Inversion of Geothermal Reservoir Models Using the Adjoint Method and Randomized Low-Rank Matrix Approximation Algorithms

Abstract

Solving the inverse problem of finding suitable parameters for a numerical model describing a high-enthalpy geothermal system is time-consuming, as it involves running numerous nonlinear simulations until model outputs match observations. In geothermal modelling the most popular inversion algorithms are basic implementations of the Levenberg-Marquardt method, which use derivatives of model outputs to improve model parameters iteratively. However, their computational cost scales poorly with the number of parameters Nm, since they use at least Nm Å1 nonlinear simulations and finite differencing to estimate model derivatives. Choosing Nm large is, therefore, impractical with these methods, which limits their flexibility. For reducing the computational cost of inverting highly-parameterized geothermal models, this study considered adjoint and direct methods, as well as randomized matrix algorithms. As demonstrated, the adjoint and direct methods can, by themselves, reduce the cost of standard derivative-based methods for inverting geothermal natural-state and production models. This is especially the case for natural-state models since they require solving expensive transient to steady-state simulations but evaluating model derivatives with the adjoint or direct methods only requires solving linear problems for the final natural-state time-step. For improved efficiency of highly-parameterized inversions, we proposed combining randomized low-rank matrix algorithms with adjoint and direct methods to generate approximate Levenberg-Marquardt updates. As discussed, the proposed randomized Levenberg-Marquardt variants are suited to high performance parallel computing as they do not require serial iterations to generate model updates and can, therefore, outperform previously proposed approximate Levenberg-Marquardt methods, which use iterative methods. To improve the accuracy and performance of the proposed inversion methods we developed new randomized low-rank matrix algorithms which improve on previous algorithms. These improved algorithms include generalized subspace iteration and block Krylov algorithms, and single-pass streaming algorithms. These algorithms can be applied to a wide range of problems which involve forming low-rank matrices. Finally, we advised and demonstrated how these new randomized low-rank algorithms can be used to accelerate inversion and uncertainty quantification of highly-parameterized geothermal models. The results suggest that the proposed randomized methods are useful for inversion of geothermal models, and this may also be the case for other inverse problems.