THE MODELING ERROR OF WELL TREATMENT FOR UNSTEADY FLOW IN POROUS MEDIA

. In petroleum engineering, the well is usually treated as a point or line source, since its radius is much smaller than the scale of the whole reservoir. In this paper, we consider the modeling error of this treatment for unsteady ﬂow in porous media.

1. Introduction. In many practical applications, especially in resistivity welllogging in petroleum exploitation, a kind of boundary value problem with equivalued surface is formulated. It is a kind of nonlocal boundary value condition, which can be also used to give mathematical descriptions for other problems in physics and mechanics (cf. [11,10,5,6]). From the physical perspective, the equivalued surface boundary value condition corresponds to a source.
In resistivity well-logging, we are most concerned about two quantities: the bottom-hole pressure(BHP) and the rate of production(injection). The BHP is often difficult to calculate since the well radius is much smaller than the scale of the whole reservoir. In practical calculation, the variation of solutions near the well is quite large, and in finite element procedure, it is necessary to have a refined partition of elements near the well. This causes a complexity in computation [12]. To get rid of this difficulty, the well can be approximately regarded as a point and corresponding boundary value problems with equivalued boundary surface on the well boundary can be approximately replaced by the boundary value problems with Dirac function. Our focus is to estimate the modeling error from the above approximation for unsteady flow. Of course, numerical well model for unsteady flow will be studied in the subsequent paper.
There have been some previous works on the steady flow in porous media. In [5,6], the authors considered the limit behavior of solutions for elliptic problems with equivalued surface boundary value conditions as the radius of well tends to zero. In [13], the authors presented the modeling error analysis for steady flow.
In this paper, we consider unsteady flow in porous media with a well. Let Ω be a bounded domain in R 2 with smooth boundary Γ. Denote by B(x 0 , δ) the disk 2172 TING ZHANG centered at x 0 with radius δ > 0 occupied by a well and by Ω δ = Ω \ (B(x 0 , δ)). We consider the following governing equation for unsteady flow in porous media where u δ is the pressure, K the permeability. On the well boundary γ = ∂B(x 0 , δ), two quantities are of particular importance in practical applications as we have already emphasized as before: the well bore pressure u δ | γ and the well flow rate γ K(x) ∂u δ ∂ν ds, where ν is the unit outer normal to ∂Ω δ . The boundary condition to be imposed on γ is either the following mixed boundary condition which fixes the well flow rate q 0 (t) or the Dirichlet boundary condition which fixes the well bore pressure α(t) (a function with respect only to t) , we consider the following two equivalued surface boundary value problems: which is a fixed rate well; and which is a fixed pressure well. In these problems, there exist two separated scales O(1) and δ, where O(1) represents the typical length of the reservoir, δ is the radius of the well, and O(1) δ. The radius δ is so small that the well is usually treated as an 'point' source. Then the approximation of problem (1.1) and (1.2) are as follows where q 0 (t) is the well flow rate mentioned in (1.1); and where the well flow rateq 0 (t) is unknown and it is implicitly determined through the condition (u −φ)(x 0 , t) +φ| γ = α(t) which is actually a first-kind Volterra integral equation with respect toq 0 (t) (will be addressed later in Section 4) and is an approximation of the boundary condition u δ | γ = α(t). Herẽ Note thatq 0 (t) depends on δ which is determined by the condition (u−φ)(x 0 , t)+ φ| γ = α(t). Problem (1.4) is not dependent on δ onceq 0 (t) is solved by appropriate method.
Problems (1.1) and (1.2) are the parabolic boundary value problems with nonlocal equivalued surface. They are different from the usual parabolic equations with the typical boundary conditions. The existence and uniqueness of weak solutions to problem (1.1) can be found in [11].
For fixed rate well, the most significant work is that A.Damlamian and Ta-Tsien Li discussed the relationship of solutions between problem (1.1) and problem (1.3) in 1982. Their main result is the following theorem only for the fixed rate well, which can be found in [2,3].
To the best of our knowledge, there are few literature discussing the fixed pressure well, even if the convergence of the solutions u δ to u in a certain sense as δ → 0.
