Dynamic Upscaling from the Pore to the Reservoir Scale

We propose a dynamic three-stage upscaling methodology from the pore scale to the reservoir scale. In traditional upscaling approaches, simulations at smaller scales are used to compute effective transport properties that are used as look-up values in a larger scale simulation. In the proposed approach, simulations at different scales are performed simultaneously. Effective properties are found directly from a simulation at a smaller scale. This method avoids errors due to incorrect boundary conditions or saturation paths. Pore scale network modeling, assuming that capillary forces dominate, is used to find core-scale properties. Standard grid-based finite-difference methods are used to compute flow at the core-to-reservoir-grid-block scale, where capillary, viscous and gravity forces are all significant. At the field scale, viscous and gravity forces dominate and streamline-based simulation is the simulation method of choice. Preliminary results are presented for coupling a bundle-of-tubes three-phase pore-scale model with a one-dimensional finite difference simulation. We show that if we ignore capillary pressure, the pore-scale model can give elliptic regions, which represent regions of the saturation space where the wavespeeds are complex. This leads to unconditionally unstable solutions to the flow equations. With capillary pressure, stable solutions are found, but the flow paths are exceptionally sensitive to the capillary pressure even though on the field-scale the capillary pressure is much smaller than the viscous pressure drop between wells. Introduction and Proposed Upscaling Approach Reservoir engineering predictions of flow are made over length scales of several kilometers, while the relative permeabilities and capillary pressures that describe the macroscopic behavior are controlled by the pore structure and wettability at the micron scale. The standard method for accommodating these nine orders of magnitude in length scales is as follows. Experiments to measure capillary pressure and relative permeability are measured on the core scale – typically on samples approximately 10 cm long. Pore-scale network modeling can also be used to predict relative permeability from knowledge of the pore structure. Network models generally represent samples at most a few cm across. The corescale predictions or measurements, however, depend on saturation path (particularly in threephase flow) and, for systems with small-scale heterogeneity, the boundary conditions on the sample will also affect the results. These relative permeabilities may then be used to simulate flow in one or more representative sections of the reservoir on a scale of 10s to 100s meters. The upscaled properties are then used in a field-scale reservoir simulation. However, the upscaled relative permeabilities are sensitive to the boundary conditions used to compute them, which may not be representative of the field-scale conditions. It is not practical to overcome the errors in upscaling by performing directly a simulation based on the representation of the reservoir at the pore scale, or even at the scale of a few centimeters. First, the reservoir description would include a huge amount of detail that would be impossible to know with any certainty. Second, the computation of a pressure field, which represents the most time consuming step of reservoir simulation, is at present limited to at most a few million grid blocks, which is many orders of magnitude smaller than the number Proceedings of 21 Annual International Energy Agency Workshop, Edinburgh, UK, 19-22 September 2000. 2 of pores, or even core-sized blocks in a reservoir. There are three distinct length scales in reservoir simulation from the pore scale upwards, based on the balance of viscous, gravity and capillary forces. Typical capillary forces are around 10 – 100 kPa in reservoir settings (Dullien, 92). Since the microscopic origin of capillary pressure is the interface between two fluids, the magnitude of the capillary force is largely independent of the size of the system. Typical pressure gradients due to viscous or gravity forces are order 10 – 100 kPa/m. At the reservoir scale – several 1000s of meters – viscous and gravity forces dominate over capillary pressure. Most upscaling research has concentrated on this limit, where the capillary pressure is small, and the principal concern of upscaling is to accommodate the effects of permeability variations at the sub-grid-block scale. For two recent reviews on upscaling see Christie (96) and Barker and Dupouy (99). For displacement-type simulations, where the effects of capillary pressure can be neglected, streamline-based simulation (Batycky et al., 97) can be used to capture the effects of heterogeneity and well placement efficiently and accurately. However, at the scale of single grid block in a streamline simulation – 10s meters – capillary, gravity and viscous forces are all of the same order of magnitude. To compute effective relative permeabilities at this scale requires a simulation that accounts for all the relevant physical forces. Conventional gridbased simulation is preferred in these cases. A single grid-block in the grid-based simulation may be a few centimeters across. At this scale capillary forces dominate. This is the scale at which experimental measurements of relative permeability and capillary pressure are made. The simulation method of choice to predict relative permeability is network modeling, where flow in a lattice of pores is simulated. For recent papers that demonstrate the current capabilities of these models see, for instance, Bakke and Øren (97), and Fenwick and Blunt (98). We propose a dynamic simulation method that couples together all three length scales: streamline-based simulation at the field scale, with effective flux functions derived from gridbased simulation at the meter scale, which uses fluxes computed directly from network modeling. The approach is shown schematically in Figure 1. The reservoir to meter scale simulation is a generalization of the nested gridding method of Gautier et al. (99), and the dual mesh method of