The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear
partial differential equations, the pressure-velocity equation and the concentration equation. In this paper, we present a mixed finite volume element
method (FVEM) for the approximation of the
pressure-velocity equation. Since modified method of characteristics (MMOC) minimizes the grid orientation effect, for the approximation of the concentration
equation, we apply a standard FVEM combined with MMOC. A priori error estimates in L∞(L2) norm are derived for velocity, pressure and concentration.
Numerical results are presented to substantiate the validity of the theoretical results.
1. Introduction
A mathematical model describing miscible displacement of one incompressible fluid by another in a horizontal porous medium reservoir Ω⊂ℝ2 with boundary ∂Ω of unit thickness over a time period of J=(0,T] is given by
(1)u=-κ(x)μ(c)∇p∀(x,t)∈Ω×J,(2)∇·u=q∀(x,t)∈Ω×J,(3)ϕ(x)∂c∂t+u·∇c-∇·(D(u)∇c)=g(x,t,c)=(c~-c)q∀(x,t)∈Ω×J,
with boundary conditions
(4)u·n=0∀(x,t)∈∂Ω×J,(5)D(u)∇c·n=0∀(x,t)∈∂Ω×J,
and initial condition
(6)c(x,0)=c0(x)∀x∈Ω,
where x=(x1,x2)∈Ω, u(x,t)=(u1(x,t),u2(x,t)) and p(x,t) are, respectively, the Darcy velocity and the pressure of the fluid mixture, c(x,t) is the concentration of the fluid, c~ is the concentration of the injected fluid, μ(c) is the concentration dependent viscosity of the mixture, κ(x) is the 2×2 permeability tensor of the medium, q(x,t) is the external source/sink term that accounts for the effect of injection and production wells, and ϕ(x) is the porosity of the medium. Further, D(u)=D(x,u) is the diffusion-dispersion tensor
(7)D(u)=ϕ(x)[dmI+|u|(dlE(u)+dt(I-E(u)))],
where dm is the molecular diffusion, dl and dt are, respectively, the longitudinal and transverse dispersion coefficients, E(u) is the tensor that projects onto u direction, whose ijth component is given by
(8)E(u)=uiuj|u|2;1≤i,j≤2,|u|2=u12+u22,
and I is the identity matrix of order 2. g(x,t,c)=(c~-c)q is a known function representing sources denoted as g(c) for convenience and c0(x) represents the initial concentration. For physical relevance 0≤c0(x)≤1 and n denotes the unit exterior normal to ∂Ω. The following compatibility condition is imposed on the data:
(9)∫Ωq(x,t)dx=0∀t∈J,
which can be easily derived from (1)-(2) and (4). Equation (9) indicates that, for an incompressible flow with an impermeable boundary, the amount of injected fluid and the amount of fluid produced are equal. We also assume that the functions ϕ, μ, κ, and q are bounded; that is, there exist positive constants ϕ*, ϕ*, μ*, μ*, κ*, κ*, q*, D* such that
(10)0<ϕ*≤ϕ(x)≤ϕ*,0<μ*≤μ(x,c)≤μ*,hhhhhhhh0<κ*≤κ(x)≤κ*,(11)|q(x)|≤q*,D(x,u)≤D*.The authors in [1–3] have discussed mathematical theory and existence of a unique weak solution of the above system (1)–(6) under suitable assumptions on the data. The pressure-velocity equation is elliptic type while the concentration equation is convection dominated diffusion type. Since, in the concentration equation only velocity is present, one would like to find a good approximation of the velocity. Therefore, for approximating velocity, it is natural to think of some mixed methods, which provide more accurate solution for the velocity compared to the standard finite element methods. Earlier, Douglas et al. [4, 5], Ewing and Wheeler [6], and Darlow et al. [7] have discussed the mixed finite element method for approximating the velocity as well as pressure and a standard Galerkin method for the concentration equation. They have also derived optimal error estimates in L∞(L2) norm for the velocity and concentration. Recently, Kumar [8] has discussed a mixed and discontinuous FVEM for incompressible miscible displacement problems in porous media.
Since the concentration equation is a convection dominated diffusion equation, the standard numerical methods fail to provide an accurate solution for the concentration and, therefore, suitable numerical methods have been proposed in the past for the approximation of the concentration equation. The standard numerical schemes fail to provide a physically relevant solution because most of these methods suffer from grid orientation effects. The other way to minimize the grid orientation effect is to use modified methods of characteristics (MMOC). Douglas and Russell [9] introduced and analyzed MMOC for the approximation of convection dominated diffusion equations. The authors in [10–12] studied MMOC combined with Galerkin finite element methods for incompressible miscible displacement problems.
The basic idea behind the modified method of characteristics for approximating the concentration equation (3) is to set the hyperbolic part, that is, ϕ(∂c/∂t)+u·∇c, as a directional derivative.
Set (see Figure 1)
(12)ψ(x,t)=(|u(x,t)|2+ϕ(x)2)1/2=(u1(x,t)2+u2(x,t)2+ϕ(x)2)1/2.
The characteristic direction with respect to the operator ϕ(∂/∂t)+u·∇ is the unit vector
(13)s(x,t)=(u1(x,t),u2(x,t),ϕ(x))ψ(x,t).
The directional derivative of the concentration c(x,t) in the direction of s is given by
(14)∂c∂s=∂c∂tϕ(x)ψ(x,t)+u·∇cψ(x,t).
This implies that ψ(x,t)(∂c/∂s)=ϕ(x)(∂c/∂t)+u·∇c.
Direction of ψ(x,t).
Hence, (3) can be rewritten as
(15)ψ(x,t)∂c∂s-∇·(D(u)∇c)=(c~-c)q∀(x,t)∈Ω×J.Since (15) is in the form of heat equation, the behavior of the numerical solution of (15) should be better than (3) if the derivative term ∂c/∂s is approximated properly.
We choose the same time steps for pressure and concentration for simplicity. However, the analysis can be extended to the case when different time steps are chosen for velocity and concentration through minor modifications.
Let 0=t0<t1<⋯tN=T be a given partition of the time interval [0,T] with the time step size Δt. For very small values of Δt, the characteristic direction starting from (x,tn+1) crosses t=tn at (see Figure 2)
(16)xˇ=x-un+1ϕ(x)Δt,
where un+1=u(x,tn+1).
An illustration of the definition xˇ.
This suggests us to approximate the characteristic directional derivative at t=tn+1 as
(17)∂c∂s|t=tn+1≈cn+1-c(xˇ,tn)Δs=cn+1-c(xˇ,tn)((x-xˇ)2+(tn+1-tn)2)1/2,
where cn+1=c(x,tn+1).
Using (16), we obtain
(18)ψ(x,t)∂c∂s|t=tn+1≈ϕ(x)cn+1-cˇnΔt,
where c˘n=c(xˇ,tn).