In this paper, we consider the error estimate for this modeling treatment. Before stating our main results, we need the following hypotheses (H2) |q 0 (t)| + |q 0 (t)| ≤ C, ∀t ∈ [0, T ], with C > 0 independent of δ, (H3) (Compatibility) α(0) = 0, q 0 (0) = 0. The main results are as follows. For fixed rate well and homogeneous media, we get For fixed pressure well and homogeneous media, we obtain and (H3) is satisfied. Let u δ be the solution of (1.2) andq δ 0 (t) = − γ K 0 ∂u δ ∂ν ds. Let (u,q 0 (·)) be the solution of (1.4) andq 0 (·) be bounded uniformly with respect to δ. Then for sufficiently small δ > 0, there exists a positive constant C independent of δ such that For fixed rate well and heterogeneous media, we have For fixed pressure well and heterogeneous media, we get Theorem 1.5. Let the assumptions (H1) and (H3) be satisfied, and Let (u,q 0 (·)) be the solution of (1.4) andq 0 (·) be bounded uniformly with respect to δ. Then for sufficiently small δ, there exists a positive constant C independent of δ such that Remark 1. We guess that the constant C = C(p) in Theorems 1.4 and 1.5 has the form as C(p) = c 1 p β for some β > 0 and c 1 is independent of p and δ. If so, we may take p = | ln δ|, then δ −3/p = e 3 for δ < 1. Hence, the result in Theorem 1.4 becomes that for sufficiently small δ > 0, and the results in Theorem 1.5 turn into that for sufficiently small δ > 0, This paper is organized as follows. In Section 2, some preliminary works are introduced; then in Section 3-Section 6, we give the proof of Theorems 1.2-1.5 respectively.
2. Some preliminary works. We first introduce the following lemma which plays an important role in the proof of the main results.
then there exists a constant C depending only on λ such that To prove our main results, we first derive the equation for error between problems (1.1) and (1.3).
Let φ be the solution of In order to derive the estimate for the error term v δ , we always split it into two and v 2 satisfies the problem where B 0 = B(x 0 , δ).
3. Proof of Theorem 1.2. We consider the case of the fixed rate well and homogeneous media, i.e. K ≡ K 0 . Our plan is as follows: To prove Theorem 1.2, we first estimate v 1 L ∞ (Q δ ) in (2.1). By maximum principle, it is bounded by the boundary value at the well: By the regularity theorem [1], we easily get that By the Maximum Principle, we get that Step 2. Next we give the bound of the flow rate γ K 0 ∂v2 ∂ν ds for v 2 in (2.2). We just aim at estimating the term − γ K 0 ∂v1 ∂ν ds and B0 ∂u ∂t dx owing to the formula: To estimate the first term at the right hand side of (3.1), we introduce the corresponding steady problem of (2.1) Multiplying (2.1) by the solution ρ 1 of (3.2) and integrating on Ω δ , we get Multiplying (3.2) by the solution v 1 of (2.1) and integrating on Ω δ , we have Hence, Thanks to ρ 1 | γ = 1, we have For problem (3.2), by Maximum Principle and the Hopf's Lemma, we have 0 ≤ ρ 1 ≤ 1 in Ω δ , and ∂ρ 1 ∂ν > 0 on γ.
Then we have by Lemma 2.1 that To bound the first term at the right hand side of (3.3), we know by the compatibility thatṽ 1 By the regularity of z ∈ C 2 (Ω × (0, T )) and Maximum Principle, we get that ∂v1 ∂t L ∞ (Q δ ) ≤ Cδ. Combining with (3.3) and (3.4), we obtain For the second term at the right hand side of (3.1), The regularity that z ∈ C 2 (Ω × (0, T )) implies For the second term, we have where we have used the hypothesis (H2) and the elementary inequality 1 − e −x ≤ x, for x ≥ 0. From (3.5), (3.6) and (3.7), we obtain the bound for the flow rate of v 2 , γ K 0 ∂v 2 ∂ν ds ≤ Cδ (1 + | ln δ|) .
