Conclusion

invading fluid decreases monotonically from the left stateS = 1 to the right state S = 0 in a rarefaction wave followed by a compressive shock. Figure 4.9(b), 4.9(d), 4.9(f), 4.9(h), 4.9(j), and 4.9(l) present saturation behavior of the in-vading fluid using the Brinkman VE-model with the permeabilities κ₁ = 1 and κ3 = 0, 0.2,0.4, 0.6, 0.8, 1, respectively. In these figures saturation decays mono-tonically from the left state S = 1 to a plateau value S^∗ < 1 in a rarefaction wave then to the right stateS = 0 in an undercompressive shock. This saturation behavior is expected according to the traveling wave analysis in [48, 49] as the ver-tical mixing between the two layers is very low. In Figure 4.9(f), 4.9(h), and 4.9(j) saturation profile is not monotone, which is a consequence of the higher amount of vertical flow from the upper layer to the lower. In these figures saturation rarefacts from the left stateS= 1 to an intermediate state (S_m <1), which is smaller than the plateau valueS^∗. This wave is then followed by an undercompressive shock to a higher valueSu > Sm and then by a compressive shock to the left state S = 0.

Figure 4.9 also shows that the spreading speed of the invading fluid using the Brinkman VE-model is smaller than that using the VE-model.

Figure 4.7 and 4.9 show the ability of the Brinkman model and the VE-model to describe the vertical dynamics in the domain. However, the Brinkman VE-model is able to capture the phenomenon of saturation overshoots, in contrast to the VE-model and, consequently, it provides sharper estimates on the invasion speed. This ability is a consequence of the higher order terms in the model, which leads to a higher computational complexity compared to the VE-model.

the model’s advantages, we performed different numerical tests and concluded the following aspects:

1. Numerical solutions of the Brinkman two-phase flow model converge to the corresponding numerical solution of the Brinkman VE-model as the geomet-rical parameter γ →0.

2. The computational complexity of the Brinkman VE-model is much less than that of the Brinkman two-phase flow model.

3. The accuracy of the Brinkman VE-model is limited to low viscosity ratios (M ≤5).

4. Including Brinkman’s equation in the two-phase flow model and in the extended VE-model allows describing the phenomenon of saturation over-shoots.

5. Decreasing the geometrical parameter γ = ^thickness_length of a porous medium re-duces the effect of saturation overshoot. This phenomenon can be explained as follows: decreasing the geometrical parameter γ in the Brinkman two-phase flow model increases the vertical velocity at the wetting front. Hence, the vertical exchange of mass increases, such that saturation decreases to values, at which saturation overshoots might be not expected to occur.

Chapter 5

Well-posedness of the Brinkman Vertical Equilibrium Model

We proposed in Chapter 4 the Brinkman VE-model to describe fluid flows in saturated flat porous media that are macroscopically heterogeneous. The proposed model is a third-order pseudo-parabolic differential equation of saturation alone that explicitly accounts for the vertical dynamics in the medium. The model has several numerical advantages over the Brinkman two-phase flow model and the VE-model. In addition to this, the higher order terms in the proposed model provide a dissipative effect, such that weak solutions in the distributional sense can be defined. Therefore, we investigate in this chapter the well-posedness of the Brinkman VE-model.

We consider the Brinkman VE-model (4.30), (4.31) and (4.32) under the as-sumptions κ := 1, D(S) := 1, and 1 =2 := 1. We also set β = β1 =β2. Then, the model reduces to

∂tS+∂x

f(S)U[S]

+∂z

f(S)W[S]

−∆S−β∆∂tS = 0, (5.1) where

U[S] = λ_tot(S) R1

0 λ_tot(S)dz, W[S(·, z,·)] =−∂_x Z z

0

U[S(·, r,·)]dr, (5.2) for allz ∈(0,1) in the domain Ω×(0, T). The definition of the velocity components U and W in (5.2) implies that the velocity V = (U, W)^T is incompressible

∇ ·V= 0. (5.3)

The Brinkman VE-model (5.1), (5.2), and (5.3) is associated with the initial and boundary conditions

