A finite element solution procedure for porous medium with fluid flow and electromechanical coupling

A FINITE ELEMENT SOLUTION PROCEDURE FORPOROUS MEDIUM WITH FLUID FLOWAND

ELECTROMECHANICAL COUPLING

MILOS KOJIC,1* NENAD FILIPOVIC,1 SNEZANA VULOVIC1 AND SRBOLJUB MIJAILOVIC2

1Faculty of Mechanical Engineering, University of Kragujevac, 34000 Kragujevac, Serbia, Yugoslavia2Harvard School of Public Health, Cambridge, MA 02139, U.S.A.

SUMMARY

We consider a coupled problem of the deformation of a porous solid, ¯ow of a compressible ¯uid and theelectrical ®eld in the mixture. The governing equations consist of balance of the linear momentum of solidand of ¯uid, continuity equations of the ¯uid and current density, and a generalized form of Darcy's lawwhich includes electrokinetic coupling. The compressibility of the solid and the ¯uid are taken into account.We transform these equations to the corresponding ®nite element relations by employing the principle ofvirtual work and the Galerkin procedure. The nodal point variables in our general formulation are dis-placements of solid, ¯uid pore pressure, relative velocity of the ¯uid and electrical potential. Derivation ofthe FE equations is presented for small displacements and elastic solid, which can further be generalized tolarge displacements and inelastic behaviour of the solid skeleton.According to this formulation we can include general boundary conditions for the solid, relative velocity

of the ¯uid, ¯uid pressure, current density and electrical potential. The dynamic-type non-symmetric systemof equations is solved through the Newmark procedure, while in the case of neglect of inertial terms we usethe Euler method.Numerical examples, solved by our general-purpose FE package PAK, are taken from biomechanics. The

results are compared with those available in the literature, demonstrating the correctness and generality ofthe procedure presented. # 1998 John Wiley & Sons, Ltd.

KEY WORDS porous medium; ¯uid ¯ow; electromechanical coupling; FEM

1. INTRODUCTION

Finite element solution procedures of coupled problems, which include deformation of solid and¯uid ¯ow through a porous medium, have been the subject of investigation of many authors.1±5

Implementation of these methods has been directed to geomechanics and to biomechanics. Manyimportant practical problems such as settlement of foundation under loading and with water ¯owthrough porous solid skeletons have been solved numerically, con®rming the correctness of thetheoretical basis and numerical (FE) procedures. Application of the FE analysis to these coupledproblems in biomechanics is still in the development phase.

Various approaches have been adopted, starting basically from the same fundamentalequations, in formulation of the governing relations for ¯ow through porous deformable media.

CCC 1069±8299/98/040381±12$17.50 Received 17 January 1997# 1998 John Wiley & Sons, Ltd. Accepted 10 November 1997

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

Commun. Numer. Meth. Engng, 14, 381±392 (1998)

* Correspondence to: Milos Kojic, Faculty of Mechanical Engineering, University of Kragujevac, 34000 Kragujevac,Serbia, Yugoslavia.

There are di�erences in some details related to compressibility of solid material. In our formula-tion we follow approach of Lewis.1 In transformation of the basic relations to the FE formula-tion, some authors use displacements of solid and ¯uid pressure as the nodal point variables(u±p formulation), like Lewis1 and Siriwardane,2 leading to a symmetrical system of equations;Simon3 proposes two formulations: (i) displacements of solid and relative displacements of¯uid Ð with a symmetric system; and (ii) u±p formulation Ð with a non-symmetric system;Gajo4 uses displacements of solid and ¯uid and pressure with a symmetric system of equations.Small displacements are considered in References 1±4, with material non-linearity of the solid,1,2

while in Reference 5 a large displacement formulation is given.In our presentation we give in some detail the derivation of dynamic FE equations for small

displacements and linear material behaviour, following Simon et al.3 and Lewis et al.1 for the¯uid continuity equation. This approach relies on the balance of the linear momentum of solidand ¯uid, and the continuity of ¯uid, which takes into account the compressibility of solid and of¯uid. The paper represents a generalization of the approach given in Kojic et al.6 We includeelectrokinetic coupling by using linearized electrokinetics and a continuity equation for thecurrent density according to References 7 and 8. From the electrokinetic coupling follows acorrection of the equation of balance of linear momentum for ¯uid.

