In this article we generalize the concept of the pseudosteady-state productivity index for the case of multiple wells producing from or injecting into a closed rectangular reservoir of constant thickness. The work complements the analytical study by RodrÍguez and Cinco-Ley 1 for systems produced at constant flowing pressures. Wells are represented by fully penetrating vertical line sources located arbitrarily in a homogeneous and isotropic reservoir. The multiwell productivity index (MPI) is a square matrix of dimension n, where n is the number of wells. The MPI provides a simple, reasonably accurate and fast analytical tool to evaluate well performance without dividing the cluster into single-well drainage areas. The MPI approach is used to obtain approximate analytical solutions for constant (but possibly different) wellbore flowing pressures, and to visualize the resulting pressure field. In addition, the skin factor trace technique is introduced as a tool to monitor a cluster of wells. The MPI technique is illustrated using a synthetic example taken from Ref. 2, as well as two field cases.
Although it has been known from the inception of reservoir engineering studies that any change in the condition of a well (damage, stimulation, choke change) modifies the production characteristics of the well-reservoir system, only the pioneering work of RodrÍguez and Cinco-Ley 1 approached this problem using analytical techniques. The constant flowing wellbore pressure solution of Ref. 1 has been improved in several ways by Camacho-V. et al., 2 including the possibility of allowing wells starting to produce at different times.
Similar to the work in Refs. 1 and 2, we consider liquid flow within a homogeneous reservoir of constant thickness. This work is, however, restricted to pseudosteady-state flow conditions. The results can be used for any type of wellbore condition so long as the pseudosteady-state approximation is acceptable.
Pseudosteady-state flow is the (idealized) finite-acting portion of the constant-rate solution for a bounded reservoir,3 and this lends itself to a simpler description. The physical meaning is that all elementary portions of the reservoir contribute to the overall production rate by the same amount. Depletion is a parallel shift of the pressure distribution with time.
While, formally, the pseudosteady state is a limiting case of the constant-rate solution, solution-gas drive reservoirs spend most of their life in a series of states, and closely resemble this flow condition.3
The basic concept in this article is the multiwell productivity index (MPI) matrix, which relates the production rate vector to the pressure drawdown vector. The pseudosteady-state flow condition ensures the uniqueness of the MPI matrix for a given system. (Without the assumption of the pseudosteady-state flow condition, this uniqueness would be lost, even for a single-well system.4)
We consider a rectangular homogeneous reservoir of uniform thickness, h, porosity, ?, permeability, k, and no-flow outer boundaries. The single-phase fluid viscosity, ?, and the total compressibility, c t are considered constant. The wells are represented by line sources.
The four corner points of the rectangle are located at (0, 0), (xe, 0) and (xe, ye) and (0, ye). In any given time interval, the number of wells, n, their locations (xwj, ywi), the wellbore radii, rwj, and the skin factors, sj, are considered constant. Fig. 1 shows a schematic of the reservoir.
For the case where only one well is on production (say, the jth one), Ozkan5 (see also Ref. 6, page 107) gives the pressure distribution in the reservoir during the pseudosteady state as
where the influence function a[ ] (a dimensionless drop in pressure) is given by
with xD and yD defined as x/xe and y/xe, respectively. In the Appendix we introduce several computational simplifications to ensure a fast and reliable approximation of the infinite sum in Eq. 2. Also shown in the Appendix is the relation of the influence function to the Dietz7 shape factor.
By superpositing, for the n-well system with n production/injection wells, we have
where the qj production rates are "constant" for the given time interval. All prior information is contained in the average pressure, , and, as such, we do not need to specifically account for the initial pressure distribution or the production history.
The two basic vector quantities we would like to relate are the pressure drawdown vector,
and the surface production-rate vector,