A Computationally-Efficient Method for Direct Long-Time Solutions of the Diffusivity Equation in Closed Systems

by
Kun Sang Lee, Ph.D.

University of Texas at Austin, 1995
Supervisors: Mark A. Miller
Kamy Sepehrnoori

A closed reservoir with wells producing a constant flowrate will exhibit pseudosteady- state flow behavior after the end of short-lived infinite acting and transition flow periods. In this work, a new approach for directly calculating reservoir flow behavior in pseudosteady state is presented. Pseudosteady state is mathematically defined by fixed mass flux conditions at all boundaries. This condition can be formulated into a spatially-dependent differential equation in terms of pseudopressure which includes the effects of changes of fluid properties with pressure and gravity. The derivation is based on application of the divergence theorem of Gauss. With a single calculation, this approach yields complete pseudopressure and flux distributions for all time after the onset of pseudosteady state. Using a finite-element calculation, this study seeks to evaluate pseudosteady-state behavior of complex reservoir systems in various flow conditions and geometries.

Verification of the approach is performed by comparing pseudopressure drawdown of regularly-shaped systems in linear or radial geometries against values reported in the literature. Applications of the pseudosteady-state concept to a wide range of reservoir problems are discussed. The effects of shape-determining quantities on flow in commonly encountered 3D systems are investigated. Geostatistically-generated heterogeneous systems are modeled to investigate the effects of geostatistical parameters and permeability distributions on the long-time performance of reservoirs under nonuniform flow. Long-time behavior of depleting gas reservoirs is also examined by introducing an approach using an overall system material balance equation. For the simulation of field-scale problems with multiple wells of differing production rates, a well model based on the near wellbore approximation of pseudopressure during pseudosteady state is introduced to reduce the concentration of elements near wells.

Results from the study show that this approach provides a fast and accurate methodology for modeling the long-time behavior of various types of reservoirs under depletion.

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