The Mechanics of Porous Flow Applied to Water-flooding Problems

Abstract

The flow of liquids through porous media is known to follow Darcy's lawwhich states that the velocity of flow is proportional to the pressuregradient. This law is but a statement of the facts obtained from experimentalstudies and is, therefore, un controversial. It should be mentioned, however, that it applies only when the flow is viscous in nature and that at excessivevelocities which tend to produce turbulence, the velocity is proportional tosome power of the pressure gradient, the exponent ranging from 1 to 0.5 as thedegree of turbulence increases. However, experiments with media havingpermeabilities corresponding to those of actual producing horizons showconclusively that the actual flow of liquids in a producing horizon must beviscous except perhaps in the immediate vicinity of a well producing at a highrate. Therefore, one is justified in assuming Darcy's law to hold for cases ofpractical inrest, as will be implicitly assumed in this discussion. Given this basic law of flow it is possible, theoretically to obtain solutionsto any problem of viscous flow of dead liquids 1 by the usual methods ofpotential theory. However, problems of practical interest often present suchcomplex geometrical configurations of sources and sinks (driving sources andoutput wells) as to make the analytical solutions extremely difficult or evenimpossible. It is useful, therefore, to note that Darcy's law is preciselyequivalent to the law of electrical conduction, so that the pressuredistribution in steady state porous flow of liquids is exactly the same as thepotential distribution in an electrical conducting medium. This analogy permitsthe use of electrical models wherein the potential distributions may bemeasured with far greater facility and accuracy than is possible in actualfluid-sand models. Such electrical models were used in determining the pressuredistribution about complex arrays of wells, an example of which will be shownlater in the discussion of the 5-spot flood. One phase of the porous flow problems which is of considerable practicalinterest involves the tracing of an advancing fluid front.