Stabilized Mixed Finite Element Method for …...Received December 2015, Accepted September 2016 No. 15-CSME-121, EIC Accession 3886 ABSTRACT Darcy ﬂow is a steady-state model for

STABILIZED MIXED FINITE ELEMENT METHOD FOR TRANSIENT DARCY FLOW

Shahab U. Ansari1, Masroor Hussain1, Sarvat. M. Ahmad2, Ahmar Rashid1 and Suleman Mazhar3

1Faculty of Computer Science and Engineering, Ghulam Ishaq Khan Institute, Topi, Pakistan2Faculty of Mechanical Engineering, Ghulam Ishaq Khan Institute, Topi, Pakistan

3Faculty of Computer Science, Information Technology University, Lahore, PakistanEmail: [email protected]

Received December 2015, Accepted September 2016No. 15-CSME-121, EIC Accession 3886

ABSTRACTDarcy flow is a steady-state model for laminar flow of a fluid through a porous medium. The present workproposes an extended model of laminar Darcy flow by introducing dynamic pressure and velocity to theclassical formulation. The solution of the proposed time-space model is attained by discretizing the problemwith a stabilized mixed Galerkin method in space and a forward Euler method in time. The resulting matrixequation is well-posed and is solved using the conjugate gradient (CG) method. The error analysis of thenumerical solutions confirms convergence to the actual model.

Keywords: Darcy flow; transient flow; stabilized mixed method; underground water flow.

MÉTHODE DE STABILITÉ DES ÉLÉMENTS FINIS MIXTES POUR UN ÉCOULEMENTTRANSITOIRE SELON LA LOI DE DARCY

RÉSUMÉL’écoulement Darcy est un modèle stationnaire pour l’écoulement laminaire d’un fluide à travers un milieuporeux. Le présent travail propose un modèle étendu pour l’écoulement laminaire de Darcy en introduisant lapression dynamique et la vitesse à la formulation classique. La solution du modèle espace-temps proposé estobtenue par discrétisation du problème dans l’espace avec la méthode mixte-stabilisée de Galerkin et dansle temps avec la méthode d’Euler. L’équation matricielle résultante est bien posée (définie) et est résolue enutilisant la méthode du gradient conjugué (CG). L’analyse des erreurs des solutions numériques confirme laconvergence du modèle en question.

Mots-clés : écoulement transitoire; stabilisée méthode mixte; écoulement de l’eau souterraine.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 1, 2017 85

1. INTRODUCTION

Darcy flow is a mathematical model for steady-state laminar (Reynolds number ≤ 1) flow of a fluid througha porous medium [1]. For given properties of the fluid and the porous medium, the Darcy model postulatesa direct relationship between the fluid flux (or the velocity) and the pressure gradient. The model hasbeen applied to a number of simulations, such as groundwater flow, flow of petroleum in reservoirs andunderground solute transport [2–6]. The classical Darcy flow model assumes constant velocity at a givenpoint in space with respect to time. In many applications, the velocity varies with time at a given spatial pointdue to some structural and environmental changes [7–9], for example, pressure drawdown in radial flow ofpetroleum at the vicinity of production well, transient flow in the horizontal well through the fracturedformation of shale reservoir, saturated flow of groundwater in aquifer and plasma filtration through arterialwalls embedded in vascular tissues exhibiting transient flow [10–12]. To capture this transient behavior, theoriginal Darcy model needs to be extended to incorporate time dependency of the flow.

