþ1;kþ1

i

� �þ ð1� hÞ pn

iþ1 � pni

� �h

0@

1A

þ ji

h pnþ1;kþ1i � pnþ1;kþ1

i�1

� �þ ð1� hÞ pn

i � pni�1

� �h

0@

1A

¼ hf hi ð33Þ

The mechanical equation is the same as fixed-strain method.The diagram of the fixed-stress split is illustrated in Fig. 3. The sta-bility condition for different values of h yields [18]:

For 0:5 6 h 6 1 : Unconditionally stable ð34Þ

For 0 < h < 0:5 : s 6 1Mþ a2

kþ 2l

� �h2

2 1� 2hð Þ �Kqg

� � ð35Þ

Eq. (35) puts a limit on the time step for 0 < h < 0.5, beyondwhich numerical instabilities occur.

3.3. Explicitly coupled approach

In this method, only a single-pass is performed betweenmechanical and flow equations. As in iterative methods, all opera-tor splits including the drained, undrained, fixed-strain and fixed-

stress splits could be considered. This approach can save computa-tional cost in comparison with iteratively coupling, but the accu-racy of this method should be investigated.

3.4. Loosely coupled approach

In subsidence problems the fluid flow parameters (pressure)change over shorter time frames than mechanics parameters (dis-placement) which progress more slowly. So pressure variations im-pose restriction on the growth of time step size throughout thecourse of simulation [22]. For this reason, in loosely coupled meth-ods, the flow equation is calculated for each time step but mechan-ical updates are performed after multiple flow time steps. Threemethods are studied to determine controls to perform the mechan-ical calculation. These methods are: the constant step size, porepressure method and the local error method.

3.4.1. Constant step sizeConstant step size is the simplest method in which the mechan-

ics time step is fixed during the simulation. For example for thetime step s of flow equation, the step size of mechanical equationis a constant coefficient of s. It is obvious that accuracy and cost ofthe method depends on this coefficient.

3.4.2. Pore pressure methodIn the pore pressure method, the difference between pore pres-

sures of two subsequent mechanical steps should be within thedetermined tolerance. But when the changes are greater than theallowable tolerance the values of pore pressures will be passedto the mechanical equation for an updated calculation [22].

3.4.3. Local error methodIn the local error method the control is based on the changes in

displacements. In this way, the mechanical equation is solved forone time step of size 2s and two time steps of size s. The resultingsolutions are uc and uf, respectively. Then the relative error isdetermined as follows [22]:

er ¼ kuf � uckL2

kuf kL2ð36Þ

Table 1Hydraulic and mechanical parameters of the porous medium.

Parameters Values

Permeability of sand �Ksand ¼ 10�5 m=sPermeability of clay �Kclay ¼ 10�8 m=sPorosity / = 0.375Fluid compressibility b = 4.4 � 10�4 MPa�1

500 R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505

The relative error is compared to the goal local error (erg), whichis defined according to the physics of the problem. If er < 0.5erg,then the next time step will be increased, else if erg P er P 0.5erg,the time step is kept constant, else if 2erg P er P erg, the solution isaccepted for this time step but the next step will be decreased andat last if 2erg < er, the solution is not correct for this time step andshould be repeated with half time step [22].

Lame coefficients l ¼ 40 MPa; k ¼ 40 MPaBiot coefficient a = 1.0

4. Numerical results

In this section numerical experiments are performed in order toinvestigate the stability and convergence behaviors. Two test casesare used to study stability. In the first case the stability and accu-racy of the finite volume consolidation model is verified againstmultilayered porous media and is compared to a standard FE mod-el. Because of known analytical solution of Terzaghi’s problem, it isused as the second case to study the stability behaviors of the fouroperator splits [14]. Also accuracy, the rate of convergence and theefficiency of iteratively coupled, explicitly coupled and loosely cou-pled methods are investigated. In this study, all of the numericalcodes have been written in MATLAB 6.0 software and have beensimulated using a serial code on an Intel� Core™ i5 CPU M540@2.53 GHz processor with 6 GB RAM.

