Non-uniqueness, Numerical Artifacts, and Parameter Sensitivity in Simulating Steady-State and Transient Foam Flow Through Porous Media

Abstract

The uniqueness and sensitivity of foam modeling parameters are crucial for simulating foam flow through porous media. In the absence of oil in the porous medium, the local-equilibrium foam model investigated in this work uses three parameters to describe the foam quality dependence: $fmmob,\, fmdry$, and $epdry$. Even for a specified value of $epdry$, in some cases, two pairs of $fmmob$ and $fmdry$ values can experimentally match measured transition foam quality ($f_\mathrm{g}^{t}$) and transition foam apparent viscosity ($\mu _\mathrm{foam,app}^t$). This non-uniqueness can be broken by limiting the solution such that $fmdry$ is smaller than the transition water saturation ($S_\mathrm{w}^t$). In addition, a three-parameter fit using all experimental data of apparent viscosity versus foam quality was developed to simultaneously estimate $fmmob,\, fmdry$, and $epdry$. However, a better strategy is to conduct and match a transient experiment, in addition to steady-state experiments, in which a gas displaces the surfactant solution at 100 % water saturation. This transient foam quality scans the entire range of fractional flow, and the values of the foam parameters that best match the experiment can be uniquely determined. The numerical artifact of pressure oscillations in simulating this transient foam process was investigated by comparing the finite difference algorithm with the method of characteristics. Sensitivity analyses indicated that the estimated foam parameters were highly dependent on the parameters used for the water and gas relative permeabilities. In particular, the water relative permeability exponent and connate water saturation are important.

Appendix

In addition to Eq. (1), the material balance of 1-D transient foam flow through porous media is governed by

$$\begin{aligned} {\phi } \frac{\partial S_\mathrm{w}}{\partial t}+\frac{\partial u_\mathrm{w}}{\partial x}&= 0 \end{aligned}$$

(19)

$$\begin{aligned} {\phi } \frac{\partial S_\mathrm{g}}{\partial t}+\frac{\partial u_\mathrm{g}}{\partial x}&= 0. \end{aligned}$$

(20)

If the dimensionless variables $t_\mathrm{D} =\frac{u^{BC}t}{{\phi } L},\, x_\mathrm{D} =\frac{x}{L}, f_\mathrm{w}=\frac{u_\mathrm{w} }{u^{BC}}$, and $f_\mathrm{g} =\frac{u_\mathrm{g}}{u^{BC}}$ are used, we can use the following partial differential equation for the gaseous phase:

$$\begin{aligned} \frac{\partial S_\mathrm{g} }{\partial t_\mathrm{D} }+\frac{\partial f_\mathrm{g} }{\partial x_\mathrm{D}}=0. \end{aligned}$$

(21)

The gas fractional flow curve ($f_\mathrm{g}-S_\mathrm{g}$) is plotted in Fig. 12 using Eqs. (1–5).

Fig. 12

Gas fractional flow curve and location of the shock front. $fmmob={47196}, fmdry={0.1006}$ and $epdry={500}$. The remaining parameters used are shown in Table 2

As shown in Figure 12, a shock front will result if 100% gas displaces 100% surfactant solution. The shock saturation is determined by drawing a straight line from the initial condition ($S_{\mathrm{g,IC}}$ = 0), which is tangential to the fractional flow curve. In the case in Fig. 12, we observe $S_{\mathrm{g,shock}}$ = 0.9182. The wave velocities and saturation profiles can be constructed based on Figure 12 using Eq. (22):

$$\begin{aligned} \left. \frac{\mathrm{d}x_\mathrm{D} }{\mathrm{d}t_\mathrm{D} }\right| _{S_\mathrm{g} =a} =\left. \frac{\mathrm{d}f_\mathrm{g} }{\mathrm{d}S_\mathrm{g} } \right| _{S_\mathrm{g} =a} \end{aligned}$$

(22)

If the saturation “a” in Eq. (22) is smaller than $S_{\mathrm{g,shock}}$, then the wave velocity at $S_\mathrm{g} =a$ is equal to the shock velocity (Table 2).

