A Generalized, Transient Flow Model For Linear, Compartmentalized Systems
Authors: Rahman N.M. Anisur , Ambastha A.K. 
Published: March 2010

Abstract
Fluid flow in narrow reservoirs or aquifers is predominately linear. However, this linear-flow system may possess variations in rock and fluid properties and/or presence of faults due to a number of geological phenomena. Such systems are modelled as linear, compartmentalized systems.
In this study, a new generalized transient-flow model has been developed analytically for an n-region compartmentalized system. Each compartment or region may have distinct rock and fluid properties. A time-dependent production rate from each compartment is modelled in a general way. The conditions at the extreme boundaries are taken as non-homogeneous, time-dependent, Cauchy-type that can be modified to Dirichlet- or Neumann-type as a special case. A possible situation of poor communication between neighboring compartments has been incorporated into the model by considering the presence of a thin skin at each interface. A generalized solution for the dimensionless pressure has been derived using an integral-transform technique.
This solution deals directly with situations like production rates and extreme boundary conditions being time-dependent without the need for using the principle of superposition in time or Duhamel's principle. Some limiting cases of this new solution have been used for validation purposes. Several advantages and practical applications of this model are also discussed.

Introduction
The flow of fluids through narrow reservoirs or aquifers is predominantly linear. However, this linear-flow system may possess variations in rock and fluid properties and/or faults due to geological phenomena. Levorsen(1) mentioned that the extent of a reservoir boundary may be sharp or it may be gradationa, as is more often the case. This heterogeneous character of a reservoir makes it difficult to model. To alleviate the mathematical difficulties of modelling such a system, the concept of compartmentalization may be used. With this concept, a heterogeneous system is assumed to be comprised of a number of homogeneous compartments. The communication of fluid between adjoining compartments may be hindered due to the presence of faults or any other barriers. Ehlig-Economides and Economides(2) have pointed out that there are several depositional environments showing possible oil- and gas-reservoir geometries that result in predominantly linear flow. These formations, which generally have long, narrow shapes, may be the result of river meander point bars, oxbow lakes, river channels, or tectonic breccias. However, such systems may also be compartmentalized due to variations of rock and fluid properties and the presence of faults and, therefore, can be modelled as linear, compartmentalized systems.
In the literature, a number of studies of linear, one-dimensional flow systems are related to homogeneous reservoirs or aquifers(3, 4, 5). Nutakki and Mattar(6) generated transient pressure responses using the line-source solution with the principle of superposition for an infinitely long, narrow, homogeneous channel. They also developed a method to determine the time to the end of the radialflow period and the beginning of the linear-flow period. Ehlig- Economides and Economides(2) have presented the techniques for interference, drawdown and buildup analysis for an elongated linear flow system.