Parameter estimation for the coupled Biot-Lamé-Navier problem in subsurface modeling

We present a parameter estimation method for a quasi-stationary multiphysics problem in subsurface modeling. Specifically, the modeling part for the state equation is based on the coupled Biot-Lamé-Navier system in order to describe the interaction between some pay-zone and a non-overlapping non-pay regime. The coupled system for the state is formulated within a variational monolithically-coupled framework. We propose an iterative stochastic ensemble method (ISEM) to estimate material coefficients such as permeability coefficients in the pay-zone or Lamé parameters in the non-pay zone. ISEM is based on stochastic estimation of gradients using an ensemble of directional derivatives within a Gauss-Newton iteration. The resulting update equation resembles the update step in ensemble Kalman filter. However, the inverse of the output covariance matrix in the update equation is regularized using standard truncated singular value decomposition. The proposed algorithm treats the forward simulator as a blackbox and avoids explicit derivation of the adjoint equations, which is major task for nonstationary systems. Our forward formulation and parameter estimation method are validated by some numerical tests including an extension of Mandel's problem.