Use of Complex Pore Pressure Transients to Measure Permeability of Rocks
Authors: Boitnott G.N. 
Published: December 2007

Abstract
While techniques for measuring permeability of core samples are well developed, many suffer from drawbacks in terms of ease of automation, duration of measurement, and limitations in range for a particular sample geometry and experimental set-up. Recent advances in computer automation of laboratory equipment have made it possible to control complex transients in pore pressure which can be used to measure permeability of core samples in a simple system. These transients can be tailored to have characteristics which have various advantages over traditional methods. Here we concentrate on one particular advantage: the ability to extend the use of transient techniques to samples with higher permeabilities. Through example measurements, we illustrate the ability to increase, by over an order of magnitude, the range of measurable permeabilities for a particular system. Methods of data reduction using frequency domain and time domain minimization are compared, and the relative advantages and drawbacks of each method are discussed.

Introduction
While transient methods for measuring permeability are well developed, a common problem is that they are difficult to apply for high permeability rocks when using liquids as the pore fluid. Traditional transient methods such as the pulse decay method and the sinusoidal oscillation technique each offer their own advantages, but both break down at high permeabilities due to system limitations in controlling and detecting short duration transients. Here we explore the use of complex (arbitrary) transients for permeability measurements and illustrate how they can extend the measurement range of a given system. To facilitate the discussion, we use as an example the adaptation of the particular rock testing apparatus. While the discussion will be specific to this apparatus, the results are relevant to any test configuration which is limited at high permeabilities by the finite slew rate of the pore pressure controller.
The specific system used here is shown schematically in Fig. 1. The confining pressure and the pore pressure are hydraulically servo-controlled with standard pressure intensifiers. The pore pressure is controlled at one end of the sample (referred to as the up-stream end), while the other end of the sample (the down-stream end) is connected to a fixed volume of pore fluid. Permeability is measured by controlling a perturbation in pore pressure at the upstream end and measuring the pressure response at the down-stream end.

Nature of the Response Function
In the general case, the response of such a system is controlled by two rock properties. the permeability and the specific storage. Analytical solutions for specific transients have been derived in a number of studies, mostly associated with issues concerning measurements on low permeability rocks.
In order to apply transient methods to high permeability rocks, the down-stream volume must become large in order to keep the equilibration time long enough to ensure proper measurement precision. This typically leads to the condition where the down-stream system storage is large compared to that of the sample (i.e. system volume is large compared to sample volume). In this case, the response of the system simplifies greatly, becoming insensitive to sample storage. For this special case, the pressure response (P) to a step function in pressure of amplitude (Po) at time t=0 follows the relation
(1)
where m is a constant dependent on fluid and system properties and sample geometry and k is the permeability. Likewise for the sinusoidal perturbation, the transfer function for this special case can be written in the frequency domain as
(2a)
(2b)
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