Hemivariational inequality for contaminant reaction-diffusion model of recovered fracturing fluid in the wellbore of shale gas reservoir

The goal of this paper is to study the dynamic behavior of recovered fracturing fluid and the reaction-diffusion phenomenon of contaminants in the wellbore of shale gas reservoir during the early stage of fracturing fluid flowback. First, by applying various physical constitutive laws, a complicated recovered fracturing fluid model which consists of a stationary incompressible Stokes equation with a nonmonotone and multivalued friction law and a reaction-diffusion equation with the Neumann boundary condition, is established. Then, we use the variational techniques and the theory of multivalued analysis to formulate a new recovered fracturing fluid model. The latter is a variational system which consists of a hemivariational inequality with constraint driven by an elliptic variational equation. The system involves mutually coupled velocity and the concentration fields. Finally, under some mild assumptions, we employ a surjectivity theorem for pseudomonotone operators, monotonicity arguments and Schauder fixed point theorem to prove the existence of a weak solution to the recovered fracturing fluid model.(c) 2022 Elsevier B.V. All rights reserved.