The Multiscale Mixed Finite-Element Method (MsMFEM)

Basic idea: Use a mixed finite-element method on a coarse scale with special basis that satisfy local flow problems and thereby account for subgrid variations. This way, an approximate fine-scale solution is constructed at the cost of solving a coarse-scale problem.

MsMFEM is formulated using two grids, a fine underlying grid on which the media properties are given, and a coarse simulation grid where each block can consists of an arbitrary connected collection of cells from the fine grid. In this sense, the method is very flexible and can be applied to almost any grid, structured or unstructured, and can easily be built on-top-of existing pressure solvers.

MsMFEM offers fast, accurate, and robust pressure solvers for highly heterogeneous porous media and can be used for

• direct simulation on high-resolution grid models with multimillion cells
• fast simulation of multiple (stochastic) realisations of reservoir heterogeneity
• model reduction to provide instant simulation of flow responses

Read more ...

Adjoint Multiscale Mixed Finite Elements

Basic idea: Combine MsMFEM with a non-uniform coarsening approach for fast approximate simulations in optimization loops using adjoint based gradients. Using initially computed multiscale basis functions and a static coarse saturation grid, the communication mappings between pressure and saturation solvers can be precomputed and allow for fast multiple simulations.

Finding e.g., optimal rates for the planning of reservoir water flooding typically requires a large number of simulations. Even though the number of states in a model may be huge, the dynamic behavior is typically of much lower order. As a result the potential for model reduction is large. Using the combined MsMFEM / flow-based coarsening approach enables

• multiple fast approximate simulations without initial tuning / snapshots from full simulations
• coarse grid communication mappings computed once prior to optimization loop
• obtaining gradients from adjoint with similar computational complexity as forward coarse model

Discretization on Complex Grids

Grid models representing geological formations are often highly complex and have unstructured connections. The most widespread gridding approaches are based on extrusion of 2D shapes (corner-point and 2.5D PEBI grids), but recently there has been an increasing focus on fully unstructured grids containing general polyhedral cells. In either case, cells can have quite irregular shapes and an (almost) arbitrary number of faces. This requires flexible discretization methods. The industry-standard method is the two-point flux approximation (TPFA) method, which may exhibit strong grid-orientation effects.

Grid-orientation effects can be drastically reduced by using a modern mixed, multipoint, or mimetic discretization method. We have worked on making mixed and mimetic methods suitable for industry-standard grids. We have developed compact implementations for general unstructured grids that can be used as a robust and accurate fine-grid solver or as a building block for a multiscale flow solver. In addition, we have developed methodology for accurate representation faults and their hydraulic properties.

Read more ... 

Causality-Based Solvers

Fast, accurate, and robust solution of advection dominated transport equations

• time-of-flight and single-phase tracer flow
• multiphase and multicomponent flow

A grid-based alternative to streamline simulation that is mass-conservative and avoids problems with mapping and choice of representative streamline distribution. Can be used for fast computation (and visualization) of flow patterns.

Basic idea: Use methods from graph theory to compute an optimal reordering that renders the (non)linear system of discretized transport equations in a lower-triangular form. A solution can then be computed by a Gauss-Seidel method, i.e., in an optimal cell-by-cell or block-by-block fashion. For nonlinear equations, one obtains local control over the nonlinear iterations, which gives orders-of-magnitude reduction in runtimes and increases the feasible time-step sizes. Combining the reordering idea with compact higher-order discontinuous Galerkin discretisations gives highly accurate solvers with very favourable scaling.

Read more..

Coarsening by Amalgamation of Cells

Provides an optimal model reduction method for transport solvers, which significantly reduces the runtime but preserves the flow structures and the accuracy of integrated responses (production curves, etc).

Basic idea: Amalgamate cells into coarse blocks based on a set of admissible and feasible directions. The feasible directions are given by a set of flow indicators and determine how the blocks adapt to the underlying flow patterns. The admissible directions determine the basic shape and regularity of the coarse blocks in the absence of flow adaption.

The resulting flow-adapted grids offer good accuracy for a low computational cost and are ideal to create efficient multi-fidelity transport solvers to accompany multiscale pressure solvers. Alternatively, one can create grids that adapt to geological features such as facies, rock types, etc.

We have developed a general algorithmic framework that includes many different coarsening methods as special cases. 

Read more...

Streamline Methods

Basic idea: By solving a porous media transport equation along streamlines, we transform a three-dimensional problem to a set of one-dimensional problems. These can be solved faster and more accuratly, by unconditionally stable, highly efficient methods like front-tracking.

A streamline is a curve that is at any time tangent to the flow direction. In a typical simulation using streamlines, a sequential splitting approach is used. First, a pressure equation is solved, yielding a velocity field. Second, streamlines are traced from injectors to producers according to this velocity field. In this process, time-of-flight coordinates are computed, and finally, the transport equation is solved in the one-dimensional time-of-flight grid along each streamline.

Streamline methods are fast and efficient for problems dominated by pressure-driven flow, and usage examples include

• quick characterization of well drainage and flooding regions
• efficient transport in simulation of large, heterogenous domains
• computation of breakthrough times and water cut.

Read more...