||In the talk, we present our recent developments of the modelling of a planar hydraulic fracture in an inhomogeneous reservoir. The hierarchy of models includes the Enhanced Pseudo 3D (EP3D) model  coupled with the proppant transport, the Planar 3D model under the modification of the Implicit Level Set Algorithm (ILSA) , and the Planar 3D Biot model that accounts for the effect of poroelasticity .
In the EP3D model, the three-dimensional fracture is modeled in terms of quantities that are averaged along the fractureâ s height dimension. This model is computationally fast, but is limited to a certain shape of the fracture, and takes into account the layered structure of the reservoir only in terms of the confining stress difference. This model is used for the development of the coupling procedure with the proppant transport module. For the latter we use the one-speed transport model with the effective viscosity varying with particle concentration. We demonstrate effects of the Saffman-Taylor instability development, proppant bridging, breakage of the proppant plug due to the displacement instability.
The Planar 3D model describes the fracture development in a layered reservoir, where the effects of the poroelasticity are neglected. The model is implemented using a modification of the ILSA approach . In particular, the model accounts for the inhomogeneity of the elastic properties of the reservoir (only the layered structure of the reservoir is allowed), and is able to simulate cases of fracture local closure due to the fluid re-distribution and/or leak-off.
The most advanced Planar 3D Biot model describes the fully coupled interaction of the stresses with the fluid filtration. The numerical model is implemented using the Finite Element Method and allows us to account for an arbitrary inhomogeneity of all physical characteristics of the reservoir. In particular, we show that in the case of the layered structure of the formation, where the layers differ only by permeability, the fracture propagation can demonstrate counter-intuitive non-monotonical behavior. Both, Planar 3D and Planar 3D Biot models, are thoroughly verified by comparison with fast approximate solution for the radial fracture , and matched with the existing experimental data .
Finally, we present results of the multi-parameter and multi-objective optimization for the Net Present Value and the Fracture Production depending on the applied flow rate, volume of fluid and proppant, and other characteristics of the fracturing process. The optimization is based on the fast algorithms for estimation of fractureâ s characteristics , on the module for computation of the production of the multiply-fractured wellbore, and on application of genetic algorithms  for construction of the Pareto front in the space of objective functions.
The work was supported by the Ministry of Science and Education of Russian Federation (grant 2016-220-05-2642).
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