A Mathematical Model for Pressure Distribution in a Bounded Oil Reservoir Subject to Single-Edged and Bottom Constant Pressure

Abstract

Well test analysis of a horizontal well is complex and difficult to interpret. Most horizontal well mathematical models assume that horizontal wells are perfectly horizontal and are parallel to the top and bottom boundaries of the reservoir. As part of effort towards correct horizontal well test analysis, the purpose of this study is to develop a mathematical model using source and Green’s functions for a horizontal well completed in an oil reservoir at late time flow period, where the reservoir is bounded by an edge and bottom constant pressure boundaries. The purpose of the derivation is to understand the effects of well completion, well design and reservoir parameters on pressure and pressure derivative behavior of the well at late flow time, when all these external boundaries are presumed to have been felt. If the model is applied for well test analysis therefore information like reservoir natural permeability distribution, actual external boundary types and even the well completion performance will be decidable easily. Dimensionless variables were used to derive throughout the derivations. Results of the derivation show that the dimensionless pressure and dimensionless pressure derivatives increase with increase in dimensionless well length. This means that higher well productivity is achievable with extended well length when the reservoir is surrounded partially by constant pressure boundaries. Furthermore, the models show that higher directional permeabilities would also encourage higher well productivity at late flow time. The dimensionless pressure derivative will, as a result of a constant dimensionless pressure, potentially collapse gradually to zero at the moment the dimensionless pressure begins to exhibit a constant trend. Finally, the dimensionless pressure and dimensionless pressure derivatives vary inversely with the reservoir dimensionless width at late flow time.