Fig. 5. Layered column consolidation test: Finite volume and standard finiteelement numerical solutions for the pore pressure variation.

Fig. 6. Layered column consolidation test: Finite volume and standard finiteelement numerical solutions for the displacement.

4.1. Layered column consolidation test

This multilayered problem and its standard finite element solu-tion were presented by Ferronato et al. [14]. It consists of a fluid-saturated column of height 15 m with 10 m of clay on top and5 m of sand on bottom as shown in Fig. 4. The sand and clay prop-erties are listed in Table 1. In this test simulation, clay permeabilityis 1000 times smaller than sand while the elastic properties are thesame. The values of Lame coefficients in Table 1 correspond toYoung’s modulus of 100 MPa and Poisson’s ratio of 0.25. There isno displacement at the bottom. Also, pore pressure variation of9810 Pa is prescribed on the bottom boundary. The top boundaryis traction-free and drainage is allowed on this boundary [14].The simulated pore pressure at different times is shown in Fig. 5.

For the sake of comparison, the same number of nodes of 30 andtime step of 0.1 s, which have been used in Ref. [14] for a standardFE model, is chosen. Fig. 5 illustrates that standard FE methodshows pronounced initial pressure oscillations at the sand–clayinterface. This numerical instability is caused as the nodal based fi-nite element method with the equal-order approximations of pres-sure and displacement (linear interpolation) is employed in thisproblem, with slightly compressible fluid flow and the discontinu-ity of pressure at the sand–clay interface. These oscillations of porepressure which exhibit in finite element modeling of consolidationproblem have been observed and discussed by [30–33]. By

Fig. 4. Sketch of the problem setup for the layered column consolidation test.

contrast, the vertical displacement is stable and accurate in boththe FE and FV models (Fig. 6). This problem illustrates that the usedFVM is stable and with the same number of unknowns shows nooscillations at the sand–clay interface. This also leads to higheraccuracy for fluxes at the interfaces.

4.2. Terzaghi’s consolidation test

In this example Terzaghi’s (1-D) consolidation problem is usedto compare the various coupling strategies [14]. Because of thehigh computational cost of the fully coupled scheme, it is desirableto develop sequential solution methods that converge to the fullycoupled approach. Also the finite volume discretization is usedon the staggered grids to derive stable discretization with highaccuracy for the Biot system. So in this case the stability, accuracyand efficiency of these coupling strategies are investigated.

Terzaghi’s problem consists of a fluid-saturated column ofheight L with a constant loading PL on top. There is a drainage

R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505 501

boundary on top. The basement is fixed and no-flow boundary con-dition is applied at bottom (Fig. 7). The load is applied instanta-neously at time t = 0 yielding initial pressure p0(x) and acorresponding settlement u0(x). Assuming the x-axis positivedownward, the analytical solution yields [14]:

pðx; tÞ ¼ 4p

p0

X1m¼0

12mþ 1

exp�ð2mþ 1Þ2p2ct

4L2

" #sin

ð2mþ 1Þpx2L

�

ð37Þ

uðx;tÞ¼ cMp0 ðL�xÞ�8Lp2

X1m¼0

1

ð2mþ1Þ2exp

�ð2mþ1Þ2p2ct

4L2

" #cos

ð2mþ1Þpx2L

� ( )

þu0

ð38Þ

where

p0ðxÞ ¼aM

Ku þ 4l=3PL ð39Þ

u0ðxÞ ¼1

Ku þ 4l=3PLðL� xÞ ð40Þ

with Ku ¼ kþ 2l=3þ a2M the undrained bulk modulus,cM ¼ ðkþ 2lÞ�1 the vertical uniaxial compressibility, andc ¼ �K=½qgðM�1 þ a2cMÞ� the consolidation coefficient. A homoge-neous sandy column with the height of 15 m is simulated. Hydrau-lic and mechanical properties for the sand layer except b are givenin Table 1. The parameter b left unspecified to test the stability ofoperator splits for different values of the coupling strength. The pre-scribed load PL is 104 Pa [14]. The column is discretized into 60nodes. The time integration is performed with a first-order implicitscheme and a constant time step s = 0.1 s. The simulation proceedsuntil steady state conditions are attained.

