This paper presents a new general analytic formulation forpressure-transient behavior of commingled systems (layered reservoirspressure-transient behavior of commingled systems (layered reservoirs withoutcrossflow). The formulation includes the effects of surface and downholeflow-rate variations and of wellbore storage resulting from the wellborevolume's location below the flow-rate measuring point (at any location in thewellbore, including the surface and sandface). The method can be applied to avariety of layered reservoirs. Each individual subzone (reservoir) in thesystem can be different, with different initial and outer-boundary conditions, and each zone can start to produce at a different time. The well completion foreach layer can also be different.

New Laplace domain solutions are presented for partially penetrating slantedwells and partially penetrated wells with and without a gas cap. The solutionfor slanted wells is based on that of Cinco-Ley et al. but includes the correcteffective wellbore radius for the case of an anisotropic formation. Solutionsto a few selected commingled systems are also presented to explore theapplication of the formulation.


Hydrocarbon reservoirs that lie on top of each other are usually separatedby shale zones or nonpermeable or semipermeable formations. The Sadlerochit, Shublik, and Sag River formations of the Prudhoe Bay field are good examples of such a system (Fig. 1). These layers do not communicate in terms of fluid flowthrough the formation but may be produced by the same wellbore. These types ofreservoirs are called commingled systems. The wellbore in commingled systemsmay be vertical, horizontal, inclined, fractured, or partially penetrated. Individual layers may be homogeneous, heterogeneous, or fractured and can havedifferent initial and outer-boundary conditions: infinite extent, constantpressure, no flow, or mixed. pressure, no flow, or mixed. During the last 30years, many papers have appeared in the petroleum literature about the behavior of commingled layered petroleum literature about the behavior of commingledlayered reservoirs. With a few exceptions, most of these papers assume thateach layer is a radial system that is either infinite or bounded by no-flow orconstant-pressure conditions at the drainage radius. Ehlig-Economides and Joseph conducted an extensive survey on layered systems, and Mavor and Walkupused the parallel-resistance concept to present solutions for commingledreservoirs in which the initial pressure present solutions for commingledreservoirs in which the initial pressure of each reservoir or layer is thesame.

This paper presents generalized analytical solutions for commingled systemsin which each reservoir or layer can have a different initial pressure ordifferent initial pressure distribution. The formalism is combined withwellbore storage in a way that allows the initial wellbore pressure and initiallayer pressures at the sandface to be different from each other. Theformulation is in the Laplace transform domain and allows the response of theentire system to be computed if individual layer solutions are known. For thisreason, we also present new Laplace space solutions for single layers for somewell/reservoir systems of interest.

Mathematical Model

Let us assume that n reservoirs or layers with different initial pressuredistributions are commingled so that they have a common pressure distributionsare commingled so that they have a common wellbore pressure, pw. If we let qibe the flow rate from the ith layer and q be the total flow rate, then wehave


q may or may not be constant, but even when it is constant, the individualflow rates will generally vary as functions of time. In the Laplace domain Eq. 1 becomes


In the ith reservoir, the relationship between flow rate (input) andpressure (output) at the wellbore can be described as a convolution pressure(output) at the wellbore can be described as a convolution operation (see the Appendix).


where phi wi(t) is the contribution from the initial pressure distribution of the ith reservoir, and Gwi(t) is the impulse response of the ith reservoir. In the Laplace domain this becomes


Solving Eq. 4 for the average of qi gives


Substituting for avg of qi in Eq. 2 and solving for the avg of pw yields


Note that Eq. 6 may be written as

(7) where


and (9)

Thus, in the time domain the solution is of the convolution form


where phi(t) and G(t) are inverse Laplace transforms of the avg of phi(s)and the avg of G(s), respectively.

The convolution integral (Eq. 10) and its Laplace transform (Eq. 7) providea general framework for treating commingled reservoirs of any type witharbitrary initial pressure distributions, time-varying flow rates, andarbitrary boundary conditions. The only requirement is the ability to solve theindividual layer problems in the Laplace domain. Note, however, that in somecases (particularly when initial layer pressures are different), the flow rateqi (t) in one or more of the layers may be negative, even if the total flowrate q(t) is positive. In these cases, the formulation is correct only if eachlayer positive. In these cases, the formulation is correct only if each layerhas the same fluid viscosity.

The above formulation uses Gwi(t) and phi wi(t) and their Laplace transformsas the basic quantities to describe each layer and the effect of its initialconditions. Precise definitions of these quantities are given in the Appendix. The quantity Gwi(t) is the impulse response function of a single layer, and isrelated to the usual constant-rate response by



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