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 ge neral solution developed includes the effect of wellbore storage and s kin. We derived the Laplace space solution in terms of individual laye r source/Green's functions. We used the Green's function to handle the inhomogenous initial boundary conditions resulting from the effects o f unequal initial pressures. The model consists of an n-layer, comming led reservoir (i.e., layered reservoir without formation crossflow) wi th unequal, uniform initial pressures (i.e., independent of space). Fo r each layer, we assumed homogeneous and isotropic rock properties, co nstant fluid properties, and homogeneous outer boundary conditions. We present the development and analysis of the Laplace space solution fo r three special cases: (1) pre-production, (2) constant-rate drawdown with equal initial pressure, and (3) constant-rate drawdown with unequ al initial pressure. The solutions were programmed to develop a semi-a nalytical simulator capable of modeling these special cases. Aly and L ee(1) presented a description and verification of the new semi-analyti cal simulator used to model the wellbore performance of multilayered r eservoirs with unequal initial pressures. The semi-analytical simulato r proved to be faster than a conventional three-dimensional, finite-di fference 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 condition s, and different initial pressures. The semi-analytical simulator was used to model the reservoir performance for two, three-, five- and n-l ayer reservoir cases from the literature. The results were verified by comparing them to the results generated using a finite-difference sim ulator. The agreement was excellent for all the tested cases. On the b asis of the mathematical development, Aly and Lee(1) designed a new we ll test, the Pre-Production Well Test or PPWT. The PPWT is performed e arly in the life of a reservoir when the information is most needed fo r planning production schedules and making economic decisions concerni ng 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 pre ssure performance from the total system (in two- or three-layer system s). Crossflow in the wellbore from one layer to another will cause the pressure signal. The crossflow is due to the differential pressure be tween the layers. One important advantage of the pre-production well t est is that there is no production at the surface during the test, Thu s, the environmental impact caused by flaring oil or gas during a conv entional 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 rat e measurements are required. The DEM requires that the wellbore pressu re be measured until one boundary is felt (i.e., the late-transient re gion must be reached).