Modelling Thousands of Fractures Using Analytic Elements

Understanding and modelling hydraulic fractures and fracture networks have a fundamental role in mapping the mechanical behaviour of rocks. A problem arises in the discontinuous behaviour of the fractures and how to accurately and efficiently model this. We present a novel approach for modelling many cracks randomly using analytic elements placed under plane strain conditions in an elastic medium. The analytic elements allow us to model the assembly computationally efficiently and up to machine precision. The crack element is the first step in the development of a model suitable for investigating the effect of fissures on tunnels in rock. The model can be used to validate numerical models and more.The solution for a single hydraulic pressurized crack in an infinite domain in plane strain was initially developed by Griffith (1921). We demonstrate that it is possible, by using series expansions in terms of complex variables, based on the Muskhelisvili-Kolosov functions, to generalize this solution to the case of an assembly of non-intersecting pressurized cracks. The solution consists of infinite series for each element Strack & Toller (2020). The expressions for the displacements and stress tensor components approach the exact solution, as the number of terms in the series approaches infinity.We present the case where two cracks approach each other orthogonally to less than 1/2000th of the cracks length. We show the effect of increasing the number of terms in the expansion and how this influences the precision, demonstrating that the result approaches the exact solution. We also present a case with 10,000 cracks; the coefficients are determined using an iterative solver. By using analytic elements, we can both present the corresponding stress and deformations field for the global scale and for small scales in the close proximity of individual cracks.ReferencesGriffith, A. A. (1921). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 221(582-593):163–198.Strack, O. D. L. and Toller, E. A. L. (2020). An analytic element model for highly fractured elastic media, manuscript submitted for publication in International Journal for Numerical and Analytical Methods in Geomechanics.