Transient Flow To Finite Conductivity Vertical Fractures
Authors: Barker Benjamin J. , Ramey Henry J. 
Published: December 2007

Abstract
A finite element model was used to study the behavior of a well intersecting a finite conductivity vertical fracture at the center of a closed square reservoir. The two dimensional flow of single phase fluid was assumed to follow Darcy's law. The fracture and formation were represented by regions of uniform permeability and porosity. The well history prior to the boundary influence was found to be a prior to the boundary influence was found to be a function only of the dimensionless fracture conductivity, (kfw)D= (kf/k) (w/x f).
A period of pseudoradial flow, during which conventional analytic methods may be applied, develops following the early transient period. This period may not develop before the onset of pseudosteady state when xf/xe &gt;1/2 and (kfw)D &gt;. Fracture length and conductivity have little effect on the time of development of pseudosteady state flow. Graphs of log PD vs. log tDf for various values of (kfw)D have few distinguishing characteristics to aid type curve matching of field data. Data from the pseudoradial flow period give a good estimate of formation permeability and wellbore skin effect when matched with any of several curves. Calculations of fracture length are not reliable, however.

Introduction
Large hydraulic fracturing operations have become commonplace, creating a need for better understanding of the flow near fractured wells. The expense of these operations increases the importance of diagnostic tools for analyzing the resulting fracture. Type curve matching of transient pressure data yields an apparent fracture length, pressure data yields an apparent fracture length, and is most often applied assuming that kf/k is infinite. However, analyses of a number of recent field tests using this technique produced apparent fracture lengths much smaller than were predicted by design calculations and by knowledge of treatment volumes. It has been speculated that this may result from limited fracture conductivity causing a deviation from ideal behavior. The non-ideal nature of both natural and induced fractures has long been recognized. Muskat in 1937 presented an analysis of pressure and fluid entry presented an analysis of pressure and fluid entry distribution along a fracture under steady state conditions. Prats studied steady state flow using conformal mapping of an elliptic fracture and reservoir. The limiting case yielded the solution for a fractured well in a circular reservoir. Prats deduced the effective well radius to be one-fourth the tip-to-tip fracture length when (kfw)D &gt;2/pi.
Because the steady state flow of fluids following Darcy's law is described by La Place's equation, numerous potentiometric model studies have been undertaken. The difficulty of finding appropriate materials has caused discrepancies between results, some of which conflict with analytically derived limits.
These studies of steady and pseudosteady state flow are the basis of fracture design charts. It is important that consistent and correct data be used in these charts and for transient test analysis in order to optimize fracture design and to successfully evaluate the job.
Transient flow to a finite conductivity fracture was modeled by Scotts, who used the analogy between heat and mass flow. He confirmed Prats' results at very high fracture conductivity, but found less agreement when (kfw)D less than 2/pi. Scott attributed this to the scale limitation imposed by his heating and thermo-couple wires.
Pressure test analysis in fractured wells has relied Pressure test analysis in fractured wells has relied primarily on numerical and analytic studies of primarily on numerical and analytic studies of infinitely conducting fractures. The works of Russell and Truitt and Gringarten, et al. have made possible the use of short time transient well test data possible the use of short time transient well test data for evaluation of both reservoir permeability and apparent fracture length. The latter showed that a graph of dimensionless pressure vs. the log of dimensionless time has the slope 1.151 in the absence of a boundary effect.