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Capillarity Approximation of Conservation Laws with Discontinuous Fluxes

Lorenzo di Ruvo

Department of Mathematics University of Bari Via E. Orabona 4 70125 Bari (Italy) EMAIL: diruvo@dm.uniba.it

Padova 2012 joint work with

• Prof. Giuseppe Maria Coclite (University of Bari)
• Prof. Siddhartha Mishra (ETH Zürich)

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 1 / 20

Two–Phase Flow Motion in a one dimensional porous medium

A one dimensional porous medium. Flow of two phases in this porous medium. Fluids are immiscible. Fluids are incompressible. The density of the two fluids is constant. Wetting phase w. Nonwetting phase n. The mass flow rate per unit volume injected or produced is zero. The mass conservation for each phase: ∂t(φSw) + ∂x(vw) = 0, t > 0, x ∈ R, ∂t(φSn) + ∂x(vn) = 0, t > 0, x ∈ R.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 2 / 20

φ ≡ porosity. Sw, Sn ≡ saturations of the two phases. vw, vn ≡ velocities. The saturation defines the fraction of the pore volume occupied by a phase. Sl(t, x) = V l(t, x) Vp , l = w, n. Therefore, Sw + Sn = 1. The porosity denotes the fraction of the volume available for flow. It depends on the pressure. φ = φ0[1 + cR(p − p0)].

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 3 / 20

cR ≡ rock compressibility. p ≡ pressure. φ0 ≡ the porosity at the pressure p0. We assume φ = 1. The mass conservation, and the saturation conservation give vw + vn = q. q ≡ total flow rate (specified, for instance, by boundary conditions). We are able to describe the motion of the system. unknowns: Sw, Sn, vw, vn. equations: 4.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 4 / 20

Darcy’s Law

The phase velocities are modelled on Darcy’s Law. For the motion of the single-flow in a porous medium, in 1856, Darcy discovered the following law:

Darcy’s Law

The volumetric flow rate of a homogeneous fluid through a porous medium is proportional to the pressure gradient and to the cross-sectional area normal to the direction of flow and inversely proportional to the viscosity

• f the fluid.

v = −K µ (∂xp − ρg). K ≡ rock permeability. (It can vary (discontinuously) in space) µ ≡ viscosity of fluid g ≡ accelaration due to gravity

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 5 / 20

Two-Phases Flow. vw = Kλw(∂xpw − ρwg), vn = Kλn(∂xpn − ρng). λl = kl

r(Sl)

µl

l = w, n: phase mobility. kl

r ≡ relative permeability.

kl

r ≤ K

l = w, n, kl

r = kl r(Sw)

l = w, n. Adding vw and vn, we have Kλw(∂xpw − ρwg) + Kλn(∂xpn − ρng) = q.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 6 / 20

Capillary pressure

Capillary Pressure

The capillary pressure is the difference between the pressure pw and pn, on the contact surface of the two fluids. pc = pw − pn. pc is a decreasing function of Sw Following the model proposed in:

• R. Helmig, A. Weiss and B. I. Wohlmuth

Dynamic capillary effects in heterogeneous porous media. Comp. Geosci., 11 (2007), 261-274.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 7 / 20

we consider pdc = psc(Sw) + τ∂tSw. pdc ≡ dynamic capillary pressure. psc ≡ static capillary pressure. τ ≥ 0 dynamic factor. psc(Sw) : [0, 1] → [0, ∞). psc(Sw) is continuously differentiable. psc(Sw) is strictly decreasing. pdc depends on the dynamics of moving fronts.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 8 / 20

PDE in Sw

τ = 1. Thanks to the mass conservation, Darcy’s Law, and the dynamic capillary pressure, we have ∂tSw + ∂x

• λwq

λw + λn − λwλnK(ρw − ρn)g λw + λn

• = − ∂x
• Kλwλn∂xpsc

− ∂x

• Kλwλn∂2

txSw

. The capillary pressure is supposed to be very small. We can rescale the previous PDE with a small scale ν. ∂tSw + ∂x

• λwq

λw + λn − λwλnK(ρw − ρn)g λw + λn

• = − ν∂x
• Kλwλn∂xpsc

− ν2∂x

• Kλwλn∂2

txSw

.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 9 / 20

PDE in u

Sw = u, 0 ≤ u(t, x) ≤ 1. we denote f (k, u) = λwq λw + λn − λwλnK(ρw − ρn)g λw + λn , g(l, u) = −Kλwλn dpsc du , h(m, u) = −Kλwλn. PDE in u. ∂tu + ∂xf (k, u) = ν∂x(g(l, u)∂xu) + ν2∂x(h(m, u)∂2

txu).

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 10 / 20

Cauchy Problem

Let t ≥ 0, x ∈ R.

                                

∂tuν + ∂xf (kν, uν) = ν∂x(g(ℓν, uν)∂xuν) + ν2∂x(h(mν, uν)∂2

txuν),

∂tkν = ν∂2

xxkν,

∂tℓν = ν∂2

xxℓν,

∂tmν = ν∂2

xxmν,

uν(0, x) = u0,ν(x), kν(0, x) = k0,ν(x), ℓν(0, x) = ℓ0,ν(x), mν(0, x) = m0,ν(x). (uν, kν, ℓν, mν) is a smooth solution, 0 ≤ uν ≤ 1.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 11 / 20