S(·,·,0) =S⁰, in Ω,

S = 0, on ∂Ω×[0, T]. (5.4)

Remark 5.1. Associating the Brinkman VE-model (5.1), (5.2), and (5.3) with Dirichlet boundary conditions is possible because the higher order terms in the model are linear. We choose a zero Dirichlet boundary condition in (5.4) to sim-plify the analysis. However, the analysis throughout the chapter can be extended to Dirichlet boundary conditions of the formS =S_D on∂Ω×[0, T] such thatS_D has appropriate regularity and satisfiesS_D =S⁰ on∂Ω at timet= 0. In addition, this condition is still physically valid by assuming a larger domain with dry boundaries.

Remark 5.2. Proving the existence of a weak solution S ∈ H¹(0, T;H¹(Ω)) for the Brinkman VE-model (5.1) and (5.2) with the initial and boundary condition (5.4) yields that W[S]∈ L²(Ω×(0, T)). With this information, we can solve the equationW =w−β₂∆w for the vertical velocity w from the previous chapter such that the zero Neumann condition on w and the periodicity condition on ∆w in equation (4.6) are satisfied.

The aim of this chapter is to prove existence and uniqueness of a weak solution for the Brinkman VE-model (5.1) and (5.2) in the bounded domain Ω_T := Ω× (0, T), where Ω = (0,1)². We do this in the following steps: in Section 5.1, we approximate the time derivatives in the model using the backward difference quotient then apply Galerkin’s method to the resulting elliptic problem. After that, we prove the existence of a sequence of discrete solutions for the approximated problem. In Section 5.2, we show that the sequence of discrete solutions fulfills a set of a priori estimates. These estimates are used in Section 5.3 to conclude a strong convergence of the sequence in the space L²(Ω_T). Then, we prove that the limit of the sequence is a weak solution of the Brinkman VE-model. Section 5.4 shows the boundedness of the weak solutions in the space L^∞(Ω_T). After that, we prove in Section 5.5 the uniqueness of the weak solutions for the Brinkman VE-model when the fractional flow function and the horizontal velocity are linear.

Finally, Section 5.6 summarizes the chapter.

5.1 Preliminaries and Assumptions

This chapter is concerned with finding weak solutions S ∈ H¹(0, T;H¹(Ω)) for the Brinkman VE-model. Thus, it is not known a priori whether saturation is bounded (S ∈ [0,1]) or not. Therefore, we extend in this chapter the domain of the fractional flow functionf and the total mobility λ_tot from [0,1] toRsuch that f(S) =f(1) for all S ∈(1,∞), f(S) =f(0) for all S ∈(−∞,0), λ_tot(S) =λ_tot(1) for all S ∈ (1,∞), and λ_tot(S) = λ_tot(0) for all S ∈ (−∞,0). Throughout this chapter, the following assumptions on the initial boundary value problem (5.1), (5.2), (5.3), and (5.4) hold.

Assumption 5.3. 1. The spatial domain Ω⊂R² is open, connected, bounded with Lipschitz continuous boundary ∂Ω and 0< T <∞.

2. The initial condition satisfies S⁰ ∈H₀¹(Ω).

3. The fractional flow function f is Lipschitz continuous, bounded, nonnegative and monotone increasing, such that there exist numbers M, L > 0 with f ≤ M, f⁰ ≤L.

4. The total mobility function λ_tot is Lipschitz continuous, bounded, strictly positive, such that there exist numbers a, M, L > 0 with 0 < a < λ_tot ≤ M and |λ⁰_tot| ≤L.

Note that the numbersM, L > 0 are chosen large enough such that Assumption 5.3(3) and 5.3(4) hold.

Definition 5.4. (Weak Solution) A functionS∈H¹(0, T;H₀¹(Ω)) is called a weak solution of the Brinkman VE-model (5.1) and (5.2) with the initial and boundary conditions (5.4) whenever the following conditions hold,

1. U[S], W[S]∈L²(Ω_T) and Z T

0

Z

Ω

∂_tSφ−f(S)U[S]∂_xφ−f(S)W[S]∂_zφ+∇S· ∇φ

dx dz dt +β

Z T 0

Z

Ω

∇∂_tS· ∇φ dx dz dt= 0, (5.5) for all test functions φ∈L²(0, T;H₀¹(Ω)).

