A Generalized Transient-Pressure Model for Areally-Compartmentalized Reservoirs
Authors: Rahman N.M. Anisur , Ambastha A.K. 
Published: December 2007

Abstract
A compartmentalized reservoir is considered to have a number of hydraulically-connected compartments. In this study a generalized, transient-pressure model for areally-compartmentalized reservoirs is developed analytically. A compartmentalized system with two-dimensional, parallelepiped compartments has been considered. The interface-boundaries between adjoining compartments are due to low-permeability vertical barriers or partially-communicating faults and are modeled as thin skins. The conditions at the extreme boundaries of the reservoir are considered to be time-dependent to deal with any changing situations at these boundaries. An integral-transformation technique has been used to derive the solution for the dimensionless pressure in terms of dimensionless time and space. The new analytical solution has been validated by comparing simplified cases with those available in the literature. A number of practical applications of the new model are also discussed.

Introduction
Due to the geologic processes, most hydrocarbon reservoirs are subject to heterogeneity. As evident in the literature, some of these reservoirs are compartmentalized. A compartmentalized reservoir is considered to have a number of hydraulically-connected compartments. The interfaces between adjoining compartments are usually partially-communicating due to the presence of faults or low-permeability barriers. This kind of compartmentalized behaviour becomes prominent in the pressure history at late times when the other hydraulically connected compartments start contributing. For the sake of understanding the long term behaviour of a compartmentalized reservoir, it is important to characterize the intercompartment fluid flow as a function of time. Running a reservoir simulator to understand the behaviour of a compartmentalized reservoir is not realistic before identifying and quantifying major flow units and barrier resistance. The main objective of this study is to develop a simple analytical model that would be able to provide useful information about a compartmentalized reservoir which could be used for the purpose of designing development schemes.
Presently, a compartmentalized reservoir is analyzed by using models that are based on material balance techniques ignoring the reservoir geometry. The amount of fluid that communicates between any two neighbouring compartments is estimated based on the difference in average pressures of the compartments. However, the amount of communicating fluid can be estimated more effectively if the reservoir is modeled for the transient-pressure responses.
The mathematical formulation developed in Ref. 2 is based on the assumption that the fluid flow between adjoining compartments occurs under steady-state conditions. Therefore, this model could be used where the hydraulic diffusivity is very high so that transient effects dissipate very quickly. In Ref. 3, an analytical model has been developed based on single-phase material balance equations. Here, the concepts of boundary-pressure time delay and desuperposition are used to predict bottom-hole pressures over different flow regimes. Diagnostic techniques are presented in Ref. 4 for detecting and quantifying poorly drained compartments using material balance equations and a numerical simulator.
A generalized transient-pressure model for compartmentalized systems in linear flow has been developed in Ref. 5. Here, poor hydraulic communication between adjoining compartments is modeled as thin skins. The mathematical model for two-dimensional compartmentalized systems as developed in this study is an extension of the model presented in Ref. 5.
It is important to have a transient-pressure model that is capable of providing useful information about compartmentalized systems with areal extent. In this study a generalized, transient-pressure model for areally-compartmentalized reservoirs is developed analytically. A compartmentalized reservoir with two-dimensional, parallelepiped compartments has been considered. Each compartment may have a number of line-sink wells producing at time-dependent rates.
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