This paper will present a mathematical and numerical model to simulate temperature and pressure behavior and their interference including the effect of fracturing fluid compressibility during a minifrac pressure test. The numerical solution and the approximate analytical solution of the Perkins-Kern-Nordgren1,2 (PKN) model were employed to calculate the fracture geometry, namely fracture length, L, and width, w, before shut-in. After shut-in, linear elastic theory was used to calculate the fracture width from simulated pressure.
The results of the model indicate that the fracture temperature changes significantly during a minifrac test, and the pressure is higher and declines at a slower rate for a compressible fluid compared with an incompressible fluid. The fluid loss coefficient estimated from the pressure decline by the standard analysis will be low if the fluid is compressible. This effect tends to be more severe if the fluid is confined (i.e., the fluid loss coefficient is low). In extreme cases of a highly compressible fluid and a low fluid loss coefficient, the model predicts that pressure can actually increase after shut-in. Most importantly, it is observed from this work that the error in estimating the fluid loss coefficient, using the standard analysis, decreases when shut-in time increases. Therefore, the error in estimating the fluid loss coefficient can be minimized by using pressure data from later in the shut-in period.