Summary
A particularly efficient reservoir simulator can be obtained by combining a recent multiscale mixed finite-element flow solver with a streamline method for computing fluid transport. This multiscale-streamline method has shown to be a promising approach for fast flow simulations on high-resolution geologic models with multimillion grid cells. The multiscale method solves the pressure equation on a coarse grid while preserving important fine-scale details in the velocity field. Fine-scale heterogeneity is accounted for through a set of generalized, heterogeneous basis functions that are computed numerically by solving local flow problems. When included in the coarse-grid equations, the basis functions ensure that the global equations are consistent with the local properties of the underlying differential operators. The multiscale method offers a substantial gain in computation speed, without significant loss of accuracy, when basis functions are updated infrequently throughout a dynamic simulation.
In this paper, we propose to combine the multiscale-streamline method with a recent "generalized travel-time inversion" method to derive a fast and robust method for history matching high-resolution geocellular models. A key point in the new method is the use of sensitivities that are calculated analytically along streamlines with little computational overhead. The sensitivities are used in the travel-time inversion formulation to give a robust quasilinear method that typically converges in a few iterations and generally avoids much of the time-consuming trial-and-error seen in manual history matching. Moreover, the sensitivities are used to enforce basis functions to be adaptively updated only in areas with relatively large sensitivity to the production response. The sensitivity-based adaptive approach allows us to selectively update only a fraction of the total number of basis functions, which gives substantial savings in computation time for the forward flow simulations.
We demonstrate the power and utility of our approach using a simple 2D model and a highly detailed 3D geomodel. The 3D simulation model consists of more than 1,000,000 cells with 69 producing wells. Using our proposed approach, history matching over a period of 7 years is accomplished in less than 20 minutes on an ordinary workstation PC.
Introduction
It is well known that geomodels derived from static data only—such as geological, seismic, well-log, and core data—often fail to reproduce the production history. Reconciling geomodels to the dynamic response of the reservoir is critical for building reliable reservoir models. In the past few years, there have been significant developments in the area of dynamic data integration through the use of inverse modeling. Streamline methods have shown great promise in this regard (Vasco et al. 1999; Wang and Kovscek 2000; Milliken et al. 2001; He et al. 2002; Al-Harbi et al. 2005; Cheng et al. 2006). Streamline-based methods have the advantages that they are highly efficient "forward" simulators and allow production-response sensitivities to be computed analytically using a single flow simulation (Vasco et al. 1999; He et al. 2002; Al-Harbi et al. 2005; Cheng et al. 2006). Sensitivities describe the change in production responses caused by small perturbations in reservoir properties such as porosity and permeability and are a vital part of many methods for integrating dynamic data.
Even though streamline simulators provide fast forward simulation compared with a full finite-difference simulation in 3D, the forward simulation is still the most time-consuming part of the history-matching process. A streamline simulation consists of two steps that are repeated:solution of a 3D pressure equation to compute flow velocities; andsolution of 1D transport equations for evolving fluid compositions along representative sets of streamlines, followed by a mapping back to the underlying pressure grid.
The first step is referred to as the "pressure step" and is often the most time-consuming. Consequently, history matching and flow simulation are usually performed on upscaled simulation models, which imposes the need for a subsequent downscaling if the dynamic data are to be integrated in the geomodel. Upscaling and downscaling may result in loss of important fine-scale information.
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