This paper is the third of a series of papers which discuss the application of the direct type curve synthesis technique for interpreting pressure transient tests in various reservoir systems. This new technique analyzes log-log plots of pressure and pressure derivatives without using type curve matching. This new approach is applied in this paper to the case of vertically fractured wells in closed systems. Both uniform flux fracture and infinite conductivity fracture cases have been investigated using log-log plots of pressure and pressure derivative functions.
For the uniform flux fracture case, pressure derivative plots for various xe/xf ratios reveal three dominant flow regimes. During early times the flow of fluids is linear and can be identified by a straight line of slope 0.5. The linear flow line is used to calculate the half-fracture length. The infinite acting radial flow regime, which can be identified by a horizontal straight line, is dominant for xe/xf 8. This flow regime is used to calculate permeability and skin. The third straight line, which corresponds to the pseudosteady flow regime, has a unit slope. This line is used to calculate the drainage area and shape factor. Several simple equations have been derived using the time of intersection, and slopes of these three lines. These equations can be used for verification purposes or for calculating reservoir parameters if one of the flow regimes is missing such as in short tests.
For the infinite conductivity fracture case, pressure derivative plots reveal a fourth dominant flow regime, called here the bi-radial flow. This flow regime, which corresponds to the transition period between the early-time linear flow regime and the infinite acting radial flow regime, can be identified by a straight line of slope 0.36. The bi-radial flow regime can be used to calculate the half-fracture length, in the absence of the linear flow line. The permeability can also be obtained from this regime if the radial flow regime is not dominant, such as when xe/xf 8. A step-by-step procedure is provided for calculating reservoir parameters of a vertically fractured well inside a closed system.
Tiab and Puthigai extended the application of the pressure derivative function to vertically fractured wells inside an infinite reservoir. They advocated the use of type-curve matching for interpreting this function. Both the uniform-flux and infinite conductivity fracture models were investigated. Wong, Harrington and Cinco-Ley analyzed the behavior of the pressure derivative curves for a well with a finite-conductivity fracture, skin and wellbore storage during bilinear flow conditions. Chukwu used the type curve matching technique to analyze the pressure and pressure derivatives of vertically fractured wells inside closed systems.