Knut-Andreas Lie

Open-source MATLAB implementation of consistent discretisations on complex grids

Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains add-on modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

Reexamining CO2 Storage Capacity and Utilization of the Utsira Formation

Presented at: ECMOR XIV - 14th European conference on the mathematics of oil recovery, Catania, Italy, 8.-11. September 2014

Well-posedness of the Single-cell Transport Problem for Two-phase Flow with Polymer

A sequential splitting of the pressure and transport equations applied to a compressible two-phase flow with polymer leads to a considerable speed-up of the simulation, see [1]. To avoid excessive limitation on the size of the time step, the transport equations are solved implicitly. By using an iterative transport solver, in which the transport equations are solved cell-by-cell from upstream we can further decrease the computation time significantly. Such approach requires a robust singlecell transport solver. The single-cell problem consists of computing the saturation and the polymer concentration in a cell, given the total flux, the saturation, and the polymer concentration in the neighbouring cells. We derive a splitting and a discretization of the mass-conservation equations for which the single-cell problem is always well defined - for any time step size. We are now able to handle the compressible case, which requires a careful choice of the pressure equation and the segregation case, which requires to use a mixed upwind/downwind evaluation of the polymer concentration in the computation of the numerical flux.

Impact of Top-reservoir Morphology on CO2 Sequestration

Models used for evaluating CO2 plume behaviour in the subsurface often employ simplified geological reservoir descriptions. Experiences from the petroleum industry show, however, that geological heterogeneities significantly influence fluid flow. The present study addresses the need for evaluating the impact of realistic geology on CO2 behaviour in the subsurface. We here demonstrate the effect of adding realistic complexity to the top reservoir morphology. A sensitivity matrix consisting of combinations of depositional and structural irregularities creating relief along the top of a reservoir was set up and the resulting models run in a fluid flow simulator monitoring CO2 plume dynamics. Results demonstrate the interaction between specific geological features and resulting plume behaviour and added retention capacity. Our study highlights the need to include realistic geology in models forecasting migration behaviour in subsurface reservoir.

Accurate Discretization of Vertically-averaged Models of CO2 Plume Migration

When CO2 is injected into a deep formation, it will migrate as a plume that moves progressively higher in the formation, displacing the resident brine. The invasion front is driven by gravity, and the upward movement of the plume is limited by a low-permeable caprock. Several authors have recently proposed to make a sharp-interface assumption and only describe the plume migration in a vertically-averaged sense. For inhomogeneous permeability, the plume migration is then described by a system of conservation laws with spatially discontinuous flux. If one disregards dissolution and residual trapping, the system reduces to a scalar conservation law with a spatially dependent flux function, which may exhibit different solutions depending on the entropy condition that is enforced to pick a unique solution. We propose a certain set of assumptions that lead to the so-called minimum-jump condition and derive the corresponding solutions to the Riemann problem. Solutions to this problem are fundamental when developing accurate Godunov-type schemes. Here, we take a slightly different approach and present an unconditionally stable front-tracking method, which is optimal for this type of problem. Moreover, we verify the well-known observation that a standard upstream mobility discretization can give wrong solutions in certain cases.