The empirical results established by R. D. Wyckoff and H. G. Botset on the flow of gas‐liquid mixtures through unconsolidated sands have been formulated into basic differential equations governing the motion of general heterogeneous fluids through porous media under both steady state and transient conditions. This formulation is based upon a representation of the porous medium as having a macroscopically local structure defined by the liquid saturation, or volume composition of the gas‐liquid mixture, this saturation in turn determining the separate permeabilities of the medium to the liquid and gas phases. The steady state solutions of these equations are derived for the cases of linear, radial, and spherical flow, and the distributions of the pressure, permeability, and saturation are given graphically. It is found that the properties of these flow systems change but little from the corresponding ones for homogeneous fluids as long as the pressure exceeds about half the saturation pressure of the gas, except for the fact that the liquid saturation is very approximately equal to the equilibrium value and the liquid permeability has a value very near to its equilibrium value. The drop in liquid permeability and saturation is highly localized about the outflow surfaces and in those regions where the pressure is very much less than the saturation pressure, the increase in the pressure gradients above their normal homogeneous fluid values being also largely confined to these regions. For the study of the early stages of transient types of flow an analytical theory is derived, in the case of the linear system, based upon a representation of the transient as a continuous succession of steady states. For the investigation of the complete history of the linear transient system a numerical method is presented in which is developed a stepwise integration of the simultaneous partial differential equations for the pressure and liquid saturation, after the replacement of the various derivatives by their appropriate differences. The specific problem is treated in which a linear column of sand of unit length filled with liquid saturated with an ideal gas to a pressure of 10 units is suddenly exposed at one terminal to a pressure of 1 unit, which is thereafter permanently maintained at that value while the other end is permanently kept closed. The results of the calculations for this problem are given graphically in the form of sets of curves showing the history of the saturation and pressure distributions within the flow channel, the time variation of the flux from the system, and the time variation of the gas‐liquid ratio associated with the liquid efflux. It is found that the liquid saturation at the time of physical depletion of the system, corresponding to an equalization of the pressures to that maintained at the outflow terminal, is quite uniform, being only 5 percent less at the outflow terminal than at the closed terminal of the linear column. The gas‐liquid ratio is found to increase monotonically with the time. The bearing of these results on such problems as well spacing and gas recycling in the production of oil from underground reservoirs, and the manner of treatment of other typical heterogeneous fluid systems so as to include both the deviations from the ideal behavior of the free gas phase and the effect of gas segregation are discussed in detail.

## REFERENCES

*Les fontaines publiques de la ville de Dijon*(1856).

**1**, 27 (1931).

*physical*velocity, but is simply the volume fluid flux per unit

*macroscopic*area of the porous medium.

*F*as $k/k0,$ are used here because it is found empirically (cf. preceding paper) that the curves $Fg$ and $Fl$ will then be practically independent of the nature of the sand (if uncon‐solidated).

*inflow*terminal of the system, so that $\u2202p/\u2202x<0.$

*some*variation of the permeabilities and saturations along the flow column, as is explicitly required by Eq. (16), as a consequence of the relation between

*p*and $\psi (\rho ).$

*p*is measured relative to some unit pressure, which is to be considered as multiplied into the right side of the equation for

*t̄*.

*R*as weighted with respect to the liquid flux, i.e., $R\u2009=\u2009(\u222bRdQl)/Ql\u2009=\u200931.94$ to $t\u0304\u2009=\u20090.15$ gives an added check on the calculations, since the fractional recovery,

*P*, till the average pressure has fallen to

*p̄*, which should be given by: $P\u2009=\u2009Ri(pi\u2212p\u0304)/piR\u0304,$ thus has the value 0.279, lying just between the values 0.269 and 0.284 given by the left and right sides of Eq. (38).