Verdière and Guérillot (96). Gautier et al. took a fine-scale reservoir description and divided it into coarse grid blocks each containing several fine cells. For a known saturation distribution at the fine scale, the transmissivity (mobility times permeability) of each coarse grid in each coordinate direction was computed by a direct solution of the pressure field using simple boundary conditions. These coarse-scale transmissivities were then used to compute the pressure at the field scale. The coarse grid pressure and the fluxes across each coarse grid face from the field-scale solution were then used as boundary conditions to compute the pressure within each coarse cell. As a result a continuous and conservative, albeit approximate, velocity field is derived for the whole finescale description. Gautier et al. then traced streamlines through the fine grid description, and used streamline-based simulation to transport fluid along the streamlines. Verdière and Guérillot (96) proposed a similar approach, but the simulation was a conventional grid-based method. In both of these methods, fluid is transported explicitly on a fine grid, but the pressure is computed using an approximate dual-scale method. Proceedings of 21 Annual International Energy Agency Workshop, Edinburgh, UK, 19-22 September 2000. 3 Figure 1. Schematic of the multi-scale dynamic upscaling approach. In the proposed method, streamline-based simulation is performed only on the coarse cells. The streamline method uses the transmissivities between coarse blocks. These will be derived from a simulation within each coarse block using grid-based simulation. After each time-step in the field-scale simulation, the meter-scale simulations in each coarse grid block will be performed, using the flux and pressure boundary conditions mentioned above. The saturation change in the each coarse grid will track the saturation change observed at the larger scale. Then effective transmissivities of each coarse cell will be computed, and these will be used to find the field-scale pressure for the next time-step, as in the nested gridding approach. Then the next time-step in the streamline simulation is performed as above. In this technique the two scales perform separate but simultaneous simulations. The pore-to-meter scale simulation will be a generalization of the work of Heiba et al. (86), and Fenwick and Blunt (98). Heiba et al. coupled an analytical two-phase network model, based on a Bethe lattice, to a one-dimensional simulator at the macroscopic scale. Fenwick and Blunt developed a three-phase network model. They showed that different saturation paths lead to very different relative permeabilities. It was not obvious given specified inlet Pore-scale modeling. An explicit representation of the pore space is used to simulate flow for a sequence of saturation changes given by the largerscale simulation. At the reservoir grid block scale conventional simulation is used. The fluxes in each grid block are derived from network modeling, and the overall flux is given to the streamline-based simulation. At the field scale streamline-based simulation uses flux functions computed from the smaller scale simulation. Proceedings of 21 Annual International Energy Agency Workshop, Edinburgh, UK, 19-22 September 2000. 4 and initial boundary conditions what the saturation path would be, and consequently, the correct relative permeabilities to use. They developed a self-consistency procedure to overcome this problem. They guessed a saturation path and used this in their network model to compute three-phase relative permeabilities. These relative permeabilities were then used in a one-dimensional finite-difference code. The saturation path derived from this simulation was then fed back into the network model to simulate the relative permeabilities again. This iterative procedure continued until no further change in saturation path or relative permeabilities were found. We propose a more direct approach, similar to that used by Heiba et al. A three-dimensional, three-phase finite difference simulator will be run. This simulator will use capillary pressures and relative permeabilities derived from pore-scale modeling. Once the saturation field is updated after one time step in the finite difference code, the same saturation change will be imposed on the pore-scale model. The relative permeability of each phase and the capillary pressure for the updated saturation will be computed in the pore-scale model and used in the next step of the larger-scale simulation. As described the method appears to use pore-level description for all parts of the reservoir, which would require an unfeasibly large amount of computer time. Instead, smaller scale simulation will be confined to a few representative rock types and structures, or to regions where the saturation is changing rapidly. By running separate models at each scale of the simulation, it is possible to use effective transmissivities from one block in several regions of the field. While this involves some approximations, and resembles standard upscaling approaches, it is necessary to allow the method to be run in a practical amount of time. At present, this research is at a preliminary stage. Below we will present results for one aspect of the model: coupling an analytic pore-scale bundle-of-tubes model with a onedimensional simulator. A Bundle-of-Tubes Model and Elliptic Regions We consider flow in bundles of cylindrical pores of different size (van Dijke et al., 00). Within each bundle the tubes are parallel to the flow direction. The bundles are aligned in series along the flow direction, and at the junction of two bundles there is pressure communication between the tubes. We assume that only one phase may occupy each tube at a time. Thus each bundle has some fraction of the tubes filled with water, oil and gas. However, the phase occupancy of the tubes can vary with distance, since different bundles will have different phase occupancies. The conductance of each tube C is defined by: ( ) x g P C Q ρ μ − ∇ = (1) where Q is the volume of fluid flowing per unit time and l is the tube length. We find C from Poiseuille’s law: