Steady-State Upscaling: From Lamina-Scale to Full-Field Model
- Gillian Pickup (Heriot-Watt U.) | P.S. Ringrose (Statoil) | Ahmed Sharif (Roxar)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2000
- Document Type
- Journal Paper
- 208 - 217
- 2000. Society of Petroleum Engineers
- 7.2.2 Risk Management Systems, 4.3.4 Scale, 5.3.1 Flow in Porous Media, 5.2.1 Phase Behavior and PVT Measurements, 5.6.2 Core Analysis, 5.1.5 Geologic Modeling, 5.5 Reservoir Simulation, 5.5.2 Core Analysis, 5.4.2 Gas Injection Methods, 5.4.1 Waterflooding, 4.6 Natural Gas, 2.4.3 Sand/Solids Control, 5.5.3 Scaling Methods
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In recent years, the impact of small-scale permeability structure on hydrocarbon recovery has been demonstrated, and upscaling procedures, such as the geopseudo method, have been developed to scale-up from the lamina scale using the hierarchy of geological length scales. However, upscaling is very time consuming, so many engineers still input rock curves into large-scale simulations. In this study, we show how the process may be speeded up using steady-state methods for calculating the pseudofunctions.
Two examples are used to demonstrate the method. The first is a three-stage scale-up of a water flood in a fluvio-aeolian model. Capillary equilibrium was assumed for the first two stages, and viscous-dominated steady state for the third stage. Two different wettability cases were examined¾water-wet and intermediate-wet. The effect of using the small-scale pseudo relative permeabilities depends on both the nature of the heterogeneities and on the wettability. In this study the recovery was reduced when small-scale pseudos were included, especially in the water-wet case.
The second case study involved gas injection into the oil leg of a tidal deltaic reservoir. Scale-up was performed in two stages: (a) from the lithofacies scale to the geological model, and (b) from the geological model to the full-field simulation model. The viscous-dominated steady-state method was used in both cases. The results showed that the effect of the fine-scale heterogeneities was of the same order as the effect of the coarse-scale heterogeneity, indicating that (even in a system where capillary pressure is negligible) the fine-scale structure can be important.
In a number of geological environments, fine-scale sedimentary structures, such as lamination, give rise to high permeability contrasts over small length scales. Such structures may have a significant effect on hydrocarbon recovery.1-3 For example, experimental floods of slabs of aeolian rock containing laminae show that oil may become trapped between laminae due to the action of capillary forces.3 Therefore, for an accurate estimate of recovery, the effects of small-scale sedimentary features should be scaled up by calculating effective relative permeability and capillary pressure curves (pseudofunctions). The geopseudo method4 has been developed to scale up from the lamina scale (millimeter to centimeter) using a hierarchy of geological length scales. For example, ...Corbett et al.4 demonstrated this using a two-dimensional (2D) cross-sectional model and the Kyte and Berry5 dynamic upscaling method. However, the simulations required to calculate the pseudofunctions are time consuming, especially in three dimensions (3D). Also, problems may occur because dynamic upscaling techniques are not robust 6,7 and the resulting pseudos must be carefully vetted. In addition, dynamic pseudofunctions are case dependent, so different pseudos have to be generated for different flood rates and directions. Because of these disadvantages, many engineers ignore the fine-scale structure, and still apply rock curves to large gridblocks (tens of meters horizontally). However, scale-up can frequently be speeded up and simplified by using steady-state assumptions.
At the smallest scales, pseudofunctions may be calculated more quickly and easily using the capillary equilibrium upscaling method.8-10 When the injection rate is very low, the fluids will approach capillary equilibrium over small distances (several tens of centimeters), so this is a valid approximation. At larger scales, pseudo calculations can be speeded up if we can use viscous-dominated steady state. This may seem to be an inappropriate procedure, because the flood is unlikely to be in a steady state and, in addition, steady-state methods do not compensate for numerical dispersion as the Kyte and Berry method5 does. However, calculations show that the errors due to numerical dispersion are not severe, provided there is a sufficient number of grid blocks between the injector and producer.11
The aim of this article is to demonstrate how steady-state scale-up methods may be applied in multistage scale-up procedures. We briefly outline steady-state scale-up methods for the capillary-equilibrium limit (CL) and viscous-dominated limit (VL). Then we describe multistage scale-up in two field examples, first a model of part of a fluvio-aeolian reservoir, and then a tidal deltaic reservoir. At each stage, the steady-state assumption is used to calculate the pseudos. Because of the large numbers of grid blocks involved, complete testing of the scale-up method is unfeasible. However, in both cases the final level of upscaling has been validated.
Steady-State Scale-Up Methods
The most common method for performing single-phase scale-up is to assume steady-state, linear flow across a model (see, for example, Ref. 12). The pressure equations are set up, assuming material balance and Darcy's law, and then solved using appropriate boundary conditions. The effective permeability is calculated by applying Darcy's law across the whole system. The boundary conditions used here were periodic boundaries to obtain full permeability tensors12 for models with tilted laminations, and no-flow boundaries in other cases. Since single-phase scale-up is routinely performed, no further discussion of the method is given here.
Two-Phase Capillary Limit.
In the capillary equilibrium scale-up method, we assume that, within the region considered, the fluids have come into capillary equilibrium. This is only valid when the viscous pressure gradient is negligibly small. However, it is a reasonable assumption over small distances (<30 cm) when the flow rate is slow (~30 cm/D, or less). The method has been described in detail by many authors, for example, Pickup and Sorbie.13 However, we reproduce the main steps for completeness here. We assume that the relative permeability and capillary pressure curves are known for each lithofacies type present in a model.
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