Compared to the conforming finite element methods (FEM), the finite volume methods are conservative in nature, and, hence, they preserve the physical conservative properties. In a mixed FVEM, one uses two different kinds of grids: a primal grid and a dual grid (see Figures 3 and 4). Mixed FVEM can also be thought of as a Petrov-Galerkin method. The analysis of these methods is based on the tools borrowed from the mixed FEM. Using a transfer operator which maps the trial space to the test space, the mixed covolume methods can be put in the framework of mixed FEM. This transfer operator plays a vital role in deriving the optimal error estimates. Earlier, Chou et al. [13, 14] have discussed and analyzed mixed covolume or FVEM for the second-order linear elliptic problems in two-dimensional domains. The standard FVEM can also be considered as a Petrov-Galerkin finite element method in which the trial space is chosen as C0 piecewise linear polynomials on the finite element partition of the domain and the test space, as piecewise constants over the control volumes are to be defined in Section 2. For more details on finite volume methods, we refer to [15–18] and references there in. In this paper, we present a mixed FVEM for approximating the pressure-velocity equations (1)-(2) and (4) and a standard FVEM combined with MMOC for the approximation of the concentration equations (15) and (5)-(6). This paper is organized as follows. In Section 2, FVE approximation procedure is discussed. A priori error estimates for velocity, pressure, and concentration are presented in Section 3. Finally in Section 4, the numerical procedure is discussed and some numerical experiments are presented.
Primal grid 𝒯h and dual grid 𝒯h*.
Primal grid 𝒯h and dual grid 𝒱h*.
2. Finite Volume Element Approximation
Let U={v∈H(div;Ω):v·n=0on∂Ω}. Note that (1)-(2) with boundary condition (4) has a solution for pressure, which is only unique up to an additive constant. The nonuniqueness of (1)-(2) may be avoided by considering the following quotient space:
(19)W=L2(Ω)ℝ.
Multiply (1) and (2) by v∈U and w∈W, respectively, and integrate over Ω. A use of Green’s formula and v·n=0on∂Ω yields the following weak formulation: Find (u,p):J¯→U×W satisfying
(20)(κ-1μ(c)u,v)-(∇·v,p)=0∀v∈U,(∇·u,w)=(q,w)∀w∈W.
We use a mixed FVEM for the simultaneous approximation of velocity and pressure in (1)-(2) and a standard FVEM for the approximation of the concentration in (15). For this purpose, we introduce three kinds of grids: one primal grid and two dual grids.
Let 𝒯h={T} be a regular, quasiuniform partition of the domain Ω- into closed triangles T; that is, Ω¯=∪T∈𝒯hT¯. Let hT=diam(T) and h=maxT∈𝒯hhT. Let the trial function spaces Uh and Wh associated with the approximation of velocity and pressure, respectively, be the lowest order Raviart-Thomas space for triangles defined by
(21)Uh={vh∈U:vh∣T=(a+bx,c+by)∀T∈𝒯h},Wh={wh∈W:wh∣Tisaconstant∀T∈𝒯h}.
Let us define the discrete norm for vh=(vh1,vh2)∈Uh as
(22)∥vh∥1,h2=∥vh∥(L2(Ω))22+|vh|1,h2,
where |vh|1,h2=∑T∈𝒯h∥∇υh1∥0,T2+∥∇υh2∥0,T2. For υh∈Uh, it is straight forward to check that
(23)∥vh∥1,h≤C∥vh∥H(div;Ω),
where C is a constant independent of h. For vh∈Uh the inequality
(24)∥vh∥(L∞(Ω))2≤C(log1h)1/2∥vh∥1,h
also holds true when Ω is in ℝ2 and the triangulation 𝒯h is quasiuniform and can be proved using the same arguments as in the proof of Lemma 4 in [19, pp. 67].
The dual grid 𝒯h* consists of interior quadrilaterals and boundary triangles, which are constructed by joining barycenter to the vertices. For the construction of the dual grid 𝒯h* we refer to [14]. In general, let TM* denote the dual element corresponding to the midside node M. The union of all the dual elements/control volume elements forms a partition 𝒯h* of Ω-. The test space Vh is defined by
(25)Vh={vh∈(L2(Ω))2:vh∣TM*isaconstantvectorjjjj(L2(Ω))2∀TM*∈𝒯h*andvh·n=0on∂Ω},
where TM* denotes the dual element corresponding to the midside node M. For connecting our trial and test spaces, we define a transfer operator γh:Uh→Vh by
(26)γhvh(x)=∑i=1Nmvh(Mi)χi*(x)∀x∈Ω,
where Nm is the total number of midside nodes and χi*’s are the scalar characteristic functions corresponding to the control volume TMj* defined by
(27)χi*(x)={1,ifx∈TMi*,0,elsewhere.
Multiplying (1) by γhvh∈Vh, integrating over the control volumes TM*∈𝒯h*, applying the Gauss’s divergence theorem, and summing up over all the control volumes, we obtain
(28)(κ-1μ(c)u,γhvh)-∑i=1Nmvh(Mi)·∫TMi*pnTMi*ds=0∀vh∈Uh,
where nTMi* denotes the outward normal vector to the boundary of TMi*. Set
(29)b(γhvh,wh)=-∑i=1Nmvh(Mi)·∫∂TMi*whnTMi*dsggggggggg∀vh∈Uh,∀wh∈Wh.
Then, the mixed FVE approximation corresponding to (1)-(2) can be written as follows: find (uh,ph):J¯→Uh×Wh such that, for t∈(0,T],
(30)(κ-1μ(ch)uh,γhvh)+b(γhvh,ph)=0∀vh∈Uh,(∇·uh,wh)=(q,wh)∀wh∈Wh,
where ch is an approximation to c obtained from (34).
Now, we introduce a dual mesh 𝒱h* based on 𝒯h which will be used for the approximation of the concentration equation. For construction we refer to [18].
For applying the standard FVEM to approximate the concentration, we define the trial space Mh on 𝒯h and the test space Lh on 𝒱h* as follows:
(31)Mh={zh∈C0(Ω-):zh∣T∈P1(T)∀T∈𝒯h},Lh={wh∈L2(Ω):wh∣VP*isaconstant∀VP*∈𝒱h*}l,
where VP* is the control volume associated with node P. Again, we define a transfer function Πh*:Mh→Lh by
(32)Πh*zh(x)=∑j=1Nhzh(Pj)χj(x)∀x∈Ω,
where Nh is the total number of vertices and χj’s are the characteristic functions corresponding to the control volume VPj* given by
(33)χj(x)={1,ifx∈VPj*,0,elsewhere.
For obtaining a finite volume approximation ch to the concentration c, we multiply (15) by Πh*zh∈Mh, integrate over the control volumes VPj*∈𝒱h*, and apply the Gauss’s divergence theorem. Then we sum up over the control volumes to obtain the FVE approximation ch corresponding to the concentration equation (15) as a solution ch:J¯→Mh such that for t∈(0,T],
(34)(ψ∂ch∂s,Πh*zh)+ah(uh;ch,zh)+(chq,Πh*zh)=(c~q,Πh*zh)∀zh∈Mh,ch(0)=c0,h,
where c0,h is an approximation to c0 to be defined later and the bilinear form ah(v;·,·) is defined by
(35)ah(v;χ,ϕh)=-∑j=1Nh∫∂VPj*(D(v)∇χ·nPj)Πh*ϕhds,nPj being the unit outward normal to the boundary of VPj* with v∈U, χ∈H1(Ω), and ϕh∈Mh.
