Additional Use of Well Test Analytical Solutions for Production Prediction
Authors: Bourgeois Marcel J. , Wilson Michael R. 
Published: December 2007

Abstract
One easy way to perform multi-well production prediction is to use empirical decline-curves. These have little theoretical justification, other than in very simple cases. The other alternative is to run full-field gridded simulators, which give accurate results, but which need both complicated algorithms, and numerical know-how to be used.
Ref. 1 developed a new, intermediate approach, as it described an analytical method to predict production, in a more rigorous manner than decline-curve analysis, but simpler than with a numerical simulator. It generalized for this purpose the theory of the Laplace transform to multi-well problems, accounting for their interferences. It allowed a prescribed flowing pressure or flowrate history for each well, and used the assumptions of slightly compressible fluid flow.
This paper extends the existing theory to more general constraints than just p or q for each well. It also adapts the above theory to non-linear problems such as single-phase gas flow, using pseudo-pressure to account for changes in volume factor and viscosity, and a material balance correction to account for changes in compressibility. Turbulence can be treated as a rate-dependent skin.
Some implementation choices, which enable to treat complicated constraints or geometries, are shown as well.
Three examples are shown, two of which are field cases. New type-curves for multi-layer or laterally heterogeneous reservoirs are shown, and compared either with known solutions or with numerical simulations.

Extensions of existing theory
Basically, the relation which links pressure and flowrate for a single-well reservoir, as sketched in the Figure below, has been generalized for n wells.
If the number of wells n in the reservoir is larger than 1, the solution requires a matrix inversion, as detailed in Ref. 1. The reservoir can then be considered as a linear system with n entries and n outputs.
We develop here a way to extend the capabilities of our analytical approach of production prediction to complicated constraints, or to single-phase gas flow.
We also verify that our approach degenerates to known solutions is symetric cases (App. C).

Drawdown vs. flowing pressure constraint
In most operated fields, the regulation is performed on pressure rather than on flowrate. Thus it makes sense to predict the flowrate for a given bottom hole flowing pressure pwf. But in low-permeability and/or closed reservoirs, this flowing pressure will decline with time, and the decline in flowing pressure is usually not known in advance. Moreover, a constant pwf constraint will often result in very high early time flowrates, which are not physical, because the tubing would not allow for such high flowrates.
Thus it can be interesting to apply constraints other than just q(t) or pwf(t).
Considering that during the plateau rate, the well flowing pressure and the average reservoir pressure will decline together, it makes sense to predict the reservoir flowrate if the wells are operated such that Pa = pavg(t) - pwf(t), which I will call drawdown, is kept constant.