A Review of a Pulse Technique for Permeability Measurements (1980)

Abstract The pulse method presented by Brace et al.1 for permeability measurement makes use of transient fluid flow in porous media. The upstream reservoir, the specimen, and the downstream reservoir are initially in equilibrium. Then a small pressure pulse is applied in the upstream reservoir. The subsequent pressure change in both reservoirs is measured, and an exponential curve fit is imposed. Hence, the natural log of pressure relates linearly to time, and the slope coefficient is a known function of the permeability of the sample. The method is based on the theoretical consideration of a transient Laplace equation. The transient term is ignored by assuming the coefficient a is negligibly small. This assumption yields linear pressure distribution along the specimen in the flow direction at any time t. Subsequently, Brace et al. modeled the phenomenon by two-segment finite difference equations. In each segment, the pressure distribution is linear. The analysis results in the exponential pressure change in both reservoirs. However, this important characteristic is not always observed, as shown by Lin.2 Various factors – such as specimen size, upstream and downstream reservoir volume, etc. – control the pressure behavior. Lin's discussion on the influence of these factors based on his finite difference calculations is helpful in designing an experiment, but the discussion is not extended to establish a clear-cut criterion for a valid experiment. Although, in general, the method has been applied properly, it is necessary to clarify the limiting conditions. The importance of such conditions is realized particularly in low-permeability measurements where a long-term experiment is inevitable. The mathematical development is followed by numerical analysis and subsequent physical interpretation of the result. Due to the limitation of space, the knowledge of the original work by Brace et al. is assumed in the following discussion. The equation given by Brace et al. is the one-dimensional transient flow equation. The solution of this equation has been given in Ref. 3 in the form of an infinite series. In this study, a different approach is taken to find the exponential pressure decay prescribed by Brace et al. and associated constraints. We presented in Ref. 4 the solution of this equation in the Laplace domain, and at x = 0 (upstream reservoir) it becomes for a unit pulse. The exponential terms may be expanded by a power series in the quantity 2aLs. When 2aLs&lt;1, it may be approximated adequately by the first three terms, and this first approximation results in an exponential pressure decay in the time domain. On the other hand, if λ1s/a&gt;&gt;1 (the second approximation), the bracketed quantity becomes unity. Since λ1 has to be far greater than a to yield this condition for sufficiently large range of s, Eq. 1 becomes simply a reciprocal of s. The pressure behavior is a step function in the time domain. If Brace et al.'s assumption, a = 0, is accepted, the condition for the first approximation is satisfied in a sufficiently large range of s. However, it is necessary for the second condition not to hold simultaneously. To find this condition, let the second condition be rewritten as s≫a/λ1. If a/λ1 is sufficiently large, this condition is not satisfied for a sufficiently large range of 5 with the upper bound (a/λ1)2. Using this upper bound, one can find 2aLs&lt;2a2L/λ1=2δ1&lt;1. When this condition is satisfied, it can be shown that the pressure decay has a time constant expressed