Remark 1.
The three grids are introduced each for the pressure, velocity, and concentration variables. This is to balance the number of unknowns and the equations in the coupled systems (30) and (34).
To approximate the concentration at any time, say tn+1, we use the approximation to the velocity at the previous time step. The fully discrete scheme corresponding to (30) and (34) is defined as follows. For n=0,1…N, find (chn,phn,uhn)∈Mh×Wh×Uh such that
(36)ch0=Rhc(0),(37)(κ-1μ(chn)uhn,γhvh)+b(γhvh,phn)=0∀vh∈Uh,(38)(∇·uhn,wh)=(qn,wh)∀wh∈Wh,(39)(ϕchn+1-c^hnΔt,Πh*χh)+ah(uhn;chn+1,χh)+(qn+1chn+1,Πh*χh)=(qn+1c~n+1,Πh*χh)∀χh∈Mh,
where c^hn=ch(x^,tn)=ch(x-(uhn/ϕ)Δt,tn) and Rhc is a projection of c onto Mh which will be defined in (45).
Note that in (17). we use the following notation for the exact velocity:
(40)cˇn=c(xˇ,tn)=c(x-un+1ϕΔt,tn).
3. Error Estimates3.1. Error Estimates for Velocity
For a given c, the auxiliary functions (u~h,p~h):[0,T]→Uh×Wh are defined as follows:
(41)(κ-1μ(c)u~h,vh)-(∇·vh,p~h)=0∀vh∈Uh,(∇·u~h,wh)=(q,wh)∀wh∈Wh.
For a proof of existence and uniqueness of the discrete solution of (41), we refer to [20, pp. 52]. For uh and u~h, the following error estimates can be obtained (see [14]):
(42)∥uh-u~h∥(L2(Ω))2≤C(h∥u~h∥(L2(Ω))2+∥c-ch∥∥u~h∥(L∞(Ω))2).
Now, since concentration depends on the velocity and vice versa, to derive the error estimates for the concentration, we also need error estimates for the velocity. For elliptic problems, the authors in [14] have derived error estimates for mixed covolume method by using Raviart-Thomas projection and L2 projection. In the similar way, for a given c, the following theorem can be shown but the proof is long so we omit it here.
Theorem 2.
Assume that the triangulation 𝒯h is quasiuniform. Let (u,p) and (uh,ph), respectively, be the solutions of (1)-(2) and (30). Then, there exists a positive constant C independent of h but dependent on the bounds of κ-1 and μ such that
(43)∥u-uh∥(L2(Ω))2+∥p-ph∥≤C[∥c-ch∥+h(∥u∥(H1(Ω))2+∥p∥1)],(44)∥∇·(u-uh)∥≤Ch∥∇·u∥1.
3.2. Error Estimates for Concentration
We write c-ch=(c-Rhc)+(Rhc-ch), where Rh:H1(Ω)→Mh is the projection of c defined by
(45)A(u;c-Rhc,χh)=0∀χh∈Mh,
where
(46)A(u;ϕ,χh)=ah(u;ϕ,χh)+(qϕ,χh)+(λϕ,χh)∀χh∈Mh.
The function λ will be chosen such that the coercivity of A(u;·,·) is assured.
We use frequently the following trace inequality [21, pp. 417]: for w∈H1(T),
(47)∥w∥∂T2≤C(hT-1∥w∥T2+hT|w|1,T2),
where ∥w∥∂T2=∫∂Tw2ds. Further, we need the following inverse inequalities (see [22, pp. 141]): ∀χh∈Mh(48)∥χh∥j,∞≤Ch-1∥χh∥1,j=0,1,ggggg∥χh∥1≤Ch-1∥χh∥.
Using the properties of Πh* operator, it is easy to see that, for T∈𝒯h and ϕh∈Mh the following holds true:
(49)∫T(ϕh-Πh*ϕh)dx=0,∫∂T(ϕh-Πh*ϕh)ds=0.
Now, using (49), we have the following lemma [17, pp. 317].
Lemma 3.
There exists a positive constant C independent of h such that
(50)jjjjjjj|ϵh(ϕ,χ)|≤Chi+j|ϕ|Wpi|χ|Wqj∀χ∈Mhwithi,j=0,1,1p+1q=1,
where
(51)ϵh(ϕ,χh)=(ϕ,χh)-(ϕ,Πh*χh)∀χh∈Mh.
Also note that, by the usual interpolation theory, the operator Πh* has the following approximation property [23, pp. 466]:
(52)jj∥χ-Πh*χ∥0,k≤Chβ|χ|s,k,0≤β≤s≤1,1≤k≤∞.
We also recall two well-known results (Lemmas 4 and 5), which will be used in the the proof of Lemmas 7–9.
Lemma 4 (see [24, pp. 240]).
The operator Πh* has the following properties.
With ∥|ϕh|∥=(ϕh,Πh*ϕh)1/2, the norms ∥|·|∥ and ∥·∥ are equivalent on Uh; that is, there exist positive constants C7 and C8, independent of h, such that
(53)C7∥ϕh∥≤∥|ϕh|∥≤C8∥ϕh∥∀ϕh∈Mh.
Πh* is stable with respect to the L2 norm; that is, there exists a positive constant C independent of h such that
(54)∥Πh*χh∥≤C∥χh∥∀χh∈Mh.
Lemma 5 (see [25, pp. 1871]).
Assume that χh,ϕh∈Mh. Then one has
(55)ah(u;χh,ϕh)=a(u;χh,ϕh)+∑T∈𝒯h∫∂T(D(u)∇χh·n)(Πh*ϕh-ϕh)ds+∑T∈𝒯h∫T∇·(D(u)∇χh)(ϕh-Πh*ϕh)dx.
Moreover, for χh∈Mh,
(56)ah(u;χh,χh)≥a(u;χh,χh)-Ch|χh|12.
Remark 6.
Note that (55) also holds true for χ∈H1(Ω).
Since D is uniformly positive definite, we obtain following from (56):
(57)ah(u;χh,χh)≥(α-Ch)|χh|12.
Choose h0>0 such that, for 0<h<h0, (α-Ch)=α0>0 and hence
(58)ah(uh;χh,χh)≥α0|χh|12∀χh∈Mh.
Now, we derive the error bound in H1 and L2 norms for c-Rhc. Let Ih be the continuous interpolant onto Mh satisfying the following approximation properties. For ϕ∈Hk+1(Ω) with k=0,1, we have [22]
(59)∥ϕ-Ihϕ∥j≤hk+1-j∥ϕ∥k+1j=0,1.
Moreover, if ϕ∈W2,∞(Ω), then
(60)∥ϕ-Ihϕ∥1,∞≤Chlog(1h)∥ϕ∥2,∞.
Lemma 7.
There exists a positive constant C independent of h such that
(61)∥c-Rhc∥1≤Ch∥c∥2,
provided c∈H2(Ω), for a.e. t∈(0,T].
Proof.
