Two-Point Determinations of Permeability and PV vs. Net Confining Stress

Abstract

Summary. Empirical equations are presented that accurately fit permeability, PV, or porosity data vs. net confining stress. Each of these equations has four adjustable parameters. With little loss of accuracy, however, two of the coefficients can be preset. Consequently, permeability, PV, or porosity measurements need to be made at only two confining stressese.g., at 1,500 and 5,000 psi [10.3 and 34.5 MPa] to define the stress dependence within close tolerances completely. With the techniques described, economical measurements of these stress-dependent properties can be made on a routine basis. The results can be used to calculate PV compressibility and to estimate productivity declines resulting from permeability reduction as deep, high-pressure reservoirs are drawn down. Introduction Many Workers have shown that the permeability, PV, and porosity of a rock sample decrease with increasing confining or "overburden" stress. The decrease is most rapid at low stresses and becomes progressively more gentle at higher stresses. Fig. 1 shows three curves of the permeability ratio at a given stress to permeability at zero net stress vs. net stress. Net stress in this case and throughout this paper is defined as the difference between isostatic (or hydrostatic) confining stress and the average pore pressure. Isostatic stresses rarely apply in an actual reservoir environment because the vertical stress is generally greater than lateral stresses. It is difficult to justify using triaxial OT even biaxial stresses in the laboratory, however, unless measurements are being made on whole-core samples, The principal stress can be applied along the axis of a whole-core sample by a hydraulic ram, and the two orthogonal lateral stresses (approximated as being equal) can be applied by pressure on a rubber sleeve. Core plugs, on the other hand, are generally cut perpendicular to the vertical axis, and it is not easy to simulate reservoir stresses for plugs cut in this orientation. Two different stresses cannot easily be applied to the rubber sleeve unless a mechanical stress is superimposed on the hydraulic pressure. Only core plugs were measured in this study, and because of the difficulty of applying appropriate triaxial stresses, only isostatic stresses were used. Teeuw has shown how to translate hydrostatic compaction data into uniaxial formation compactions if the Poisson ratio of the rock is known or can be estimated accurately. Andersen and Andersen and Jones have presented experimental data comparing the two stress modes. The main difficulty with obtaining compaction data and permeability-stress data is that measurements must be made at several stresses to define the curves adequately. Furthermore, in the past, each of these measurements has been a tedious undertaking. hence, they have been expensive. This paper will demonstrate how these curves can be generated from measurements at only two stresses and will explore the magnitude of errors that may be generated. Jones and Owens developed a permeability-stress correlation that permits a two-point fit, modified by Ostensen The equations presented in this paper should be more applicable over a wider stress range than the previous correlations. Experimental Measurements All measurements in this study were made with an unsteady-state Klinkenberg permeameter-porosimeter on 1-in. [2.5-cm] -diameter sandstone and conglomerate core plugs ranging in length from 1.6 to 3 in. [4.0 to 7.6 cm]. The operation principle of this instrument is similar to that described in Ref. 34. However, this instrument has been modified in several respects from the one described. First of all, the capability of making PV measurements has been added by incorporating an internal valve to the outlet end of the core holder. When this valve is open, gas (helium) can exit from the plug for a Klinkenberg permeability measurement. When the valve is closed, it seals the effluent end of the core plug, and helium is expanded from a chamber of accurately known volume into the core plug, which was initially filled with helium at atmospheric pressure. The PV of the plug is calculated from the ratio of the gas pressure initially in the reservoir before expansion to the pressure in the reservoir plus core plug after expansion, from a Boyle's law calculation. The second change was to increase the strength of the core holder so that it can accommodate 10,000 psig [69 MPa] sleeve pressure. Additionally, it has been provided with a pneumatically driven ram that applies an axial stress to the plug equal to the sleeve pressure. Sleeve pressures up to 10,000 psig [69 MPa] are applied through a pressure intensifier. The third major modification was to automate the instrument fully. Up to 18 core plugs can be loaded into a carousel. The length, diameter, and core plug identification are typed into a microcomputer that controls the instrument. The confining stresses, up to a maximum of eight, are then entered. After all the plugs have been loaded and the information entered, the operation is completely automatic, The final output includes the Klinkenberg permeability, Klinkenberg slip factor, Forchheimer inertial coefficient. PV, and porosity for each confining stress selected. If two or more stresses have been selected to be run on a particular plug, then permeability-vs.-stress and PV-vs.-stress curve fits are made automatically. In the instrument described earlier, the starting pressure for permeability measurements was 100 psig [689 kpa]. In the new instrument, the starting pressure has been increased to 250 psig [1.7 MPa]. The main reasons for the increase were to improve the accuracy of PV measurements, to increase the accuracy of the inertial term, and to increase speed in the permeability measurements. The minimum upstream pressure for the permeability measurements has been increased from 1 to 20 psig [7 to 138 kpa]. Another change was to provide three helium reservoirs instead of one. The three reservoirs consist of a manifold, which has a relatively small volume, and a small tank and a large tank that can be selectively connected to the manifold through valves. The manifold alone is used as the reservoir for the Boyle's law expansion for PV and for permeability measurements with tight core plugs. The small tank is connected to the manifold for permeability measurements on somewhat higher-permeability plugs. SPEFE P. 235^