Analysis of Transient Pressure Response Near a Horizontal Well-A Coupled Diffusion-deformation Approach
Authors: Li Y. , Wong R.C.K. , Yeung K.C. 
Published: March 2010

Abstract
Analysis of transient pressure response near a horizontal well has been based largely upon the analysis of mathematical solutions for the diffusivity equation. The diffusivity equation satisfies the principle of mass conservation, but does not consider the principle of equilibrium. No deformation properties of reservoir are included in the diffusivity equation. However, in some deformable reservoirs, the stress and deformation caused by injection or production could be so significant that it might affect the pressure response. This paper presents analyses of transient pressure response near a horizontal well using a coupled diffusion-deformation method. The results are compared with those obtained from the single diffusivity equation. Implications on practical applications such as well testing are addressed.

Introduction
Reliable information about in situ reservoir conditions is important in many phases of petroleum engineering. Much of the information can be obtained from well testing by measuring the variations in wellbore pressure that result from changes in the operating condition of the well. For example, pressure transients measured at an injecting well (under constant injecting rate) after it is shut in can be used to estimate the permeability of the reservoir. Well testing theory is based largely upon the solutions for the diffusivity equation, the differential equation for fluid flow in porous medium. Traditionally, it is assumed that the permeability, porosity and compressibility of fluid are constant, and pressure gradient is small(1,2). In fact, the change in fluid pressure is related to the change in the volume of fluid and reservoir rock. In order to obtain a better insight into the reservoir properties, two mechanisms should be considered in the interaction between the interstitial fluid and the porous rock:an increase of pore pressure induces a dilation of the rock; andcompression of the rock causes a rise of pore pressure, if the fluid is prevented from escaping the pore network. On the other hand, if pore pressure induced by compression of the rock is allowed to dissipate through diffusive fluid mass transport, further deformation of the rock progressively takes place. The earliest theory to account for this coupled diffusion-deformation mechanism was developed by Biot(3) who proposed a theory for three-dimensional consolidation of soils. Terazghi(4) developed a model for one-dimensional consolidation of soils. Rice and Cleary(5) linked the theory of poroelasticity to the concepts in rock mechanics with the full account of fluid and solid grain compressibility's.
The objective of this paper is to study the effects of coupled diffusion-deformation mechanisms and compressibility's of fluid and grain on the pressure response near a horizontal well. Numerical cases in oil sand and sandstone reservoirs are generated to investigate the above effects, using a coupled numerical simulator. Then, the results are compared with those from the well testing method.

Analysis Methods
Single Diffusion Equation (Well Testing Analysis Technique)
In 1967, Matthews and Russell(2) published their monograph dealing with well testing and analysis, which became a standard reference for many petroleum engineers. The basic fluid diffusion equation is a combination of the law of conservation of mass, Darcy's law, and state equation.