Responses of Commingled Systems With Mixed Inner and Outer Boundary Conditions Using Derivatives

Abstract

Summary This work presents a spectrum of possible derivative responses in wells producing commingled reservoirs. Three issues pertaining to commingled reservoir buildup behavior are studied:derivative responses for commingled systems, with emphasis on unequal layer radii;pressure distributions in the layers with a view toward understanding backflow, particularly the origin of rate reversals during buildup; andan application of Peaceman's probe radius formula to obtain pressure distributions in each layer at shut-in. Several approaches are developed from Issue 1 to estimate the areal extent of the smallest layer. Some general information that should be useful, or at least pertinent, to the design of production logging surveys is derived from Issue 2. Issue 3 also has application in miscible flooding for the location of areas below minimum miscibility pressure (MMP). Introduction The algorithm that Spath et al.1 presented is used to study the transient behavior of commingled systems consisting of bounded and unbounded layers with contrasting layer skin factors or contrasting fracture stimulations. Well responses are presented for such systems over all flow periods, including times after which the bounded layers have begun to "deplete." We focus first on the two-layer system composed of a bounded and an unbounded layer. A rigorous explanation for the reservoir-storage period identified in Refs. 2 and 3 is provided. Requirements for the bounded nature of the reservoir to be evident are deduced. We note that the flow capacity of the unbounded layer may not be evident on the drawdown response but can be determined from the buildup response. Diagnostic procedures to identify the bounded layer from buildup tests are presented. All our observations are independent of the well condition. Results concerning layer rates are also presented. Methods to relate layer rates during flow and buildup periods are derived. We show that rate reversals that occur during the buildup tests can be explained by studying the evolution of pressure profiles in each of the layers. These reversals reach a maximum as producing time increases and cease if wells are produced long enough. These observations should also be useful in designing production logging tests (PLT's). Observations for two-layer systems are then extended to three-layer systems and to commingled systems of different areal extent. The last portion of this work examines applications of the probe-radius concept developed by Peaceman4 to layered systems.5 We discuss procedures to obtain pressure distributions in each layer at shut in. In light of our observations about the relation between wellbore pressure and layer-average pressures, this procedure is invaluable in obtaining depletion levels in each layer.