Recently, a stable scheme for a transient flow of a non-Newtonian fluid saturating a porous mediumwas examined using the Darcy–Brinkman–Forchheimer model [13]. The simultaneous approximation ofthe velocity and the thermal boundary layers are obtained using explicit finite difference scheme at eachtime step until reaching steady state solution. In a study, an inverse method based on local approximatesolutions (LAS) is proposed to invert transient flows in heterogeneous aquifers. The stabilized inversesolution includes parameters (hydraulic conductivities and specific storage coefficients) and unknown ini-tial and boundary conditions of the fluid flow [14]. Silva et al. [2011] proposed a stabilized solution ofthree-dimensional simulation of compressible viscoelastic flows with moving free surfaces [15]. The flowequations were derived from Navier–Stokes incompressible equations and solved using space-time discon-tinuous Galerkin finite element method. Flows and transfers in natural discrete fracture networks (DFNs)have also gained attention recently. The modeling of three-dimensional transient pressure diffusion in frac-ture network is also studied using quasi-steady-state method [9]. An analytical solution of transient flow inporous media is determined using Laplace transform followed by analytic element method (AEM) [16]. TheLaplace transform is first applied to the original problem resulting in time-independent Helmholtz equationwhich is solved by AEM and the solution is inverted numerically back into the time domain. Jia et al. [2010]proposed a characteristic stabilized finite element method for transient Navier–Stokes equation employinglowest equal-order conforming finite element subspace [17]. In another study, an integral method for ana-lyzing transient fluid flow through a porous medium with pressure-dependent permeability is investigated[18]. One-dimensional approximate analytical solutions have been obtained for linear and radial flow by anintegral-solution technique. In this work, the transient Darcy flow is assumed laminar (non-turbulent) whichresults in a system of linear algebraic equations following finite element discretization.

The governing partial differential equations (PDEs) of Darcy flow involve unknown pressure and ve-locity of the fluid. The mixed finite element methods have been successfully used to solve such PDEs ofmultiple variables [19]. The mixed finite element methods (MFEMs) represent a classical saddle-point op-timization problem that results in spurious solution and oscillations in the approximations. Therefore, thestability of the mixed finite element solutions is of prime importance. The instability in the numerical so-lutions also arises in convection-dominated fluid flows. Stabilized MFEMs have addressed the instabilityof the solutions by adding perturbation terms to mathematical formulation [20, 21]. Some of the well-known stabilization techniques include Streamline-Upwind/Petrov–Galerkin, Galerkin/Least-Squares andPressure-Stabilizing/Petrov–Galerkin [22–24]. Tezduyar et al. proposed a modified Galerkin/least-squaresformulation using bilinear and linear equal-order interpolation velocity-pressure elements for the computa-tion of steady and unsteady incompressible flows [21]. A variational multiscale approach is adopted using anarbitrary Lagrangian–Eulerian (ALE) formulation for incompressible flows with stable stratification. Suchformulation is suitable for applications involving moving boundaries, such as fluid-structure interaction

86 Transactions of the Canadian Society for Mechanical Engineering, Vol. 41, No. 1, 2017

Fig. 1. A 2D porous domain � with boundary 0, showing fluid flow due to negative pressure gradient in x-direction.

(FSI) [25–28]. Masud et al. (2002) proposed another stabilized mixed formulation by adding a self-adjointresidual form of Darcy’s law to the mixed formulation [29]. This stabilized mixed formulation significantlysimplifies the problem of incompatibility between solutions and allows the use of standard finite elementspaces [30–32].

The objective of this study is to develop a transient Darcy flow model based on stabilized mixed Galerkinmethod proposed by Masud et al. [2002]. The transient Darcy flow, an initial boundary value problem, isidentical to heat conduction problem described by a diffusion equation [33]. Based on the heat diffusionmodel, a stabilized mixed transient Darcy model which consists of a time-dependent damping term in addi-tion to the stiffness matrix is derived. The mixed formulation is discretized in space and time simultaneouslyusing the Galerkin method and the forward difference method, respectively. The proposed transient model isasymptotically stable due to the following factors: (i) symmetric positive definite stiffness matrix resultingfrom the stabilized Galerkin method [29], and (ii) negative gradient of the vector of unknown coefficientswith respect to time in damping term [34]. The emerging system of linear equations is well-posed andis subsequently solved by the CG method for unknown pressure and velocity of the fluid. The numericalresults of the transient model is validated by an analytical model proposed by Mongan [35]. The partialresults of the stabilized mixed Galerkin transient method of sequential and parallel implementations havebeen published in [36, 37].

The rest of the paper is outlined as follows: Section 2 describes the formulation of the stabilized mixedGalerkin method for transient Darcy flow. Section 3 provides the numerical results for convergence rateand transient effect in Darcy flow approximations. Section 4 contains a brief discussion of the results, andSection 5 concludes the paper with some future enhancements.