The coercivity and boundedness of A(u;·,·) with (45) yield
(62)∥Ihc-Rhc∥12≤CA(u;Ihc-Rhc,Ihc-Rhc)≤CA(u;Ihc-c,Ihc-Rhc)≤C∥c-Ihc∥1∥Ihc-Rhc∥1,
and hence
(63)∥Ihc-Rhc∥1≤C∥c-Ihc∥1,
where C depends on the bound of D(u) given in (11). Combine the estimates (63) and (59) and use the triangle inequality to complete the proof.
For deriving the L2 error bounds for c-Rhc, we need the following lemma.
Lemma 8. Let ϵa(u;ϕ,χh)=a(u;ϕ,χh)-ah(u;ϕ,χh). There exists a positive constant C such that, for ϕ∈H1(Ω) and χh∈Mh,
(64)|ϵa(u;c-Rhc,ϕh)|≤Ch2(|g|1+|ψ∂c∂s|1+∥c∥2)|ϕh|1∀ϕh∈Mh.
Using (55) (see also Remark 6), we find that
(65)|ϵa(u;c-Rhc,ϕh)|≤|∑T∈𝒯h∫T∇·(D(u)∇(c-Rhc))(ϕh-Πh*ϕh)dx|+|∑T∈𝒯h∫∂T(D(u)∇(c-Rhc)·n)(ϕh-Πh*ϕh)ds|=J1+J2.
To bound J1, first we use the fact that Rhc is linear on each triangle T to obtain
(66)J1=|∑T∈𝒯h∫T∇·(D(u)∇(c-Rhc))gggggggg∑T∈𝒯h×(ϕh-Πh*ϕh)dx|=|∑T∈𝒯h∫T(∇·(D(u)∇c)-(∇·D(u))·∇Rhc)gggggggg∑T∈𝒯h×(ϕh-Πh*ϕh)dx|.
Now use (3), (49), and (50) to obtain
(67)J1≤|∑T∈𝒯h∫T(-g+ψ∂c∂s)(ϕh-Πh*ϕh)dx|+|∑T∈𝒯h∫T[(∇·D(u)-(∇·D(u))T)·∇Rhc]gggggggg∑T∈𝒯h×(ϕh-Πh*ϕh)dx|≤Ch2(|g|1+|ψ∂c∂s|1+∥c∥2)|ϕh|1,
where (∇·D(u))T denotes the average value of ∇·D(u) on triangle T.
Based on the analysis in [25, pp. 1873], we estimate J2 as follows. Note that an appeal to the continuity of ∇c·n with (49) yields
(68)J2=|∑T∈𝒯h∫∂T((D-D¯T)∇(c-Rhc)·n)gggggggg∑T∈𝒯h×(ϕh-Πh*ϕh)ds|,
where D=D(u) and D¯T is a function such that, for any edge of a triangle T∈𝒯h,
(69)D¯T(x)=D(xc),x∈E,
and xc is the midpoint of E. Since |D(x)-D¯T|≤ChT∥D∥1,∞, we use trace inequalities (47) and (61) to arrive at
(70)J2≤Ch|∑T∈𝒯h∫∂T(∇(c-Rhc)·n)(ϕh-Πh*ϕh)ds|≤Ch(∑T∈𝒯h∫∂T|(∇(c-Rhc)·n)|2)1/2×(∑T∈𝒯h∫∂T|ϕh-Πh*ϕh|2)1/2≤Ch(hT-1/2∥c-Rhc∥1+hT1/2∥c∥2)×(hT-1/2∥ϕh-Πh*ϕh∥+hT1/2|ϕh|1)≤Ch2∥c∥2|ϕh|1.
Substitute the estimates of J1 and J2 in (65) to complete the rest of the proof.
Lemma 9.
There exists a positive constant C independent of h such that
(71)∥c-Rhc∥≤Ch2(∥c∥2+|g|1+|ψ∂c∂s|1),
provided c∈H2(Ω),u·∇c∈H1(Ω), and ∂c/∂t∈H1(Ω) for t∈(0,T] a.e.
Proof.
To obtain optimal L2 error estimates for c-Rhc, we now appeal to Aubin-Nitsche duality argument. Let ϕ∈H2(Ω) be a solution of the following adjoint problem:
(72)-∇·(D(u)∇ϕ+uϕ)+λϕ=c-RhcinΩ,(D(u)∇ϕ+uϕ)·n=0on∂Ω,
which satisfies the elliptic regularity condition
(73)∥ϕ∥2≤C∥c-Rhc∥.
Multiply the above equation by c-Rhc and integrate over Ω. An integration by parts and a use of (45) yield
(74)∥c-Rhc∥2=a(u;ϕ,c-Rhc)-(u·∇ϕ,c-Rhc)-(∇·uϕ,c-Rhc)+λ(ϕ,c-Rhc)=[a(u;c-Rhc,ϕ-ϕh).+(u·∇(c-Rhc),ϕ-ϕh).+λ(c-Rhc,ϕ-ϕh)].+ϵa(u;c-Rhc,ϕh)∀ϕh∈Mh=I1+I2.
For I1, use (61) to find that
(75)|I1|=|a(u;c-Rhc,ϕ-ϕh)ggl+(u·∇(c-Rhc),ϕ-ϕh)ggl+λ(c-Rhc,ϕ-ϕh)|≤C∥c-Rhc∥1∥ϕ-ϕh∥1≤Ch∥c∥2∥ϕ-ϕh∥1.
The bound for I2 follows from Lemma 8 and hence
(76)|I2|≤Ch2(|g|1+|ψ∂c∂s|1+∥c∥2)|ϕh|1.
Substitute (75) and (76) in (74) to find that
(77)∥c-Rhc∥2≤C[h∥c∥2∥ϕ-ϕh∥1|ψ∂c∂s|1hhhh+h2(|g|1+|ψ∂c∂s|1+∥c∥2)|ϕh|1].
Now choose ϕh=Ihϕ in (77). Then use elliptic regularity condition (73) with (59) to obtain
(78)∥c-Rhc∥≤Ch2(∥c∥2+|g|1+|ψ∂c∂s|1),
and this completes the proof.
A use of inverse inequalities (48), (59), (60), and (63) yields
(79)∥Rhc∥1,∞≤C∥c∥2,∞.
Lemma 10.
There exists a positive constant C such that ∀θ∈Mh,
(80)|ah(u;Rhc,θ)-ah(uh;Rhc,θ)|≤C(∥u-uh∥(L2(Ω))2+h∥∇·(u-uh)∥)|θ|1.
Proof.
Note that
(81)|ah(u;Rhc,θ)-ah(uh;Rhc,θ)|=|∑i=1Nh∫∂Vi*(D(u)-D(uh))∇Rhc·niΠh*θds|=|∑T∈𝒯hKT|,
where KT=∑l=13∫∂Vl*∩T(D(u)-D(uh))∇Rhc·nlθlds and θl=θ(Pl); see Figure 5. For each triangle T, KT can be written as
(82)KT=∑l=13∫MlB(D(u)-D(uh))∇Rhc·nlhhhhhh×(θl+1-θl)ds(θ4=θ1).
Using the Cauchy-Schwarz inequality and (79), we obtain
(83)KT≤∑l=13|θl+1-θl|×∫MlB|(D(u)-D(uh))∇Rhc·nl|ds≤C∑l=13|θl+1-θl|∥D(u)-D(uh)∥(L2(MlB))2×2×(meas(MlB))1/2.
