Thomas A. Hewett

Theory for the semi-analytical calculation of oil recovery and effective relative permeabilities using streamtubes

Abstract A semi-analytical method has been developed for calculating oil recovery in two and three dimensions, and for calculating effective relative permeabilities for coarse grids. The calculations are based on the assumption that the effects of a changing mobility field can be accounted for by using fixed streamtube geometries with flowrates updated to account for the changing mobility distribution. The single-phase pressure distribution from a numerical solution of Laplaces equation is used to calculate the pressure distribution for a two-phase flow based on a mapping of the solution of the Buckley-Leverett equation onto the streamtubes derived from the single-phase solution. The displacement calculations for oil recovery are based on theory previously developed by Dykstra and Parsons, extended to include the effects of spatially varying permeability and continuously changing mobilities, as occurs in solutions of the Buckley-Leverett equation for typical values of the mobility ratio. This idea has also been extended to the calculation of effective relative permeabilities for coarse-grid simulation and finally establishes the proper rules for averaging the results of fine-grid numerical simulations of two-phase flow for the definition of effective two-phase flow properties on coarse grids. These calculations have been generalized to three-dimensional flows by the simple device of conceptually inserting a gridded plane across the flow and defining each streamtube at that location as those streamlines which pass through any one of the grid cells. When combined with time-of-flight calculations from the gridded plane to both the producer and injector, the distribution of pore volume along each streamtube can be calculated. This information, combined with a tabulation of the single-phase, steady-state pressure distribution along each streamtube, provides all of the information needed for the semi-analytical calculation of oil recovery and effective flow properties in three-dimensional flows.

Challenges in reservoir forecasting

The combination of geostatistics-based numerical geological models and finite difference flow simulation has improved our ability to predict reservoir performance. The main contribution of geostatistical modeling has been more realistic representations of reservoir heterogeneity. Our understanding of the physics of fluid flow in porous media is reasonably captured by flow simulators in common usage. Notwithstanding the increasing application and success of geostatistics and flow simulation there remain many important challenges in reservoir forecasting. This application has alerted geoscientists and physicists that geostatistical/flow models in many respects, are, engineering approximations to thereal spatial distribution andreal flow processes. This paper reviews current research directions and presents some new ideas of where reserach could be focused to improve our ability to model geological features, model flow processes, and, ultimately, improve reservoir performance predictions.

A review of current trends in petroleum reservoir description and assessment of the impacts on oil recovery

Abstract This paper gives a general review of the problems associated with realistic representation of reservoir heterogeneities and their impacts on current trends in reservoir simulation. The basic three-phase flow equations used in reservoir engineering are summarized and some of the uncertainties associated with the properties of two-phase relative permeabilities are reviewed. Techniques for statistically generating representations of heterogeneity are discussed, and one of the most popular methods, sequential indicator simulation, is described. The approaches available for averaging both absolute and relative permeabilities in representative elements of volume are summarized, and examples are given of some of their attributes. For large reservoirs, a prediction procedure based on a synthesis from streamline solutions is sometimes attractive. The concluding section identifies topics where more research is needed for further progress in this rapidly evolving area.

Upscaling, Gridding, and Simulation Using Streamtubes

This paper presents an innovative upscaling methodology based on the semi-analytical simulator developed in the first part. The methodology generates a coarse grid based on streamtubes and isobars, whose upscaled properties are accurately calculated using the properties of each streamtube that constitutes the coarse block. Beyond the calculation of upscaled static properties of the grid, the methodology also calculates upscaled relative permeability curves, or pseudofunctions, using the semi-analytical streamtube method to perform this without requiring any significant additional time. Tests showed that the results from the streamtube coarse grid had an excellent agreement with the fine grid solution. In contrast, simulations with a coarse grid upscaled with conventional techniques failed in many situations. Although this upscaling methodology is heavily based on a fixed streamtube simulation method, it provided good results even for situations in which this streamtube simulator is not supposed to work, such as displacements with favorable mobility ratios, and for problems with gravity and compressibility.

Considerations affecting the scaling of displacements in heterogeneous permeability distributions

Dispersion and convection scaling can be used to model 1D miscible and immiscible flows, respectively, but these scaling techniques fail in multidimensional heterogeneous permeability distributions. The length dependence observed for dispersion induced by correlated heterogeneity in miscible displacement processes precludes the definition of an effective dispersion coefficient. Effective relative permeabilities can be defined that will reproduce the results of the cross-sectional simulations from which they were derived, but complications arise when they are used in areal models

Sequential Upscaling Method

This paper presents a new method for scaling up multiphase flow properties which properly accounts for boundary conditions on the upscaled cell. The scale-up proposed does not require the simulation of a complete finely-gridded model, instead it calls for assumptions allowing the calculation of the boundary conditions related to each block being scaled up. To upscale a coarse block, we have to assume or determine the proper boundary conditions for that coarse block. To date, most scale-up methods have been based on the assumption of steady-state flow associated with uniform fractional flows over all the boundaries of the coarse block. However, such an assumption is not strictly valid when we consider heterogeneities. The concept of injection tubes is introduced: these are hypothetical streamtubes connecting the injection wellbore to all inlet faces of the fine grid cells constituting the block to be scaled up. Injection tubes allow the capturing of the fine-scale flow behavior of a finely-gridded model at the inlet face of the coarse block without having to simulate that fine grid. We describe how to scale up an entire finely-gridded model sequentially using injection tubes to determine the boundary conditions for two-phase flow. This new scale-up method is able to capture almost exactly the fine-scale two-phase flow behavior, such as saturation distributions, inside each isolated coarse-grid domain. Further, the resultant scaled-up relative permeabilities reproduce accurately the spatially-averaged performance of the finely-gridded model throughout the simulation period. The method has been shown to be applicable not only to viscous-dominated flow but also to flow affected by gravity for reasonable viscous-to-gravity ratios.

Fast 3D Reservoir Simulation and Scale Up Using Streamtubes

This paper presents an implementation of a semianalytical method for oil recovery calculation in heterogeneous reservoirs that is both fast and accurate. The method defines streamline paths based on a conventional single-phase incompressible flow calculation. By calculating the time-of-flight for a particle along a streamline and assigning a volumetric flux to each streamline, the cumulative pore volume of a streamtube containing the streamline can be calculated. Subsequently, the streamtube geometries are kept constant and the effects of the time varying mobility distribution in two-phase flow are accounted for by varying the flow rate in each streamtube, based on fluid resistance changes along the streamtube. Oil recovery calculations are then done based on the 1D analytical Buckley–Leverett solution. This concept makes the method extremely fast and easy to implement, making it ideal to simulate large reservoirs generated by geostatiscal methods. The simulation results of a 3D heterogeneous reservoir are presented and compared with those of other simulators. The results shows that the new simulator is much faster than a traditional finite difference simulator, while having the same accuracy. The method also naturally handles the upscaling of absolute and relative permeability. We make use of these upscaling abilities to generate a coarse curvilinear grid that can be used in conventional simulators with a great advantage over conventional upscaled Cartesian grids. This paper also shows an upscaling example using this technique.