A constrained version for the stereology inverse problem: Honoring power law and persistences of the fracture traces exposed on arbitrary surfaces

We present a stereological study in a cave setting that is part of a karstic carbonate system located in the São Francisco Craton, Brazil. Using a Lagrangian approach, a constrained version of the nonlinear inverse problem of stereology is solved. Besides the classical demand of fitting the histogram of fracture traces measured on arbitrary exposed surface, it is imposed that the solution honors also measures of surface intensity (P21) and power law exponent obtained from fracture traces on the same exposed surfaces. Estimates of volumetric intensities (P32) of conjugate fracture pairs might be also imposed to be close values. The resulting cost functional is minimized using the Particle Swarm Optimization (PSO) method. The implemented version of PSO furnishes the best solution and a set of suboptimal quasisolutions, from which the solution uncertainty is evaluated. A key aspect of the implemented approach is that all terms composing the cost functional are normalized, obtaining as a result, robustness for the weighting parameters to changes in the input data. The resulting stochastically simulated discrete fracture networks honor all statistical observations but, in general, do not reproduce the positions of the observed fracture traces. The methodology is applied to synthetic and field data examples. All obtained solutions are stable and geologically reliable.