This paper presents the development, validation, and application of new approximate (semi-analytical) solutions for the wellbore pressure and fractional flowrate responses of commingled layered reservoirs — without interlayer crossflow (crossflow is only permitted in the wellbore, not in the reservoir). These formulations use "basis" pressure drop relations for each layer as a mechanism to create multilayer solutions in the Laplace domain — and due to the choice of these basis relations; the formulation can be analytically inverted into the real domain. This process provides a direct (albeit approximate) solution for multilayer reservoir systems. The basis relations include a "constant" pressure case and "linear" pressure case. In addition to these methods, we also provide a "Total Pressure/Rate Averaging" (TPRA) formulation which is used to provide an average rate and pressure response for each layer.

These semi-analytical solutions can be used to construct well test or production profiles in a simple computational environment (e.g., a spreadsheet). These solutions can also be used to diagnose well test and production data in terms of layered reservoir behavior. The "linear" pressure case (non-zero intercept) gives the best performance — however, from a practical standpoint, the algebra involved limits this approach to 2-layer reservoir case. The TPRA formulation is very consistent, but not as accurate as the other solutions — however; the TPRA formulation is easily be extended to n-layers.


The primary objectives of this work are:

  • To develop approximate (semi-analytical) solutions for the wellbore pressure and fractional flowrate responses of commingled layered reservoirs--without interlayer crossflow (crossflow is only permitted in the wellbore, not in the reservoir).

  • To apply these new solutions for the analysis/modeling of well performance behavior in multilayer reservoir systems.


We begin with a brief discussion of the concepts of pressure transients and differential depletion which exist in the multilayer reservoirs. We consider a layered reservoir to be a geologic formation that consists of horizontally continuous, homogen-eous, and isotropic layers which differ markedly in permeability and porosity, but not necessarily in gross lithologic features. Thus, a reservoir that is lithogically homogeneous on a macroscopic scale is considered layered if it could be divided into laterally continuous layers of differing permeabilities and porosities. For the purpose of this study, a layered reservoir system is defined as a reservoir that is composed of two or more layers of differing physical characteristics, such as permeability, porosity, thickness, outer radius, and skin. The term "layer" is defined as a reservoir body that has a particular combination of permeability and porosity which allows fluid to flow. Each layer could be separated by impermeable barriers where no interlayer crossflow occurs -- this is also known as a commingled reservoir, where crossflow is only permitted in the wellbore (not in the reservoir).

Fig. 1 shows a schematic of an actual multilayer reservoir system which consists of a sand and shale sequence [Gringarten et al (1981)]. Fig. 2 illustrates the schematic multilayer reservoir model used in this study. The reservoir consists of two or more layers with different physical characteristics such as permeability, thickness, porosity, skin, and outer radius. These layers are unconnected except at the well. Each layer of the reservoir is assumed to be homogeneous and isotropic, and is filled with a fluid of small and constant compressibility. The reservoir is initially at a uniform pressure and at all times is produced in such a manner that the total production rate measured at initial reservoir conditions is held constant. Gravity and capillary pressure effects are assumed to be insignificant.

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