It has been proved in [26, pp. 332] that the matrix D is uniformly Lipschitz; that is, there exists a constant C such that for u and v∈(L2(Ω))2,
(84)∥D(u)-D(v)∥(L2(Ω))2×2≤C∥u-v∥(L2(Ω))2.
A use of the trace inequalities (47) and (84) yields
(85)KT≤C∑l=13|θl+1-θl|∥u-uh∥(L2(MlB))2hT1/2≤ChT1/2∑l=13|θl+1-θl|[hT-1/2∥u-uh∥Tggggggggggggggggl+hT1/2∥∇·(u-uh)∥T].
Now, using Taylor series expansion, we find that
(86)|θl+1-θl|≤hT[|∂θ∂x|+|∂θ∂y|]≤[(|∂θ∂x|2+|∂θ∂y|2)hT2]1/2=|θ|1,h,T,l=1,2,3.
Let |ϕh|1,h=(∑T∈𝒯h|ϕh|1,h,T2)1/2. Since θ is linear on each triangle, seminorms |·| and |·| are identical.
Substitute (86) in (85) and use the estimates for KT to arrive at
(87)|ah(u;Rhc,θ)-ah(uh;Rhc,θ)|≤C(∥u-uh∥(L2(Ω))2+h∥∇·(u-uh)∥)|θ|1,
and this completes the rest of the proof.
A triangular partition.
Now, we prove our main theorem.
Theorem 11.
Let cn and chn be the solutions of (3) and (39) at t=tn, respectively, and let ch(0)=c0,h=Rhc(0). Further assume that Δt=O(h). Then, for sufficiently small h, there exists a positive constant C(T) independent of h but dependent on the bounds of κ-1 and μ such that
(88)max0≤n≤N∥cn-chn∥(L2(Ω))22≤C[h4(∥(c-c~)q∥L∞(0,T;H1)2∥ψ∂2c∂t∂s∥L∞(0,T;H1)2ffffffffffffhhl+∥ψ∂c∂s∥L∞(0,T;H1)2+∥c∥L∞(0,T;H2)2ffffffffffffhhl+∥∇·u∥L∞(0,T;H1)2+∥ct∥L∞(0,T;H2)2ffffffffffffhhl+∥ψ∂2c∂t∂s∥L∞(0,T;H1)2)+(Δt)2(∥ut∥L2(0,T;(L2(Ω))2)2+∥(∇·u)t∥L2(0,T;L2)2∥ψ∂2c∂t∂s∥L∞(0,T;H1)2ffhhhhffhh+∥∂2c∂τ2∥L2(0,T,L2)2)∥ψ∂2c∂t∂s∥L∞(0,T;H1)2+h2(∥u∥L∞(0,T;(H1(Ω))2)2+∥p∥L∞(0,T;H1)2)].
Proof.
Write cn-chn=(cn-Rhcn)-(chn-Rhcn)=ρn-θn. Since the estimates of ρn are known, it is enough to estimate θn.
Multiplying (3) by Πh*χh and subtracting the resulting equation from (39) at t=tn+1, we obtain
(89)(ϕchn+1-c^hnΔt,Πh*χh)+ah(uhn;chn+1,χh)-ah(un+1;cn+1,χh)+(qn+1chn+1,Πh*χh)-(qn+1cn+1,Πh*χ)=(un+1·∇cn+1+ϕ∂cn+1∂t,Πh*χh)∀χh∈Mh.
Choose χh=θn+1 in (89) and use the definition of Rh to obtain
(90)(ϕθn+1-θnΔt,Πh*θn+1)+ah(uhn;θn+1,θn+1)=(θn+1qn+1,Πh*θn+1)+(ρn+1,θn+1)+(ρn+1qn+1,θn+1-Πh*θn+1)+[ah(un+1;Rhcn+1,θn+1)-ah(uhn+1;Rhcn+1,θn+1)]+(un+1·∇cn+1+ϕ∂cn+1∂t-ϕ(cn+1-cˇn)Δt,Πh*θn+1)+(ϕ(ρn+1-ρˇn)Δt,Πh*θn+1)-(ϕ(θn-θ^n)Δt,Πh*θn+1)+(ϕ(R^hcn-Rˇhcn)Δt,Πh*θn+1)=T1+T2+T3+T4+T5+T6+T7+T8.
To estimate T1, T2, and T3, we use the Cauchy-Schwarz inequality, boundedness of q, and (54) to obtain
(91)|T1|=|(θn+1qn+1,Πh*θn+1)|≤C∥θn+1∥2,|T2|=|(ρn+1,θn+1)|≤C∥ρn+1∥∥θn+1∥,|T3|=|(ρn+1qn+1,θn+1-Πh*θn+1)|≤C∥ρn+1∥∥θn+1∥.
To bound T4, we use Lemma 10 to obtain
(92)|T4|=|ah(un+1;Rhcn+1,θn+1)hh-ah(uhn+1;Rhcn+1,θn+1)|≤C(∥un+1-uhn∥(L2(Ω))2hhhh∥un+1-uhn∥(L2(Ω))2+h∥∇·(un+1-uhn+1)∥)|θn+1|1≤C(∥un+1-un∥(L2(Ω))2hhhh+∥un-uhn∥(L2(Ω))2hhhh+h∥∇·(un+1-un)∥hhhh∥un+1-un∥(L2(Ω))2+h∥∇·(un-uhn)∥)|θn+1|1.
Since
(93)|un+1-un|2=|∫tntn+1utds|2≤Δt∫tntn+1|ut|2ds,
hence,
(94)∥un+1-un∥(L2(Ω))22≤Δt∥ut∥L2(tn,tn+1;(L2(Ω))2)2,
and, similarly,
(95)∥∇·(un+1-un)∥L2(Ω)2≤Δt∥∇·ut∥L2(tn,tn+1;L2(Ω))2.
Hence,
(96)|T4|≤C[(Δt)1/2(∥ut∥L2(tn,tn+1,(L2(Ω))2)hhhhhhhhhh∥ut∥L2(tn,tn+1,(L2(Ω))2)+h∥(∇·u)t∥L2(tn,tn+1,L2(Ω)))gggg+∥un-uhn∥(L2(Ω))2gggg∥ut∥L2(tn,tn+1,(L2(Ω))2)+h∥∇·(un-uhn)∥]|θn+1|.
Using the Cauchy-Schwarz inequality and (54), we obtain
(97)|T5|=|(un+1·∇cn+1+ϕ∂cn+1∂t-ϕ(cn+1-c^n)Δt,Πh*θn+1)|≤C∥un+1·∇cn+1+ϕ∂cn+1∂t-ϕ(cn+1-c^n)Δt∥∥θn+1∥.
Let σ(x)=[ϕ(x)2+un+1(x)2]1/2, so that
(98)ϕ∂cn+1∂t+un+1·∇cn+1=σ∂cn+1∂τ,
where τ approximates the characteristic unit vector s. Now using the same arguments given in [12], we have the following bound:
(99)∥σ∂cn+1∂τ-ϕcn+1-cˇnΔt∥2≤CΔt∥∂2c∂τ2∥L2(tn,tn+1;L2)2.
