An Eulerian moving front algorithm with weak-form tip asymptotics for modeling hydraulically driven fractures

The coupled equations for a propagating hydraulic fracture (HF) exhibit a multi-scale structure for which resolution on all length scales becomes computationally prohibitive. Asymptotic analysis is able to identify the dominant physical process active at the computational length scale. This paper describes a novel algorithm that uses weak-form tip asymptotics on a rectangular Eulerian mesh to solve the problem of a propagating HF. The location of the fracture front is determined within each tip element by matching the volume associated with the known asymptotic solution to the flux of fluid into the given element. Even if the fracture front is curved, the algorithm is able to capture the solution on a relatively coarse rectangular mesh by implementing a weak form of the tip asymptotic solution, based on averaging the volume over a tip element to determine the fracture widths at tip element centers. The fracture is divided into a 'channel' region made up of elements that are filled with fluid and a 'tip' region comprising partially filled elements. An iterative procedure is used to determine the fracture widths and fluid pressures in the channel region and the fracture font locations in the tip regions. The algorithm is tested by analyzing an HF propagating in a viscosity-dominated regime within an impermeable homogeneous elastic material for which a similarity solution is available. The numerical results show close agreement with the exact solution even though a relatively coarse mesh is used.