Table 2 Parameters used to model foam in this work

According the definitions of the local foam apparent viscosity (Eq. (17)) and the average foam apparent viscosity (Eq. (18)), the following relationship is obtained:

$$\begin{aligned} {\overline{\mu }}_{\mathrm{foam,app}}&= \frac{k\left( {p_{\mathrm{out}} -p_{\mathrm{in}} } \right) }{\left( {u_\mathrm{w} +u_\mathrm{g} } \right) L} \nonumber \\&= \frac{k}{L}\int \limits _0^L {\frac{1}{u_\mathrm{w} +u_\mathrm{g} }\cdot \frac{\mathrm{d}p}{\mathrm{d}x}\mathrm{d}x} \nonumber \\&= \frac{k}{L}\int \limits _0^L {\frac{1}{-\frac{kk_\mathrm{rw} }{\mu _\mathrm{w} }\cdot \frac{\mathrm{d}p}{\mathrm{d}x}-\frac{kk_{\mathrm{rg}}^\mathrm{f} }{\mu _\mathrm{g} }\cdot \frac{\mathrm{d}p}{\mathrm{d}x}}\cdot \frac{\mathrm{d}p}{\mathrm{d}x}\mathrm{d}x} \nonumber \\&= \int \limits _0^1 {\frac{1}{\frac{k_{\mathrm{rw}} }{\mu _\mathrm{w} }+\frac{k_{\mathrm{rg}}^\mathrm{f} }{\mu _\mathrm{g} } }}\mathrm{d}x_{\mathrm{D}} \nonumber \\&= \int \limits _0^1 {\mu _{\mathrm{foam,app}} \mathrm{d}x_\mathrm{D} }. \end{aligned}$$

(23)

At a specific time $t_\mathrm{D} =t_0$, both $k_{\mathrm{rw}}$ and $k_{\mathrm{rg}}^\mathrm{f}$ are functions of $S_\mathrm{g}$. The saturation profile is already known by computing the wave velocities. Thus, Eq. (23) can be approximated by numerical integration using available data points:

$$\begin{aligned} {\overline{\mu }}_{\mathrm{foam,app}} =\int \limits _0^1 {\frac{1}{\frac{k_{\mathrm{rw}} }{\mu _\mathrm{w} }+\frac{k_{\mathrm{rg}}^\mathrm{f} }{\mu _\mathrm{g} }}\mathrm{d}x_\mathrm{D} \approx \sum _{i=1}^n {\frac{1}{\frac{k_{\mathrm{rw}} \left( {S_{\mathrm{g},i} } \right) }{\mu _\mathrm{w}}+\frac{k_{\mathrm{rg}}^\mathrm{f} \left( {S_{\mathrm{g},i} } \right) }{\mu _\mathrm{g} }}\Delta x_{\mathrm{D},i} } }. \end{aligned}$$

(24)

Equation (24) is used to calculate the average foam apparent viscosity in the MOC solution. For FD simulations, the average foam apparent viscosity is approximated by the pressure difference between the first and last grid blocks:

$$\begin{aligned} {\overline{\mu }}_{\mathrm{foam,app}} =\frac{k\left( {p_{NX} -p_1 } \right) }{\left( {u_\mathrm{w} +u_\mathrm{g} } \right) L}\cdot \frac{NX}{NX-1}. \end{aligned}$$

(25)

In steady-state foam calculations, the transition water saturation $S_\mathrm{w}^t$ can be calculated without knowing the values of the foam modeling parameters ($fmmob,\, fmdry$, and $epdry$). According to Eqs. (8) and (9), one can obtain

$$\begin{aligned} k_{\mathrm{rw}} (S_\mathrm{w}^t )=\frac{\mu _\mathrm{w} (1-f_\mathrm{g}^t )}{\mu _{\mathrm{foam,app}}^t } \end{aligned}$$

(26)

Equating Eq. (4) with (26) at the transition water saturation yields

$$\begin{aligned} S_\mathrm{w}^t =S_{\mathrm{wc}} +(1-S_{\mathrm{gr}} -S_{\mathrm{wc}} )\left[ {\frac{\mu _\mathrm{w} (1-f_\mathrm{g}^t )}{k_{\mathrm{rw}}^0 \mu _{\mathrm{foam,app}}^t }} \right] ^{\frac{1}{^{n_\mathrm{w}}}}. \end{aligned}$$

(27)