History Matching Production Data Using Analytical Solutions for Linearly Varying Bottomhole Pressure
- J.P. Spivey (S.A. Holditch & Assocs., Inc.) | J.H. Frantz Jr. (S.A. Holditch & Assocs., Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Eastern Regional Meeting, 8-10 November, Charleston, West Virginia
- Publication Date
- Document Type
- Conference Paper
- 1994. Society of Petroleum Engineers
- 5.6.4 Drillstem/Well Testing, 5.2.1 Phase Behavior and PVT Measurements, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 2.5.2 Fracturing Materials (Fluids, Proppant), 5.5.8 History Matching
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Performance of gas wells is often modeled using analytical solutions which arc based on the assumption of constant pressure production at the wellbore. A variable pressure history can be modeled by using superposition of these constant pressure solutions. Unfortunately, each pressure change results in a spike in the resulting production rate. In practice, wells are often operated such that pressure declines slowly and smoothly until line pressure is reached. Thus, superposition of constant pressure solutions does not accurately model production in real wells. In this paper we present a procedure for calculating production rate and cumulative production using superposition of solutions for bottomhole pressure which varies linearly with time. These solutions for linearly varying bottomhole pressure may be easily obtained from the constant pressure solution for the same reservoir geometry. Rates calculated using superposition of linear pressure solutions model actual production data more faithfully than do those calculated using superposition of constant pressure solutions.
Wells are often operated in a manner such that pressure declines slowly and smoothly until line pressure is reached. This may be done for a variety of reasons. First, there may be an early period of curtailment due to deliverability being higher than sales contract commitments. Second, controlled flowback is often used to prevent proppant crushing and embedment in hydraulically fractured wells. Finally, cleanup of mud filtrate or hydraulic fracture fluids may result in increasing production rate and decreasing effective back pressure during the first few days of production of a well.
Figs. 1, 2, and 3 show pressure and production histories for three different wells that exhibit characteristic behavior. The production histories show several similarities as well as some interesting differences. For all three wells, the pressure decreased initially, then leveled off at a fairly constant rate. The period in which the pressure was decreasing ranged from approximately one month for Well B to as much as six months for Wells A and C. There are several disadvantages to using constant pressure solutions to model rates and pressure histories like these. These disadvantages can be reduced or eliminated through the use of the linear pressure solutions presented in this paper.
In this paper, we present a general method for obtaining the solution to the diffusivity equation for a linearly varying wellbore pressure from the constant pressure solution for the same reservoir geometry. We also present the necessary equations for modeling an arbitrary, piecewise linear pressure history using superposition of these linear pressure solutions.
In the next section, we review the literature on solutions to the diffusivity equation for boundary conditions other than constant rate or constant pressure. In the following section, we discuss the problems that arise when using superposition of constant pressure solutions and illustrate the benefits to be gained by using the new linear pressure solution. In the remaining sections, we present the linear pressure solution in terms of the constant pressure solution in both the time domain and the Laplace domain. We also show how to apply the linear pressure solution to gas reservoirs. We then validate the linear pressure solution by comparison with the constant pressure solution.
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