In this paper, we develop a simple, closed form approximation for the Laplace transform solution for the case of a well with a finite conductivity vertical fracture in an infinite-acting reservoir. Our hybrid solution is based on a coupling of the solution for a trilinear finite conductivity vertical fracture model (which does not model radial flow) and the solution for a uniform flux/infinite conductivity vertical fracture (which does model pseudoradial flow). These solutions are readily obtained from the literature.
Overall, we consider our solution to be valid for and we show that our solution gives less than 1 percent error in both PD and PD' for. We suggest that our hybrid solution is not valid for and do not recommend its use for under any circumstances.
We have verified this solution against four different solutions given in the literature. Each comparison was excellent which suggests that our simplified solution is more than adequate for practical applications. In particular, we provide verification for constant rate and constant pressure production for values of between 0.25 and 10,000. We also show that our solution is capable of producing very accurate derivative functions.
In addition, by reproducing the literature solutions so well, we also verified that individual flow regimes (formation/fracture bilinear flow, formation linear flow, and pseudoradial flow) are all modeled accurately by our new solution.
Our motivation for this paper is to provide the technical audience with an accurate, but computationally simple, approximation for the Laplace transform solution for the case of a well with a finite conductivity vertical fracture in an infinite-acting reservoir.
We have the following objectives for this work
To develop a rigorous closed form solution that can be used to accurately model a single well with a finite conductivity vertical fracture in an infinite-acting homogeneous reservoir.
To provide evidence of systematic verification to warrant application of this method for general practice.
To illustrate the application of this approximate solution to generate solutions for wells produced at both constant rates and constant pressures.
To identify and discuss any limitations of this solution and to give recommendations for computational considerations.
The original development of the uniform flux/infinite conductivity solutions for a vertically fractured well was performed by Gringarten, et al. The most important aspect of that work is that it provides working relations and theoretical considerations for fractured wells as well as the foundation and motivation for later efforts regarding finite conductivity vertical fracture solutions.
Prior to the development of solutions for a well with a finite conductivity vertical fracture, the application of the uniform flux and infinite conductivity solutions often drew heated debate as to which was the most appropriate physical model. It is generally agreed that the uniform flux and infinite conductivity are "best case" scenarios whereas the finite conductivity model is seen as the most appropriate for general practice.
However, the uniform flux and infinite conductivity vertical fracture solutions do have a theoretical tie to the finite conductivity fracture case. As such, it is theoretically consistent to use the uniform flux and infinite conductivity vertical fracture solutions to develop relations for finite conductivity vertical fracture solutions. Such is the case in our work. As background for the development of our solutions, we need to familiarize ourselves with the solutions for the uniform flux and infinite conductivity cases and the computational aspects of these solutions..
Gringarten, Ramey, and Raghavan Real Space Solution:
From Gringarten, et al. we have the following result for the constant rate behavior of an infinite conductivity or uniform flux vertical fracture