4.2.1. Iterative and explicit solutionsStability, accuracy and rate of convergence of the explicit meth-

od as a type of iterative solution (single pass only) and iterativemethods are studied in this section.

4.2.1.1. Stability. For stability analysis, first we assumeb = 4.4 � 10�2 MPa�1 which means M ¼ 60:6 MPa, so the couplingstrength is smaller than 1. For this value of b, stability conditionsfor drained and fixed-strain methods are satisfied and as is shownin Figs. 8 and 9, all of the sequential methods converge to the fullycoupled and analytical solutions.

By assuming b ¼ 4:4� 10�4 MPa�1 ðM ¼ 6:06 GPaÞ, the cou-pling strength will be greater than 1, so the drained and fixed-

Fig. 7. Sketch of the problem setup for the Terzaghi consolidation test.

strain splits are unstable. On the other hand, the undrained andfixed-stress splits are stable and there is good agreement betweenthe analytical and numerical solutions for pore pressure, as isshown in Fig. 10. Similar findings have been reported by Kimet al. [17,18].

It should be noticed that the value of b = 4.4 � 10�2 MPa�1 hasthe same order of magnitude as the values of fluid compressibilityin the Refs. [17,18]. As Eqs. (24) and (30) show, the stability criteriaof drained and fixed-strain splits not only depend on the value offluid compressibility but also on the values of drained bulk modu-lus and Biot coefficient. So in a porous media with high values ofdrained bulk modulus and low values of Biot coefficient (e.g. inthe range of 0.4–0.8), these two conditional stable splits could beused even with low compressible fluids.

4.2.1.2. Accuracy and rate of convergence. To enable the possibilityof comparison of accuracy and rate of convergence between allthe operator splits, b is assumed 4.4 � 10�2 MPa�1. The rate of con-vergence of these methods are compared with each other bymeans of iterations per time step, total mechanical iterations andCPU time. In the first row of Table 2 the number of iterations forconvergence at time t = 60 s is calculated. The convergence crite-rion is the L2-norm of the difference between the solution vectors

at successive iterations which defined as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1 pnþ1;kþ1

i � pnþ1;ki

� �2r

for pressure andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1ðunþ1;kþ1i � unþ1;k

i Þ2

qfor displacement. For L2-

norm = 10�3 and 10�6, calculations show that the drained andfixed-strain splits converge in equal number of iterations and therate of convergence of these methods are lower than other meth-ods, such that when the tight convergence criteria is considered,CPU time for these two methods exceed the CPU time of the fullycoupled approach. While the fixed-stress and undrained methodsare unconditionally stable, the fixed-stress split is faster and con-verges in just one iteration. For this test the root mean square errorof pore pressures (RMSE) is calculated to compare the sequentialmethods with the fully coupled solution. RMSE has widely beenused for evaluation the numerical methods performances [5]. Re-sults show that even when four operator splits are converged,the fixed-stress split achieves lower RMSE values than other meth-ods. So the fixed-stress method is the best operator split in com-parison with other sequential methods because of itsunconditional stability, accuracy and rate of convergence. The re-sults certify the findings of Kim et al. [17,18].

The influences of mesh size and time step are also studied byconsidering the numbers of nodes of 60, 120, and 240, and simula-tion time steps in the range of 0.001 s to 1.0 s. The convergenceproperties of the different schemes of iterative and explicit meth-ods have been compared with the analytical solution using RMSEof the pore pressure. Fig. 11 shows RMSE of the pressure for itera-tive and explicit schemes in a double log–log plot for various meshsizes and time steps.

The source of these errors includes two parts: first is due to theerror between fully coupled methods and sequential methods, asshown in Table 2 for iterative and explicit approaches, and the sec-ond is induced by the difference between fully coupled and analyt-ical solution. The first source of error depends on convergencecriteria and the number of iterations, as discussed above, and thesecond is because of the numerical model itself.

