A Comprehensive Analytical Approach to Multistage Formation Fracturing and Production

An analytical model is developed for multistage fracturing for any number of fractures in a given area, and subsequent oil and water production for the entire life of the operation, from initial fracture fluid injection to the end of production. The analytical model results were in good agreement to those produced by two commercial simulators for the many cases investigated. The starting point is extensions of the Carter equation for a single fracture for the injection period, and calculation of the fracture width using analytical equations. The fracture conductivity is calculated using Carman-Kozeny equation and its modifications for single-size and two-size packings. Both infinite and finite conductivity fractures are considered. These results are then used to calculate the pressure distribution for complex branched fractures in the Laplace domain. Following that, the solution is superimposed for two dissimilar fractures. Theoretically, this can be done for any number of fractures at arbitrary spacings. The paper gives the results for two fractures. The final solution is numerically inverted to obtain the pressures, which are used to compute the respective flow rates. For flowback, Buckley-Leverett and Darcy equations are used for production using a set of oil-water relative permeability curves. Several cases of flow are considered. The overall analytical results are compared with those produced by two commercial simulators ran in tandem (for coverage of the entire fracture and production history, not feasible with one simulator), showing very good agreement for the many cases considered. The proposed analytical approach is useful for providing a global answer to the performance of a multistage fracture operation and providing insights into the mechanics of each stage of the operation. Being analytical, the answer obtained is a continuum.