2. The weak incompressibility property Z T

0

Z

Ω

(U[S]∂xφ+W[S]∂zφ) dx dz dt= 0, (5.6)

holds for all test functions φ∈L²(0, T;H₀¹(Ω)).

3. S(., .,0) =S⁰ almost everywhere.

Remark 5.5. Note that the integralRz

0 λ_tot S(·, r,·)

dr, z ∈(0,1)in the definition of the velocity componentsU, W is an integral over a set of measure zero. However, it is well-defined in the trace sense as the Trace theorem 2.12 implies the existence of a bounded linear operator T : H¹(Ω) → L² {x} ×(0, z)

, for almost all x and z∈(0,1), and a constant C such that

kT Sk_L²({x}×(0,z)) ≤ kSk_H¹_(Ω).

Remark 5.6. If the velocity component W would be Lipschitz continuous with respect to S, then the well-posedness of the model (5.1), (5.2), (5.3), and (5.4) follows [26].

Lemma 5.7. If Assumption 5.3(4) holds, then the velocity components U and W satisfy the properties:

1. U is bounded, such that kU[Q]k_L^∞_(ΩT) ≤ ^M_a for any function Q∈L²(Ω_T).

2. For any functions Q₁, Q₂ ∈L²(Ω_T), the horizontal velocity U satisfies kU[Q₁]−U[Q₂]k_L2(ΩT) ≤ 2M L

a² kQ₁−Q₂k_L²_(ΩT).

3. For any function Q∈L²(0, T;H¹(Ω)), the component W satisfies the growth condition

kW[Q]k_L²_(ΩT))≤ 2M L

a² k∂_xQk_L2(ΩT).

Proof. 1. Using Assumption 5.3(4) we have kU[Q]k_L^∞_(ΩT) =

λtot(Q) R1

0 λ_tot(Q(·, z,·))dz L^∞(ΩT)

≤ M a .

2. Using the chain rule and the triangle inequality, we have kU[Q₁]−U[Q₂]k_L²_(ΩT)

=

λ_tot(Q₁) R1

0 λ_tot(Q₁)dz − λ_tot(Q₂) R1

0 λ_tot(Q₂)dz L²(ΩT)

,

≤

λ_tot(Q₁)R1

0 λ_tot(Q₂)−λ_tot(Q₁) dz R1

0 λtot(Q1)dzR1

0 λtot(Q2)dz _L2(Ω

T)

+

R1

0 λ_tot(Q₁)dz λ_tot(Q₂)−λ_tot(Q₁) R1

0 λ_tot(Q₁)dzR1

0 λ_tot(Q₂)dz L²(ΩT)

,

≤ M a²

Z 1 0

λ⁰_tot(Q) Q₂ −Q₁ dz

L²(ΩT)

+ M L

a² kQ₂−Q₁k_L2(ΩT), for some Q ∈ L²(Ω_T). Note that the first term in the above inequality is constant in the vertical direction. Applying Jensen’s inequality, then Fubini’s inequality to this term yields

kU[Q₁]−U[Q₂]k_L²_(ΩT) ≤ 2M L

a² kQ₂−Q₁k_L2(ΩT).

3. Using the chain rule and the triangle inequality, we have for anyz ∈(0,1) kW[Q]k_L²_(ΩT))=

−∂_x Rz

0 λ_tot(Q(·, r,·))dr R1

0 λ_tot(Q(·, r,·))dr L²(ΩT)

,

=

Rz

0 λ⁰_tot(Q(·, r,·))∂_xQ(·, r,·)drR1

0 λ_tot(Q(·, r,·))dr R1

0 λ_tot(Q(·, r,·))dr2

L²(ΩT)

+

R1

0 λ⁰_tot(Q(·, r,·))∂xQ(·, r,·)drRz

0 λtot(Q(·, r,·))dr R1

0 λ_tot(Q(·, r,·))dr2

L²(ΩT)

,

≤M L a²

Z z 0

∂_xQ(·, r,·)dr L²(ΩT)

+

Z 1 0

∂_xQ(·, r,·)dr L²(ΩT)

! .

