Performance of a horizontal well in a bounded anisotropic reservoir: Part I: Mathematical analysis

To enhance the productivity of horizontal wells, it is of necessity to ensure that they perform optimally. This requires an understanding of how the reservoir's geometry, anisotropy and well design affect the pressure response. Mathematical formulations can be used to simulate pressure response in the wellbore and the data obtained can be analysed to obtain well and reservoir parameters that can aid performance and evaluation. In this study, a mathematical model that can be used to approximate pressure response in a horizontal well is formulated, and a detailed mathematical analysis that can be used to obtain well and reservoir parameters are provided. A horizontal well inside a rectangular drainage volume with sealed boundaries is considered and the effect of each boundary on pressure throughout its productive life is studied. In the analysis, investigations on how the reservoir parameters can be approximated over a given period of production are conducted. This is achieved by identification of the appropriate source and Green's functions. These source functions allow us to formulate a mathematical model for dimensionless pressure. Considering the diagnostic plots for both dimensionless pressure and dimensionless pressure derivative, mathematical analysis studies the possible behaviour of the plots. Analysis indicates that the reservoir anisotropy can be approximated during the infinite-acting flow at early times when other parameters are known. Further, when the first boundary is felt, in this case the vertical boundary, the horizontal permeability can be approximated during the transition flow periods at middle times. Finally, at late times when all the boundaries have been felt and a pseudosteady state flow is evident, reservoir dimensions can be approximated. These results can significantly improve well test analysis and enhance the performance evaluation of a horizontal well.