0 < ν ≤ 1. f , g, h : R2 → R are smooth functions. 0 < α ≤ g(·, ·), h(·, ·). f (k, ·) is genuinely nonlinear for every k ∈ R. u ∈ [0, 1] → f (k, u) is not affine on any nontrivial interval for every k ∈ R. Moreover, we assume k0,ν, ℓ0,ν, m0,ν ∈ C∞(R) ∩ W 1,1(R), u0,ν ∈ C∞(R) ∩ L1(R) ∩ L∞(R), 0 ≤ u0,ν ≤ 1, and that there exist k, ℓ, m ∈ BV (R) ∩ L1(R), u0 ∈ L1(R) ∩ L∞(R), 0 ≤ u0 ≤ 1,

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 12 / 20

such that u0,ν → u0, k0,ν → k, ℓ0,ν → ℓ, m0,ν → m, a.e. and in Lp(R) uν,0L2(R) ≤ u0L2(R) , ν, k0,νL∞(R) ≤ kL∞(R) , k0,νL2(R) ≤ kL2(R) , ∂xk0,νL1(R) ≤ TV (k), ν, ℓ0,νL∞(R) ≤ ℓL∞(R) , ℓ0,νL2(R) ≤ ℓL2(R) , ∂xℓ0,νL1(R) ≤ TV (ℓ), ν, m0,νL∞(R) ≤ mL∞(R) , m0,νL2(R) ≤ mL2(R) , ∂xm0,νL1(R) ≤ TV (m), ν, 1 ≤ p < ∞.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 13 / 20

Theorem

There exist a sequence {νn}n ⊂ (0, 1], νn → 0, a sequence {uνn}n∈N of solutions of the previous Cauchy problem, and a distributional solution u of

• ∂tu + ∂xf (k(x), u) = 0,

t > 0, x ∈ R, u(0, x) = u0(x), x ∈ R. such that uνn → u, in Lp

loc((0, ∞) × R), 1 ≤ p < ∞, and a.e. in (0, ∞) × R.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 14 / 20

Proof

Murat’s Lemma

Let Ω be a bounded open subset of R2. Suppose the sequence {Ln}n∈N of distributions is bounded in W −1,∞(Ω). Suppose also that Ln = L1,n + L2,n, where {L1,n}n∈N lies in a compact subset of H−1

loc (Ω) and {L2,n}n∈N lies in

a bounded subset of Mloc(Ω). Then {Ln}n∈N lies in a compact subset of H−1

loc (Ω).

• G. M. Coclite, K. H. Karlsen, S. Mishra, and N. H. Risebro

Convergence of vanishing viscosity approximations of 2 × 2 triangular systems of multi-dimensional conservation laws. (2009).

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 15 / 20

Let us show that the following family {∂t|uν − c| + ∂x(sign (uν − c) (f (k, uν) − f (k, c)))}ν>0 (1) is compact in H−1

loc ((0, ∞) × R).

c is a fixed real number. Let {(ην, Qν)}ν>0 be a family of maps such that ην ∈ C∞(R), Qν ∈ C∞(R2), ∂uQν(k, u) = ∂uf (k, u)η′

ν(u),

η′′

ν ≥ 0,

ην − η0L∞([0,1]) ≤ ν,

• η′

ν − η′

• L1([0,1]) ≤ ν,
• η′

ν

• L∞([0,1]) ≤ 1,

ην(0) = Qν(k, 0) = 0. It is an approximation of (1).

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 16 / 20

From the equation, we obtain ∂t|uν − c| + ∂x(sign (uν − c) (f (k, uν) − f (k, c))) =I1,ν + I2,ν + I3,ν + I4,ν + I5,ν + I6,ν + I7,ν + I8,ν + I9, where I1,ν =ν∂x

η′

ν(uν)g(ℓν, uν)∂xuν

,

I2,ν = − νη′′

ν(uν)g(ℓν, uν)(∂xuν)2,

I3,ν =ν2∂x

η′

ν(uν)h(mν, uν)∂2 txuν

,

I4,ν = − ν2η′′

ν(uν)h(mν, uν)∂xuν∂2 txuν,

I5,ν = − (η′

ν(uν)∂uf (k, uν) − ∂kQ(k, uν))∂xkν,

I6,ν =∂t(η0(uν) − ην(uν)), I7,ν =∂x(Q0(k, uν) − Qν(k, uν)), I8,ν =∂x(Qν(k, uν) − Qν(kν, uν)), I9 =sign (−c) ∂x(f (k, 0) − f (k, c).

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 17 / 20

The paper shows that {I5,ν}ν>0 is bounded in L1((0, T) × R) for each T, I6,ν → 0, I7,ν → 0 in H−1

loc ((0, ∞) × R),

{I8,ν}ν>0 is compact in H−1

loc ((0, ∞) × R),

I9 ∈ Mloc((0, ∞) × R). We have to prove that I1,ν → 0, I3,ν → 0 in H−1((0, T) × R), for each T, {I2,ν}ν>0, {I4,ν}ν>0 are bounded in L1((0, T) × R) for each T.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 18 / 20

We need the following result:

L2− estimate

uν(t, ·)2

L2(R) + νµ1

t

∂xuν(s, ·)2

L2(R) ds

+ ν3µ2

t

• ∂2

txuν(s, ·)

• 2

L2(R) ds + νµ3

t

∂tuν(s, ·)2

L2(R) ds

≤Ct + A k2

L2(R) + u02 L2(R) .

µ1, µ2, µ3, C, A are positive constants. STEPS OF THE PROOF The equation is multiplied by uν + B∂tuν (B is a positive constant). Integrate by parts on R. Use the Young’s inequality. Choose the constant B in a suitable way.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 19 / 20

Thank you for your attention.

Lorenzo di Ruvo (Bari) Capillarity Approximation of Conservation Laws with Discontinuous Fluxes Padova 2012 20 / 20