2. STABILIZED MIXED GALERKIN FORMULATION FOR TRANSIENT DARCY FLOW

Let � ∈R2 be the porous medium with sufficiently smooth boundary 0 and T > 0 be the time. Let κ be thepermeability of the medium and µ be the viscosity of the fluid flowing through �. A schematic diagram ofthe porous fluid flow is shown in Fig. 1. For laminar flow, the Reynolds number of the fluid is assumed tobe ≤ 1.0. If v is the average velocity of the fluid and ∇p is the pressure gradient, then the transient Darcyflow is given by the following PDEs [33]:

µ

κv+∇p = 0 in �× (0,T ) (1)

S∂p

∂t+∇ ·v= ϕ in �× (0,T ) (2)


v ·n= ψ on 0× (0,T ) (3)

Equation (1) is the fundamental Darcy’s law (also known as momentum equation) and Eq. (2) representsthe mass balance equation with a source/sink ϕ. The parameter S is the storativity of the fluid definedas the product of porosity and compressibility of the fluid and the medium [38]. It is assumed that κ,µand S are strictly positive functions and the negative sign is used to counter the negative gradient of thepressure. Equation (3) assigns the normal component of the velocity a nonzero function ψ . The givenfunctions of fluid flow, ϕ and ψ must be constrained so that

∫�ϕd�=

∫0ψd0 for steady-state is satisfied.

Equations (1–3) describe an IBVP with the following initial and boundary conditions:

p(0, t) = 0 (Dirichlet condition)

p(x,0) = 0 (initial condition)

Let the solution spaces be V ∈H(div,�) and P ∈H 1(�) for velocity and pressure, respectively. Mathe-matically, these solution spaces are defined as

V =H(div,�)= {v | v ∈ (L2(�))2,∇ ·v ∈ L2(�)} (4)

andP =H 1(�)= {p | p ∈ L2(�),∇p ∈ L2(�)} (5)

The mixed variational formulation of the governing equations can be realized using weighted residualmethod (

w,µ

κv)+ (w,∇p)+ (q, p)+ (q,∇ ·v)= (q,ϕ), for ∀w ∈ V,∀q ∈ P (6)

In the above equation, (·, ·) denotes the L2(�) inner product. Applying integration by parts on the secondterm in Eq. (6) gives (

w,µ

κv)− (∇ ·w,p)+ (q, p)+ (q,∇ ·v)= (q,ϕ) (7)

The discrete solution spaces can be approximated by finite dimensional subspaces Vh ∈ V and Ph∈ P as

Vh = {vh | vh ∈ V,vh ∈ (C0(�))2,vh|�e ∈ (Pk(�e))1} (8)

andPh= {ph | ph ∈ P,ph ∈ C0(�),ph|�e ∈ P l(�e)} (9)

The discrete weak formulation for vh ∈ Vh and ph ∈ Ph is written as(wh,

µ

κvh)− (∇ ·wh,ph)+ (qh, ph)+ (qh,∇ ·vh)= (qh,ϕ),∀wh

∈ Vh,∀qh ∈ Ph (10)

The above equation is an example of a saddle-point problem. To ensure stability, a self-adjoint residualform of Darcy’s law is added to Eq. (10) [29](

wh,µ

κvh)− (∇ ·wh,ph)+ (qh, ph)+ (qh,∇ ·vh)

+12

((−µ

κwh+∇qh

),κ

µ

(µκ

vh+∇ph))+α

2

(∇wh,

µ

κh2∇ ·vh

)= (qh,ϕ)+

α

2

(∇ ·w,

µ

κh2ϕ

)∀wh∈ Vh,qh ∈ Ph (11)


The parameter α is an arbitrary constant and h is the length of the smallest edge in an element. Thediscrete approximation of the solution in finite-dimensional subspaces using Galerkin method is defined as

vh =N∑i=1

aiNi (12)

ph =

N∑i=1

biNi (13)

where N is the number of nodes in an element, and a and b are the unknown coefficient vectors for the basis(interpolation) functions Ni of the finite subspaces. Substituting (12) and (13) in Eq. (11) results in a matrixequation of the following form:

Ceui+Keui = fe (14)

where Ce is a capacity matrix, ui is the time derivative of unknown vector at the ith node, Ke is the stiffnessmatrix, and fe is the source vector that includes Dirichlet and Neumann boundary conditions as well.

