We present a new streamline-based simulator applicable to field scale flow. The method is three dimensional (3D) and accounts for changing well conditions that result from infill drilling and well conversions, heterogeneity, mobility effects, and gravity effects. The key feature of the simulator is that fluid transport occurs on a streamline grid rather than between the discrete gridblocks on which the pressure field is solved. The streamline grid dynamically changes as the mobility field and boundary conditions dictate. A general numerical solver moves the fluids forward in space and time along each streamline. Multiphase gravity effects are accounted for by an operator-splitting technique that also requires a numerical solver. Because fluid transport is decoupled from the underlying grid, the method is computationally efficient and very large timesteps can be taken without loss in solution accuracy.
We present results of the streamline-based simulator applied to tracer, waterflooding, and first-contact miscible (FCM) displacements in two and three dimensions. Where possible, comparisons with conventional methods indicate that the streamline model minimizes numerical diffusion and is up to two orders of magnitude faster. We also demonstrate the efficiency of the method on a field-scale, million-gridblock, 36-well waterflood that includes a pattern-modification plan to improve oil recovery. Last, we present results of the method applied to the House Mountain waterflood in Canada.
The use of streamlines and streamtubes to model convective displacements in heterogeneous media has been presented many times since the early works of Muskat,1–3 Fay and Prats,4 and Higgins and Leighton.5–7 Important contributions to the field were also made by Parsons,8 Martin and Wegner,9 Bommer and Schechter,10 Lake et al.,11 Emanuel et al.,12 and Hewett and Behrens.13,14
Streamline methods have recently resurfaced as a viable alternative to traditional finite-difference methods for large, heterogeneous, multiwell, multiphase simulations.15–27 The efficiency of the method has made it an ideal tool for ranking equiprobable reservoir images28; rapid assessment of production strategies, such as infill drilling and gas injection29; computing upscaled component flux properties for compositional simulation30; and integration with production data for reservoir characterization31. The method has also allowed for solution of fine-scale models [on the order of 106 gridblocks] on standard computer resources, thus reducing the need for significant upscaling.
In this paper, we present advances on our previous work where we mapped analytical solutions along streamlines.19,22 Although the streamline paths were updated periodically to account for changing mobility fields, the method could not account for changing well conditions or gravity - two key phenomena that must be modeled in general field-scale simulations.
We account for these mechanisms by mapping one-dimensional (1D) numerical solutions along streamlines, as first proposed by Bommer and Schechter.10 In doing so, nonuniform initial conditions that appear along recalculated streamline paths, resulting from changing well and mobility conditions, can be moved forward in space and time correctly. Streamline paths are updated, and the transport process repeated. The grid on which the pressure field is solved is effectively decoupled from the streamline grid used to transport fluids. There is no longer a global grid Courant-Friedrichs-Lewy (CFL) condition to restrict timestep size. Furthermore, grid-orientation and numerical-diffusion effects are minimized. Finally, operator splitting is used to account for gravity in multiphase flow.32,33 After moving fluids convectively along streamlines, fluids are then moved vertically along 1D gravity lines. Bratvedt et al.24 presented a similar operator-splitting technique in the context of their front-tracking method.
Our application of streamlines to field-scale reservoir simulation is a combination of four existing ideas:
updating the streamline paths to account for changing mobility field and well conditions,9,15,19
numerical solutions along streamlines,10 and
including gravity effects in multiphase flow by use of operator splitting.23,24,32,33 Using streamlines and gravity lines decouples the 3D transport problem into multiple 1D problems and leads to a very fast and accurate method applicable to a wide range of field conditions.
In this section we outline the streamline method. The Appendix gives a detailed discussion on how to trace the streamlines.
The streamline method is an IMPES method. Ignoring capillary and dispersion effects, the governing equation for pressure p, for incompressible, multiphase flow is given by
where D=a depth below datum. Total mobility, ?t, and total gravity mobility, ?g, are defined as
where krj=relative permeability of Phase j, µj=phase viscosity, ?j=phase density, g =gravity acceleration constant, and np=number of phases present. We also require a material-balance equation for each Phase j35:
The total velocity, ut, is derived from the 3D solution to the pressure field (Eq. 1) and application of Darcy's law. The phase fractional flow is given by
and the phase velocity resulting from gravity effects is given by
Eqs. 1 and 3 form the IMPES set of equations in the formulation of the streamline simulator. We confine our discussion to the solution of these equations for two-phase flow.