Optimal Well-Placement Using Probabilistic Learning

Abstract

A new method based on manifold sampling is presented for formulating and solving the optimal well-placement problem in an uncertain reservoir. The method addresses the compounded computational challenge associated with statistical sampling at each iteration of the optimization process. An estimation of the joint probability density function between well locations and production levels is achieved using a small number of expensive function calls to a reservoir simulator. Additional realizations of production levels, conditioned on well locations and required for evaluating the probabilistic objective function, are then obtained by sampling this jpdf without recourse to the reservoir simulator.

Physics‐constrained non‐Gaussian probabilistic learning on manifolds

“…The PLoM allows for generating additional realizations for M ≫ N without using the computational model, but using only the training data set, which we denote by D N . These additional realizations allow, for instance, a cost function to be evaluated, in which is a given measurable real‐valued mapping on as well as constraints related to a nonconvex optimization problem and this, without calling the mathematical/computational model.…”

“…The PLoM allows for generating additional realizations {(q ar , w ar ), = 1, … , M} for M ≫ N without using the computational model, but using only the training data set, which we denote by D N . These additional realizations allow, for instance, a cost function J(w) = E{ (Q, W)|W = w} to be evaluated, in which (q, w)  →  (q, w) is a given measurable real-valued mapping on ℝ n q × ℝ n w as well as constraints related to a nonconvex optimization problem 39,41,42 and this, without calling the mathematical/computational model. Sometimes, additional information in the form of statistics may become available, synthesized from partial measurements, published data, or numerical simulations.…”

“…This paper is devoted to this framework: learning from a high‐dimensional small data set, without invoking the Gaussian assumption. The first ingredient of the proposed approach is the probabilistic learning on manifolds (PLoM) presented in the work of Soize and Ghanem, for which complementary developments and applications with validation are presented in other works. This PLoM allows for constructing a generated data set made up of M additional independent realizations of a non‐Gaussian random vector X , defined on a probability space , with values in , for which the probability distribution P X ( d x ) is unknown, using only an initial data set D N (training set), which is a small data set, made up of N independent realizations { x j , j =1,…, N } of X with x j = X ( θ j ) for θ j ∈Θ and such that M ≫ N .…”

Design optimization of a scramjet under uncertainty using probabilistic learning on manifolds

“…For d/8 and d/16, the 28 observables shown in Tables (1), (2), (4), and (5) are included in the manifold detection process. For the case d/32, given the small number of training samples (maximum of 23), only 9 observables were included in the analysis, consisting of the 4 QoIs and 5 controls, shown in Tables (4) and (5), respectively. The value of ν, representing the dimension of the decorrelated observations is also shown in table (3).…”

“…One rational formalism for accounting for these assumptions throughout the design process is to explore the robustness of an optimal solution to perturbations in these assumptions. Probabilistic modeling provides an effective procedure for characterizing these assumptions and has typically been implemented through either parametric procedures where model parameters are described as random variables [1,2,3,4,5] or within a non-parametric framework where the model itself is described as a random operator [6]. Irrespective of how a non-deterministic problem is implemented, it requires the numerical exploration of a statistical ensemble, thus quickly exacerbating the computational burden of an already massive exercise.…”