The above formulation is suitable for general use, since we can implement all boundaryconditions appearing in the applications: boundary conditions for solid, relative ¯uid velocity,given current density and electrical potential.

In the next Section we summarize the fundamental equations of the coupled problem describedabove, and then, in Section 3 we derive the basic FE equations for linear dynamics. In Section 4some typical numerical examples from biomechanics are presented, and ®nally we give someconcluding remarks in Section 5.

2. FUNDAMENTAL EQUATIONS

We present here the fundamental equations, supposing ®rst that displacements of the solid aresmall, which represent the basis for the FE formulation in Section 3. The assumptions are that thesolid is linear elastic, with compressible material of the skeleton, and that the ¯uid is com-pressible. Also, we consider the dynamic problem and take into account inertial forces of thesolid and the ¯uid. We ®rst present all governing equations in the case when there is no electro-kinetic coupling and then give corrections due to this e�ect.The balance equation of the solid in the case of no electric coupling can be written in the form

�1 ÿ n�LTss � �1 ÿ n�rsb � kÿ1nq ÿ �1 ÿ n�rs �u � 0 �1�

where ss is stress in the solid phase, n is porosity, k is the permeability matrix, rs is the density ofthe solid, b is the body force per unit mass, q is the relative velocity of the ¯uid, uÈ is theacceleration of the solid material and LT is the di�erential operator.

The balance equation of the ¯uid phase (no electrokinetic coupling) is

ÿnHp � nrfb ÿ kÿ1nq ÿ nrf _vf � 0 �2�

Commun. Numer. Meth. Engng, 14, 381±392 (1998) # 1998 John Wiley & Sons, Ltd.

382 M. KOJIC ET AL.

where p is the pore ¯uid pressure, rf is the density of the ¯uid and v.f is the acceleration of the ¯uid.

This equation is also known as the generalized Darcy law. Both equilibrium equations are writtenper unit volume of the mixture. Combining equations (1) and (2) we obtain

LTs � rb ÿ r�u ÿ rf _q � 0 �3�

where s is the total stress, which can be expressed in terms of ss and p as

s � �1 ÿ n�ss ÿ nmp �4�and r� (17 n)rs � nrf is the density of mixture. Here m is a constant vector de®ned asmT� {1 1 1 0 0 0} to indicate that the pressure contributes to normal stresses only. Also, wehave taken into account that the pressure has a compressive character, while tensional stressesand strains are considered positive. In further analysis we use the e�ective stress s0, de®ned as

s0 � s � mp �5�which is relevant for the constitutive relations of the solid. Using the de®nition of relative velocityq as the volume of ¯uid passing in unit time through unit area of the mixture (also called theDarcy velocity), i.e.

q � n�vf ÿ _u� �6�we transform (3) to a form

ÿHp � rfb ÿ kÿ1q ÿ rf �u ÿ rf

n_q � 0 �7�

The next fundamental equation is the constitutive relation of the solid,

s0 � CE�e ÿ ep� �8�

where CE is the elastic constitutive matrix of the solid skeleton, e is total strain and ep is thedeformation of the solid material due to pressure,1

ep � ÿm

3K s

p �9�

Here Ks is the bulk modulus of the solid grains.Now we write the continuity equation for the ¯uid, which takes into account the compress-

ibilities of the solid and ¯uid, as well as the elastic constitutive law for the solid skeleton,1

HTq � m

T ÿ mTCE

3K s

� �_e � 1 ÿ n

K s

� n

K f

ÿ mTCEm

9K2s

!_p � 0 �10�

Now we consider electrokinetic coupling. The basic relations, which encompasses both Ohm'sand Darcy's laws as special cases, are

qj

� �� ÿk11 k12

k21 ÿk22

� �HpHf

� ��11�

# 1998 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng, 14, 381±392 (1998)

A FINITE ELEMENT SOLUTION PROCEDURE 383

where f is the electrical potential, k11 is the (short-circuit) Darcy hydraulic permeability, k22 is theelectrical conductivity, and k12 and k21 are electrokinetic coupling coe�cients that are equal byOnsager reciprocity, according to References 7 and 8. Next, we generalize the equation of balanceof linear momentum for the ¯uid, (7), to include electrokinetic coupling. The resistance force Fw

for the ¯uid follows from the ®rst equation of system (11),

Fw � ÿkÿ111 q � kÿ111 k12Hf �12�

and the equation of balance of linear momentum, (8), now changes to

ÿHp � rfb ÿ kÿ111 q � k

ÿ111 k12Hf ÿ rf �u ÿ rf

n_q � 0 �13�

Note that this equation corresponds to isotropic conditions of the solid skeleton, and represents aform of the generalized Darcy law.

