In this thesis the problem of a hydraulic fracture in the plane is studied as a problem coupling elasticity and lubrication equations. In the first part of the thesis the formulation as a differential equation is introduced, showing all the steps and hypotheses to obtain the model problem. The elasticity equation is then written as a Boundary Integral Equation using the analytical knowledge of the fundamental solution of the differential operator and the use of a representation formula that expresses the solution of the problem through integrals defined on the surface of the spatial domain. After introducing the weak formulation, the equation is discretized using the Boundary Element Method. The lubrication equation, after introducing its weak formulation, is approximated through the application of the Finite Element Method. The equations that regulate the movement of the fronts of the spatial domains of the two equations are then introduced. The one for the fluid domain front is obtained directly from the mass balance equation while the one for the fracture domain front is based on a viscous regularization of the fracture propagation problem. An in-depth analysis is aimed at the quadrature formulas used for the numerical evaluation of the singular integrals of the matrix resulting from the discretization using boundary elements. Particular shape functions to be used in the mesh elements that contain the fracture fronts are also analyzed in order to evaluate, through a closed formula, the Stress Intensity Factors. A numerical study on the classical fracture problem in a linear elastic medium shows the correctness and efficiency of these implemented techniques. Part of the thesis is dedicated to study stability estimates: theoretical results obtained starting from a problem involving a simpler equation are shown, both when the fronts are considered fixed and when they can move. The last part of the thesis is devoted to show the numerical results of the coupled problem: it is shown how the crack opening evolves over time and how the pressure intensity varies. A study is conducted on the convergence of the energy norm and L2 norm of the fracture opening. Finally, possible future developments and open problems are presented.