Abstract
A new analytical solution is presented for the laboratory pulse decay permeability problem. With this solution, pulse decay permeability problem. With this solution, permeability of a core sample can be calculated from the permeability of a core sample can be calculated from the decay rate of a pressure pulse applied to one end of the sample. This development permits rapid. accurate measurement of permeability in samples such as tight gas sands, limestones, and shales.
Introduction
Because of its usefulness in measuring very low permeability. the pulse decay technique has been permeability. the pulse decay technique has been discussed often in the literature. In this technique, a small pore pressure pulse is applied to one end of a jacketed sample, and the pressure vs. time behavior is observed as the pore fluid moves through the sample from one reservoir to another. Brace et al. cave an approximate solution to this problem with the assumption of a linear pressure gradient at all times. This simplification leads to a predicted exponential pressure vs. time decay. By means of numerical solutions, Lin and Yamada and Jones have shown that the Brace solution can lead to significant errors in calculating permeability. These numerical solutions. however. are inconvenient to use and require considerable computer programming time. We present an analytical solution based on realistic assumptions and boundary conditions.
Experimental Technique
To understand the theoretical problem more thoroughly, a short description of the experiment is desirable. Fig. 1 is a schematic of the system. Initially, both valves are open and pressure is constant throughout the system. Next, Valve 1 is closed, and the pressure is changed slightly in the large Reservoir 1. Valve 1 remains closed for a few minutes to allow thermal effects to diminish (particularly important if the pore fluid is (as). Valve 2 then is closed, and, at time equal zero. Valve 1 is opened. A small differential pressure between the reservoirs will be indicated by the p transducer and will decrease with time. Pressure in Reservoir 1 remains constant during the decay. After the differential pressure has decreased by approximately 20%, Valve 2 is opened to terminate the decay. This accelerates the equilibration of pressure so that the next measurement can be made. pressure so that the next measurement can be made. Theory
As stated earlier, the pressure in Reservoir 1 remains essentially constant during the decay (t 0) because the volume of Reservoir 1, V1, is much greater than the pore volume, Vp, or the volume of Reservoir 2, V2. It can be assumed that fluid viscosity, is independent of position, x, in the sample and that fluid density, p, position, x, in the sample and that fluid density, p, permeability, k, and porosity, are dependent only on permeability, k, and porosity, are dependent only on fluid pressure, P. By combining Darcy's law with the one-dimensional diffusion equation we obtain
,..................(1)
where B is fluid compressibility, Bs, is rock compressibility, and Bk is the dependence of permeability on pore pressure. The magnitude of the nonlinear terms pore pressure. The magnitude of the nonlinear terms with respect to the linear ones is equal to (Bk + B)P0, where P0 is the pressure pulse amplitude. Because (Bk - B ) = 10 -2 bar - 1 (Ref. 8) and P0=1 bar, the product is small, and, hence, nonlinear terms can be product is small, and, hence, nonlinear terms can be ignored. If we further assume that the equation of flow is
------- = --- --------, ......................(2)
SPEJ
P. 719
Homogenization of a dual-permeability problem in two-component media with imperfect contact
