Semi-analytical Solution for Multiple Layer Reservoir Problems with Multiple Vertical, Horizontal, Deviated and Fractured Wells
Authors: Gilchrist John Philip , Busswell Geoff Stuart , Banerjee Raj , Spath Jeffrey Bernard , Thambynayagam Raj Michael 
Published: February 2016

Abstract
We present new semi-analytical solutions to the multiple layer reservoir problem, both in real-time and Laplace space. Assuming a vertically stacked system of layers, an analytic solution within each layer can be derived using a method of integral transforms. We fully account for crossflow between layers by coupling these analytic solutions together and solving for the flux field at the layer interfaces. The time evolution of these flux fields is governed by a Volterra integral equation.
Within a multiple layer reservoir scenario our semi-analytical solutions are applicable to partially penetrating vertical, horizontal, deviated and fractured wells, where fractures can have infinite or finite conducting properties. Using the principle of superposition our model fully accounts for interference effects between wells as well as multiple rate effects. Using solutions in Laplace space we are able to model naturally fractured reservoirs, wellbore storage, non-Darcy flow factors as well as constant well pressure production.
We present case two case studies comparing our solutions with that generated using a commercial finite difference simulator. We find excellent accuracy with huge gains in model building and execution times. Our solution is suitable for a number of applications. These include pressure/rate transient interpretation, rapid field development studies and integration with real-time software.

Introduction
Over the years reservoir simulation has become standard practice in the oil and gas industry for the management of oil and gas reservoirs. Typically such simulators are based on numerical methods such as the finite difference or finite element method. The advantage of such an approach is that a wide variety of physical processes can be modeled in the reservoir e.g. rock layering effects, faults, geomechanics, etc. While such simulators will continue to play a crucial role for reservoir engineers it is well known that to use these tools requires much time and expertise. Indeed, even for a well trained engineer a full history matching study can take months to perform. Also, because of the nature of the numerical modeling process, issues such as gridding, time-stepping and convergence can often become problematic.
Other tools exist which allow engineers to perform "quick studies" of an oil or gas field. Typically, such methods are either based on material balance (Tracy1, Reudelhuber and Hinds2, Hobson and Mrosovsky3) or decline curve analysis techniques (Johnson and Bollens4, Power and Pishny5, Arps6). While such approaches require very lightweight workflows and can be extremely useful, they are based on very simplistic physics and fail to fully account for transient effects and interference between wells. There is clearly a need for an alternative technique that allows very fast model building and execution, similar to a material balance or decline curve technique, but which is based on solving the same set of equations as is done numerically when using a simulator.
Analytical solutions to the diffusivity equation, which describes the pressure diffusion in a porous medium for a slightly compressible fluid, provide the basis for such a technique. They have been successfully used for well testing applications for many years (Gringarten and Ramey7, Cinco-Ley et al.8, Raghavan9, Al-Khalifah et al.10, Gomes and Reza11) and are generally applicable to a single well. The application of analytical solutions to full field reservoir problems, where multiple well interference effects and reservoir boundaries are fully accounted for, have been presented for porous media with homogeneous and anisotropic permeability by Busswell et. al.12, Thompson et. al.13 and Ozkhan and Raghavan14.
A more difficult problem involves addressing reservoir problems with inhomongeneous and anisotropic permeability, where wells penetrate multiple layers and the flow from each is commingled during production. In general the pressure and production behavior can be classified under two types; layers that are separated by an impermeable barrier resulting in no crossflow in the formation (commingled systems), and layers that do communicate via crossflow in the formation.