A New Test for Determination of Individual Layer Properties in a Multilayered Reservoir

Abstract

Summary. Many wells penetrate several layers, and flow from each layeris often commingled during production or injection. The behavior of pressure transients for layered systems has been studied in detail for pressure transients for layered systems has been studied in detail for two types of systems:layers that are separated by impermeable barriers(no formation crossflow) andlayers that communicate in the reservoir. In this work, the general problem of n homogeneous layers, in which anytwo adjacent flowing layers may or may not be crossflowing in theformation, is solved analytically. Properties, including permeability, porosity, layer thickness, the wellbore skin factor, and the vertical porosity, layer thickness, the wellbore skin factor, and the vertical permeability between crossflowing layers, are distinct for each flowing permeability between crossflowing layers, are distinct for each flowing layer. Wellbore storage effects are also included, and solutions areprovided for both infinite-acting and bounded systems. The key to finding provided for both infinite-acting and bounded systems. The key to finding the individual layer properties in a given reservoir lies in theinterpretation of the transient flow rates from each layer following achange of the total flow rate for the well. Pressure transients are alsoused in the analysis. The limiting forms of the analytic solution areused to identify certain characteristic patterns in the flow-ratetransients that provide the means for determining the individual layerproperties. The methods used in this analysis are entirely new. An properties. The methods used in this analysis are entirely new. An example of the interpretation method is provided. Introduction Interest in the behavior of multilayered reservoir systems hasprompted a great number of studies in the last 25 years. Only in prompted a great number of studies in the last 25 years. Only in the last 5 years, however, have testing methods begun to providemore than a qualitative grasp on the reservoir description. Table 1 indicates the scope of various information on thissubject, listing the relevant papers that have been published inthe petroleum literature. A certain oscillation in the perceptionof the multilayered system is apparent. In particular, themodeling of interlayer formation crossflow has alternated with afocus on "commingled" systems. The latter terminology hasgenerally been reserved for multilayered systems with nocommunication in the formation, other than through the well. Hence, a distinction must be made between formation crossflow, which can occur only when a nonzero vertical permeability existsbetween two adjacent layers, and wellbore crossflow, which canoccur between any two layers that are penetrated by the wellbore. The earliest rigorous study found for the commingledreservoir system was that of Lefkovits et al. This study, whichaddressed an arbitrary number of layers with distinct layerproperties, including thickness, porosity, permeability, and properties, including thickness, porosity, permeability, and skin, provided an analytic model and a host of practicalobservations that have served as the basis for much of thework that has followed. Of particular interest were thepresentations of both pressure and layer flowrate transients, presentations of both pressure and layer flowrate transients, the latter of which will be shown in this work to be essentialdata for evaluation of individual layer properties in a well test. For the commingled system, the model development was extendedby Tariq and Ramey, whose contributions included the introductionof wellbore storage and, together with Ref. 36, the first use inthe petroleum literature of the Stehfest algorithm for numericalinversion of Laplace transforms. Since that time, use of thisalgorithm has resulted in numerous practical applications in thewell test literature, including this work. The other class of models for multilayered systems involvesinterlayer formation crossflow. The earliest studies used partialdifferential equations written in two-dimensional cylindricalcoordinates (r-z symmetry). The first known formulation of thistype was offered for the no-flow bounded system by Jacquard. TheJacquard solution was computed for two layers by Pottier, whoprovided observations of the following: drawdown pressure provided observations of the following: drawdown pressure transients for various layer thickness ratios, comparisons with thecommingled two-layer pressure transients, layer flow-ratetransients for the commingled system, and radial pressuredistributions vs. time for the communicating layers and thepressure gradient between them. pressure gradient between them. Drawing from the terminology for dual-porosity systems, formation crossflow in the Jacquard formulation can be calledtransient interlayer flow, An early study reported byPolubarinova-Kocina and recent models published in Refs. 27 Polubarinova-Kocina and recent models published in Refs. 27 through 30 and 35 have presented the interlayer flow by apseudosteady-state approximation that reduces the partial pseudosteady-state approximation that reduces the partial differential equations to a system of onedimensional (radial)equations. Refs. 27 and 28 provided observations similar to thoseof Pottier and extended the formulation to an arbitrary number oflayers. SPEFE P. 261