Two-phase porous-media flow

Immiscible fluids share pore space, with each phase's Darcy flux determined by its pressure, viscosity and relative permeability. Fluid saturation specifies how much pore space each phase occupies. A capillary pressure couples their pressures, while their individual mass conservation laws determine displacement and saturation evolution.

When a nonwetting phase recedes from pores, surface tension and pore geometry can retain a residual saturation $s$. If $h$ is the mobile invaded thickness and $M$ its maximum past value, stored phase volume per plan area is $\varphi [(1-s)h+sM]$. Newly invading pores have $M=h$; during recession $M$ remains fixed. With mobile Darcy velocity $u$, conservation gives advance speed $u/\varphi$ and recession speed $u/[\varphi (1-s)]$. The latter is faster for $0<s<1$.

For capillary residual trapping of an initial triangular current, the method of characteristics translates its leading and trailing faces at $v_A=u/\varphi$ and $v_R=u/[\varphi (1-s)]$. Their intersection has height $a[L-(v_R-v_A)t/2]$. Extinction occurs at $t_{\ast }=2L/(v_R-v_A)$ and position $x_{\ast }=2L/s$. These formulas require $0<s<1$; at zero residual saturation the triangle translates indefinitely.

For the triangular current with capillary retention, the initial trailing face is already the maximum for $0<x<L$. Further upslope, the maximum occurs when the moving crest passes. HenceThe final trapped phase has saturation $s$ below this envelope. Its volume is $\varphi s\int M_{\infty }dx=\varphi a{L}^2$, exactly the initial mobile volume. Maximum invaded thickness is distinct from residual saturation: $s$ multiplies occupied pore volume rather than shrinking the geometrical envelope.

Injecting water to displace oil through a porous reservoir. Spatial permeability contrasts cause preferential flow and early water breakthrough, while unswept slower regions retain oil. The recovery also depends on fractional flow, capillary pressure and phase mixing; a passive streamline model alone does not capture all two-phase effects.

The one-dimensional mass conservation equation for saturation when total Darcy flux is constant and capillary pressure and gravity are neglected. It is a scalar conservation law with dimensional characteristic speed $(Q/\varphi )F^{\prime }(s)$. Shock speeds obey a Rankine-Hugoniot condition; physically admissible solutions satisfy an entropy selection. Capillary diffusion regularizes the saturation transition.

Eliminating phase pressures with $p_c=p_n-p_w$ gives $\varphi s_t+QF(s{)}_x=(D_c{s}_x{)}_x$. For decreasing capillary pressure, $D_c\ge 0$. This nonlinear diffusion broadens a saturation shock; it can degenerate where a phase mobility vanishes. Endpoint conservation retains the limiting Rankine-Hugoniot condition.

When a rarefaction wave attaches to a saturation shock, its terminal characteristic speed must match the Rankine-Hugoniot condition speed. This gives a tangent from the initial state $(s_0,F(s_0))$ to the fractional-flow curve at $s_s$. It selects the upstream shock saturation; the actual dimensional shock speed still includes total flux divided by porosity.

The fraction of total advective Darcy flux carried by one phase when capillary and gravity corrections are omitted. For two phases it is ${\lambda }_w/({\lambda }_w+{\lambda }_n)$. This quantity lies between zero and one for nonnegative phase mobilities. Its nonlinear dependence on fluid saturation determines characteristic speeds and shocks in the Buckley-Leverett equation.

The fraction of pore volume occupied by a fluid phase. For a completely fluid-filled two-phase pore space the saturations sum to one. Residual saturation refers to fluid remaining immobile under the modeled displacement; it can impose zero relative mobility at the endpoint.