Hence, T5 is bounded as follows:
(100)|T5|≤C(Δt)1/2∥∂2c∂τ2∥L2(tn,tn+1;L2)∥θn+1∥.
To bound T6, we proceed as follows:
(101)|T6|=|(ϕ(ρn+1-ρˇn)Δt,Πh*θn+1)|≤|(ϕ(ρn+1-ρˇn)Δt,θn+1-Πh*θn+1)|+|(ϕ(ρn+1-ρˇn)Δt,θn+1)|=I1+I2.
Now I1 can be written as
(102)I1≤|(ϕ(ρn+1-ρn)Δt,θn+1-Πh*θn+1)|+|(ϕ(ρn-ρˇn)Δt,θn+1-Πh*θn+1)|.
Following the proof lines of Theorem 4.1 in [12], it can be shown that, for f∈H1(Ω), there exists a positive constant C independent of h and Δt such that
(103)∥f-fˇ∥≤CΔt∥∇fn∥.
A use of (103) yields
(104)∥(ρn-ρˇn)Δt∥≤C∥∇ρn∥.
It is easy to show that
(105)∥(ρn+1-ρn)Δt∥≤C(Δt)-1/2∥∂ρ∂t∥L2(tn,tn+1,L2).
Using (52), (104), (105), and the Cauchy-Schwarz inequality, we obtain
(106)I1≤C[∥θn+1-Πh*θn+1∥∥∂ρ∂t∥L2(tn,tn+1,L2)hhhhh×((Δt)-1/2∥∂ρ∂t∥L2(tn,tn+1,L2)+∥∇ρn∥)]≤Ch[(Δt)-1/2∥∂ρ∂t∥L2(tn,tn+1,L2)ggggl∥∂ρ∂t∥L2(tn,tn+1,L2)+∥∇ρn∥]∥∇θn+1∥.I2 can be bounded as follows:
(107)I2≤|(ϕ(ρn+1-ρn)Δt,θn+1)|+|(ϕ(ρn-ρˇn)Δt,θn+1)|=J1+J2.
A use of Cauchy-Schwarz inequality yields
(108)J1≤C(Δt)-1/2∥∂ρ∂t∥L2(tn,tn+1,L2)∥θn+1∥.
In order to bound J2, we use the following result.
If η∈L2(Ω) and ηˇ(x)=η(xˇ) with xˇ=x-r(x)Δt, for a nonzero function r(x) such that r and ∇·r are bounded, then
(109)∥η-ηˇ∥-1≤C∥η∥Δt,
where C is a positive constant independent of h and Δt; for a proof, we refer to [9, pp. 875]. Using (109), we obtain
(110)J2≤C∥(ρn-ρ^n)Δt∥-1∥θn+1∥1≤C∥ρn∥∥θn+1∥1.
This implies that
(111)I2≤C((Δt)-1/2∥∂ρ∂t∥L2(tn,tn+1,L2)+∥ρn∥)∥θn+1∥.
Now, using (106), (111), and (101), we obtain the following bound for T6:
(112)|T6|≤C((Δt)-1/2∥∂ρ∂t∥L2(tn,tn+1,L2)hhhhh∥∂ρ∂t∥L2(tn,tn+1,L2)+h∥∇ρn∥+∥ρn∥)∥θn+1∥.
Using the Cauchy-Schwarz inequality, we obtain
(113)|T7|≤|((θn-θ^n)Δt,Πh*θn+1-θn+1)|+|((θn-θ^n)Δt,θn+1)|≤∥(θn-θ^n)Δt∥∥Πh*θn+1-θn+1∥+∥(θn-θ^n)Δt∥-1∥θn+1∥1.
We use (103) and (109) to bound ∥(θn-θ^n)/Δt∥ and ∥(θn-θ^n)/Δt∥-1, respectively. For this, we need that uhn and its first derivative are bounded.
First let us make an induction hypothesis. Assume that there is a constant say K*≥2K with ∥u~hn∥(L∞(Ω))2≤K such that
(114)∥uhn∥(L∞(Ω))2≤K*,
where u~hn is the projection of uhn at t=tn defined in (41). To bound ∥∇·uhn∥∞, we use inverse inequality and (114):
(115)∥∇·uhn∥∞Δt≤Ch-1Δt∥uhn∥∞≤C,
where we have used the assumption that Δt=O(h).
Using (103), we have
(116)∥θn-θ^nΔt∥≤C(K*)∥∇θn∥.
Using (109), we have
(117)∥θn-θ^nΔt∥-1≤C(K*)∥θn∥.
Now, using (52), (116), inverse inequality (48), and (117), we obtain the following bound for T7:
(118)|T7|≤C(K*)∥θn∥∥θn+1∥1.
To bound T8, we use maximum norm estimate of ∇Rhc (see (79)):
(119)∥R^hcn-RˇhcnΔt∥≤∥Rhc∥1,∞∥un-uhn∥≤C∥un-uhn∥(L2(Ω))2,
And, hence, using (54), T8 is bounded as follows:
(120)T8≤C∥θn+1∥∥un-uhn∥(L2(Ω))2.
Substitute the estimates of T1,T2⋯T8 in (90) and use nonsingular property of ϕ and a kick-back argument with the Young’s inequality to obtain
(121)1Δt[(θn+1,Πh*θn+1)-(θn,Πh*θn+1)]≤C(K*)[Δt-1∥∂ρ∂t∥L2(tn,tn+1,L2)2hhhhhhhhh+∥θn+1∥2+∥θn+1∥12hhhhhhhhh+∥θn∥2+∥ρn+1∥2hhhhhhhhh+Δt(∥ut∥L2(tn,tn+1;(L2(Ω))2)2∥∂2c∂τ2∥L2(tn,tn+1,L2)2hhhhhggggghhhhhl+∥(∇·u)t∥L2(tn,tn+1;L2)2hhhhhggggghhhhlh+∥∂2c∂τ2∥L2(tn,tn+1,L2)2)hhhhhhhhh+∥un-uhn∥(L2(Ω))22hhhhhhhhh+h2∥∇·(un-uhn)∥2hhhhhhhhh∥∂ρ∂t∥L2(tn,tn+1,L2)2+∥ρn∥2+h2∥∇ρn∥2].
Using (43) and (44), we obtain
(122)∥un-uhn∥(L2(Ω))2≤C[∥cn-chn∥+h(∥un∥(H1(Ω))2+∥pn∥12)]≤C[∥θn∥+∥ρn∥+h(∥un∥(H1(Ω))2+∥pn∥12)],∥∇·(un-uhn)∥≤Ch∥∇·un∥1.