As shown in Fig. 11 the explicit method shows higher value ofRMSE, because the error between explicit and fully coupled meth-ods is larger than the error between iterative and fully coupled. Astime step decreases, explicit methods converge to the iterativemethods. Also by refining the mesh size for each profile, the valuesof RMSE become closer to each other and the errors of pressure de-crease as Dt. L2-norm for iterative methods is 10�3 for the results

Fig. 8. Terzaghi’s consolidation test. (a) Analytical, fully coupled and fixed-strain and (b) Analytical, fully coupled and fixed-stress solutions for coupling strength smaller than1 (M = 60.6 MPa).

Fig. 9. Terzaghi’s consolidation test. (a) Analytical, fully coupled and undrained and (b) Analytical, fully coupled and drained solutions for coupling strength smaller than 1(M = 60.6 MPa).

Fig. 10. Terzaghi’s consolidation test. (a) Analytical, fully coupled and undrained and (b) Analytical, fully coupled and fixed-stress solutions for coupling strength greater than1 (M = 6.06 GPa).

502 R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505

given in Fig. 11, however for L2-norm of 10�6 the same trend isfollowed.

4.2.2. Loosely coupled solutionFor loosely coupled approach, three step size selection algo-

rithms are investigated to determine their advantages anddisadvantages.

4.2.2.1. Stability. The mechanical response affects the flow equationthrough mechanical variables which could be the volumetric strainor mean total stress [20]. Utilizing each of volumetric strain ormean total stress will represent the flow equation as it is infixed-strain or fixed-stress methods. So the stability conditions of

fixed-strain and fixed-stress methods are also valid in this couplingstrategy.

4.2.2.2. Accuracy and rate of convergence. The results of these threemethods are presented in Table 3. Since in this problem, mean totalstress is considered constant, the results for fixed-stress split inloosely coupled method is identical to the iterative and explicitmethods. So only the effect of volumetric strain on the flow equa-tion (i.e. fixed-strain split) has been investigated in Tables 3 and 4.

For the constant step size method, two step size of 1 and 5 s arecompared with each other. As the result shows, by increasingmechanical step size to 5 s, the error is increased significantly.For the pore pressure method two values of 5 and 20 Pa are consid-ered as the pore pressure errors. It means that pore pressure

Table 2Accuracy and run-time information for iterative and explicit methods.

Comparisonfactors

Explicit methods Iterative methods Fully coupledmethod

L2-norm = 10�3 L2-norm = 10�6

DRa UDb FSNc FSSd DR UD FSN FSS DR UD FSN FSS

Iterations pertime step

1 1 1 1 13 3 13 1 23 4 23 1 –

Mechanicaliterations

600 600 600 600 8585 1890 8585 600 14622 2657 14622 600 –

CPU time (s) 0.4 0.4 0.4 0.4 5.1 1.1 5 0.4 8.8 1.6 8.4 0.4 2.9RMSE (Pa) 0.1259 0.1896 0.1259 4.9E�10 4.2E�05 2.5E�09 0.0021 4.9E�10 4.7E�08 4.3E�10 4.6E�06 4.9E�10 0

a Drained.b Undrained.c Fixed-strain.d Fixed-stress.

(a)

(b)

Fig. 11. Terzaghi’s consolidation test. RMSE of pore pressure of (a) Drained and undrained and (b) Fixed strain and fixed-stress solutions vs. time step for various mesh sizes att = 60 s.

Table 3Accuracy and run-time information for constant step size, pore pressure and localerror methods for loosely coupled solutions.

Comparisonfactors

Constant stepsize method

Pore pressuremethod

Local errormethod

Mechanics timestep

Pore pressureerror

Goal local error(erg)

1 s 5 s 5 Pa 20 Pa 5 � 10�4 10�4

CPU time (s) 0.29 0.27 0.31 0.28 0.3 0.35RMSE (Pa) 1.2631 6.4052 1.8207 8.2882 15.7045 4.1852Mechanical

iterations60 12 67 17 16 78

R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505 503

change between two successive mechanical steps is limited to thevalues of 5 and 20 Pa. As the results in the Table 2 show, in thismethod greater errors than constant step size method will occurduring the simulation despite of increase in the number ofmechanical iterations. This issue has been caused because of thegrowth of mechanical time steps. In this problem the time intervalon which the output of flow equation is sent to the mechanicalproblem, reaches to 1.5 s and 5.8 s for the error values of 5 and20 Pa respectively.

