Horizontal flow and capillarity-driven redistribution in porous media

A recent macroscopic mixture theory for two-phase immiscible displacement in porous media has introduced percolating and nonpercolating phases. Quasi-analytic solutions are computed and compared to the traditional theory. The solutions illustrate physical insights and effects due to spatiotemporal changes of nonpercolating phases, and they highlight the differences from traditional theory. Two initial and boundary value problems are solved in one spatial dimension. In the first problem a fluid is displaced by another fluid in a horizontal homogeneous porous medium. The displacing fluid is injected with a flow rate that keeps the saturation constant at the injection point. In the second problem a horizontal homogeneous porous medium is considered which is divided into two subdomains with different but constant initial saturations. Capillary forces lead to a redistribution of the fluids. Errors in the literature are reported and corrected.

Figure 1

Typical capillary pressure vs water saturation relation $P_c(S;S_{\mathbb{W}0},S_{20},S_{40})$ for primary imbibition (solid), secondary imbibition (dashed), primary drainage (dash-dotted), and secondary drainage (dotted). The corresponding initial saturations $S_{\mathbb{W}0},S_{20},S_{40}$ are listed in Table , while the parameter values are listed in Table .

Figure 2

Parameter functions for characteristic processes. From top to bottom the fractional flow function $f_{\mathbb{W}}\left(S\right)$, its derivative $d{f}_{\mathbb{W}}\left(S\right)/dS$ and the macroscopic capillary diffusion coefficient $D\left(S\right)$ are shown. Solid curves show primary imbibition functions, dashed curves show secondary imbibition functions, dash-dotted curves show primary drainage, and dotted curves show secondary drainage processes.

Figure 3

Saturation profiles for primary imbibition processes (initial condition A) with three different values of $r$ at $t=1000\text{s}$. Solid curves correspond to $r=1$, dashed curves to $r=0.8$, and dash-dotted curves to $r=0$. The saturation profiles of the Buckley-Leverett problem with flux decreasing as $\sim 1/\sqrt{t}$ are shown for comparison as dotted curves. The profile $S_2=0$ coincides with the $x$ axis. The profile $S_{\mathbb{W}}$ has values in the saturation range from 0 to $0.7$, and the profile $1-S_4$ has values in the range from $0.8$ to 1.

Figure 4

Saturation profiles for secondary imbibition processes (initial condition B) with three different values of $r$ at $t=1000$ s. Solid curves correspond to $r=1$, dashed curves to $r=0.8$, and dash-dotted curves to $r=0$. The saturation profiles of the Buckley-Leverett problem with flux decreasing as $\sim t^{-1/2}$ are shown for comparison as dotted curves. The profile $S_2$ has values in the range from 0 to $0.15$. The profile $S_{\mathbb{W}}$ has values in the saturation range from $0.15$ to $0.7$, and the profile $1-S_4$ has values in the range from $0.81$ to 1.

Figure 5

Comparison of saturation profiles for initial conditions B and C with three different values of $r=0$ (left profiles), $r=0.8$ (middle profiles), and $r=1.0$ (right profiles) at $t=1000\text{s}$. Solid curves correspond to initial conditions of type B and dashed curves to initial conditions of type C. The profiles $S_2$ have values in the range from 0 to $0.15$. The profiles $S_{\mathbb{W}}$ have values in the saturation range from $0.15$ to $0.7$, and the profiles $1-S_4$ have values in the range from $0.81$ to 1.

Figure 6

Saturation profiles for the Philip problem for (a) a primary redistribution and (b) a secondary redistribution. Solid curves show profiles at $t=1000\text{s}$ and dashed lines show the initial conditions. Profile $S_2$ has values in the range from 0 to $0.15$. Profile $S_{\mathbb{W}}$ has values in the saturation range from $0.15$ to $0.7$, and profile $1-S_4$ has values in the range from $0.81$ to 1.

Figure 7

Flux profiles at $t=1000\text{s}$ for the Philip problem. The solid curve corresponds to a primary redistribution and the dashed curve to a secondary redistribution.