Papers »

Universal meshes for the simulation of hydraulic fracture

We describe our approach to simulating hydraulic fractures in two-dimensions, based on the use of Universal Meshes. The non-linear coupling between the fluid pressure and the crack opening, and the presence of two moving boundaries, the crack tip and the fluid front, present a challenge to simulations. Often in simulating hydraulic fracture, a displacement discontinuity approach is adopted. While this approach has been highly successful for simplified cases, there would be great difficulty in extending it to more general problems, for example when it is desired to model the effects of poroelasticity or the intersection of hydraulic fractures with pre-existing natural fractures. Finite-element-based approaches are attractive in the simulation of hydraulic fractures because of their ability to easily
handle inhomogeneities in the material and more general geometries.

Constructing finite-element-based approximations for hydraulic fracture problems faces a crucial obstacle though: a suitable mesh is needed over the faces of a possibly-curved-crack to solve for the pressure distribution in the fracturing fluid. Since the crack itself is part of the solution, it is not possible to a priori know where the crack will be and hence where to construct such mesh. Standard solutions for crack propagation, such as cutting elements as in the extended finite element method, lead to very irregular meshes over the crack surfaces not suitable for computation. Such meshes can lead to accuracy and conditioning problems.

To this end, we have introduced the idea of a Universal Mesh. A Universal Mesh is one that can be used to mesh a class of geometries by slightly perturbing some nodes in the mesh, and hence the name universal. In this way, as the crack evolves, the Universal Mesh is always deformed so as to exactly mesh the crack surface. The advantages of such an approach are: (a) the crack faces are exactly meshed with a conforming mesh at all times, and the quality of the surface mesh is guaranteed to be good, (b) apart from duplicating degrees of freedom when the crack grows, the connectivity of the mesh and the sparsity of the associated stiffness matrix remains unaltered; this has the positive effect of enabling efficient iteration over the crack geometry, needed for the satisfaction of Griffith's criterion at the crack tip.


This is joint work with: Ramsharan Rangarajan, Michael Hunsweck, Yongxing Shen, Maurizio Chiaramonte, and Leon Keer.

Author(s):

Adrian Lew    
Stanford University
United States