Unconventional reservoirs exhibit ultra-low matrix permeability, typically in the order of nano-Darcies. To produce hydrocarbons from these reservoirs economically, hydraulically-fractured horizontal wells are commonly used. During production and reservoir depletion, the permeability changes due to decrease in pore pressure and the subsequent increase in effective stress. Similarly, conductivity (kfwf) of the hydraulic fractures could also change under closure stress. The reservoir simulation models used in the industry may consider such variations in transport as a function of stress using mechanistic stress-dependent permeability models, however analytical models used extensively in the industry may not consider such variability. Further, these analytical models often consider production occurring under constant bottom hole pressure (BHP) condition.
In this paper, we tested the accuracy of such an analytical model, also known as A√k plot, in rate transient analysis (RTA). The A√k plot is used to estimate the effective hydraulic fracture surface area A of the well's hydraulic fractures contributing to production. Hence, it is used to measure quality of the well's completion and the extent of its stimulated draining volume. In measuring the quality, however, it is assumed that the matrix has an average permeability k that stays constant during the production. This may be a reasonable assumption for some of the tight gas formations but shale formations have stress-dependent quantities with impact on gas transport such as the matrix permeability, fracture permeability, and fracture width. Hence, ideally these quantities should be treated dynamically during the RTA analysis.
For the study we used a single-well reservoir flow simulation model including a horizontal well with exactly known effective fracture surface area. The reservoir, completion parameters and the fluid properties are taken from an independent study on a shale gas well's production-rate history-matching, optimization, and forecasting. The history-matching of this study used the bottom hole pressure history of one year and a half of gas production under natural flow conditions (no artificial lift). The calibration included reservoir and hydraulic fracture permeability values changing as a function of mechanical stresses induced dynamically by the withdrawal of the fluids. Using the forward simulation results, we showed that the A√k plot works for wells with infinite conductivity hydraulic fractures and for production under static conditions, i.e., constant permeability for the matrix and constant conductivity for the fractures. However, the plot yields significant error in the calculated fracture surface area (larger than 40%) when we consider dynamic matrix and fracture permeability conditions. The error in the estimated fracture surface area is greater in the presence of dynamic fracture conductivity under closure stress, compared to dynamic matrix permeability under overburden stress. Furthermore, the error in calculated area is more pronounced in ultra-low permeability formations and for the wells with finite (and limited) conductivity hydraulic fractures. The error in the estimated area is also significant, when the well experiences a non-constant BHP. We propose a modified RTA method to consider these dynamic conditions in order to minimize the errors in the calculated area. As part of the modification, we propose a simple weighted-averaging of the permeability to use in the RTA analysis. This approach correctly recovers the area under 1-3% accuracy for any dimensionless fracture conductivity (CfD) condition and stress dynamic matrix permeability. Interestingly, the results show that the fractures can control flow not only during fracture linear flow and bi-linear flow but also during the formation linear flow regime.
