Succession-of States Model for Calculating the Long-Time Performance of Depletion Reservoirs
Authors: Lee K.S. , Miller M.A. , Sepehrnoori K. 
Published: December 2007

Abstract
A bounded reservoir with wells producing at constant rate will exhibit pseudosteady-state behavior after the end of typically short-lived infinite-acting and transition flow periods. This study develops a new approach for directly calculating pseudosteady-state flow behavior without solving the full time-dependent form of the diffusivity equation. This approach can be applied to the linearized forms of the diffusivity equation for either single-phase liquid or gas flow. A finite-element method is used which allows for spatially-dependent reservoir properties, complex reservoir geometries, and multiple wells.
The first part of this paper presents a verification of the approach by comparing results for some regularly-shaped systems against full-transient solutions reported in the literature. For the simulation of field-scale problems with multiple wells of differing production rates, a well model based on a near-wellbore approximation of the pseudopressure distribution during pseudosteady-state is introduced to reduce the concentration of elements near wells. The second part of the paper demonstrates application of the direct pseudosteady-state concept to actual reservoir problems. To account for rate changes during extended production periods, the pseudosteady-state equation was solved successively for each flow period and combined with an overall reservoir material balance analysis.
Results from this study show that this approach provides a fast and accurate method for modeling the long-time behavior of various types of reservoirs under depletion conditions. The approach is particularly applicable to single-phase volumetric gas reservoirs.

Introduction
One or more wells produced at constant rate in a closed reservoir will exhibit monotonic pressure decline. Pressure decline behavior can be characterized as occurring in four distinct regimes. The first regime (early time) is dominated by wellbore storage effects. During this time, pressure response is largely determined by the unloading of fluids from the wellbore with little actual response due to flow into the wellbore from the reservoir. This time generally lasts from a few minutes to a few hours.
The second flow regime is sometimes called infinite-acting time, during which well response is essentially the same as that of a well being produced from an infinite reservoir. For radial flow, infinite-acting time is characterized by a semilog straight line of well flowing pressure vs. time. Most pressure transient tests are analyzed during this time period. Infinite-acting time ends when the well response begins to deviate from infinite-acting behavior due to a nearby barrier, discontinuity, or reservoir property change in the vicinity of the well. Once the first nearby boundary affects the well response, it enters what is sometimes called transition time.
Stabilized time begins at the end of transition time, after all reservoir boundaries, discontinuities, etc. are fully felt and the well approaches a quasi-steady flow regime sometimes called semisteady or pseudosteady-state flow. The onset of stabilized time may be anywhere from a few hours to even many months. For most reservoirs stabilized time begins at times that are much shorter than the life of the reservoir. The day-to-day operation of most wells in depletion reservoirs is governed by relationships during this time.
This paper addresses the depletion behavior of closed systems during pseudosteady state. For a well produced at constant rate, pseudosteady state is characterized by pressures in the reservoir declining everywhere in the reservoir at the same rate. This means that the pressure profile lowers uniformly throughout the reservoir and that mass fluxes everywhere are constant with time. P. 589