Production rate of multi-fractured wells modeled with Gaussian pressure transients

Abstract

This study presents new pressure transient solutions, illustrated with some examples of the vast practical application potential. Gaussian pressure transients (GPT) are derived here to quantify the temporal and spatial propagation of instantaneous pressure changes in porous media, as initiated from cylindrical sources (vertical wells) and planar sources (hydraulic fractures). After solving the scalar pressure field in the reservoir space, and adequately accounting for the interference of the various pressure fronts by mathematical integration and superposition, the resulting pressure gradients solve for the velocity field in the reservoir space. Unique for GPT solutions is that the well rate, unlike in the traditional well-testing equations, does not appear as an input. Applying Darcy's Law, the fluid flux from the reservoir into the well and hydraulic fractures can be directly computed from the GPT solutions. The closed-form production-forecasting model can be implemented either in matrix-coded flow-visualizations of pressure depletion and flow paths for reservoir sections or in grid-less spreadsheet solutions to instantaneously generate production profiles for wells in any type of fluid injection/extraction project (water production, geothermal energy extraction, hydrocarbon production, and fluid disposal wells). Additionally, the Gaussian method also is suitable for physics-based decline curve analysis. The practical examples included in this study are for Eagle Ford shale oil and Marcellus dry gas wells. The hydraulic diffusivities are constrained by the field data, and range between 2.36 x10−10 and 3.48 x10−10 m2 s−1 for the Eagle Ford Formation; for the Marcellus the range is 3.64x10−9 to 5.67x10−8 m2 s−1. The breakthrough solution method of Gaussian pressure transients is placed in the context of past and present modeling approaches for shale plays developed with multi-fractured wells.