Applying Jensen’s inequality, Fubini’s inequality and the fact that k∂_xQk_L2(ΩT) is constant in the vertical direction yields

kW[Q]k_L²_(ΩT))≤2M L a²

Z 1 0

k∂_xQk_L2(ΩT) dr≤ 2M L

a² k∂_xQk_L2(ΩT),

For N ∈ N, ∆t := T /N, and any t ∈ (0, T) we use the backward difference quotient S(t)−S(t−∆t)

∆t to approximate the time derivative∂_tS. Then, equation (5.1) is approximated by

S(t)−S(t−∆t)

∆t +∂x

f(S(t))U[S(t)]

+∂z

f(S(t))W[S(t)]

−∆S(t)

−β∆S(t)−∆S(t−∆t)

∆t = 0. (5.7)

Weak solutions of the approximated model (5.7) and (5.2) are expected to belong to the Hilbert space V(Ω) := H₀¹(Ω), for almost all t ∈ (0, T). Since H₀¹(Ω) is separable [1], it has a countable orthonormal basis

{wi}i∈N⊂V(Ω). (5.8)

By applying Galerkin’s method to (5.7), the solution spaceV(Ω) is projected into a finite dimensional space Vm(Ω) spanned by a finite number of the orthonormal functionswi, i= 1, ..., m. For positive integers m, N, we search a function

S_m^N(x, z, t) :=

m

X

i=1

c^N_mi(t)w_i(x, z), (5.9)

where the unknown coefficients c^N_m,i ∈ L^∞((0, T)), i = 1, . . . , m, are chosen such that for almost allt ∈(0, T)

Z

Ω

S_m^N(t)−S_m^N(t−∆t)

wi−∆tf(S_m^N(t)) U[S_m^N(t)]∂xwi+W[S_m^N(t)]∂zwi

dx dz +

Z

Ω

∆t∇S_mⁿ(t) +β∇(S_m^N(t)−S_m^N(t−∆t))

· ∇w_idx dz = 0, (5.10) holds for all i= 1, ..., m, with

U[S_m^N(t)(x, z)] = λ_tot S_m^N(t)(x, z) R1

0 λ_tot S_m^N(t)(x, r) dr, W[S_m^N(t)(x, z)] =−∂_xRz

0 U[S_m^N(t)(x, r)]dr,

(5.11)

for almost all t ∈ (0, T) and (x, z) ∈ (0,1)². The function S_m^N is also required to satisfy the weak incompressibility relation

Z

Ω

V[S_m^N]· ∇w_idx dz = 0, for all i= 1, ..., m. (5.12)

The initial data is chosen to be

S_m^N(t) =S_m⁰, for t∈(−∆t,0], (5.13) whereS_m⁰ is the L²-projection of the initial dataS⁰ to the finite dimensional space V_m(Ω).

To prove the existence of a weak solution of the discrete problem (5.10), (5.11), and (5.12) we need the following technical lemma on the existence of zeros of a vector field [25].

Lemma 5.8. (Zeros of a vector field, [25]) Let r > 0 and v : Rⁿ → Rⁿ be a continuous vector field, which satisfies v(x)·x≥0 if |x|=r. Then, there exists a point x∈B(0, r) such that v(x) = 0.

Lemma 5.9. For any m, N ∈ N and for almost all t ∈ (0, T), if S_m^N(t−∆t) ∈ V_m(Ω) is known, then problem (5.10), (5.11), and (5.12) has a solution S_m^N(t) ∈ V_m(Ω) that satisfies

Z

Ω

(S_m^N(t)−S_m^N(t−∆t))φ dx dz−∆t Z

Ω

f(S_m^N)U[S_m^N]∂_xφ+f(S_m^N)W[S_m^N]∂_zφ dx dz +

Z

Ω

∆t∇S_m^N +β∇ S_m^N(t)−S_m^N(t−∆t)

· ∇φ dx dz = 0, (5.14) for all φ∈V_m(Ω).

