An algorithm to compute pressure distributions in commingled reservoirs that are produced via complex completion systems is described. We show that single layer solutions for systems of interest can be readily combined to obtain pressure distributions in commingled reservoirs for combinations of rock type. Completion schemes and outer boundary conditions for each layer may be different. Theoretical claims are substantiated by considering example applications.


This communication presents a stable and robust algorithm to compute responses at wells that produce commingled reservoirs. We take advantage of the unique feature of commingled reservoir production for the constant terminal pressure solution and present an algorithm to determine the well response for constant or variable rate production. Expressions to compute sand-face layer rates are also presented. The advantages of the algorithm we present are:

  • any combinations of wellbore and outer boundary conditions can be incorporated and the combinations can be different in each layer,

  • characteristics of each layer including rock type can be different.

  • existing codes for single layer systems can be used with minor modifications to compute commingled reservoir responses, and

  • approximate solutions for various situations of interest can be derived. Our intent is similar to that in Refs. 1–3, in that this algorithm will enable analysts to obtain commingled reservoir responses for interactive analysis and history matching purposes.

The efficacy of the algorithm has been tested by comparing responses with standard solutions. 4–7 In general solutions are in agreement to at least three digits.

Theoretical Considerations

We consider the flow of a slightly compressible fluid of constant viscosity in a commingled reservoir. Each layer is assumed to be a uniform porous medium. However, the properties, rock type, completion conditions, location of boundaries and the boundary condition (closed, constant pressure) in each layer are entirely arbitrary. The initial pressure in each layer is assumed to be different.

Solutions discussed in this work will be presented for convenience in dimensionless form. The Van Everdingen-Hurst8 definitions will be used in this work. For describing properties of the entire reservoir system we use thickness averaged permeability. kh and thickness averaged porosity compressibility product oct i.e.:

Equation (1) (Available in full paper) and Equation (2) (Available in full paper)

Here J is the layer index and n is the total number of layers.

To outline the algorithm, we will for simplicity assume that the initial pressure in each layer is identical> It is well-known that if a commingled reservoir is produced at a constant pressure, then the layers are effectively decoupled and production from each layer is independent of the other layer.9 Thus, if qD(tD) is the total production rate from the commingled reservoir based on kh and qD is the flow rate from layers J based on kJhJ then

Equation (3) (Available in full paper)

The algorithm takes advantage of Eq. 3 and Dubamel's theorem which is given by

Equation (4) (Available in full paper)


Equation (5) (Available in full paper)

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