Data-driven modelling with coarse-grid network models

We propose to use a conventional simulator, formulated on the topology of a coarse volumetric 3D grid, as a data-driven network model that seeks to reproduce observed and predict future well responses. The conceptual difference from standard history matching is that the tunable network parameters are calibrated freely without regard to the physical interpretation of their calibrated values. The simplest version uses a minimal rectilinear mesh covering the assumed map outline and base/top surface of the reservoir. The resulting CGNet models fit immediately in any standard simulator and are very fast to evaluate because of the low cell count. We show that surprisingly accurate network models can be developed using grids with a few tens or hundreds of cells. Compared with similar interwell network models (e.g., Ren et al., 2019, 10.2118/193855-MS), a typical CGNet model has fewer computational cells but a richer connection graph and more tunable parameters. In our experience, CGNet models therefore calibrate better and are simpler to set up to reflect known fluid contacts, etc. For cases with poor vertical connection or internal fluid contacts, it is advantageous if the model has several horizontal layers in the network topology. We also show that starting with a good ballpark estimate of the reservoir volume is a precursor to a good calibration.publishedVersio

Full Approximation Scheme for Reservoir Simulation

Simulation of multiphase flow and transport in porous rock formations give rise to large systems of strongly coupled nonlinear equations. Solving these equations is computationally challenging because of orders of magnitude local variations in parameters, mixed hyperbolic-elliptic governing equations, grids with high aspect ratios and strong coupling between local and global flow effects. The state-of-the-art solution approach is to use a Newton-type solver with a algebraic multigrid preconditioner for the elliptic part of a linearized system. Herein, we discuss the use and implementation of a Full Approximation Scheme (FAS) in which algebraic multigrid is applied on a nonlinear level. By use of this method, global and semi-global nonlinearities can be resolved on the appropriate coarse scale. Improved nonlinear convergence is demonstrated on standard benchmark cases from the petroleum literature. The method is implemented in the solver framework of the open-source Matlab Reservoir Simulation Toolbox (MRST).&nbsp

GAWPS: A MRST-based module for wellbore profiling and graphical analysis of flow units

Several graphical methods have been developed to understand the stratigraphy observed in wells and assist experts in estimating rock quality, defining limits for barriers, baffles, and speed zones, and in particular, delineating hydraulic flow units. At present, there exists no computational tool that bundles the main graphical methods used for defining flow units. This paper introduces an add-on module to the MATLAB Reservoir Simulation Toolbox that contains computational routines to carry out such graphical analyses, both qualitatively and quantitatively. We also describe a new secondary method defined as the derivative of the stratigraphic modified Lorenz plot, which we use to classify depth ranges within the reservoir into barriers, strong baffles, weak baffles, and normal units, based on flow unit speed over those depths. We demonstrate the capabilities of the “Graphical Analysis for Well Placement Strategy” module by applying it to several case studies of both real and synthetic reservoirs.Cited as: Oliveira, G. P., Rodrigues, T. N. E., Lie, K.-A. GAWPS: A MRST-based module for wellbore profiling and graphical analysis of flow units. Advances in Geo-Energy Research, 2021, 6(1): 38-53. https://doi.org/10.46690/ager.2022.01.0

Efficient Reordered Nonlinear Gauss-Seidel Solvers With Higher Order For Black-Oil Models

The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-condi\-tioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (\textit{i}) a sequential formulation that decouples flow and transport into separate subproblems, and (\textit{ii}) a highly efficient Gauss--Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process. This method has been successfully applied to real-field problems using the standard first-order finite volume discretization. Here, we extend the idea to first-order dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.Comment: Comput Geosci (2019

An implicit local time-stepping method based on cell reordering for multiphase flow in porous media

We discuss how to introduce local time-step refinements in a sequential implicit method for multiphase flow in porous media. Our approach relies heavily on causality-based optimal ordering, which implies that cells can be ordered according to total fluxes after the pressure field has been computed, leaving the transport problem as a sequence of ordinary differential equations, which can be solved cell-by-cell or block-by-block. The method is suitable for arbitrary local time steps and grids, is mass-conservative, and reduces to the standard implicit upwind finite-volume method in the case of equal time steps in adjacent cells. The method is validated by a series of numerical simulations. We discuss various strategies for selecting local time steps and demonstrate the efficiency of the method and several of these strategies by through a series of numerical examples.publishedVersio

Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies

Authors final draft post-refereeing.Many numerical methods for systems of convection-diffusion equations are based upon an operator splitting formulation, where convective and diffusive forces are accounted for in separate substeps. We describe the nonlinear mechanism of the splitting error in such numerical methods in the onedimensional case, a mechanism that is intimately linked to the local linearizations introduced implicitly in the (hyperbolic) convection steps by the use of an entropy condition. For convection-dominated flows, we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts, unless the splitting step is of the same magnitude as the diffusion scale. To compensate for the potentially damaging splitting error, we propose a corrected operator splitting (COS) method for general systems of convection-diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces. In particular, COS produces viscous shocks with correct structure also when the splitting step is large. A front tracking method for systems of conservation laws, which in turn relies heavily on a Riemann solver, constitutes an important part of our COS strategy. The proposed COS method is successfully applied to a system modelling two-phase, multicomponent flow in porous media and a triangular system modelling three-phase flow