Data-space approaches for efficient uncertainty quantification in subsurface flow problems
Keywords: Energy Resources Engineering.

Uncertainty quantification for subsurface flow problems is typically accomplished through the use of model inversion procedures in which multiple posterior (history-matched) geological models are generated and used for flow predictions. These procedures can be demanding computationally, and it is not always straightforward to maintain geological realism in the resulting history-matched models. In some applications, it is the flow predictions themselves (and the uncertainty associated with these predictions), rather than the posterior geological models, that are of primary interest. This is the motivation for the data-space inversion (DSI) procedures developed in this work. In the DSI framework, an ensemble of prior model realizations, honoring prior geostatistical information and hard data at wells, are generated and then (flow) simulated. The resulting reservoir responses (e.g., time-series of flow rate data at wells, and/or limited spatial saturation fields) are assembled into data vectors that represent prior `realizations' in the data space. The conditional distribution of data variables given observed data is then constructed within a Bayesian framework. This distribution is directly sampled using a data-space randomized maximum likelihood method. Due to the non-Gaussian characteristics of the data variables, we introduce pattern-based mapping operations, or histogram transformation, along with principal component analysis. These treatments allow us to represent the data variables using a set of low-dimensional variables that are closer to multivariate Gaussian, which is shown to improve the performance of DSI. We present extensive numerical results for two example cases involving oil-water flow in a bimodal channelized system and oil-water-gas flow in a Gaussian permeability system, in which the quantities of interest (QoI) are time-series data at wells. DSI results, with pattern-based mapping operations, for uncertainty quantification (e.g., P10, P50, P90 posterior predictions) are compared with those obtained from a strict rejection sampling (RS) procedure. Reasonable agreement between the DSI and RS results is consistently achieved, even when the (synthetic) true data to be matched fall near the edge of the prior distribution. Computational savings using DSI are very substantial in that RS requires O(10^5--10^6) flow simulations, in contrast to 500 for DSI, for the cases considered. We then apply the DSI procedure, with the histogram transformation treatment for data reparameterization, for naturally fractured reservoirs (NFRs), represented as general discrete-fracture-matrix (DFM) models. This DSI procedure is first tested on two-dimensional DFM systems involving multiple fracture scenarios. Comparison with an approximate rejection sampling procedure for this case indicates the DSI results for the P10, P50 and P90 responses are again consistent with RS results. The DSI method is then applied to a realistic NFR that has undergone 15 years of primary production and is under consideration for waterflooding. To construct the DSI representation, around 400 prior DFM models, which correspond to different geologic concepts and properties, are simulated. Two different reference `true' models, along with different data-assimilation durations, are considered. In all cases, the DSI predictions are shown to be consistent with the forecasts from the `true' model, and to provide reasonable quantification of forecast uncertainty. Finally, we investigate the application of DSI to quantify the uncertainty associated with carbon storage operations, in which the QoI is the spatial distribution of CO2 saturation in the top layer of a storage aquifer, and the observed data are pressure and CO2 saturation measurements from a few monitoring wells. We also introduce a procedure to optimize the locations of monitoring wells using only prior-model simulation results. This approach is based on analytical DSI results, and determines monitoring well locations such that the reduction in expected posterior variance of a relevant quantity is maximized. The new DSI procedure is applied to three-dimensional heterogeneous aquifer models involving uncertainties in a wide range of geological parameters, including variogram orientation, porosity and permeability fields, and regional pressure gradient. Multiple monitoring scenarios, involving four to eight monitoring wells, are considered in this evaluation. Application of DSI with optimal monitoring wells is shown to consistently reduce the posterior variance in predictions of the average CO2 saturation in the top layer, and to provide detailed saturation fields in reasonable correspondence with the `true' saturation distribution.