Finally, we can use the continuity equation for the current density in the form

HTj � 0 �14�

Substituting the current density j from the second equation of system (11) into (14), we obtain

k21HTHp ÿ k22H

THf � 0 �15�

3. DERIVATION OF FINITE ELEMENT EQUATIONS

In this Section we transform the fundamental relations of Section 2 into the ®nite elementequations. We present derivations assuming small displacements of solid and elastic material.These equations can be generalized to large displacements of the solid and inelastic deformationof the material by a standard procedure.9±12

First, by employing the principle of virtual work and supposing that the material is elastic,from equation (4) we obtain the following equation:Z

V

deTCEe dV �

ZV

deTCEm

3K s

ÿ m

� �p dV �

ZV

duTr�u dV �ZV

duTrf _q dV

�ZV

duTrb dV �ZA

duT t dA �16�

Next, in the case of electrokinetic coupling we multiply equation (13) by the interpolation matrixHT

q for the relative velocity of ¯uid q, and integrate over the ®nite element volume according tothe Galerkin method. The resulting equation is

ÿZV

HTqHp dV ÿ k

ÿ111 k12

ZV

HTqHf dV �

ZV

HTq rfb dV ÿ k11

ZV

HTq q dV

ÿZV

HTq rf �u dV ÿ

ZV

HTq

rfn

_q dV � 0 �17�


384 M. KOJIC ET AL.

Further, we multiply the continuity equation (10) by the interpolation matrix HTp for pressure

(which is vector-column) and obtain

ZV

HTpH

Tq dV �

ZV

HTp m

T ÿmTCE

3K s

� �_e dV �

ZV

HTp

1 ÿ n

K s

� n

K f

ÿ mTCEm

9K2s

!_p dV � 0 �18�

Finally, we multiply the continuity equation (10) by the interpolation matrix HTf for electrical

potential and integrate over the volume V,

k21

ZV

HTfH

THp dV ÿ k22

ZV

HTfH

THf dV � 0 �19�

Note that (usually) in practical applications interpolation functions for displacements Hu andfor relative velocities Hq are quadratic, while Hp for pressure and Hf for electrical potential arelinear.

We employ the standard procedure of integration over the element volume in equations (16)±(19) and use the Gauss theorem. The resulting FE system of equations is

muu 0 0 0

0 0 0 0

mqu 0 0 0

0 0 0 0

266666664

377777775

�u

�p

�q

�fffff

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;�

0 0 cuq 0

cpu cpp 0 0

0 0 cqq 0

0 0 0 0

266666664

377777775

_u

_p

_q

_fffff

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

�

kuu kup 0 0

0 0 kpq 0

0 kqp kqq kqf

0 kfp 0 kff

266666664

377777775

u

p

q

fffff

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;�

fu

fp

fq

ff

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;�20�

The matrices and vectors in this equation are:

muu �ZV

HTu rHu dV mqu �

ZV

HTq rfHu dV

cuq � mTqu �

ZV

HTu rfHq dV cpu � ÿ

ZV

HTp m

T ÿ mTCE

3K s

� �B dV

cpp � ÿZV

HTp

1 ÿ n

K s

� n

K f

ÿ mTCEm

9K2s

!Hp dV cqq �

ZV

HTq

rfnHq dV



kuu �ZV

BTC

EB dV kup � c

Tpu �

ZV

BT CEm

3K s

ÿ m

� �Hp dV

kpq �ZV

HTp;xHq dV kqp �

ZV

HTqHp;x dV

kqq �ZV

HTq kÿ1Hq dV kqf � ÿkÿ111 k12

ZV

HTqHf;x dV

kfq � k21

ZV

HTf;xHq;x dV kff � ÿk22

ZV

HTf;xHf;x dV

fu �ZV

HTu rb dV �

ZA

HTu t dA fp �

ZA

HTp n

Tq dA

fq �ZV

HTq rfb dV ff �

ZA

HTfn

Tj dA

�21�

In these expressions n is the normal vector to the boundary and B is the strain±displacementtransformation matrix.

