The shape of fractures in an elastic medium under different stress distributions have been well studied in the literature, however, the fracture closure process during unloading has not been investigated thoroughly. The fracture surface is normally assumed to be perfectly smooth so that only the width and stress intensity at the fracture tip are critical in the analysis. In reality, the creation of a fracture in a rock seldom produces smooth surfaces and the resulting asperities on the fracture surfaces can impact fracture closure. Correctly modeling this fracture closure behavior has numerous applications in structural engineering and earth sciences. In this article, we present an approach to model fracture closure behavior in a 2D and 3D elastic media. The fracture surface displacements under arbitrary normal load are derived using superposition method. The contact stress, deformation, and fracture volume evolution can be estimated in a computationally efficient manner for various fracture surface properties. When compared with the traditional integral transform methods used to model fracture closure, the superposition method presented in this study produces comparable results with significant less computation time. In addition, with the aid of parallel computation, large scale fracture closure and contact problems can be successfully simulated using our proposed dynamic fracture closure model (DFCM) with very modest computation times.

This work presents a model for refracturing horizontal wells employing diverting agents. The goal is to quantitatively model the placement of diverting agents and fracture propagation, when re-stimulating the entire wellbore with multiple clusters of perforations accepting fluid. The model can be used to help estimate the outcome of refracturing and provide recommendations for refrac treatment design. Three types of perforations are identified before a refract operation: propped-depleted fractures, unpropped-depleted fractures and new perforations that have not been previously fractured. In the simulation, fractures are discretized into segments for leak off, pressure drop and width calculations. Fracture width and propagation are correlated to pressure by PKN fracture model. During refrac, the fluid distribution is calculated by a flow resistance model, and is shown to gradually divert from low pore-pressure zones to high pore-pressure zones by diverting agent stages. This diversion process is reflected in the increase of fracturing pressure for each successive stage in the refrac treatment. When bottom-hole pressure reaches a critical value, the propped-depleted fractures will widen. Two simple test scenarios are presented to show the fluid diversion process. A field refracturing treatment in the Haynesville Shale is simulated next. For the field case, a complete simulation workflow of the initial fracturing treatment, production history match and refracturing treatment was completed. The production history was simulated using a coupled geomechanical reservoir model to account for the pore pressure and stress change. The refracturing treatment simulation was performed using the model presented in this paper. The modeled bottom-hole pressure (BHP) during the refracturing treatment is compared with field data and matched the field BHP trend very well. This match can be expected to provide an estimate of the fluid diversion process. The estimated propped fracture length after refrac can be used for production prediction. Questions such as the amount of diverting agent to use, the desired diverting agent permeability as well as maximum diverting agent stages are explored based on the simulation results.

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