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.

1.
Cf. preceding paper.
2.
H. Darcy, Les fontaines publiques de la ville de Dijon (1856).
3.
M. Muskat and H. G. Botset 1, 27 (1931).
4.
In fact, is not to be considered as a physical velocity, but is simply the volume fluid flux per unit macroscopic area of the porous medium.
5.
The choice of the liquid saturation is, of course, based only upon convenience. The use of its complement, the gas saturation, would evidently be equally definitive from a physical point of view.
6.
These definitions of the term ρ and the functions 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).
7.
Although these functions Fg and Fl(ρ) are, of course, the products of definite physical factors and of the properties of the sand structure, attempts to calculate them by considerations of the capillary structure of the medium must obviously involve such extreme simplifications as to dispel all confidence in their numerical implications. While it is most helpful to construct a physical picture for qualitatively explaining the variations of the Fl and Fg curves, it is clear that a quantitative treatment of such models of the microscopic flow system is even more remote from any practical applicability than the many fruitless attempts that have been made to calculate the normal homogeneous fluid permeability from grain size distributions. Furthermore, the practical situation with respect to the study of heterogeneous fluid systems, at least in the field of oil production, demands the recognition of the fact that sands do have the properties they are found with, which must therefore be determined by direct measurements, so as to permit an analysis of their macroscopic behavior under practical field conditions, and that even a satisfactory calculation of what the sand properties would have been if they had a certain idealized structure will have at best only an academic interest.
8.
The minus sign on the right sides of Eqs. (13) and (14) simply anticipates the choice of x = 0 as the inflow terminal of the system, so that ∂p/∂x<0.
9.
In a strict sense, however, there must be 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 ψ(ρ).
10.
M.
Muskat
and
H. G.
Botset
,
Physics
1
,
27
(
1931
),
and
M.
Muskat
,
Physics
5
,
71
(
1934
).
11.
Cf. last reference, in particular section II, D.
12.
It is also assumed that the pressure p is measured relative to some unit pressure, which is to be considered as multiplied into the right side of the equation for .
13.
Because of the lack of automatic recording equipment it was unfortunately impossible to obtain the detailed transient history of this type of flow system empirically with which to check the calculations directly.
14.
There is the limitation, however, that the closed boundary must also be an equi‐pressure surface, as the result either of its natural geometrical symmetry, as in strictly radial or spherical flow systems, or of a dynamical symmetry induced by the surrounding and neighboring flow systems.
15.
In the whole range of t̄(0<t̄≤0.15) there were 65 intervals of Δt̄, for each of which complete ρ(x̄) and p(x̄) curves were calculated.
16.
That the actual variation in ultimate saturation is even less than shown in Fig. 7 is also indicated by the direct production experiment described in the preceding paper. (Cf. Table III.)
17.
The average of R as weighted with respect to the liquid flux, i.e., R = (∫RdQl)/Ql = 31.94 to t̄ = 0.15 gives an added check on the calculations, since the fractional recovery, P, till the average pressure has fallen to , which should be given by: P = Ri(pi−p̄)/piR̄, 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).
18.
When the gas‐liquid ratios become excessive the additional factor of the evaporation of the liquid into the gas phase will enter the problem.
19.
Cognizance must, however, be taken of the limitation mentioned in footnote 14.
20.
The analytical expression of this condition is that the saturation is such that the liquid permeability is zero. Whether this is to mean that the saturation is actually zero at the boundary where the gas is injected or that it is still of the order of 20 percent, as is indicated by the experimental data, cannot be settled until more accurate experiments are performed in the region of very low liquid saturations. In any case it may be anticipated that either choice will give essentially the same results, and that when the very low saturations are attained throughout the main sand body, it will be necessary to take into account the evaporation of the liquid into the gas phase, as can be readily done by a slight modification of the fundamental Eqs. (9) and (10).
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