(3.9) To bound v 2,2 , we we introduce the auxiliary problem Proceed as before and obtain thatv 2,2 ≤ v 2,2 . Moreover, Hopf's Lemma implies thatv 2,2 ≤ 0 in Ω δ . Similar to argument in (3.8), we can show that v 2,2 | γ ≥ −Cδ| ln δ| (1 + | ln δ|) . (3.10) This completes the proof of Theorem 1.2, since 4. Proof of Theorem 1.3. In this section, we deal with the case of the fixed pressure well and homogeneous media i.e. K(x) ≡ K 0 . To prove Theorem 1.3, we first give some comments on the additional condition at the well boundary in which is an approximation of the fixed well bore pressure u δ | γ = α(t) in problem (1.2). Actually, (4.1) can be rewritten into a first-kind Volterra integral equation with respect toq 0 (·) as follows where the explicit form of the kernel f (·) will be derived in the following. Set z = u −φ, we know that z satisfies the problem Thus the kernel f (·) of (4.2) has the form as The well-posedness of a first-kind Volterra integral equation is a tough topic. Due to f (0) = 0 and ∂ n f ∂t n (0) = 0, n = 1, 2, · · · , the equation (4.2) falls into a class of ill-posed problem in general (see [8,7,9]). Here we do not present the wellposedness of the equation (4.2) in the classical sense. However, equation (4.2) has a special feature as which makes it quite different from general ill-posed problems. Then we will prove the solvability of the linear system which is generated by the discretization of equation (4.2). In fact, denote by a partition of [0, T ] with h = T /n, and t i = ih, ∀0 ≤ i ≤ n. Taking t i = ih (i = 1, 2, · · · , n) and discretizing the left hand side of the integral equation (4.2) by trapezoidal rule, we obtain a linear system with a lower triangular matrix 1))h), for i, j = n, and q = (q 0 , q 1 , · · · , q n−1 ) T , α = (α 0 , α 1 , · · · , α n−1 ) T , where q i = q(t i ), α i = α(t i ) (i = 0, 1, · · · , n − 1).
Take h = δ 2 , then for sufficiently small δ, the associated matrix F for the linear system of equations (4.5) is diagonally concentrated Therefore, the linear system (4.5) is stable and easy to be solved.
which means that F is almost singular and the linear system (4.5) can hardly be solved.
In the following, we give a priori error estimate for the flow rateq 0 (·) under the condition thatq 0 (·) is bounded uniformly with respect to δ.
By the Maximum Principle, we get where C = C(p) is independent of δ and α = 1 − 3 p , p > 3.
Step 2. Next we give the bound of the flow rate γ K ∂v2 ∂ν ds for v 2 in (2.2). By the similar argument to Step 2 in Section 3, we get that Step 3. Finally, we give the bound of v 2 L∞(Q δ ) in (2.2).
The argument is the same as Step 3 in Section 3, since only Maximum Principle and Hopf's Lemma are multiple used in Section 3 and the assumption for the regularity of K does not affect their application to a certain extent. Hence, 6. Proof of Theorem 1.5. In this section, we deal with the fixed pressure well and heterogeneous media. Similar to the proof of theorem 1.3, the additional condition in problem (1.4) can also be written into a first-kind Volterra integral equation with respect toq 0 (·) that t 0q 0 (y)g(t, y)dy = α(t), t ∈ [0, T ], (6.2) where the explicit form of the kernel g(·) will be derived in the following. Set z = u −φ, we know that z satisfies the problem By Green formula, the solution of problem (6.3) can be expressed as · ∇G(x − ξ, t − τ )dτ dξdy The kernel g(·) of (6.2) is Due to g(0) = 0 and ∂ n g ∂t n (0) = 0, n = 1, 2, · · · , the equation (6.2) is also ill-posed in general as (4.2). However, g(·) has the similar features as (4.4) to the kernel f (·) in (4.2), which makes the integral equation (6.2) easy to be solved numerically as (4.5)-(4.6). And all the properties ofq 0 (·) in Section 4 still hold in this section.
In the following, we give a priori estimate for the flow rateq 0 (·).