In the local error method, control is on the changes of displace-ment. As listed in Table 3 calculations are performed for two goalerrors of 10�4 and 5 � 10�4. During the total simulation time of60 s, the size of the mechanical step changes in the range of 0.1 s

Table 4Effect of mesh size on RMSE values of pore pressure for constant step size, porepressure and local error methods.

Number ofnodes

Constant stepsize method

Pore pressuremethod

Local error method

Mechanics timestep

Pore pressureerror

Goal local error(erg)

1 s 5 s 5 Pa 20 Pa 5 � 10�4 10�4

30 3.3025 8.3710 3.425 9.1211 16.5839 5.6361360 1.8469 6.9635 2.3125 8.7002 16.1674 4.7055

120 1.4918 6.6236 2.2277 8.5785 16.0959 4.5330240 1.4048 6.5426 2.2143 8.5159 16.0875 4.4958

504 R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505

at the beginning to 3.2 s towards the end. By progressing over time,during 600 s simulation, for fixed-strain method the value of RMSEof pressure due to the fully coupled method is reduced to 1.473 Pa.In this simulation, mechanical time step increases to 6.4 s atT = 600 s and as a result the mechanics solution updates 186 times.In contrast, in the constant step size method for the mechanicstime step of 5 s, although the mechanics time step is less than max-imum mechanics time step of the local error method (i.e. 6.4 s), thevalue of RMSE increases to 1.53 Pa. This means that in the local er-ror method the solution procedure is more consistent with thephysics of the problem. The main disadvantage of this method isthe reliable estimation of goal local error, because by decreasingthe goal local error, instabilities can occur in solving the problem.

Also the pore pressure method has been examined over 600 ssimulation with different pore pressure errors. This study showsthat for the pore pressure error of 7.25 Pa, RMSE of pressure dueto the fully coupled method would be 1.473 Pa. The total mechan-ical iterations for this value of pore pressure error is 163, which isless than local error method despite of the same value of RMSE.This occurs because in the pore pressure method, mechanics timestep increases in a more uniform manner.

The effects of mesh sizes on the RMSE of the pressure betweenloosely coupled methods and the analytical solution, are given inTable 4. Because two different time steps are used for flow andmechanics equation, only grid sensitivity analysis is performed.Wide ranges of mesh size have been checked and just the resultsof a few cases are given in Table 4. As shown in Table 4, by refiningmesh size slight differences occur in the result of all schemes. In or-der to examine the effect of mechanical time step, it is required tominimize the effects of the error produced by the space discretiza-tion, so more refined meshes are used. In the constant step sizemethod, the ratio of RMSE for mechanics time step of 5 s to 1 s is4.66 for 240 nodes, which is close to the ratio of mechanics timesteps. Similar results could be obtained for the other two schemes.In pore pressure method the ratio of RMSE for error value of 20 to5 Pa is 3.85, and the ratio of mechanics time step for these values oferror is 3.87 at t = 60 s. Also for the local error method, the ratio ofRMSE for two goal errors of 5 � 10�4 to 10�4 is 3.58, where in thiscase the ratio of mechanics time step is equal to 12:8 s

3:2 s ¼ 4:00. Theseall demonstrate the first-order convergence of solutions with re-spect to the mechanics time step.