For temporal discretization of (14) the forward Euler formula is used. In this method, the time derivativeof unknown vector ui is computed iteratively by using the first-order forward difference. According to thisformula, the value ui at time n+1 is evaluated as

un+1i =

un+1i −uni1t

(15)

where 1t is the uniform discrete time step required for numerical time integration. Substituting (15) in (14)gives an iterative solution as follows:

11t

Ceun+1i +Keun+1

i = fe+11t

Ceuni (16)

Restructuring Eq. (28) for unknown vector u at time n+1 can be explicitly computed as

un+1i =

(11t

Ce+Ke

)−1(fe+

11t

Ceuni

)(17)

The solution of the above equation includes a short-lived transition period between two steady states forthe Darcy flow in response to certain structural and/or environmental changes. The system of linear algebraicequations in (17) is numerically solved using the CG method.

3. NUMERICAL RESULTS

In the simulation study, quadrilateral and triangular linear elements are used. The convergence study in-cludes quadrilateral meshes of 64, 400, 1600 and 1000 elements and triangular meshes of 400, 10000 and40000 elements. The samples of quadrilateral and triangular meshes are shown in Figs. 2a and 2b, respec-tively. The medium to be discretized is assumed to be isotropic and homogeneous with constant permeabilityκ as a function spatial coordinates. The fluid is considered as incompressible with constant viscosity µ thatdoes not depend on pressure variations. The flow is considered as laminar (non-turbulent, i.e., R ≤ 1), wherethe inertial flow is negligible as compared to viscous flow. The contribution of gravity to the flow is assumedto be zero. The boundary conditions for the current study are assumed to be zero for the Darcy pressure. Thenumerical integration involved with stiffness and damping matrices is computed by applying 2× 2 Gaussquadrature rule for quadrilaterals and 4-point quadrature rule for triangles [39]. In the numerical approxi-mations, three degrees of freedom – Darcy pressure and x- and y-components of the Darcy velocity – are


(a) (b)

Fig. 2. Meshing using (a) 8×8 mesh of 4-node quadrilaterals and (b) 16×16 mesh of 3-node triangles.

Table 1. Parameters used in the approximation and analytical solutions.Viscosity of water, µ (poise at 20◦C) 1.0Permeability of sand, κ (cm/sec) 1.0Density of water, ρ (gm/cm3) 0.998Stability factor, α 10Number of capillary tubes 7000Acceleration due to gravity, g (cm/sec2) 980Time step, τ (msec) 1.7Radius of capillary tube, a (cm) 0.01Area of porous region (cm2) 4First Bessel coefficient, a 2.405Storativity, S 0.6Reynolds number, R ≤ 1.0

evaluated at each node. Finally, for the solution of system of linear equations, the CG method is employed.For a well-posed system, the CG method converges to exact solution in O(n) steps, where n is the numberof equations [40]. Table 1 shows the list of parameters used in the study for approximation and analyticalsolutions.

3.1. Convergence StudyFor three degrees of freedom (i.e., the pressure and the two components of velocity), the stiffness matrixin (14) is symmetric and positive definite due to the stability term in the mixed formulation. The dampingmatrix is symmetric and may be indefinite. The transient system in (14) is asymptotically stable if thefollowing condition is met [34]:

duidt

< 0 (18)

Thus the system of linear equations resulting from the Finite Element Modeling (FEM) discretizationis well posed with no spurious and oscillatory solution [20, 29, 36, 37]. This validates that the stabilizedmixed Galerkin algorithm is sound and complete. The ratio of the permeability κ of the medium and the fluidviscosity µ is taken as 1.0 and the mesh-dependent factor h is set to the length of the edge in quadrilateralelements and the smallest edge in triangular elements as benchmark values [29]. The exact pressure and


Fig. 3. Approximate solution using hexahedral mesh with 8000 elements: (a) pressure, (b) x-component of velocity,and (c) y-component of velocity.

