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A coupled elasto-plastic constitutive and stress-dependant permeability rock model for hydrocarbon bearing rocks is presented, and model parameters values obtained from tests done on Indiana limestone are presented. An oscillating pulse technique is presented to measure the permeability changes with stress changes. Data from comprehensive laboratory tests is used to develop an empirical relationship for permeability in terms of effective stress for Indiana limestone. The coupled constitutive-permeability rock model is implemented in a Finite Element (FE) computer program, and used to analyze the conventional triaxial compression test samples tested in the laboratory. The FE results are compared with experimental data and back predicted results obtained from the constitutive and permeability models.
Based on the coupled shear-volumetric response of rocks, observed two distinct states, termed "characteristic states," can be identified (Roegiers et al. 1991). The first one relates to the state of failure when a rock experiences progressive shear deformation at constant volume. The second state is identified with the onset of dilatancy at which the rate of volumetric strain momentarily vanishes as the rock passes from the compressive mode to the dilatant mode of deformation. The constitutive model presented in this paper explicitly incorporates both the characteristic states, which are represented mathematically as surfaces in the stress space. To describe the response of a rock in both the compressive and dilative regimes, two separate hardening surfaces and the corresponding plastic potential functions are employed. These basic elements of the constitutive model are shown in Fig. 1, and are described in the sub-sections that follow.
The first characteristic state surface is defined as the failure envelope of the rock.
The second characteristic state line, CS1-2, represents the transition of the volumetric response to the dilatant. Stress-states lying below the second characteristic state line induce only compressive volume changes.
The oscillating pulse technique used consists of applying a sinusoidal pressure wave of known amplitude and frequency to the upstream end of a sample, under a known confining pressure and pore pressure. Among the several factors which describe the behavior of the rock sample, the permeability of the sample and its storage capacity are the only unknowns. The remaining parameters can be either measured or evaluated through calibration procedures. The solution to the diffusivity equation with the above initial and boundary has been presented by Kranz et al. (1990). The solution is given in terms of R, the ratio of the amplitudes between the downstream and upstream ends, and d, the phase difference between the downstream and the upstream waves.
The procedure to obtain permeability, k, and diffusivity values from the above equations requires Eqs. (2.5) and (2.6) to be solved simultaneously for a and ?, knowing R and d from the experimental data. Once a and ? are evaluated, Eq. (2.3) can be used to evaluate permeability.
