Study of the Well-Posedness of Models for the Inaccessible Pore Volume in Polymer Flooding

Abstract

Inaccessible pore volume, also known as dead pore space, is used when simulating enhanced oil recovery by polymer injection. We show that a widely used model for inaccessible pore volume can lead to an ill-posed problem, resulting in unphysical results. By considering shock solutions of the one-dimensional problem, we derive a necessary condition that an inaccessible pore volume model must fulfill in order to obtain well-posed equations. In this derivation, we use the Rankine–Hugoniot jump condition as a selection criterion for acceptable solutions. There are other possible criteria for the one-dimensional problem, in particular $\delta$-shock solutions, which we also briefly describe, but these are challenging and impractical to use. Based on a heuristic understanding of relative permeability, we subsequently derive two modified models for inaccessible pore volume. The first model follows directly from the modeling assumptions, but it has limited applicability. If the inaccessible pore volume is larger than the irreducible water saturation, then the equations are ill-posed for convex relative permeabilities. A second model is derived by relaxing the assumption of inaccessibility, allowing a limited fraction of the polymer to enter the smallest pores. This second model fulfills our necessary condition for well-posedness for all values of the inaccessible pore volume and any choice of relative permeabilities. Through one- and two-dimensional numerical examples, the different models for inaccessible pore volume are compared. For our second suggested model, the polymer concentration is observed to stay below the maximum injected value, which is not the case for the conventional model. This enables a more stable implementation of the highly nonlinear system, and a reduction in the number of nonlinear iterations is also observed in some cases. As this suggested model is straightforward to implement into existing reservoir simulators and can be used for a wide range of polymer models, it serves as a possible alternative to the conventional model.

Appendix

We have shown that the condition (13) is a necessary condition for well-posedness. With certain assumptions on the fractional flow function, we now show that this condition also is a sufficient condition when considering shock solutions where $s_l>s_r$. In the following proposition, $s_\text {wir}$ denotes the irreducible water saturation and $s_\text {or}$ the residual oil saturation.

Proposition 1

Assume that the fractional flow function $f \in C^1(s_\text {wir},1-s_\text {or})$ has only one inflection point $\hat{s}\in (s_\text {wir},1-s_\text {or})$ and is strictly convex on $[s_\text {wir},\hat{s}]$ and strictly concave on $[\hat{s},1-s_\text {or}]$, see Fig. 7. Assume further that $(s_l, s_r)$ form a single shock where $s_l>s_r$. If the velocity enhancement factor satisfies

$$\begin{aligned} \gamma (s)\le \frac{s}{s - s_\text {wir}}, \end{aligned}$$

(41)

for all $s\in [s_\text {wir}, s_l]$, then all pairs $(z_l, z_r)$ are solvable and the model equations (9) are well-posed.

Proof

Consider a single shock pair $(s_l, s_r)$ where $s_l>s_r$. We want to show that the apparent flux $g_l$, as defined by (11), is non-positive. Then, the well-posedness follows from the discussion in Sect. 4. Assuming (41) holds, we have

$$\begin{aligned} g_l = \frac{\gamma (s_{l})f(s_{l})}{s_{l}} - \frac{f(s_l) - f(s_r)}{s_l - s_r} \le \frac{f(s_{l}) - f(s_\text {wir})}{s_l - s_\text {wir}} - \frac{f(s_l) - f(s_r)}{s_l - s_r}. \end{aligned}$$

If $s_r=s_\text {wir}$, this immediately implies $g_l = 0$. Assume in the following that $s_r>s_\text {wir}$. We will prove that $g_l < 0$ in this case by a geometric argumentation. Define first the three line segments shown in blue in Fig. 7: $\ell _{ml}$ go from $s_\text {wir}$ to $s_l$, $\ell _{mr}$ go from $s_\text {wir}$ to $s_r$, and $\ell _{rl}$ go from $s_r$ to $s_l$.

Fig. 7

Plot of the fractional flow function with the definition of $\hat{s}$ and $\theta$

It is assumed that the solution of the Riemann problem is a shock, which implies that $\ell _{rl}$ must lie above f(s) for all $s\in (s_r,s_l)$, that is,

$$\begin{aligned} \rho f(s_l) + (1 - \rho ) f(s_r) > f(\rho s_l + (1 - \rho ) s_r) \end{aligned}$$

(42)

for $\rho \in (0, 1)$. Letting $\rho$ tend to zero, this gives

$$\begin{aligned} f'(s_r) < \frac{f(s_l) - f(s_r)}{s_l - s_r}. \end{aligned}$$

(43)

If $s_r \ge \hat{s}$, then f(s) would lie above $\ell _{rl}$ for all $s\in (s_r,s_l)$ as f is strictly concave on $[\hat{s}, 1-s_\text {or}]$, which would contradict the shock assumption. Thus, $s_r < \hat{s}$, and f(s) is strictly convex on $(s_\text {wir},s_r)$. We then have

$$\begin{aligned} f(s_l) - f(s_\text {wir})&= f(s_l) - f(s_r) + f(s_r) - f(s_\text {wir})\\&< f(s_l) - f(s_r) + f'(s_r)(s_r - s_\text {wir}) \end{aligned}$$

by the strict convexity of f in $(s_\text {wir}, s_r)$. Using (43), we get

$$\begin{aligned} f(s_l) - f(s_\text {wir})&< f(s_l) - f(s_r) + \frac{f(s_l) - f(s_r)}{s_l - s_r}(s_r - s_\text {wir})\\&= \bigl (f(s_l) - f(s_r)\bigr )\frac{s_l - s_\text {wir}}{s_l - s_r} \end{aligned}$$

and it follows that $g_l<0$ for all shock pairs when $s_r>s_\text {wir}$. This concludes the proof.

We note, without proof, that for a given fractional flow function f, which fulfills the assumptions in Proposition 1, there exists a point $s^*>\hat{s}$, which is such that for any $s_l\in (s_\text {wir},s^*]$, there exists an $s_r<s_l$ such that $(s_l, s_r)$ is a single shock and, reciprocally, if $(s_l, s_r)$ is a single shock, then $s_l\le s^*$. This point is $s^*$ is defined by

$$\begin{aligned} f'(s^*) = \frac{f(s^*) - f(s_\text {wir})}{s^* - s_\text {wir}}, \end{aligned}$$

if this point exists, or otherwise by the end point $s^* = 1-s_\text {or}$. This means that if (41) is fulfilled for all $s\in [s_\text {wir},s^*]$, then the problem is well-posed. Proposition 1 considers the case $s_l>s_r$, but if $s_l<s_r$ instead, then the condition (41) should be replaced by $\gamma (s)\le \frac{s}{s - s_\text {or}}$, where $s_\text {or}$ is the residual oil saturation, and the conclusion of the proposition still holds.

Finally, the proposition shows that $g_l>0$ does not provide acceptable shock solutions to (10). This implies, under the assumptions of the proposition, that the first configuration in Fig. 2 will never occur, and so the polymer is not able to overtake a water shock front.