This paper presents analytic solutions of the pressure transient behavior of a well intersected by a finite-conductivity fracture in an infinite-acting, or in cylindrically or rectangularly bounded finite reservoirs. These solutions include the practical effects of reservoir permeability anisotropy and dual porosity behavior. Those solutions are analytic, and thus do not require discretization in space.
The analytical solutions of the finite-conductivity fracture transient behavior presented in this paper eliminate the numerical difficulties associated with other mathematically rigorous finite-conductivity fracture solutions that have been reported in the literature. Both the pressure and rats transient responses can be accurately evaluated using the finite conductivity fracture solutions presented in this paper. This is especially important for low-conductivity fractures, for which the pressure and rate transient behavior is often difficult to evaluate accurately using the solutions available in the literature.
The pressure transient behavior of finite-conductivity vertical fractures has been investigated extensively in the past few decades in order to better estimate the propped fracture geometry and conductivity resulting from hydraulic fracture well stimulation treatments. The various types of models that have been used in these investigations include both finite-difference and finite-element numerical models, real and Laplace space analytic solutions for the transient behavior of uniform flux and infinite-conductivity fractures, and real and Laplace space semi-analytic solutions for the transient behavior of finite-conductivity vertical fractures. Of particular interest in this paper are the studies pertaining to the evaluation of the transient behavior of finite-conductivity fractures using the laplace transformation technique and the Boundary Element Method
Two concurrently and separately developed solutions for the transient behavior of finite-conductivity fractures were reported by Cinco-Ley and Meng and van Kruysdijk. The model developed by Cinco-Ley and Meng considered the fracture storage effects to be negligible, while the model reported by van Kruysdijk included the fracture storage effects. Both models were developed using the Boundary Element method and assumed that the vertical fracture was of uniform fracture width, conductivity, and height. The fracture height in each of the models Was assumed to be equal to the reservoir thickness. Later, more general finite-conductivity fracture models were reported that permitted arbitrary fracture geometry and conductivity distributions. All of these semi-analytic solutions - for the transient behavior of a finite-conductivity vertical fracture require discretization in space in order to solve the Fredholm integral equations that comprise the transient solutions. This technique involves solving a system of equations numerically in order to determine the unknown flux distribution in the fracture and the wellbore pressure.
Riley, et al. presented an analytic solution of the transient behavior of an elliptical finite-conductivity fracture which does not require discretization of the fracture. The analytic solution presented by Riley et al. generally provides a somewhat more rapid evaluation procedure for the transient behavior of a finite-conductivity vertical fracture than do the semi-analytic solutions reported in Refs. 11 through 13 for the same level of numerical accuracy. However, the solution presented by Riley et al. can still be very time consuming to evaluate due to the slow convergence of the series of the solution.
The difficulty involved with evaluating the transient behavior of finite conductivity vertical fractures using the solutions presented in Refs. 11 through 13 is primarily due to the singular nature of the integral equations and the numerical evaluation procedures required to evaluate the unknown flux distribution and the wellbore pressure. At very early transient times, the principal component of the well production comes from the reservoir region nearest the wellbore. This generally requires the use of a large number of fracture elements in the fracture nearest the wellbore to accurately evaluate the unknown flux distribution.
Similarly, at very late transient times, the principal component of the well production may be from the reservoir region beyond the fracture tip. This requires a large number of elements in the fracture to accurately evaluate the flux distribution in the fracture as well.