Inversion of Lattice Network Structure Subjected to Carbonate Mercury Intrusion Capillary Pressure: Hamiltonian Monte Carlo Posterior Sampling

Abstract

The wide range of pore sizes in carbonates (differences of three to five orders of magnitude) aggravates the already difficult problem of multiple minima in pore network model (PNM) parameters. In this paper, we propose a method to estimate the PNM structural parameters. We have applied the method to four cases: one synthetic case and three carbonate rock samples. The PNM structural parameter estimates were conditioned to the mercury intrusion capillary pressure (MICP) via a stochastic inversion algorithm that uses molecular dynamics combined with stochastic steps using Hamiltonian mechanics. This algorithm, Hamiltonian Monte Carlo (HMC), allows for large moves in the phase space making use of the periodicity developed due to the Hamiltonian formulation, thus large modifications in the model parameters are possible. Additionally, we introduced a new estimator of the pore-size distribution using MICP. The statistics of the stochastic inversion found the multiple local minima and the global minimum of the misfit function in a difficult case. The results for the synthetic case showed that the volume and fraction of small pores in the pore-size distribution can lead to multiple local minima. For the three carbonate rock samples, the model parameters clearly showed that the volume exponents are greater than zero, which contradicts the usual assumption and inferences made by other methods. As the pore-size distribution becomes multi-modal, the volume exponent decreases but is greater than zero. These results suggest that when microporosity is ignored the volume exponent will systematically tend to be strongly underestimated (values close to zero). Our findings demonstrate that lattice-type PNM parameter estimation is a difficult problem because of multiple minima. The HMC method was able to escape from those multiple minima, resulting in better estimates of the PNM parameters.