A Bayesian framework for uncertainty quantification of different reservoir scales with a focus on advanced and enhanced oil recovery processes
Keywords: Energy Resources Engineering.

Almost any informed decision in the oil and gas industry depends on the forecasts of unknown quantities. The challenging part of forecasting is that forecasts are subject to uncertainties that originate from our lack of knowledge of parameters of the subsurface system. Because those parameters are an essential input for building subsurface models that are in turn input to simulators that forecast different quantities such as oil rate, it is important to have a good estimate of such parameters. Engineers rely on measured quantities like field oil rate and pressure to estimate these parameters. The process of using observations to estimate the causal factors is named an inverse problem and is challenging because it is not invertible, non-unique, and often ill-posed. The process of building a full field reservoir model involves a series of inversion processes beginning at the core scale and ending at the field scale. Because the result of each inversion process on a specific scale becomes the input for the subsequent inversion process, uncertainty propagates through scales. Every physical process in each scale is complex in itself, but the interaction of all of them is immensely difficult for the human mind to process. Quantifying and tracking uncertainties through scales is central to incorporating the effect of all uncertainties on the final decision. Traditionally, a deterministic approach is followed where parameters of a single model are modified until the simulation response is sufficiently close to the observed data. The disadvantage of this approach is that the resulting 'history matched' model is unrealistic and consequently unreliable because the applied ad hoc modifications, often, are physically inconsistent. Because the deterministic approach results in a single model, we are not able to compute any statistics to quantify uncertainty. In this dissertation we build and expand on two probabilistic approaches: the Bayesian Causal Analysis (BCA) for all scales where we compute parameter posteriors and the Bayesian Evidential Learning (BEL) protocol to quantify the uncertainty of the decision variable on the field scale. Both BCA and BEL deliver a stochastic perspective to the problem and consequently do not aim to find a single model that best matches the observed data but rather aim to identify a set of models in the vicinity of the observed data. Both approaches build a statistical relationship; in the case of BCA, between the parameters of the model and the desired data variable (output of the simulator such as oil rate versus time) and in the case of BEL between the data variable and forecast variable of interest. The statistical relationship in combination with measured data enables us to predict directly the posterior of the model parameters (BCA) or the posterior of the prediction variable (BEL) that enables us to estimate reliably the associated uncertainty. We begin by quantifying the uncertainty of a core-scale experiment from a lab experiment. Even though parameters believed to be relevant for recovery cannot be directly observed or measured in the lab, we can use the physical response measured in the lab to infer these parameters. In the case of relative permeability, we can measure the oil rate over time for a core flood and use it to infer relative permeability curves. For some reservoirs, parameterized relative permeability models are commonly used, but in other cases with advanced physical processes these models do not suffice due to their complex shape. We quantify the uncertainty of relative permeability curves that originate from an experiment that investigated a voidage-replacement-ratio of less than unity and often exhibits irregular shapes that are difficult to parameterize. We use a novel formulation of the gradual deformation algorithm to compute multiple realistic relative permeability curves that describe the measured lab data. This workflow enables the engineer to quantify the uncertainty of the relative permeabilities that can then be used as an input for the next scale. In other cases where experiments are performed to enhance the understanding of the underlying processes, a posterior distribution of parameters can reduce the risk of misinterpretation because the entire possible range of a parameter is known to the scientist. Next, we give an example of uncertainty quantification of the well scale. Produced water always contains suspended particles. If re-used for injection this water can potentially create a fracture by plugging the formation resulting in the injection pressure exceeding the breakdown pressure. This fracture will continue to grow as we inject water and potentially bypass oil by short circuiting injector/producer pairs. We first introduce a novel definition for risk for the case of a fractured water injector to answer the question: How much water can we inject and not impair oil production? Second, we explore two avenues, namely an optimization and a machine learning technique, that use observed data to reduce previously defined risk and derive posterior probability density functions for all model parameters. Both the optimization as well as the machine learning approach reduce the risk and are able to reduce the uncertainty for all model parameters. We also outline the differences of both strategies and show that the machine learning approach reduces computational time up to 70% and increases the consistency of the parameter posteriors. The posterior parameters of this chapter including the Young's modulus, the minimum horizontal stress, and the Poisson ratio describing the geomechanical model of the subsurface are used as critical input for the reservoir scale. In practice this workflow enables an engineer to propose an injection rate for a newly drilled injector given the risk attitude with the developed risk plot. Furthermore, the workflow helps and directs the engineer to select an appropriate signal, such as the bottom hole pressure, that reduces the uncertainty of the risk. Finally, we reach the field scale to provide the necessary input to make well-informed decisions. On the field scale, we aim to reduce the uncertainty in forecast variables that influences our decision at hand. Contrary to preceding scales, we do not want to compute posterior model parameters on the field scale because they do not influence our decision and also do not comprise the input for a subsequent scale. For that reason, we follow the Bayesian Evidential protocol that focuses on the forecast rather than the parameters that are used to build the models. We extend the formulation of the Bayesian Evidential Learning framework to predict the incremental oil of an enhanced oil recovery project, a polymer flood, and demonstrate those capabilities in a case study. We also develop a novel extension of the Bayesian Evidential Learning protocol to compute the value of information of a polymer pilot. The combination of decision theory and the BEL protocol enables engineers both to reduce the human bias in decision making and to assess the value of a pilot under uncertainty before the information of the pilot is collected. The main advantage of this novel work-flow is the fact that we can generate a significant number of forecasts for each decision branch so that we can derive robust statistics for good decisions. Finally, we also argue and demonstrate that the integration of the uncertainty of the fiscal model can be of paramount importance in making decisions.