A Quick Method To Determine Subsidence, Reservoir Compaction, and In-Situ Stress Induced by Reservoir Depletion
Authors: Morita N. , Whitfill D.L. , Nygaard O. , Bale A. 
Published: December 2007

Summary
This paper provides a quick method to determine subsidence, compaction, and in-situ stress induced by pore-pressure change. The method is useful for a reservoir whose Young's modulus is less than 20% or greater than 150% of the Young's modulus of the surrounding formation (where the conventional uniaxial strain assumption may not hold). In this work, a parameter study was conducted to find groups of parameters controlling the in-situ stress, subsidence, and compaction. These parameter groups were used to analyze the numerical calculation results generated by a three-dimensional (3D), general, nonlinear, finite-element model (FEM). The procedure and a set of figures showing how to calculate the in-situ stress, subsidence, and compaction induced by pore-pressure changes are provided. Example problems are also included to prevent confusion on sign convention and units. This work showed that Geertsma's results, which are based on no modulus contrast between cap and reservoir rocks, should be extended to simulate more closely "real" reservoirs, which generally have distinct property differences between the cap and reservoir rocks. Highly porous and high-pressure North Sea reservoirs and tight sand formations surrounded by soft shale often fall into this category. The application is intended for sand-production control, casing buckling problems, design of hydraulic fracturing jobs, subsidence, and estimation of PV and formation damage resulting from permeability reduction during hydrocarbon production.

Introduction
The in-situ stress induced by pore-pressure change usually has been calculated on the assumption that a rock deforms uniaxially without inducing strain along the horizontal direction. The amount of subsidence was calculated by Geertsma, with the strain nuclei method. These calculations assume that a reservoir is thin, that its depth is reasonably great, and that its rigidity is close to that of the confining formation. However, statistics of field measurements has shown that many hydrocarbon reservoirs are thick or shallow or have elastic moduli that are significantly different from those of confining formations. For example, North Sea reservoirs often have static Young's moduli that are orders of magnitude smaller than those of the surrounding rocks before they are compacted because of hydrocarbon production, although the dynamic Young's modulus calculated from sonic logs may give only three to six times modulus contrast. Some tight formations in the U.S. also have rock several times more rigid than surrounding shale. When a hydraulic fracture, a sand-control process. a subsidence-control operation, or an evaluation of formation damage resulting from permeability reduction is conducted in such a reservoir, accurate information on the in-situ stress, reservoir compaction, or subsidence induced by pore-pressure changes helps in designing such operations. This work does not use or develop new mathematical techniques, but emphasizes two important issues. First, the common practice in the oil industry is to calculate PV compressibility, reservoir compaction, and in-situ stress change on the basis of reservoir-rock property data. However, this work emphasizes that some reservoirs also require the caprock property data to evaluate these quantities. Second, a quick method to evaluate PV compressibility, reservoir compaction, in-situ stress change, and subsidence has not been published previously. Although techniques to calculate these values are available, they require long times to run sophisticated simulation models. The purpose of this work is to provide a method for quick estimation of in-situ stress, compaction, and subsidence for a reservoir having simple geometry. A quick estimation of these values is often sufficient during the reservoir development stage because accurate reservoir descriptions are not available. Such a crude estimation is essential because the decisions on downhole and surface facility designs are made during the early stages of reservoir development.
After more accurate reservoir descriptions are collected, however, we recommend that the 3D FEM be used for this work to get a better evaluation. The model can handle various complex problems, such as multilayer problems with heterogeneous rock properties, inclined reservoirs, irregular reservoirs, nonuniform pore pressure, nonlinear properties of rock, hysteresis effect of cyclic loading, and nonuniform reservoir pressure.

Assumptions and Calculation Methods
The in-situ stress is decomposed into two parts-original in-situ stress and-in-situ stress induced by pore-pressure change.
................................ (1a) and ............................ (1b) where K is the stress-ratio coefficient affected by rock grain shape, grain-size distribution, sedimentation process, present Poisson's ratio, tectonic force, temperature, and pore pressure. Delta sigma and delta sigma are in-situ stress components induced by pore-pressure change. If the pore-pressure change occurs over several years, we can reasonably assume that rock deforms elastically during the period. In addition, if the pore-pressure change is reasonably small and the state of stress is not far from hydrostatic-i.e., a small deviatoric stress-then a linear elastic deformation is a good approximation. Hence, a linear elastic deformation is assumed in this work for the calculation of delta sigma and delta sigma induced by the pore-pressure change. A disk-shaped reservoir is assumed for the present calculation as shown in Fig. 1. Although the moduli of the reservoir and the surrounding formation may vary within each formation, uniform moduli are assumed within both structures, respectively. The reservoir is located at depth D below the surface and its radius and height are r and h, respectively. More complex reservoir geometries require that data be entered directly into the 3D FEM used for the present calculations. Fig. 2 shows the finite-element meshes used for this work. The upper surface is free from a traction force, and the bottom surface is fixed to the rigid base rock. Infinite elements were used for the outer boundary. The hatched section is the reservoir and has elastic moduli different from those of surrounding formations. The pore pressure of the reservoir section is reduced to calculate the deformations and stress change of the reservoir and surrounding formations. Test runs were conducted for a well with and without a casing cemented to the borehole.
JPT
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