As we can see from (20) the nodal point variables are displacements of solid u, relativevelocities q, pressures p and electrical potential fffff. Boundary conditions include general boundaryconditions for the solid, relative velocities, surface pressures, current densities and electricpotential.

The system of equations (20) is non-symmetrical in general. In the case when inertial forces areneglected the system becomes symmetric. A standard Newmark method can be employed fortime integration of the system (20), as it is done in Reference 8. We note that in the case of noelectrokinetic coupling we drop-out terms corresponding to the electric potential fffff.

4. NUMERICAL EXAMPLES

We give two typical examples from biomechanics and demonstrate that our results agree withthose available in the cited references.13±15

4.1. Example 1. One-dimensional electrokinetic transduction

We consider electrokinetic transduction in cartilage in one dimension. Two transductions areanalysed: (i) mechanical-to-electrical; and (ii) electrical-to-mechanical. We impose on the top:

1. displacement with amplitude U0 , which elicits stress and electrokinetic response, and2. current density with amplitude J0 which generates a mechanical response.

Both the prescribed displacement and the current density have an oscillatory character, as givenin Figure 1(a).

The cartilage is constrained at the bottom surface z� 0 by no-¯ow, no-displacement andimpermeability with respect to currency; also, pressure p� 0 at z� h.


386 M. KOJIC ET AL.

Displacements and ¯uid velocities in the cartilage have an oscillatory character. In Figures 1(b),(c) and (d) we show distributions along the cartilage depth for various frequencies of excitation.They are practically the same as those reported in Reference 15, obtained analytically andexperimentally.

4.2. Example 2. Cartilage subjected to current density

In this example we analyse the response of cartilage poroelastic material subjected to currentdensity on the top surface, according to the function8

Jy � J0 cos�2px=l ÿ ot�

Figure 1(a) Caption on next page

Figure 1(b) Caption on next page



Figure 1(c)

Figure 1(d)

Figure 1(a±d). One-dimensional electrokinetic transduction in cartilage: (a) geometrical andmaterial data; (b) distributionof displacement amplitudes generated by displacement on top surface; (c) distribution of displacement amplitudesgenerated by current density on top surface; (d) distribution of ¯uid velocity amplitudes generated by current density


388 M. KOJIC ET AL.

where J0 is the amplitude, l is the wavelength and o is the angular frequency (o� 2pf,f� frequency). The medium is in®nite in the x-direction and is subjected to plane strainconditions in the z-direction. We use the length l� 3 mm in the analysis, which, for givenl� 2 mm, provides repetition of the solution in the x-direction. Boundary conditions are shownin Figure 2(a): top and bottom surfaces have zero displacements and no normal ¯uid ¯ow;current density ¯ux is zero through the bottom surface.

We have solved this example by using appropriate time steps, depending on frequency f.Solutions repeat after time period T� 1/f. Results are shown in Figures 2(b)±(e). They agree withthose in Reference 8, obtained analytically and by experimental investigation.

5. CONCLUSIONS

We have presented a general numerical procedure for FE analysis of coupled problems whichinclude deformation of a porous medium with ¯ow of compressible ¯uid, and with electrokineticcoupling. The generalized Darcy law is extended to include electrokinetic coupling. Theformulation also takes into account inertial forces of solid and ¯uid.

Figure 2(b) Caption on page 391

Figure 2(a) Caption on page 391



Figure 2(c) Caption opposite

Figure 2(d) Caption opposite


390 M. KOJIC ET AL.

Generalized nodal point variables are: displacements of solid, ¯uid pressure, relative ¯uidvelocity and electric potential. With these nodal point variables all boundary conditionsappearing in practice can be employed, including given current density.

The linear formulation which has a general form can be extended to geometrically andmaterially non-linear problems. Therefore, we have presented a methodology applicable togeneral problems in geomechanics and biomechanics. We have employed the implicit timeintegration, with equilibrium iteration in time step, and hence selection of time-step size isdictated by the excitation characteristics of the system. Although we have a non-symmetricsystem of equations with more variables per node than in other formulations in the cited refer-ences (for example, the u±p formulation), the presented approach o�ers generality and suitabilityfor applications.

We have implemented the presented algorithm into our general-purpose FE package PAK as aspecial module PAK-P16, so that 2D and 3D practical problems in geomechanics and bio-mechanics can be analysed. Numerical examples show the accuracy of our solutions, since theresults agree very well with those obtained with other FE formulations and other methods.

