Dynamic Measurements Of Capillary Pressure
Authors: Gardner G.H.F. , Acheson W.P. 
Published: December 2007

Abstract
When water invades a porous cylinder under a constant inlet pressure, and displaces air, the moment of the mass of water in the sample about an axis in the inlet face increases at a constant rate. This statement is shown to be an exact consequence of the Buckley-Leverett theory, no matter what the initial distribution of water in the sample happens to be. Furthermore, the rate of increase of the moment is theoretically a linear function of the inlet pressure. The intercept on the pressure axis is equal to a certain average value of the capillary pressure and the slope is proportional to the permeability at residual air saturation.
These results are compared with measurements made on sandstone and alundum samples. This gives an exact method of comparing the theory with observations.

Introduction
This paper describes an apparatus for the quantitative measurement of capillary and viscous forces under certain dynamic conditions in porous media. A measurement is made of the penetration of a liquid into a porous cylinder while the pressure at the inlet face is held constant. This arrangement has been employed in the past. The volume of fluid imbibed has been shown to vary linearly with the square root of the time. Also, the distance from the inlet to the edge of the advancing liquid has been shown to vary in the same way. However; these previous measurements are qualitative, or comparative, in nature because of the theoretical difficulty of relating the absolute magnitudes of the variables observed to the capillary pressure and relative permeability. The novelty of the technique described here consists in measuring the moment of the mass of fluid in the cylinder at any instant about an axis in the inlet plane, and its usefulness depends on the relation which may be exactly derived from the usual theory. It is shown below that the moment increases at a constant rate which is independent of the initial distribution of liquid in the cylinder. By measuring the rate of moment increase for two different inlet pressures, it is theoretically possible to obtain measures of the capillary pressure and relative permeability. Experimental data are given in this paper using the technique with two materials, Alundum and Berea sandstone.
On the evidence of the limited observations made, two departures from the exact theory are suspected to exist. First, the rate of moment increase at a fixed inlet pressure is not constant, but increases gradually with time. Second, the rate of moment increase is not a linear function of the inlet pressure, but increases at a faster rate. The changes required in the physical assumptions, to account for these departures, are not established beyond doubt in this paper. The purpose, rather, is to describe the apparatus and the experiments made, and point out that the discrepancies appear to exist. Further experiments would be required to elucidate the matter. However, it is suggested that the first discrepancy is caused by the uncertain contact angle as the liquid enters dry rock, while the second is caused by the finite compressibility of the residual air trapped in the material.

THEORY
We assume that the liquid is flowing into a horizontal, cylindrical, uniform, porous material and displacing air from the pore space. The viscosity of the air and the density difference between the liquid and the air are assumed to have a negligible effect. Darcy's law for the flow of liquid may be written
.........................[1]
The air pressure throughout the pore space is assumed constant, because the viscosity of the air is small compared with that of the liquid. Hence the capillary pressure Pc is related to the liquid pressure by
P = constant - p .......................[2]c