velocity are defined as follows:p = sin(πx)sin(πy) (19)

vx =−κ

µ(π cos(πx)sin(πy)) (20)

vy =−κ

µ(π sin(πx)cos(πy)) (21)

The approximate pressure and velocity are plotted in Fig. 3 for a mesh of 1600 quadrilateral elements.The numerical solution of the PDEs using MFEM introduces computational errors. Such errors are typical

in numerical methods and generally occur due to the discretization of the domain, numerical computationsand approximations of PDEs by low-order polynomials. The errors can be minimized by increasing the num-ber of elements and order of shape functions. The convergence rate of the numeric solutions is determinedusing L2 norm error for the pressure and the velocity and H 1 norm error for the pressure. The computations


Fig. 4. Convergence rate for quadrilateral elements using two stability terms as a function of mesh size.

of L2 and H 1 error norms for N number of elements are carried out using the following formulas:

‖ep‖L2 =

(N∑i=1

(pei −phei)2

)1/2

(22)

‖ev‖L2 =

(N∑i=1

(vei −vhei)2

)1/2

(23)

‖ep‖H 1 = ((ep)2+ (∇ep)

2)1/2 (24)

Figures 4 and 5 show the convergence rate of the approximate solution for quadrilateral and triangularelements. It can be inferred from the convergence study that the numerical error decreases with an increasein the number of elements and the approximated values converge to the exact values.

3.2. Transient StudyThe transient analysis is carried out on the underground water flowing through sand. The density ρ ofwater at 20◦C is about 0.998 gm/cm3 and its dynamic viscosity µ is 1.0 poise. The permeability κ of themedium is taken as 1.0 cm/s, and the storativity S for the given porous region using 7000 capillary tubes isexpected to be 0.6. The transient effect is studied for 10 time constants with time interval of approximately1.7 msec [35]. The initial velocity at t = 0 is assumed to be zero, that is, the flow starts from rest. Thenumerical transient solutions are validated using quadrilateral elements of mesh size of 400 and triangularelements of mesh size of 800. The velocity starts increasing and continues to exhibit dynamic behavioruntil it reaches a new steady-state condition. For the stabilized solution, the transient velocity is plotted inFig. 6 for quadrilateral and triangular meshes. In the transient plots, the validation of the transient velocityis achieved using Mongan’s model [35]. The empirical formula for transient velocity is given as

vemp =a2G1

8µ[1− e−t/Tc1] (25)


Fig. 5. Convergence rate for triangular elements using two stability terms as a function of mesh size.

Fig. 6. Transient Darcy velocity using (a) quadrilateral mesh of 400 elements and (b) triangular mesh of 800 elements.

where a is the radius of the cylindrical tubes, G1 is the pressure gradient step and Tc1 is the time constantdefined as follows:

Tc1 =ρ

µ

a2

α21, (26)

where ρ is the density and µ is the dynamic viscosity of the water. The parameter α1 is the first root of theBessel function that is equal to 2.405 and the value of a is set to 0.01 cm [35]. For the validation process,7000 capillary tubes are used in a region of 2 cm2, i.e., a porosity of 0.6.

3.3. Statistical Error AnalysisThe approximation model for transient Darcy flow is computed iteratively in time steps such that the error(or residual) ε(t) between the actual model and the approximation model is minimized. For a reliable


Fig. 7. Autocorrelation of the error between approximation and actual solution.

approximation, the residual should be reduced to a zero-mean white random noise η(t) as [41]

ε(t)≈ η(t) (27)

Based on Eq. (25), the accuracy of the proposed approximation model is assessed using a statistical ap-proach. In this approach, the autocorrelation of the residual between the actual solution and the solutionfrom the approximation is computed. Since pure white noise is an uncorrelated function of time, its auto-correlation function has a maximum value at zero lag (or time shift) which dies down for increasing lags.Mathematically, the autocorrelation φεε is the expected (or mean) value E of the product of the residual andthe residual delayed by τ and is estimated as follows:

φεε(τ )= E[ε(t)ε(t− τ)] for ∀τ (28)

Figure 7 presents the autocorrelation of the residual plotted as a function of lags. The autocorrelationapproaches zero with increasing lags and confines itself within the confidence bounds denoted by dottedlines.

