A new method to calculate slurry distribution among multiple fractures during fracturing and refracturing

Abstract An accurate calculation of slurry distribution among multiple perforation clusters is necessary for all multi-fracture simulators. The slurry distribution was solved by applying the Newton-Raphson Method in the past. This method requires a numerical evaluation of the Jacobian matrix and can be unstable in some cases. In this paper, we propose a new approach to calculate slurry distribution among multiple fractures, the Resistance Method. This new approach is computationally efficient, requires no Jacobian matrix calculation, and has been proved to converge locally at least linearly. The Resistance Method is derived by reforming the mass conservation and pressure continuity equations for multiple fractures into a system of nonlinear fixed-point iteration equations. In the new equation system, the fracture resistance is defined as the pressure drop per unit flow rate. And the injected slurry is distributed among multiple fractures in inverse proportion to the resistance of each fracture. This new slurry distribution method has been applied to several different fracture models. The convergence of this new slurry distribution method has been analyzed with the map contraction theory. In this paper, we show three example applications of the Resistance Method. First, the Resistance Method is implemented with an analytical PKN model, with direct comparison made with the Newton-Raphson method. Simulation results show that both methods solve the slurry distribution accurately, but we show that the Resistance Method can be more computationally efficient than the Newton-Raphson Method. Second, the Resistance Method is used to obtain the proppant distribution in multiple clusters with particle transport efficiency curves implemented implicitly. It is shown that the proppant distribution among multiple clusters can be quite different from the fluid distribution. Due to inertial effects, proppant tends to accumulate towards the toe side clusters and causes pre-mature screen-out, resulting in a heel-biased treatment. Third, the Resistance Method is used in a refracturing example problem. A synthetic refracturing treatment with 50 existing fractures, 49 new clusters and about 15 h' treatment time was simulated. The results show that the refrac treatment is heel biased with the 20 heel side clusters receiving most of the treatment and that both existing fractures and new perforations propagate during refracturing. In such simulations extending over long periods of injection and with a large number of perforation clusters it is important to have a very efficient solver to obtain the fluid and proppant distribution in a reasonable computational time frame. In this work, we developed a new method to calculate slurry distribution among multiple clusters with applications to fracturing and refracturing treatments. This new method can be integrated with any fracture model. The main advantages of this new approach include: 1) simple implementation, 2) no numerical evaluation of the Jacobian matrix, 3) computationally efficiency, and 4) easy integration of slurry distribution, proppant transport, stress shadow effects and multiple fracture propagation.