Predicting Reservoir-Condition PV Compressibility From Hydrostatics Stress Laboratory Data

Abstract

Summary. Laboratory PV reduction data can be converted into reservoir-condition formation compressibility through a nonlinear elastic model. Data demonstrate the validity of the theory for consolidated sandstone. The algorithm described requires a nonlinear fitting routine available in many computer subroutine libraries to obtain the prediction parameters. parameters. Introduction As a new oil field begins production, the reservoir engineer needs to know the volume of oil in place (OIP) as quickly as possible. Early production history is matched to the drive forces available in the reservoir to determine the size of the reservoir. If there is no waterdrive, the only force is stored in the compression of the reservoir fluids and rock by the overlying strata. The pressure of the fluid in the pore space partly supports the overburden pressure. The compressional properties of the rock are given by the net over-burden pressure, which is the difference between the overburden and pore pressures. I The pore pressure decreases as the reservoir is produced, allowing the reservoir fluids to expand and provide energy for production. In addition to fluid expansion, the formation contracts as the net overburden increases, providing additional energy for fluid production. The amount of this contraction is given by the PV compressibility. The reservoir volume can be calculated from the produced volume and the system compressibility, which is the sum of the fluid, gas, and PV compressibilities. For a reservoir with no water encroachment, the reservoir volume is given by ..........................................(1) where the system compressibility is ..........................................(2) The PV compressibility, cf, can be an important contribution to production in an undersaturated volumetric reservoir. In some cases, production in an undersaturated volumetric reservoir. In some cases, ignoring the PV compressibility can lead to almost 100% error in the calculated OIP. It is relatively easy to measure the PV compressibility under a hydrostatic load, where the pressure is the same from all directions. However, this condition is not representative of a reservoir being depleted. For example, consider a horizontal bed that is thin in comparison with both its depth of burial and its lateral extent. As the fluid pressure decreases with production, the net overburden increases, causing the formation to contract and the grains to expand. The size change of the formation in the lateral direction will be prevented, however, by contact with the surrounding rock, so the prevented, however, by contact with the surrounding rock, so the only contraction will be in the vertical direction. In the laboratory, this condition is referred to as a uniaxial-strain condition-a load is applied in one direction and the confining stress in the lateral directions is adjusted to retain a zero lateral strain condition. This test is not as easy to perform as a loading under hydrostatic stress. The experiments reported here were designed to find a conversion from the hydrostatic-stress PV reduction test. which is cheap and relatively easy to perform, to the uniaxial-strain compressibility that is closer to the reservoir condition. In this paper, a method to convert from the laboratory data to reservoir values is described. The data supporting the method have been published previously. but representative curves are reproduced here. The three-step conversion procedure can be incorporated easily into computer programs. Experimental Apparatus Experiments were performed to compare the PV changes of a sample under a hydrostatic load with those of the same sample under a uniaxial-strain loading condition. These tests were done without disturbing the sample between the two modes. Andersen and Jones reported the experimental details. Although tests on rocks under stress are often destructive to the sample, the results of these tests on consolidated sandstone samples yielded reproducible results. A friable sandstone required multiple stress cycles before the data would reproduce on successive cycles. The sandstone samples, 7.6 cm [3 in.] in diameter and about 8.9 cm [3 1/2 in.] long, were equipped with strain gauges to measure both the axial and lateral (circumferential) bulk strain as shown in Fig. 1. Results of the strain-gauge measurements are discussed in Appendix A. An elastomer sealing material enclosed the sample between endcaps, which allowed communication to the pore space. Switching from hydrostatic to uniaxial loading was done by valves external to the sample chamber. The stress was changed in a stepwise manner, and after equilibrium was reached, the changes in PV and the strains were recorded. The PV change was determined PV and the strains were recorded. The PV change was determined by displacement of pore fluid into a graduated pipette maintained at atmospheric pressure. Results The measured change in PV was normalized by dividing the measured volume change by the sample PV at the initial condition. The results from a test on Berea sandstone are shown in Fig. 2. The filled squares are the data for the sample under hydrostatic loading, and the filled circles are data for the same sample under a subsequent uniaxial-strain loading condition. Note that the test was not started at zero stress. At very low stresses, the sealing material intrudes into the pores on the surface of the sample. To avoid this intrusion that is not the result of mechanical changes of the rock, the PV measurements were started at elevated initial stresses between 1.7 and 3.5 MPa [250 and 500 psi]. Fitting Data The stress/strain behavior of most engineering materials is described by a linear relationship. When a stress is applied to a metal or to a crystalline material, the length change is directly proportional to the applied stress (below the elastic limit), which would be a straight line in a stress/strain plot, such as Fig. 2. The curvature to the data in these tests indicated that the normal linear constitutive relationship was not valid for reservoir rocks. P. 1078