4. DISCUSSION

An extended time-space Darcy model for transient flow is presented. The proposed transient model is solvedusing stabilized mixed Galerkin method. The linear algebraic equations resulting from stabilized MFEM arewell posed and numerically solved by the CG method. The numerical solutions of the transient model toa sinusoidal input are stable without any spurious oscillations. Figure 3 shows the approximations for thepressure and the two velocity components with negligible error. The approximation errors can be furtherminimized by increasing the mesh size or the order of the shape functions. Figures 4 and 5 shows L2 andH 1 error norms as the function of mesh size for quadrilateral and triangular elements, respectively. Theerror performance is almost the same in the two mesh types. It is worth noting that the boundary of thedomain is quite regular, and hence, the meshing is quite uniform in both triangular and quadrilateral elementtypes (see Fig. 2). For irregular boundary shapes, the results for the two element types may have been more


distinctive as the triangular elements are known to work well in complex geometry. The stability of theMFEM solutions deteriorates when the size of the mesh becomes too small in the case of mesh refinement.In the present work, the mesh is fixed, and therefore, there is no significant effect of mesh size on thesimulation results.

In transient study, the Darcy velocity exhibits a dynamic behavior in response to a change in pressure.Figure 6 shows the transient Darcy effect in the numerical solution obtained for quadrilateral and triangularmeshes. As shown in this figure, the velocity of the fluid approaches an asymptotic value of 3.14 cm/s inapproximately 5.0 msec. The numerical solution is computed iteratively in 10 time steps with an intervalof 1.7 msec. The dynamic Darcy model is validated with an analytical model proposed by Mongan [35].The validation confirms that the results obtained for the quadrilateral and triangular elements are similar. Inaddition, there is negligible influence of mesh type on the numerical approximations when the mesh is fixed(i.e., not refined). The study shows that the numerical error between the approximate model and the actualmodel decreases in the subsequent iterations. This is captured by computing the autocorrelation functionof the residual noise as shown in Fig. 7. The autocorrelation function indicates that the error approachesan uncorrelated white noise bounded by confidence limits. This result suggests that the proposed model ishighly accurate and reliable for practical purposes.

5. CONCLUSION

A stabilized mixed Galerkin method for two-dimensional laminar transient Darcy flow is presented. Thepurpose of this study is to address the inadequacy of steady-state Darcy model in dynamic porous flow ap-plications and devise an extended model to incorporate the transient behavior. The resulting system of linearequations from transient Darcy model is well posed and stable with no oscillations and spurious behaviorin the approximate solutions. For meshes of different sizes and types, the convergence study shows that thestabilized mixed finite element methods are very promising in solving IBVPs with acceptable error mar-gins. The L2 and H 1 error norms suggest that the steady-state solution converges to analytical solution withincreasing mesh size. The numerical errors are also insensitive to mesh type and size. Previous study hasshown that, for very small mesh sizes, the results deviate from exact solutions. The proposed transient modelof the Darcy flow is validated by an analytical model proposed by Mongan [35]. The model shows transitionin the flow of a fluid through a porous medium in response to a change in pressure gradient. This transientresponse in the Darcy flow is very crucial for accurate simulations in petroleum industry and solute trans-port. The statistical analysis of the dynamic Darcy model is carried out by computing the autocorrelationof the residual noise. The autocorrelation function confirms that the residual is an uncorrelated zero-meanwhite noise sequence implying that the proposed FEM approach yields high fidelity models. This work thusdemonstrates both static and dynamic approaches for successful modeling of the Darcy flow.

ACKNLOWLEDGEMENTS

The authors are very grateful to Dr. Arif Masud (University of Illinois at Urbana Champaign) for providingvaluable guidance and suggestions to improve this research work. The authors also extend their gratitude toGhulam Ishaq Khan Institute for supporting this research work.

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Stabilized Mixed Finite Element Method for …...Received December 2015, Accepted September 2016 No. 15-CSME-121, EIC Accession 3886 ABSTRACT Darcy ﬂow is a steady-state model for