Substitute (122) in (121) to obtain
(123)∥|θn+1|∥2-∥|θn|∥2≤C(K*)[Δt(∥θn+1∥2+∥θn∥2+∥ρn∥2∥∂2c∂τ2∥L2(tn,tn+1,L2)2ffffffffffffffffffffhfh∥θn+1∥2+∥ρn+1∥2+h2∥∇ρn∥2)fffffffffffffffhff+(Δt)2(∥ut∥L2(tn,tn+1;(L2(Ω))2)2∥∂2c∂τ2∥L2(tn,tn+1,L2)2fffffffffffffffffffhhhhhhhl+∥∇·ut∥L2(tn,tn+1;L2)2fffffffffffffffffffhhhhhhhl+∥∂2c∂τ2∥L2(tn,tn+1,L2)2)fffffffffffffffhff+∥∂ρ∂t∥L2(tn,tn+1,L2)2ffffffffffffffhfff+h2Δt(h2∥∇·un∥12+∥un∥(H1(Ω))22fffffffffffffffffhffhhhhl∥∂2c∂τ2∥L2(tn,tn+1,L2)2∥un∥(H1(Ω))22+∥pn∥12)].
Taking summation over n=0⋯m-1, we obtain
(124)∥|θm|∥2-∥|θ0|∥2≤C(K*)[∑n=0m-1{Δt(∥θn+1∥2+∥θn∥2+∥ρn∥2∥∂2c∂τ2∥L2(tn,tn+1,L2)2hhhhhhhhhhhhhhhhhl+∥ρn+1∥2+h2∥∇ρn∥2)hhhhhhhhhhhhh.hh+(Δt)2(∥ut∥L2(tn,tn+1;(L2(Ω))2)2∥∂2c∂τ2∥L2(tn,tn+1,L2)2hhhhhhhhhhhhhhhhhhhhhh.+∥∇·ut∥L2(tn,tn+1;L2)2hhhhhhhhhhhhhhhhhhhhhh.+∥∂2c∂τ2∥L2(tn,tn+1,L2)2)hhhhhhhhhhhhhhh.+∥∂ρ∂t∥L2(tn,tn+1;L2)2hhhhhhhhhhhhhhh.+h2Δt(h2∥∇·un∥12+∥un∥(H1(Ω))22hhhhhhhhhhhhhhhhhhhhl∑n=0m-1∥∂2c∂τ2∥L2(tn,tn+1,L2)2∥∇·un∥12∥un∥(H1(Ω))22+∥pn∥12)}].
Now, using discrete Gronwall’s, equivalence of the norms (53), and the estimates of ρ, we obtain
(125)∥θm∥2≤C(K*)×[∥θ0∥2+h4(∥(c-c~)q∥L∞(0,T;H1)2∥ψ∂2c∂t∂s∥L∞(0,T;H1)2jjjjjjjjjjjjjjjjjjjjjjjhj+∥ψ∂c∂s∥L∞(0,T;H1)2+∥c∥L∞(0,T;H2)2jjjjjjjjjjjjjjjjjjjjjjhjj+∥∇·u∥L∞(0,T;H1)2+∥ct∥L∞(0,T;H2)2jjjjjjjjjjjjjjjjjjjhjjjjj+∥ψ∂2c∂t∂s∥L∞(0,T;H1)2)hh+(Δt)2(∥ut∥L2(0,T;(L2(Ω))2)2∥∂2c∂τ2∥L2(0,T,L2)2jjjjjjjjjjhjhjjjjjjk+∥∇·ut∥L2(0,T;L2)2+∥∂2c∂τ2∥L2(0,T,L2)2)jhjjhh+h2(∥u∥L∞(0,T;(H1(Ω))2)2jhjjhhhh.hh∥ψ∂2c∂t∂s∥L∞(0,T;H1)2∥u∥L∞(0,T;(H1(Ω))2)2+∥p∥L∞(0,T;H1)2)].
Now, it remains to show the induction hypothesis (114). Using (23) and (24), we have
(126)∥uhn∥(L∞(Ω))2≤∥uhn-u~hn∥(L∞(Ω))2+∥u~hn∥(L∞(Ω))2≤C(log1h)1/2∥uhn-u~hn∥H(div;Ω)+K.
Using ∥∇·(uhn-u~hn)∥=0, we have
(127)∥uhn-u~hn∥(L∞(Ω))2≤C(log1h)1/2∥uhn-u~hn∥(L2(Ω))2.
Now using (42) and (125), we obtain for small h(128)∥uhn∥(L∞(Ω))2≤C(K*)log1h(h+Δt)+K≤2K.
Here, we have used Δt=O(h) and hlog(1/h)→0 as h→0 and this proves our induction hypothesis (114).
Now combine the estimates of ρ and θ and use triangle inequality to complete the rest of the proof.
Using (43) and Theorem 11, we obtain the following error estimates for velocity as well as pressure.
Theorem 12.
Assume that the triangulation 𝒯h is quasiuniform. Let (u,p) and (uh,ph) be, respectively, the solutions of (1)-(2) and (30) and let ch(0)=c0,h=Rhc(0). Further assume that Δt=O(h). Then for sufficiently small h there exists a positive constant C(T) independent of h but dependent on the bounds of κ-1 and μ such that
(129)max0≤n≤N∥un-uhn∥(L2(Ω))22≤C[h4(∥(c-c~)q∥L∞(0,T;H1)2∥ψ∂2c∂t∂s∥L∞(0,T;H1)2hhhhhhhhhh.+∥ψ∂c∂s∥L∞(0,T;H1)2+∥c∥L∞(0,T;H2)2hhhhhhhhhh.+∥∇·u∥L∞(0,T;H1)2+∥ct∥L∞(0,T;H2)2hhhhhhhhhh.+∥ψ∂2c∂t∂s∥L∞(0,T;H1)2)hhhhhhhh+(Δt)2(∥ut∥L2(0,T;(L2(Ω))2)2∥∂2c∂τ2∥L2(0,T,L2)2hhhhhhjjjjjjjjjhhhhh+∥∇·ut∥L2(0,T;L2)2+∥∂2c∂τ2∥L2(0,T,L2)2)hhhhhhhh∥ψ∂2c∂t∂s∥L∞(0,T;H1)2+h2(∥u∥L∞(0,T;(H1(Ω))2)2+∥p∥L∞(0,T;H1)2)].
4. Numerical Experiments
For our numerical experiments, we consider (1)–(6), with q=q+-q- and g(x,t,c)=c-q+-cq-, where c- is the injection concentration and q+ and q- are the production and injection rates, respectively.
Experimentally, it has been observed that the velocity is much smoother in time compared to the concentration. It was suggested in [27] that, for a good approximation to the concentration, one should take larger time step for the pressure equation than the concentration equation. Let 0=t0<t1<⋯tM=T a given partition of the time interval (0,T] with step length Δtm=tm+1-tm for the pressure equation and 0=t0<t1<⋯tN=T a given partition of the time interval (0,T] with step length Δtn=tn+1-tn for the concentration equation. We denote Cn≈ch(tn), Cm≈ch(tm), Um≈uh(tm), and Pm≈ph(tm).
If concentration step tn relates to pressure steps by tm-1<tn≤tm, we require a velocity approximation at t=tn, which will be used in the concentration equation, based on Um-1 and earlier values. We define a velocity approximation [12, pp. 81] at t=tn by
(130)EUn=(1+tn-tm-1tm-1-tm-2)Um-1-tn-tm-1tm-1-tm-2Um-2,form≥2,EUn=U0,form=1.