5. Conclusions

There are two key challenging issues which should be ad-dressed in the numerical simulation of Biot’s consolidation model.The first issue is to develop a stable numerical discretization whichremoves pressure oscillations especially along the interface be-tween the materials with different properties and the second isthe selection of a suitable coupling strategy with high accuracy,stability and efficiency. In this study the FVM has been consideredfor both fluid and mechanical equations. This method is verifiedagainst standard FEM in multilayered media. Various coupling

strategies which include fully coupled, iteratively coupled, explic-itly coupled and loosely coupled are investigated to assess theirCPU time, total mechanical iterations and the value of RMSE rela-tive to the fully coupled approach. From this study, the followingconclusions can be drawn:

(1) It is shown that the FVM removes the oscillations of the porepressure along the interface between materials with differ-ent properties at the initial stage of the process. The corre-sponding discrete scheme is locally conservative, so itpreserves the fluxes across the discontinuity, and yieldshigher accuracy for the flow and mechanics parameters.

(2) In iteratively coupled methods, drained and fixed-strainconverge in the same number of iterations and by tighteningthe convergence criteria, these two methods even becomemore time consuming than the fully coupled method. In con-trast fixed-stress scheme takes less number of iterationsthan other operator splits and gives results with a higherlevel of accuracy relative to the fully coupled method.Because of these reasons and unconditional stability of thefixed-stress split for 0:5 6 h 6 1 even in incompressible sys-tems (unlike the undrained solution), it is the best operatorsplit.

(3) Among loosely coupled schemes, the pore pressure and localerror methods which are based on variation of pressure anddisplacement respectively, show consistency with the phys-ics of the problem. In these methods with low number oftotal mechanical iterations, errors within acceptance rangecould be achieved. Because in the pore pressure methodmechanics time step increases more uniformly, this methodwould be less costly in comparison with local error method.For both of these methods, reliable estimation of goal localerror and pore pressure error is a key issue in performanceand stability of these coupling strategies.

Acknowledgment

The authors wish to thank two anonymous reviewers for theirconstructive comments which helped to improve the finalmanuscript.

References

[1] Lewis RW, Schrefler BA. The finite element method in the static and dynamicdeformation and consolidation of porous media. 2nd ed. Chichester: Wiley;1998.

[2] Biot MA. General theory of three-dimensional consolidation. J Appl Phys1941;12:155–64.

[3] Phillips PJ, Wheeler MF. A coupling of mixed and continuous Galerkin finiteelement methods for poroelasticity I: The continuous in time case. ComputGeosci 2007;11(2):131–44.

[4] Ataie-Ashtiani B, Raeesi-Ardekani D. Comparison of numerical formulationsfor two-phase flow in porous media. Geotech Geolog Eng 2010;28(4):373–89.

[5] Shahraiyni HT, Ataie-Ashtiani B. Mathematical forms and numerical schemesfor the solution of unsaturated flow equations. J Irrig Drain Eng 2011;138(1):63–72.

[6] Zienkiewicz OC, Chan AHC, Pastor M, Schrefler BA, Shiomi T. Computationalgeomechanics with special reference to earthquake engineering. Chichester: Wiley; 1999.

[7] Gambolati G, Ferronato M, Teatini P, Deidda R, Lecca G. Finite element analysisof land subsidence above depleted reservoirs with pore pressure gradient andtotal stress formulations. Int J Numer Anal Meth Geomech 2001;25(4):307–27.

[8] Gutierrez MS, Lewis RW. Coupling of fluid flow and deformation inunderground formations. J Eng Mech 2002;128(7):779–87.

[9] Gaspar FJ, Lisbona FJ, Vabishchevich PN. Staggered grid discretizations for thequasi-static Biot’s consolidation problem. Appl Numer Math 2006;56(6):888–98.

[10] Gaspar FJ, Lisbona FJ, Vabishchevich PN. A finite difference analysis of Biot’sconsolidation model. Appl Numer Math 2003;44(4):487–506.

[11] Naumovich A, Iliev OP, Gaspar FJ, Lisbona FJ, Vabishchevich PN. On numericalsolution of 1-D poroelasticity equations in a multilayered domain. Math ModelAnal 2005;10(3):287–304.

R. Asadi et al. / Computers and Geotechnics 55 (2014) 494–505 505

[12] Ewing RE, Iliev OP, Lazarov RD, Naumovich A. On convergence of certain finitevolume difference discretizations for 1D poroelasticity interface problems.Numer Methods PDEs 2007;23(3):652–71.