Figure 2(e)

Figure 2(a±e). Cartilage subjected to current density (two-dimensional in®nite medium): (a) geometrical and materialdata; (b) relative velocity ®eld, f� 0.001, l� 2 mm, t� 1 s. Maximum value is 9.3 nm; (c) magnitude of horizontaldisplacement jUx j , f� 0.1 to f� 0.001, x� 1.75 mm, l� 2 mm, t� 1000 s; (d) magnitude of vertical displacementjUy j , f� 0.1 to f� 0.001, x� 1.75 mm, l� 2 mm, t� 1000 s; (e) normal stress (syy) ®eld, f� 0.001, l� 2 mm, t� 1 s



REFERENCES

1. R. W. Lewis and B. A. Schre¯er, The Finite Element Method in the Deformation and Consolidation ofPorous Media, Wiley, 1987.

2. H. J. Siriwardane and C. S. Desai, `Two numerical schemes for nonlinear consolidation', Int. j. numer.methods eng., 17, 405±426 (1981).

3. B. R. Simon, J. S.-S. Su, O. C. Zienkiewicz and D. K. Paul, Èvaluation of u±w and u±p ®nite elementmethods for the dynamic response of saturated porous media using one-dimensional models', Int. j.numer. anal. methods geomech., 10, 461±482 (1986).

4. A. Gajo, A. Saetta and R. Vitaliani, Èvaluation of three- and two-®eld ®nite element methods fordynamic response of saturated soil', Int. j. numer. methods eng., 37, 1231±1247 (1994).

5. M. E. Levenston, E. H. Frank and A. J. Grodzinsky, `Variationally derived 3-®eld ®nite elementformulations for quasistatic poroelastic analysis of hydrated biological tissues', private communication.

6. M. Kojic, N. Filipovic and S. Mijailovic, À general formulation for ®nite element analysis of ¯owthrough a porous deformable medium', Theor. Appl. Mech. (Yugoslavian), 23, 67±82 (1997).

7. E. H. Frank and A. J. Grodzinsky, `Cartilage electromechanics Ð I. Electrokinetic transduction andthe e�ects of electrolyte pH and ionic strength', J. Biomech., 20, 615±627 (1987).

8. J. R. Sachs and A. J. Grodzinsky, Àn electromechanically coupled poroelastic medium driven by anapplied electric current: surface detection of bulk material properties', PCH, 11, 585±614 (1989).

9. M. Kojic, R. Slavkovic, M. Zivkovic and N. Grujovic, PAK Ð Program Package for Linear andNonlinear Structural Analysis, Heat Conduction, Mass and Heat Transfer (Fluid Mechanics), Laborat-ory for Engineering Software, Faculty of Mechanical Engineering, Kragujevac, 1996.

10. M. Kojic and K. J. Bathe, Inelastic Analysis of Solids and Structures, Wiley, in preparation.11. K. J. Bathe, Finite Element Procedure in Engineering Analysis, Prentice-Hall, Englewood Cli�s, N.J.,

1982.12. M. Kojic, R. Slavkovic, N. Grujovic and M. Vukicevic, Ìmplicit stress integration algorithm for the

modi®ed Cam-Clay material', Theor. Appl. Mech. (Yugoslavian), 20, 95±118 (1994).13. B. R. Simon, J. S. S. Wu, M.W. Cartlon, J. H. Evans and L. E. Kazarian, `Structural models for human

spinal motion segments based on a poroelastic view of the intervertebral disk', J. Biomech. Eng., 107,327±335 (1985).

14. Y. Kim, L. J. Bonassar and A. J. Grodzinsky, `The role of cartilage streaming potential, ¯uid ¯ow andpressure in the stimulation of chondrocyte biosynthesis during dynamic compression', J. Biomech., 28,1055±1066 (1995).

15. E. H. Frank and A. J. Grodzinsky, `Cartilage electromechanics Ð II. A continuum model of cartilageelectrokinetics and correlation with experiments', J. Biomech., 20, 629±639 (1987).

16. M. Kojic, R. Slavkovic, M. Zivkovic, N. Grujovic, N. Filipovic and S. Vulovic, PAK-P ProgramModule for FE Analysis of Flow Through a Porous Deformable Medium, Laboratory for EngineeringSoftware, Faculty of Mechanical Engineering, Kragujevac, 1996.


392 M. KOJIC ET AL.

A finite element solution procedure for porous medium with fluid flow and electromechanical coupling

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