In Situ, Vol.20, No.1, 61-92, 1996
Development of a mathematical model for multilayer reservoirs with unequal initial pressures
In this paper, we present the general solution in Laplace space for a commingled multilayer reservoir with unequal initial pressures. The general solution developed includes the effect of wellbore storage and skin. We derived the Laplace space solution in terms of individual layer source/Green's functions. We used the Green's function to handle the inhomogenous initial boundary conditions resulting from the effects of unequal initial pressures. The model consists of an n-layer, commingled reservoir (i.e., layered reservoir without formation crossflow) with unequal, uniform initial pressures (i.e., independent of space). For each layer, we assumed homogeneous and isotropic rock properties, constant fluid properties, and homogeneous outer boundary conditions. We present the development and analysis of the Laplace space solution for three special cases: (1) pre-production, (2) constant-rate drawdown with equal initial pressure, and (3) constant-rate drawdown with unequal initial pressure. The solutions were programmed to develop a semi-analytical simulator capable of modeling these special cases. Aly and Lee(1) presented a description and verification of the new semi-analytical simulator used to model the wellbore performance of multilayered reservoirs with unequal initial pressures. The semi-analytical simulator proved to be faster than a conventional three-dimensional, finite-difference commercial simulator. Also, the amount of information needed to run the model is much reduced. The semi-analytical simulator allows each layer to have different properties, different boundary conditions, and different initial pressures. The semi-analytical simulator was used to model the reservoir performance for two, three-, five- and n-layer reservoir cases from the literature. The results were verified by comparing them to the results generated using a finite-difference simulator. The agreement was excellent for all the tested cases. On the basis of the mathematical development, Aly and Lee(1) designed a new well test, the Pre-Production Well Test or PPWT. The PPWT is performed early in the life of a reservoir when the information is most needed for planning production schedules and making economic decisions concerning the life of the wells. Preproduction is the period after completion but before production of the well. Immediately after perforation, we position a pressure gauge above the top perforation to measure the pressure performance from the total system (in two-or three-layer systems). Crossflow in the wellbore from one layer to another will cause the pressure signal. The crossflow is due to the differential pressure between the layers. One important advantage of the pre-production well test is that there is no production at the surface during the test, Thus, the environmental impact caused by flaring oil or gas during a conventional well test is alleviated. In this paper we develop and present the asymptotic real-time solutions. These solutions provide the basis for development of real-time analysis methods for the PPWT. Aly er al.(2) developed the Derivative Extreme Method (DEM) for the analysis of wellbore pressures measured during the pre-production well test. The DEM determines layer properties from a single pressure profile; no rate measurements are required. The DEM requires that the wellbore pressure be measured until one boundary is felt (i.e., the late-transient region must be reached).