A degenerate parabolic system for three-phase flows in porous media

A classical model for three-phase capillary immiscible flows in a porous medium is considered. Capillarity pressure functions are found, with a corresponding diffusioncapillarity tensor being triangular. The model is reduced to a degenerate quasilinear parabolic system. A global existence theorem is proved under some hypotheses on the model data.


Introduction
We study the question of global solvability for the 2 × 2 quasi-linear parabolic system u t + f (u) x = (B(u)u x ) x , x ∈ Ω = {x ∈ R : |x| < 1}, 0 < t < T, (1.1) motivated by three-phase capillary flows in a petroleum reservoir.Here and (1.1) is a short version of To help readers gain intuition about the work, we explain how the reservoir flow equations can be reduced to (1.1).We consider one-dimensional horizontal flows of three incompressible immiscible fluids formed in phases, say, oil, gas, and water [1].The balance of masses is governed by the mass conservation equations The author is supported by the Russian RFBR Grant 03-05-65299.
where Φ denotes porosity of the porous medium, u i , ρ i , and v i are the saturation, density, and seepage velocity of the i-th phase.Since u i denotes the portion of the unite pore volume filled with the i-th phase, one has (1. 3) The momentum equations are given in the form of Darcy's law where k stands for the absolute permeability, λ i is the mobility of the i-th phase, and p i is the pressure of the i-th phase.
Let us denote ) It follows from (1.2) and (1.3) that v x = 0, so v depends on t only.Setting v ≡ 1 and eliminating the third phase, we obtain system (1.1) for the vector u = (u 1 , u 2 ) T , where f (u) := (f 1 , f 2 ) T and the matrix B is given by .11)Up to now very little is known about the functions p ij (u) both theoretically and experimentally [3,6].The same is true for the mobility functions λ i (u), i = 1, 2, 3 [11].The following properties are conventional [1]: (i) The functions λ i (u), p ij (u) and the corresponding matrix B, calculated by the formulas (1.6) and (1.8)- (1.11), should be such that system (1.1) is parabolic in a sense.
(ii) The functions λ i satisfy the conditions (1.12) Due to (1.12), system (1.1) is not parabolic on the boundary of the triangle .Since B is degenerate and not diagonal, the well-known theory of parabolic equations cannot be applied to system (1.1).For example, in the theory of Ladyzenskaya, Solonnikov and Ural'tseva [7], the matrix B is nondegenerate and it is a scalar multiple of the identity matrix.Nevertheless, system (1.1) can be analyzed in the case of special equations of state.In [3], a numerical study of system (1.1) was performed in the case of an existing potential p c (u): In [4] and [5], three-phase flows were considered with a triangular capillary diffusion tensor B, i.e., with the conditions which mean that the first and third phases do not influence the diffusion of the second phase.
As is well known by reservoir engineers, mobilities and capillary pressures can be plotted as functions of the saturations only in the case of flows where just two phases are present [9,6], that is, when u ∈ ∂ .While there are some widely accepted models which artificially prescribe the mobility functions in the interior of , such as the one proposed by H. Stone [10], the same is no longer true for the functions representing the capillary pressures.Now, the constraints (1.13) amount to a linear hyperbolic system of partial differential equations for the capillary pressures p 13 and p 23 , whose coefficients involve the mobility functions, for which we set the boundary conditions enforcing compatibility with the two-phase flow case, where the functions ϕ ij can be obtained from two-phase flows experiments, as already mentioned.The result is a consistent recipe for defining the capillary pressures in the interior of the triangle of saturations .Here, we study in detail the case when the mobilities are linear functions of the corresponding phase saturation (1.15) Below we establish by the group analysis methods that conditions (1.13) are a tool for interpolating two-phase systems to three-phase ones in the following sense.Equalities (1.14) imply that the capillary pressure p 13 is prescribed for the two-phase system phase 1-phase 3 and the capillary pressure p 23 is prescribed for the two-phase system phase 2-phase 3. Equations (1.13) can be used to recover the capillary pressures p ij in the entire phase triangle ∆ by the formulas where When the mobilities and the capillary pressures are given by (1.15)-(1.17), the vector f and the matrix B become The volume balance equation (1.3) reduces to the condition In short, the condition (1.20) reads We study system (1.1) under the restriction (1.20), with f and B given by (1.18) and (1.19).Observe that the existence theorems of H. Amann [2] are also obtained for system (1.1) (without the restriction (1.20)) under the condition B 21 = 0 but they are valid only for nondegenerate matrix B in the case when ∂f 2 ∂u 1 ≡ 0 and under the assumption that some solution's norms are finite a priori.

