M. Panzeri

Integration of Markov mesh models and data assimilation techniques in complex reservoirs

We present a methodology that allows conditioning the spatial distribution of geological and petrophysical properties of reservoir model realizations on available production data. The approach is fully consistent with modern concepts depicting natural reservoirs as composite media where the distribution of both lithological units (or facies) and associated attributes are modeled as stochastic processes of space. We represent the uncertain spatial distribution of the facies through a Markov mesh (MM) model, which allows describing complex and detailed facies geometries in a rigorous Bayesian framework. The latter is then embedded within a history matching workflow based on an iterative form of the ensemble Kalman filter (EnKF). We test the proposed methodology by way of a synthetic study characterized by the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm and its ability with respect to the standard EnKF to properly estimate model parameters and assess future reservoir production. We show the feasibility of integrating MM in a data assimilation scheme. Our methodology is conducive to a set of updated model realizations characterized by a realistic spatial distribution of facies and their log permeabilities. Model realizations updated through our proposed algorithm correctly capture the production dynamics.

Integration of Markov Mesh Models and Ensemble Data Assimilation in Reservoirs with Complex Geology

We present a methodology conducive to updating both facies and petrophysical properties of a reservoir models set characterized by a complex architecture within the context of a history matching procedure based on the Ensemble Kalman Filter (EnKF). Spatial distribution of facies is handled by means of a Markov Mesh (MM) model. The latter is adopted because of its ability to reproduce detailed facies geometries and spatial patterns and can be integrated in a consistent probabilistic Bayesian framework. The MM model is then integrated into a history matching procedure which is based on the EnKF scheme. We test the proposed methodology by means of a synthetic reservoir in the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm with respect to the standard EnKF both in terms of history matching quality and forecast prediction capabilities. The results show that the integration of MM in the data assimilation scheme allows obtaining realistic geological shapes for spatial facies distribution and an improved estimation of petrophysical properties. In addition, the updated ensemble correctly captures the production range in the long term.

Uncertainty Quantification in Scale‐Dependent Models of Flow in Porous Media

Equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale-dependent. In the Moment Equations (ME) method, exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations. Computable approximations of the MEs can be derived via perturbation expansion in orders of the standard deviation of the random model parameters. As such, their convergence is guaranteed only for standard deviation smaller than one. Here we consider steady-state saturated flow in a porous medium with random second-order stationary conductivity field. We show it is possible to identify a support scale η*, where the typically employed approximate formulations of ME yield accurate (statistical) moments of a target state variable. Therefore, at support scale η* and larger, ME presents an attractive alternative to slowly convergent Monte Carlo (MC) methods whenever lead-order statistical moments of a target state variable are needed. We also demonstrate that a surrogate model for statistical moments could be constructed from MC simulations at larger support scales and be used to accurately estimate moments at smaller scales, where MC simulations are expensive and the ME method is not applicable.

Ensemble Kalman Filter Assimilation of Transient Groundwater Flow Data: Stochastic Moment Solution Versus Traditional Monte Carlo Approach

The ensemble Kalman filter (EnKF) allows assimilating newly available data in transient groundwater and other earth system models through real-time Bayesian updating of system states (e.g., hydraulic heads) and parameters (e.g., hydraulic conductivities). Assimilating data in groundwater transient stochastic flow equations via the traditional EnKF entails computationally intensive Monte Carlo (MC) simulations. Previously we proposed a way to circumvent the need for MC through (1) an approximate direct solution of nonlocal (integrodifferential) equations that govern the space-time evolution of conditional ensemble means (statistical expectations) and covariances of hydraulic heads and fluxes and (2) the embedding of these moments in EnKF. Here we compare the accuracies and computational efficiencies of our newly proposed EnKF approach based on stochastic moment equations and the traditional Monte Carlo approach.