Improving the computational efficiency of approximate gradients using a multiscale reservoir simulation framework

We present an efficient workflow that combines multiscale (MS) forward simulation and stochastic gradient
computation - MS-StoSAG - for the optimization of well controls applied to waterflooding under geological uncertainty.
The Iterative Multiscale Finite Volume (i-MSFV), a mass conservative reservoir simulation strategy, is
employed as the forward simulation strategy. The i-MSFV method has the ability to accurately capture fine scale
heterogeneities, and thus the fine scale physics of the problem. This allows for accurate and efficient simulations
with fine-scale error estimates in a more computationally efficient coarse-scale simulation grid. The Stochastic
Simplex Approximate Gradient (StoSAG) method, a stochastic gradient technique, inspired from Ensemble
Optimization (EnOpt), is employed to compute the gradient of the objective function. Stochastic methods have
recently been shown to be very efficient in terms of gradient quality and computational cost especially for robust
optimization problems. They rely on the reservoir simulator as a black-box and perform a large number of
forward simulations to approximate a gradient. In our numerical experiments, we investigate the impact of MS
parameters such as coarsening ratio and heterogeneity contrast on the quality of the approximate MS-StoSAG
gradient. Our experiments illustrate that i-MSFV simulations provide accurate forward simulation responses for
the gradient computation, with the advantage of speeding up the workflow due to faster simulations. Speedups up
to a factor of five on the forward simulation were achieved for the examples considered in the paper. The combination
of speed and accuracy of MS forward simulation with the flexibility of the StoSAG technique (several
different parameters/responses can be easily considered) allows for a flexible and efficient optimization strategy
suitable for large-scale problems