Global existence
We consider system (1.1), (1.18), (1.19), (1.20) with the following initial and boundary conditions: where σ = const > 0 and Observe that system (1.1) decouples when k 1 = k 3 because f 2 (u) is then independent of u 1 and the equation for u 2 takes the form However, condition (1.20) written for u 2 in the form prevents us from solving equation (2.2) for u 2 independently.The solvability of the degenerate problem (1.1), (1.20), (2.1) with k 1 = k 3 was established in [5].Here, we consider the general case when (possibly) k 1 = k 3 , but the parameter It is also assumed that the vector functions u 0 (x) and d(t) take values strictly inside a triangle ∆ δ ⊂ ∆; i.e., for x ∈ Ω, 0 ≤ t ≤ T, for some δ > 0. Specifically, the first of these inequalities means that all the three phases are initially present at each point of the porous sample; i.e., the physical system is not degenerate.
Under the conditions imposed on the elements of B, system (1.1) is parabolic.Below is the main result.

, (2.1) has a solution in the class
where ε * is a constant depending on δ, T , an the number M bounding the norms of the initial and boundary data: Proof.
Step 1.We perform the change of variables (u 1 , u 2 ) → (ξ, u 2 ).Note that ξ is the relative phase saturation, since ξ = u 1 /(u 1 + u 3 ).Written in terms of the new variables, the original problem becomes ) where The advantage of system (2.8)-(2.11)over (1.1) is that the former decouples with respect to the higher derivatives: the equation for ξ does not contain u 2xx , and the equation for u 2 does not contain ξ xx .It should be stressed that A 12 vanishes at ξ = 0 and ξ = 1, and A 21 vanishes at u 2 = 0 and u 2 = 1.
Step 2. The indicated structural properties of system (2.8)-(2.11)guarantee that none of the three phases disappears.This fact in underlain by the following statement.Lemma 2.2.Let u(x, t) be a smooth solution to the degenerate parabolic equation in a bounded domain Ω ⊂ R n with the initial and boundary conditions Given δ ∈ (0, 1), let Then there is a constant δ 1 ∈ (0, 1) depending on δ, T , and The Lemma is proved by proceeding to v = 1 2 ln u/(1 − u) and considering the equivalent parabolic problem 1 + e 2v , (2.14) The coefficients ãj and Bij are calculated in terms of a j and B ij as a result of the substitution u → v.
Obviously, estimates (2.13) are equivalent to the boundedness estimate for |v|.This estimate can be easily obtained by applying the maximum principle.
Applying Lemma 1 to equation (2.10) and, then, to equation (2.8) gives the estimates where δ i depends on δ, T and J i : Moreover, δ 1 depends on δ 2 .Note that δ i does not decrease with J i .Although inequalities (2.15) are not a priori estimates, they nevertheless suggest that system (2.8)-(2.11) is nondegenerate on smooth solutions.Thus, the well-known theory of [7] can be applied to it.

Capillary pressure functions
Here we perform derivation of the capillary pressure functions (1.16) and (1.17) which agree with the hypotheses (1.13) on the diffusion capillarity tensor B.
First, we apply the group symmetry analysis to the homogeneous system for p ij (u 1 , u 2 ) B 12 = 0, B 22 = 0, which writes Application of the algorithm of group calculation [8] shows that any oneparameter group admitted by (3.1) is defined with the infinitesimal operator where the functions ζ i and η i are subject to the restrictions Hence, there is a one-parameter group with the operator The meaning of this group is that system (3.1) is invariant under the change of variables (u 1 , u 2 ) → (u 1 , u 2 ) : One can verify easily that system (3.1) has a solution depending only on the variable ξ.Indeed, given a function q 1 (ξ), the functions p 13 = q 1 (ξ), p 23 = A(ξ)q 1 (ξ)dξ solve system (3.1).Now we address the nonhomogeneous system (1.13) which writes We study these equations for p i3 (u 1 , u 2 ) in the case when the mobilities λ i are linear functions: The above analysis of the homogeneous system suggests to look for solutions in the form It follows from (3.2) that the functions q i and Q i solve the system q 2 (ξ) = q 1 (ξ)A(ξ), Assume that the capillary pressure p 13 (u) is a given function of u 1 at the part of the boundary of the triangle ∆ where u 2 = 0: Assume also that the capillary pressure p 23 (u) is a given function of u 2 at the edge where u 1 = 0 of the triangle ∆: It follows from (3.3) that ϕ 13 (u 1 ) = q 1 (u 1 ) + Q 1 (0), ϕ 23 (u 2 ) = q 2 (0) + Q 2 (u 2 ).
Then the other functions Q 1 (u 2 ) and q 2 (ξ) are defined from (3.4) as follows: Thus, we arrive at formulas (1.16) and (1.17) for the capillary pressures.We call the procedure yielding formulas (1.16) and (1.17) the method of physical interpolation since these formulas define the capillary pressures p 13 p 23 in ∆ from their values when u 2 = 0 and u 1 = 0, respectively.