[13] Naumovich A, Gaspar FJ. On a multigrid solver for the three-dimensional Biotporoelasticity system in multilayered domains. Comput Vis Sci2008;11(2):77–87.

[14] Ferronato M, Castelletto N, Gambolati G. A fully coupled 3-D mixed finiteelement model of Biot consolidation. J Comput Phys 2010;229(12):4813–30.

[15] Pao WKS, Lewis RW, Masters I. A fully coupled hydro-thermo-poro-mechanicalmodel for black oil reservoir simulation. Int J Numer Anal Meth Geomech2001;25(12):1229–56.

[16] Kihm JH, Kim JM, Song SH, Lee GS. Three-dimensional numerical simulation offully coupled groundwater flow and land deformation due togroundwater pumping in an unsaturated fluvial aquifer system. J Hydrol2007;335(1):1–14.

[17] Kim J, Tchelepi HA, Juanes R. Stability and convergence of sequential methodsfor coupled flow and geomechanics: drained and undrained splits. ComputMethods Appl Mech Eng 2011;200(23):2094–116.

[18] Kim J, Tchelepi HA, Juanes R. Stability and convergence of sequential methodsfor coupled flow and geomechanics: fixed-stress and fixed-strain splits.Comput Methods Appl Mech Eng 2011;200(13):1591–606.

[19] Jha B, Juanes R. A locally conservative finite element framework for thesimulation of coupled flow and reservoir geomechanics. Acta Geotech2007;2(3):139–53.

[20] Mainguy M, Longuemare P. Coupling fluid flow and rock mechanics:formulation of the partial coupling between reservoir and geomechanicalsimulators. Oil Gas Sci Technol 2002;57(4):355–67.

[21] Armero F. Formulation and finite element implementation of a multiplicativemodel of coupled poro-plasticity at finite strains under fully saturatedconditions. Comput Methods Appl Mech Eng 1999;171(3):205–41.

[22] Minkof SE, Kridler NM. A comparison of adaptive time stepping methods forcoupled flow and deformation modeling. Appl Math Model 2006;30(9):993–1009.

[23] Ferronto M, Gambolati G, Teatini P. Ill-conditioning of finite elementporoelasticity equations. Int J Solids Struct 2001;38(34):5995–6014.

[24] Kim J. Sequential methods for coupled geomechanics and multiphase flow.Phd thesis, Stanford University; 2010.

[25] Dean RH, Gai X, Stone CM, Minkoff SE. A comparison of techniques for couplingporous flow and geomechanics. SPE J 2006;11(1):132–40.

[26] Settari A, Mourits FM. A coupled reservoir and geomechanical simulationsystem. SPE J 1998;3(3):219–26.

[27] Naumovich A. Efficient numerical methods for the Biot poroelasticity systemin multilayered domains. Phd thesis, Technical University Kaiserslautern;2007.

[28] Ewing RE, Iliev OP, Lazarov RD. A modified finite volume approximation ofsecond-order elliptic equations with discontinuous coefficients. SIAM J SciComput 2001;23(4):1335–51.

[29] Miga MI, Paulsen KD, Kennedy FE. Von Neumann stability analysis of Biot’sgeneral two-dimensional theory of consolidation. Int J Numer Meth Eng1998;43(5):955–74.

[30] Vermeer PA, Verruijt A. An accuracy condition for consolidation by finiteelements. Int J Numer Anal Methods Geomech 1981;5(1):1–14.

[31] Murad MA, Loula AFD. Improved accuracy in finite element analysis ofBiot’s consolidation problem. Comput Methods Appl Mech Eng 1992;95(3):359–82.

[32] Murad MA, Loula AFD. On stability and convergence of finite elementapproximations of Biot’s consolidation problem. Int J Numer Methods Eng1994;37(4):645–67.

[33] White JA, Borja RI. Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients.Comput Methods Appl Mech Eng 2008;197(49):4353–66.