Effects of Some Simplifying Assumptions On Interpretation of Transient Data
Authors: Jiang Q. , Stanislav J.F. , Mattar L. 
Published: March 2010

Abstract
Fluid flow in porous media is described by the diffusional type of partial differential equation. In deriving the flow equation for the constant compressibility fluids, an important assumption is made regarding the magnitude of the pressure gradient: it is assumed that this gradient is small enough that the square of this term becomes negligible. In this study, an attempt is made to quantify the effect of this simplification. Data generated by numerical simulator and analytical solutions are compared in order to determine the conditions under which this simplification is justified. Except for extreme conditions (non-linear coefficient greater than 0.01), it is justifiable to ignore the square of the pressure gradient term in oil diffusional equation.
The drawdown solution on gas wells contains gas properties, which are pressure dependent. As the system is being depleted, the average reservoir pressure declines. Although it seems logical to consider the gas properties at the prevailing reservoir pressure, the gas viscosity and compressibility are usually evaluated at the initial pressure level. Comparison of the analytical solutions with numerical solutions obtained for a variety of typical well test conditions indicates the magnitude of inaccuracy incurred by using standard approach to data interpretation. Based on this investigation, several pressure levels for properties calculation are considered. In applying analytical equations for pseudo-steady state flow conditions, gas properties should be evaluated at an average value of the initial and current reservoir pressures.

Introduction
Well test analysis is based on expressions obtained by solving the flow (diffusion) equation for a variety of boundary conditions. The flow equation which represents the core of the pressure transient method is based on the principle of mass conservation, Darcy's law and an appropriate constitutive assumption: the constant compressibility concept for liquids and the equation of state for real gases.
The general form of the flow equation contains a nonlinear term, the square of the pressure gradient. Obviously, the occurrence of this non-linearity is mathematically inconvenient and thus should be eliminated if possible. The simplification is achieved by assuming small pressure gradients in the formation; consequently, a square of a small number can be ignored. To the best of our knowledge, no study has been reported which would determine whether or not such a simplification is indeed justified.
In this study, the effect of a non-linear term on the interpretation of transient data was evaluated by comparing the analytical solutions with numerical solutions which are generated for the identical test conditions. For gas reservoirs, a step-wise approach and a pressure averaging method for evaluating the gas properties in analytical solutions are proposed.
The one dimensional flow of a slightly compressible fluid (oil) in a porous medium can be expressed, in a dimensionless form, as follows:
Equation 1–8 (available in full paper)
The numerical simulator is based on Equations (1) and (4). The solutions are obtained for the condition of a constant production rate, the early transient and the pseudo-steady state time periods. The finite difference implicit method and the iterative Newton- Raphson procedure are applied.