Decline Curve Analysis for Naturally Fractured Gas Reservoirs: A Study on the Applicability of "Pseudo-time" and "Material Balance Pseudo-time"
Authors: Gerami Shahab , Pooladi-Darvish Mehran , Mattar Louis 
Published: February 2016

Abstract
Robust techniques for analysis of production data of single porosity gas reservoirs have been developed and widely used for many years. These methods range from the traditional Arp's decline method to modern type-curve matching. The more recent techniques are based on the use of pseudo-time for linearization of gas-flow equations, and material balance time to account for variable operating conditions.
Whereas the governing equation for gas flow in a single porosity reservoir has been successfully linearized using pseudo-time evaluated at average reservoir pressure, the linearization of the governing equation in a double porosity reservoir is problematic due to pressure differences between the matrix and fracture systems. Similarly, the suitability of the material balance time concept, which has been successfully applied to a single porosity gas reservoir, has not been demonstrated for a double porosity gas reservoir. The purpose of this work is to suggest an appropriate pseudo-time and material-balance time functions when gas is produced from a naturally fractured gas reservoir. For this purpose, the resulting equations describing the matrix-fracture flow are cast in a form similar to those proposed by Warren and Root. These are then linearized, using appropriately-defined pseudo-time and material-balance pseudo-time functions. The results are compared against those of a commercial numerical simulator for two production scenarios, including constant rate and constant pressure production over some range of reservoir parameters. The results show that pseudo-time and material balance pseudo-time allow the accurate use of traditional double-porosity type-curves for naturally fractured gas reservoirs provided that the gas properties are evaluated at average reservoir pressure as determined from the material balance equation for double porosity systems.

Introduction
Type-curves are routinely used by engineers to estimate initial hydrocarbon-in-place and hydrocarbon reserves at some abandonment conditions, as well as flowing characteristics of individual wells such as permeability and skin.
Type-curves are plots of the theoretical solution to the governing flow equation for constant-rate production, or constant-pressure production, from a well in any kind of reservoir model. Generally, the operating conditions during production from a well are not constant. Hence, to analyze real production data, one needs to develop a robust methodology to account for these changes. This problem was solved by the use of the material balance time concept. Blasingame and Lee1 showed that, with use of this concept, one could have a single solution to the governing flow equation for both types (i.e., constant-rate and constant-pressure) of boundary conditions. It was shown that the same solution applies to cases where both rate and pressure are smoothly changing with time.
Whereas the governing flow equation for a slightly compressible fluid is linear, the governing flow equation for a compressible gas is non-linear due to the pressure dependency of gas properties. By applying the pseudo-pressure and pseudo-time transformations, the non-linear governing gas flow equation may be linearalized allowing one to use the slightly compressible liquid solution for a gas system.
Generally, the methodology for type-curve analysis of production data under variable operating conditions requires: an equation to obtain the average reservoir pressure — this is generally a material balance equation;evaluation of the material balance time at average reservoir pressure; andperforming traditional well test (constant rate) analysis, by transformation of real time (or pseudo-time) to material balance time (or material balance pseudo-time) and use of rate normalized pressure-drop. Similar analysis could be performed Using production decline analysis techniques.