Matching Simulator Wellblock Pressures With Observed Buildup Pressures in Two-Layer Reservoirs
- Litlehamar Terje (Rogaland Research Inst.) | Larsen Leif (Rogaland Research Inst.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- March 1986
- Document Type
- Journal Paper
- 183 - 193
- 1986. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling, 5.6.4 Drillstem/Well Testing, 5.5.8 History Matching, 5.2.1 Phase Behavior and PVT Measurements
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An equation is given for the buildup time at which the wellbore pressure should match the kh average of the simulator wellblock pressures in history-matching procedures of two-layer reservoirs. The equation takes into account contrasts in flow capacities, porosity/compressibility/thickness products, and skin factors and makes use of fractional flow rates given by the simulator. The equation is derived for reservoirs without crossflow, but it can also be used for reservoirs with interlayer crossflow. Also included is a method to handle cases when only a single buildup pressure is recorded. To check the validity of the results, data from a three-dimensional (3D) reservoir simulator were compared with buildup data from an analytical solution for systems without crossflow and with data from a radial reservoir simulator for systems with crossflow.
An important step in reservoir studies on numerical simulators is the history matching of calculated and observed data. All reliable reservoir data should be used in this process to enhance the uniqueness of the match, but only the use of flowing or shut-in wellbore pressure is considered in this paper, with emphasis on shut-in periods. The initial pressure in an area around each well should be used as the starting point in the matching process. This pressure can be determined from a drillstem test or a repeat formation tester.
van Poollen et al.1 presented a method to compare the wellblock pressure with observed buildup data for a single-layer reservoir. Their method was based on a simple expression for the buildup time at which the wellbore pressure should match the flowing wellblock pressure at the time of shut-in. The expression was based on the assumption that the wellblock pressure equals the average pressure in a circular drainage area of the same size as the wellblock. With the same assumption, Earlougher2 presented a method to handle cases when only a single buildup pressure was recorded.
Peaceman3 later showed that the wellblock pressure equals the steady-state flowing pressure at a radial distance 0.2 ?x, where ?x is the dimension of a square wellblock. On the basis of this observation, Peaceman derived a new expression for the buildup time to be used in the matching process. He also modified the method presented in Ref. 2 for a single-point buildup pressure. Peaceman4 later extended his work to nonsquare grids and anisotropic single-layer reservoirs. Ref. 4 also contains a more accurate expression for the radial distance where the steady-state flowing pressure equals the wellblock pressure from the simulator. For a square wellblock this distance equals 0.198 ?x.
This paper extends the results in Refs. 1 through 3 from single- to two-layer reservoirs. Our method is based on a simple expression for the buildup time at which the wellbore pressure should match the kh average of the two simulator wellblock pressures. This expression takes into account contrasts in flow capacities, porosity/compressibility/thickness products, and skin factors. To derive this expression, it is assumed that Peaceman's observations concerning a single-layer reservoir can be applied to each layer of a two-layer reservoir in terms of fractional flow rates. Furthermore, the derivation is based on an infinite-acting approximation given by Larsen5 for the wellbore pressure in two-layer reservoirs without crossflow.
Although the derivations in this paper are based on the infinite-acting approximation in Ref. 5 for systems without crossflow, the results also are found to be applicable for systems with crossflow. This agrees with Prijambodo et al's.6 observation that two-layer reservoirs with or without crossflow show the same buildup behavior at early times because short buildup times are used in the history-matching procedure.
The examples show that differences in skin factors are very important for two-layer reservoirs. Actually, if the two layers have the same skin factor, then the expression given by Peaceman can be used with negligible error, even for cases with significant contrasts in layer parameters. In this case, the total flow capacity and the total porosity/compressibility/thickness product must be used.
Many examples have been included in this paper to verify the results. In these examples, data from a 3D reservoir simulator have been matched with buildup data computed from the analytical solution given in Ref. 5 for reservoirs without crossflow and with data from a radial reservoir simulator for reservoirs with interlayer crossflow.
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