Then the combined time stepping procedure is defined as follows: find C:{t0,t1,…,tN}→Mh and (U,P):{t0,t1,…,tM}→Uh×Wh such that
(131)(κ-1μ(Cm)Um,γhvh)+b(γhvh,Pm)=0∀vh∈Uh,(132)(∇·Um,wh)=(q+-q-,wh)∀wh∈Wh,m≥0,(133)(ϕCn+1-C^nΔt,Πh*χh)+ah(EUn+1;Cn+1,χh)+(q-Cn+1,Πh*χh)=(q+c-,Πh*χh)∀χh∈Mh,
where C^n=Cn(x-(EUn/ϕ)Δt).
To solve the pressure equations, that is, (131) and (132), we use the mixed FVEM, and for the concentration equation (133), we use the standard FVEM.
For the test problems, we have taken data from [28]. The spatial domain is Ω=(0,1000)×(0,1000)ft2, the time period is [0,3600] days, and viscosity of oil is μ(0)=1.0 cp. The injection well is located at the upper right corner (1000,1000) with the injection rate q+=30ft2/day and injection concentration c-=1.0. The production well is located at the lower left corner with the production rate q-=30ft2/day and the initial concentration is c(x,0)=0. For time discretization, we take Δtp=360 days and Δtc=120 days; that is, we divide each pressure time interval into three subintervals.
Test 1. We assume that the porous medium is homogeneous and isotropic. The permeability is κ=80. The porosity of the medium is ϕ=.1 and the mobility ratio between the resident and injected fluid is M=1. Furthermore, we assume that the molecular diffusion is dm=1 and the dispersion coefficients are zero. In our numerical simulation, we divide the spatial domain into 20 equal subdivisions along both x and y axis. For time discretization, we take Δtp=360 days and Δtc=120 days; that is, we divide each pressure time interval into three subintervals.
The surface and contour plots for the concentration at t=3 and t=10 years are presented in Figures 6 and 7, respectively. Since only molecular diffusion is present and viscosity is also independent of the velocity, Figure 6 shows that the velocity is radial and the contour plots for the concentration are circular until the invading fluid reaches the production well. Figure 7 shows that when these plots are reached at production well, the invading fluid continues to fill the whole domain until c=1.
Contour (a) and surface plot (b) in Test 1 at t=3 years.
Contour (a) and surface plot (b) in Test 1 at t=10 years.
Test 2. In this test we consider the numerical simulation of a miscible displacement problem with discontinuous permeability. Here, the data is the same as given in Test 1 except the permeability of the medium κ(x). We take κ=80 on the subdomain ΩL:=(0,1000)×(0,500) and κ=20 on the subdomain ΩU:=(0,1000)×(500,1000). The contour and surface plot at t=3 and t=10 years are given in Figures 8 and 9, respectively. Figures 8 and 9 show that when the injecting fluid reaches the lower half domain, it starts moving much faster in the horizontal direction on this domain compared to the low permeability domain, that is, upper half domain. We observe that one should put the production well in a low permeability zone to increase the area swept by the injected fluid.
Contour (a) and surface plot (b) in Test 2 at t=3 years.
Contour (a) and surface plot (b) in Test 2 at t=10 years.
Order of Convergence. In order to verify our theoretical results we also compute the order of convergence for the concentration for this particular test problem. We compute the order of convergence in L2 norm. To discretize the time interval [0,T], we take uniform time step Δt=360 days for pressure and concentration equations. The computed order of convergence is given in Figure 10. Note that the computed order of convergence matches with the theoretical order of convergence derived in Theorem 11.
Order of convergence in L2 norm.
Note. This paper has been presented in the International Conference on Numerical Analysis and Applications which was held at Bulgaria during June 15–20, 2012. Moreover, some of the results without proof have been published in the proceeding of that conference, kindly see [29].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor Neela Nataraj and Professor Amiya Kumar Pani (Department of Mathematics, IIT Bombay, India) for their valuable suggestions and the comments which helped improve the paper.
ChenZ.EwingR.Mathematical analysis for reservoir models19993024314532-s2.0-0032622872FengX. B.On existence and uniqueness results for a coupled system modeling miscible displacement in porous media199519438839102-s2.0-034611538810.1006/jmaa.1995.1334SammonP. H.Numerical approximations for a miscible displacement process in porous media19862335085422-s2.0-0022736889DouglasJ.Jr.EwingR. E.WheelerM. F.The approximation of the pressure by a mixed method in the simulation of miscible displacement1983171733DouglasJ.JrEwingR. E.WheelerM. F.A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media.198317249265EwingR. E.WheelerM. F.Galerkin methods for miscible displacement problems in porous media198017351365DarlowB. L.EwingR. E.WheelerM. F.Mixed finite element methods for miscible displacement in porous media19842443913982-s2.0-0021181536KumarS.A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media2012284135413812-s2.0-8486021931510.1002/num.20684DouglasJ.Jr.RussellT. F.Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element method or finite difference procedures198219871885DawsonC. N.RussellT. F.WheelerM. F.Some improved error estimates for modified method of characteristics19892614871512DuranR. G.On the approximation of miscible displacement in porous media by a method characteristics combined with a mixed method1988149891001EwingR. E.RussellT. F.WheelerM. F.Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics1984471-273922-s2.0-0021615427ChouS.-H.KwakD. Y.Mixed covolume methods on rectangular grids for elliptic problems20003737587712-s2.0-0000850660ChouS. O.-H.KwakD. O. Y.VassilevskiP. S.Mixed covolume methods for elliptic problems on triangular grids1998355185018612-s2.0-0000489496BankR. E.RoseD. J.Some error estimates for the box method19872447777872-s2.0-0023401301CaiZ.On the finite volume element method19915817137352-s2.0-000106343510.1007/BF01385651ChatzipantelidisP.Finite volume methods for elliptic PDE'S: a new approach20023623073242-s2.0-003599928210.1051/m2an:2002014KumarS.NatarajN.PaniA. K.Finite volume element method for second order hyperbolic equations2008511321512-s2.0-40649104615ThoméeV.1984New York, NY, USASpringerBrezziF.MichelF.1991New York, NY, USASpringerChatzipantelidisP.A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions19998234094322-s2.0-0033432593CiarletP. G.1978New York, NY, USANorth-HollandChouS.-H.LiQ.Error estimates in L2, H1 and L∞ in covolume methods for elliptic and parabolic problems: a unified approach2000692291031202-s2.0-0001347009LiR. H.ChenZ. Y.WuW.2000New York, NY, USAMarcel DekkerEwingR. E.LinT.LinY.On the accuracy of the finite volume element method based on piecewise linear polynomials2002396186518882-s2.0-003655636410.1137/S0036142900368873SunS.RiviéreB.WheelerM. F.A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media2002New York, NY, USAKluwer Acadmic, Plenum pressEwingR. E.RussellT. F.Efficient time stepping methods for miscible displacement problems in porous media198219167WangH.LiangD.EwingR. E.LyonsS. L.QinG.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 methods20012225615812-s2.0-003506773310.1137/S1064827598349215KumarS.Finite volume apprximations for incompressible miscible displacement problems in porous media with modified method of characterstics8236Proceedings of the International conference on Numarical Analsyis and Applications held at Bulgaria2013379386Lecture Notes in Computer Science