Proof. Before starting with the proof, we notify that S_m^N(t−∆t) for t ∈ (0,∆t]

is well-defined by the choice of the initial condition (5.13). Now, we define the vector field K:R^m → R^m, K= (k₁, ..., k_m), and c^N_m(t) = (c^N_m,1(t),· · · , c^N_m,m(t)) of the unknown coefficients in equation (5.9) such that, for almost allt ∈(0, T),

k_i(c^N_m(t)) :=

Z

Ω

(S_m^N(t)−S_m^N(t−∆t))w_idx dz

−∆t Z

Ω

f(S_m^N)U[S_m^N]∂_xw_i+f(S_m^N)W[S_m^N]∂_zw_idx dz +

Z

Ω

∆t∇S_m^N +β∇(S_m^N(t)−S_m^N(t−∆t))

· ∇widx dz, (5.15)

for all i = 1, ..., m. The vector field K is continuous by Assumption 5.3(3) and 5.3(4) Moreover, using equation (5.9), we have

K(c^N_m(t))·c^N_m(t)

= Z

Ω

S_m^N(t)−S_m^N(t−∆t)

S_m^N(t)dx dz

−∆t Z

Ω

f(S_m^N) U[S_m^N]∂_xS_m^N +W[S_m^N]∂_zS_m^N

dx dz, +

Z

Ω

∆t∇S_m^N(t) +β∇ S_m^N(t)−S_m^N(t−∆t)

· ∇S_m^N(t)dx dz. (5.16) LetF(S) := RS

0 f(q)dq, then the second term on the right side of (5.16) satisfies Z

Ω

f(S_mⁿ) U[S_m^N]∂_xS_m^N +W[S_m^N]∂_zS_m^N

dx dz = Z

Ω

f(S_m^N)V[S_m^N]· ∇S_m^Ndx dz,

= Z

Ω

V(S_m^N)· ∇F(S_m^N)dx dz, where V[S_m^N] = (U[S_m^N], W[S_m^N]). Using the assumption that S_m^N vanishes on the boundary ∂Ω by equation (5.4) and the property F(0) = 0, the weak incom-pressibility of V[S_m^N] in equation (5.12) also holds with F(S_m^N) ∈L²(0, T;H₀¹(Ω)) replacing w_i. Thus, we have

Z

Ω

V(S_m^N)· ∇F(S_m^N)dx dz = 0. (5.17) Substituting equations (5.17) into (5.16), then applying Cauchy’s inequality yields

K(c^N_m(t))·c^N_m(t)≥ 1

2kS_m^Nk²_L2(Ω)+ β

2 + ∆t

k∇S_m^Nk²_L2(Ω)− 1

2kS_m^N(t−∆t)k²_L2(Ω)

−β

2k∇S_m^N(t−∆t)k²_L2(Ω).

Equation (5.9) and the orthonormality of w_i, i∈ {1,· · · , m} yield K c^N_m(t)

·c^N_m(t)≥ 1

2 +β 2 + ∆t

|c^N_m|²− 1

2kS_m^N(t−∆t)k_L²_(Ω)

−β

2k∇S_m^N(t−∆t)k_L²_(Ω).

Note that S_m^N(t −∆t) ∈ V_m(Ω) is known. Set r = |c^N_m(t)|, we conclude that K(c^N_m(t))·c^N_m(t) ≥ 0 provided that r is large enough. Thus, Lemma 5.8 ensures the existence of a vectorc^N_m(t)∈R^m withK(c^N_m(t)) =0. Using equation (5.15) we

conclude the existence of an S_m^N(t), defined as in (5.9), that satisfies the discrete problem (5.10), (5.11), and (5.12).

Remark 5.10. The function S_m^N is defined for all t∈(0, T) and is a step function in time. This results from the structure of equation (5.10), which implies that S_m^N(t) for all t ∈ [(n−1)∆t, n∆t) is determined inductively using the given data S_m^N(t−∆t).

Im Dokument Modeling and analysis of almost unidirectional flows in porous media (Seite 91-101)