Analysis of Pressure-Buildup Data From a Well in a Multiwell System

Abstract

Summary. This work investigates the buildup response of a well located in asystem of producing wells completed in a closed, bounded reservoir. Using ananalysis method based on the pressure derivative, we show that the drainagearea of the well (at the instant of shut-in) and the flow-capacity(permeability-thickness) product can be computed directly from the derivative of pressure-buildup data even in cases where conventional semilog straightlines are not well defined. The analysis methods assume that thebuildup-surveyed well has established its own drainage area before shut-in. Fora system of wells located in a closed, bounded reservoir, this means thatpseudo-steady-state flow prevails at the instant of shut-in. We also present asecond method for estimating a well's drainage area from pressure-buildup data. The second method relies on the fact that during buildup, the well's shut-inpressure increases to a maximum and then decreases as a result of interferencefrom neighboring producing wells. It is shown that the maximum shut-inpressure, and the time at which it occurs, often can also be used to computethe well's drainage area at the instant of shut-in and the average pressure inthis drainage area. Introduction As is well known, the Miller-Dyes-Hutchinson (MDH) and Matthews-Brons-Hazebroek methods can be used to compute the volumetric averagereservoir pressure from a buildup test provided that permeability can bedetermined by semilog analysis, the drainage area of the well is known, and thewell's location in the drainage area and the shape of the drainage area areknown; i.e., the Dietz shape factor is known. Two new methods for determiningthe well's drainage area directly from buildup data are the primarycontributions of this work. Previous investigations of the buildup response of a single well in a closeddrainage area have focused, for the most part, on the existence and duration of the Horner or MDH semilog straight lines and on methods for computing theaverage reservoir pressure. For cases where pseudo-steady-state flow exists atthe instant of shut-in, the results of Ramey and Cobb and Larsen indicate thatthe duration of the Horner and MDH semilog straight lines are essentiallyidentical. In this work, we show that a longer semilog straight line can beobtained with a modified MDH plot. This modified MDH plot is identical to onesuggested by Bossie-Codreanu and is also conceptually similar to a methodsuggested by Slider. Bossie-Codreanu, however, suggested a trial-and-errorprocedure for determining the correct shut-in pressure to use in the modified MDH plot. Our work removes this restriction. In addition, Bossie-Codreanuconsidered only a single unfractured well in a closed drainage area, whereas weconsider general multiwell systems. Larsen showed that if the producing time is long enough, the duration of the Horner semilog straight line can be increased by replacing the actual producingtime, tp, by t' in the Horner time ratio, where t' corresponds to the time suchthat tAD =0.08. To obtain t', one must have a priori knowledge of the reservoirdrainage area at the instant of shut-in and the permeability. Our method forcomputing drainage area and permeability directly from buildup data enables usto compute the value of t' necessary to apply Larsen's suggestion. The principal contributions of this work can be summarized as follows. A newmethod is presented to compute the drainage area and permeability-thicknessproduct from a pressure-buildup test. The method uses the derivative of theshut-in pressure and does not require a priori knowledge of initial pressure, pi, or the shut-in pressure at the instant of shut-in, pwf, s. The method isapplicable to any reservoir/well geometry where buildup data reflect a radialor pseudoradial flow regime. Conceptually, the method can be extended to otherflow regimes, e.g., to the linear flow regime commonly exhibited by buildupdata from a fractured well. The analysis procedure can be used to analyze thebuildup pressures at a well procedure can be used to analyze the builduppressures at a well located in one of the following three systems:a single well in a closed drainage area,a system of producing wells in a closed, bounded reservoir, andan infinite multiwell pattern. The basic assumptionis that the buildup-surveyed well has established a unique drainage area beforeshut-in. For the closed, bounded reservoir cases (Cases 1 and 2), this meansthat pseudo-steady-state flow is established throughout the reservoir beforeshut-in. For Case 3. our assumption is that before shut-in, pseudo-steady-stateflow has been established in the drainage area of the well under consideration. A method to determine the average reservoir pressure is also given. This methodrequires an estimate of the Dietz shape factor but is not highly sensitive tothis estimate. It is shown that the shut-in pressure of a well located in an infinitemultiwell pattern or in a multiwell closed, bounded reservoir increases to amaximum and then decreases when nearby producing wells start to affect theshut-in pressure of the producing wells start to affect the shut-in pressure of the buildup-surveyed well. We also show that the drainage area (at the instant of shut-in) and the average pressure in this drainage area usually can becalculated accurately if the maximum shut-in pressure is observed. Basic Assumptions and Definitions Throughout this work, we assume single-phase flow of a slightly compressiblefluid of constant viscosity in a homogeneous reservoir. Furthermore, we assumethat all wells in a multiwell pattern and in a closed, bounded system produceat a constant rate, but the flow rate may vary from well to well. We define thedimensionless variables in the standard way and oilfield units are usedthroughout. The dimensionless wellbore pressure is defined by .........(1) Here, we use q1 in the pwD definition because we consider multiwell systems. Throughout this work, Well 1 and the subscript 1 refer to the buildup-surveyedwell; i.e., the well that is shut in after producing for a time tp. Thus, throughout, pwD represents the dimensionless drawdown wellbore pressure for thewell under consideration. The expression for pwD depends on the well locationin the drainage area and the flow regime (linear, radial, pseudo-steady-state, etc.) that exists during the time of interest. For pseudo-steady-state, etc.)that exists during the time of interest. For pressure buildup, thedimensionless wellbore shut-in pressure is defined pressure buildup, thedimensionless wellbore shut-in pressure is defined by ....... (2) where pws = shut-in pressure and q1 = production rate of thebuild-up-surveyed well (Well 1) before shut-in. The dimensionless times based on wellbore radius, rw, fracture half-lengthin the x direction, xf, Well 1's drainage area, A1, and the tow reservoirdrainage area, At, are defined by .....(3) SPEFE P. 101