Finite volume element method for the incompressible miscible displace-ment problems in porous media

Sarvesh Kumar1,∗, Neela Nataraj, and Amiya K. Pani1 Department of Mathematics, IIT Bombay, Powai, Mumbai, India.

We present the finite volume element method (FVEM) for incompressible miscible displacement problems in porous media.Optimal error estimates in L

∞(L2) are derived for velocity and concentration. Numerical experiments are also presented.

1 Introduction

A mathematical model describing miscible displacement of one incompressible fluid by another in a porous medium reservoirΩ in R

2 with boundary ∂Ω of unit thickness over a time period of J = (0, T ] is given by, see [4]:

u = −κ(x)

µ(c)∇p, ∇ · u = q ∀(x, t) ∈ Ω × J, (1)

φ(x)∂c

∂t+ u · ∇c −∇ · (D(u)∇c) = g(x, t, c) ∀(x, t) ∈ Ω × J, (2)

with boundary and initial conditions u · n = 0, D(u)∇c · n = 0 ∀(x, t) ∈ ∂Ω × J, c(x, 0) = c0(x) ∀x ∈ Ω.We use a mixed finite volume element method for the simultaneous approximation of the velocity and pressure in (1) and astandard finite volume element method for the approximation of the concentration in (2). Let Th = T be a regular, quasi-uniform partition of the domain Ω into closed triangles T and h = maxT∈Th

diam(T ).The trial space for velocity (Uh) and for pressure (Wh) are defined byUh = vh ∈ U : vh|T = (a + bx, c + by),vh · n = 0 on ∂Ω, ∀T ∈ Th , Wh = wh ∈ W : wh|T ∈ P0(T ) ∀T ∈ Th ,where U = H(div; Ω) and W = L2(Ω)/R. The test space Vh for velocity is defined byVh =

vh ∈ (L2(Ω))2 : vh|T∗

M∈ P0(T

∗M ) ∀T ∗

M ∈ T ∗h and vh · n = 0 on ∂Ω

, where T ∗

M is a control volume correspond-ing to the mid-side node M , for the construction of such type of control volumes, we refer to [2].

Setting b(vh, wh) = −∑Nm

i=1 vh(Mi) ·∫

∂T∗

Mi

wh nT∗

Mids, ∀vh ∈ Vh, ∀wh ∈ Wh, the mixed FVE approximation

corresponding to (1) is: find (uh, ph) : J −→ Uh × Wh such that for t ∈ (0, T ]

(κ−1µ(ch)uh, γhvh) + b(γhvh, ph) = 0 ∀vh ∈ Uh, (∇ · uh, wh) = (q, wh) ∀wh ∈ Wh. (3)

Here, the transfer operator γh : Uh −→ Vh defined as γhvh(x) =∑Nm

j=1 vh(Mj)χ∗j (x) ∀x ∈ Ω, where χ∗

j ’s are the scalarcharacteristic functions corresponding to the control volume T ∗

Mj. For applying the standard finite volume element method to

approximate the concentration, we define the trial space Mh on Th and the test space Lh on V∗h as follows:

Mh =zh ∈ C0(Ω) : zh|T ∈ P1(T ) ∀T ∈ Th

, Lh =

wh ∈ L2(Ω) : wh|V ∗

P∈ P0(V

∗P ) ∀V ∗

P ∈ V∗h

,

where V ∗P is a control volume corresponding the node P , for the construction V ∗

P , see [1].

Define ah(v; χ, ψh) = −

Nh∑j=1

∫∂V ∗

Pj

(D(v)∇χ · nPj

)Π∗

hψhds,

where Π∗h : Mh −→ Lh defined as Π∗

hzh(x) =∑Nh

j=1 zh(Pj)χj(x) ∀x ∈ Ω, χj’s are the characteristics functions corre-

sponding to V ∗Pj

. Then, FVE approximation corresponding to (2) is to find a solution ch : J −→ Mh such that for t ∈ (0, T ]

(φ

∂ch

∂t, Π∗

hzh

)+ (uh · ∇ch, Π∗

hzh) + ah(uh; ch, zh) = (g(ch), Π∗hzh) ∀zh ∈ Mh, ch(0) = c0,h, (4)

where c0,h is an approximation to c0.

2 Error Estimates

Write c − ch = (c − Rhc) + (Rhc − ch), where Rh is the Ritz projection is defined by

A(u; c − Rhc, χ) = ah(u; c − Rhc, χ) + (u · ∇(c − Rhc), χ) + (λ(c − Rhc), χ) = 0 ∀χ ∈ Mh. (5)

The function λ will be chosen to assure the coercivity of the bilinear form A(u; ψ, χ).

∗ Corresponding author E-mail: rajput@math.iitb.ac.in, Phone: +912225767481 Fax: +1 515 294 0258

PAMM · Proc. Appl. Math. Mech. 7, 2020015–2020016 (2007) / DOI 10.1002/pamm.200700090

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Theorem 2.1 Let (c,uh) and (ch,uh) be the solutions of (1)-(2) and (3)-(4) respectively, and let ch(0) = c0,h = Rhc(0).Then for sufficiently small h there exists a positive constant C(T ) independent of h but dependent on the bounds of κ−1 andµ such that

‖c − ch‖2L∞(J;L2) + ‖u − uh‖

2L∞(J;(L2(Ω))2) ≤ C(T )

[ ∫ T

0

(h4(‖c‖2

2 + ‖g‖21 + ‖u · ∇c‖2

1 + ‖φ∂c

∂t‖21 + ‖ct‖

22 + ‖gt‖

21 +

‖(u · ∇c)t‖21 + ‖φ

∂2c

∂t2‖21 + ‖u‖2

(H1(Ω))2 + ‖p‖21 + ‖∇ · u‖2

1) + h2(‖u‖2(H1(Ω))2 + ‖p‖2

1))ds

].

3 Numerical Experiments

For the test problems, we have taken data from [3]. The spatial domain is Ω = (0, 1000) × (0, 1000) ft2 and the time periodis [0, 3600] days. In the numerical simulation for spatial discretization, we divide in 25 number of division both x and y axis.For time discretization we take ∆t = 360 days. We have performed the numerical experiments for the cases (i): The matrixD is independent of the velocity u and (ii): The matrix D depends on u. The surface and contour plots for concentration attψ = 3 andψ tψ = 10 years, respectively are presented in Fig. 1 and Fig. 2 for case (i) and Fig. 3 and Fig. 4 for case (ii).

Fig. 3 Contour (a) and surface (b) plot at t = 3 years Fig. 4 Contour (a) and surface (b) plot at t = 10 years

Acknowledgements The first author is thankful to National Board for Higher Mathematics, India for providing financial support forattending ICIAM-07 which was held at Zurich, Switzerland 16-20 July 2007.

References

[1] R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations, Marcel Dekker, New York, 2000.[2] S. H. Chou, D. Y. Kwak and P. Vassilevski, Mixed covolume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal.,

35, 1850-1861 (1998).[3] H. Wang, D. Liang, R. E. Ewing, S. L. Lyons and G. Qin, An approximation to miscible fluid flows in porous media with point sources

and sinks by an Eulerian-Lagrangian localized adjoint method and mixed finite element methods, SIAM J. Sci. Comput., 22, 561-581(2000).

[4] Jim Douglas Jr. and R.E. Ewing, The approximation of the pressure by mixed method in the simulation of miscible displacement,R.A.I.R.O. Numer. Anal., 17, 17-33 (1983).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2020016