Accelerating the Solution of Geothermal Inverse Problems Using Adjoint Methods in Waiwera: a Case Studies on Kerinci and Wairakei

Abstract

Computer modelling of geothermal fields plays a key role in the management of geothermal reservoirs. The calibration of geothermal simulation models involves inferring the permeability structure of the system and the location of the deep upflow of heat and mass using noisy temperature, pressure and enthalpy measurements from wells. Mathematically, model calibration is an ill-posed inverse problem. Solving such inverse problems typically involve minimising an objective or cost function that combines a measure of the fit of the model results to the available data, and a regularization term to overcome the ill-posedness or instability of the problem. Given sufficient computational resources, inverse problems can thus be formulated and solved as general constrained optimisation problems. These methods can require a large number of simulator runs, however, and large geothermal simulation models are typically computationally expensive. Adjoint methods provide a method of speeding up the solution of inverse problems, provided model Jacobian information and other analytical model derivatives are available. Adjoint methods can use this information to provide efficient evaluation of both first derivatives and second derivatives of the cost function. The gradient of the cost function can, for instance, be determined by running one nonlinear forward model simulation and one additional linear adjoint solve. Here we discuss an application of the adjoint method to the calibration of a model of the Wairakei geothermal field constructed in the new simulator Waiwera. Waiwera is based on an improved implementation of the same basic algorithm as (AU)TOUGH2, with a fully parallel code and improved solvers. Most importantly for implementing the adjoint method, Waiwera also provides easy access to the model Jacobian.