Vegard Kippe

Modelling of Multiscale Structures in Flow Simulations for Petroleum Reservoirs

Flow in petroleum reservoirs occurs on a wide variety of physical scales. This poses a continuing challenge to modelling and simulation of reservoirs since fine-scale effects often have a profound impact on flow patterns on larger scales. Resolving all pertinent scales and their interaction is therefore imperative to give reliable qualitative and quantitative simulation results. To overcome the problem of multiple scales it is customary to use some kind of upscaling or homogenisation procedure, in which the reservoir properties are represented by some kind of averaged properties and the flow is solved on a coarse grid. Unfortunately, most upscaling techniques give reliable results only for a limited range of flow scenarios. Increased demands for reservoir simulation studies have therefore led researchers to develop more rigorous multiscale methods that incorporate subscale effects more directly.

Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels

The topic of this thesis is streamline-based integration of dynamic data for porous media systems, particularly in petroleum reservoirs. In the petroleum industry the integration of dynamic data is usually referred to as history matching. The thesis starts out by giving an introduction to streamline-based history-matching methods. Implementations and extensions of two existing methods for streamline-based history matching are then presented.The first method pursued is based on obtaining modifications for streamline-effective properties, which subsequently are propagated to the underlying simulation grid for further iterations. For this method, two improvements are proposed to the original existing method. First, the improved approach involves less approximations, enables matching of porosity, and can account for gravity. Second, a multiscale approach is applied for which the data integration is performed on a hierarchy of coarsened grids. The approach proved robust, and gave a faster and better match to the data.The second method pursued is the so-called generalized travel-time inversion (GTTI) method, which earlier has proven very robust and efficient for history matching. The key to the efficiency of this method is the quasilinear convergence properties and the use of analytic streamline-based sensitivity coefficients. GTTI is applied together with an efficient multiscale-streamline simulator, where the pressure solver is based on a multiscale mixed finite-element method (MsMFEM). To make the history matching more efficient, a selective work-reduction strategy, based on the sensitivities provided by the inversion method, is proposed for the pressure solver. In addition, a method for improved mass conservation in streamline simulation is applied, which requires much fewer streamlines to obtain accurate production-response curves. For a reservoir model with more than one million grid blocks, 69 producers and 32 injectors, the data integration took less than twenty minutes on a standard desktop computer. Finally, we propose an extension of GTTI to fully unstructured grids, where we in particular address issues regarding regularization and computation of sensitivities on unstructured grids with large differences in cell sizes.

A Method To Improve the Mass Balance in Streamline Methods

During the last decades, streamline methods have emerged as highly efficient simulation tools that are well-suited for e.g., history matching and simulation of large and complex reservoir models. Streamline methods are based on a sequential solution procedure in which pressure and fluid velocities are computed by solving a pressure equation on a grid in physical space and the fluid transport is computed by solving 1-D transport problems along streamlines. The sequential Eulerian-Lagrangian procedure is the key to the high computational efficiency of streamline methods. On the other hand, it necessitates mapping of saturations (or fluid compositions) back and forth between the Eulerian pressure grid and the Lagrangian streamlines. Unfortunately, this introduces mass-balance errors that may accumulate in time and in turn yield significant errors in production curves. Mass-balance errors might be reduced by considering higher-order mapping algorithms, or by increasing the number of streamlines. Since the computational speed scales linearly with the number of streamlines, it is clearly desirable to use as few streamlines as possible. Here we propose a modification of the standard mapping algorithm that: (i) improves the mass-conservation properties of the method and (ii) provides high-accuracy production curves using few streamlines. Mass conservation is improved by changing quantities in the transport equation locally, and we show that these modifications do not significantly affect the global saturation errors as long as a sufficient number of streamlines is used. Moreover, we propose an adaptive strategy for ensuring adequate streamline coverage. The efficiency and accuracy of the modified streamline method is demonstrated for Model 2 from the Tenth SPE Comparative Solution Project. Highly accurate production curves (compared to reference solutions) are obtained in less than ten minutes using one processor on a standard (Intel Core 2 Duo) desktop computer.

Multiscale Methods for Subsurface Flow

“There is a growing recognition that the world faces a water crisis that, left unchecked, will derail the progress towards the Millennium Development Goals and hold back human development. Some 1.4 billion people live in river basins in which water use exceeds recharge rates. The symptoms of overuse are disturbingly clear: rivers are drying up, groundwater tables are falling and water-based ecosystems are being rapidly degraded. Put bluntly, the world is running down one of its most precious natural resources and running up an unsustainable ecological debt that will be inherited by future generations.”

Multiscale Methods and Streamline Simulation for Rapid Reservoir Performance Prediction

We introduce a novel multiscale approach for reservoir simulation as an alternative to industry-standard upscaling methods. In our approach, reservoir pressure and total velocity is computed separately from the fluid transport. Pressure is computed on a coarse grid using a multiscale mixed-finite element method that gives a mass-conserving velocities on a fine subgrid. The fluid transport is computed using streamlines on the underlying fine geogrid.