Porous Media Problems
Wayne T. Ford, Maria Concepcion Fuente, Margaret C. Waid · SIAM Journal on Mathematical Analysis · 1980
Laminar isothermal fluid flow of two immiscible compressible fluid phases in a porous medium is formulated in terms of four unknown functions $\rho _1$, $\rho _2$, $S_1$ and $S_2$ in a pair of partial differential equations \[ \frac{\partial } {{\partial t}}\left[ {\phi (x,t)S_i \rho _i } \right] = \frac{\partial } {{\partial x}}\left\{ {\kappa (x,t)\sigma _i (S_i )\frac{\partial } {{\partial x}}\left[ {\Phi _i \left( {\rho _i } \right)} \right]} \right\} \] and a pair of auxiliary relations \[ S_1 = \Gamma _1 \left( {\rho _1 ,\rho _2 } \right)\quad {\text{and}}\quad S_2 = 1 - S_1 .\] The first boundary value problem is reformulated for this system as a fixed point problem involving a mapping $\prod $ wherein $S_1$ implies $S_2$, the differential equations are used to find $\rho _1$ and $\rho _2$, and $\prod (S_1 )$ is set equal to $\Gamma _1 (\rho _1 ,\rho _2 )$. The mapping $\prod $ is shown to map a subset of $H^{2 + \theta ,{{1 + \theta } / 2}} (\bar \Omega )$ into $H^{2 + \theta ,{{1 + \theta } / 2}} (\bar \Omega )$ under appropriate conditions on the coefficients and equations.