The Scaleup of Two-Phase Flow in Porous Media Using Phase Permeability Tensors

Abstract

First presented at the SPE Annual Technical Conference & Exhibition held in New Orleans, 25-28 September, 1994, as " Development and Application of a New Two-Phase Scaleup Method Based on Tensor Permeabilities". Abstract The effects of different levels of geological heterogeneity on a fluid displacement process may be captured at a larger scale using scaleup techniques. In the context of reservoir simulation, these are algorithms which should reproduce the results of fine grid calculations on a coarser grid. These techniques are referred to as pseudo-isation, the main objective being to produce pseudo-functions which can be used on the coarse grid. When carried out successfully, the pseudo-functions (e.g. pseudo relative permeabilities)incorporate the interaction between the fluid mechanics and the heterogeneity as well as correcting for numerical dispersion. These pseudo-functions also depend on the viscous/capillary and viscous/gravity ratios and are valid for the boundary conditions relevant to the particular flows. For single phase flow, the scaleup problem involves the derivation of effective permeability which, in general, is a tensor quantity. Multi-phase flow is more complex since scaled-up dynamic transport quantities must be calculated which depend on phase saturation, flow rate etc. In this paper, we present a method which extends the idea of tensor (absolute) permeability to tensor effective phase permeabilities. They are extensions of conventional functions which also include the off-diagonal phase crossflow terms which maybe important in certain systems. Two- phase tensor methods are presented which are valid (a) in the capillary equilibrium limit and (b) for arbitrary values of viscous/capillary and viscous/gravity ratios. Numerical examples of the application of these methods are presented for ripple-bedded systems where the two-phase crossflow effects are significant, and where oil trapping within the lamina structure may occur under certain conditions. The results show that it is important to use phase tensors in upscaling where gravity effects are significant, in order to generate the correct vertical flows. Introduction The modelling of two-phase flow through porous media is important for forecasting hydrocarbon recovery in oil and gas reservoirs. This process is complicated by the fact that rocks may be very heterogeneous over a wide range of length scales. If the effects of these heterogeneities are to be taken into account in full-scale simulations, a method is required for averaging flow properties from small-scale models, for incorporation in larger-scale ones. The calculation of effective flow properties (sometimes known as pseudoproperties)has been studied, from different view points, by a number of authors. Generally speaking, scaleup can only be carried out successfully when there is a separation of length-scales. In fact, many sedimentary rocks do exhibit heterogeneities at distinct length scales, e.g. laminae (em), beds (m), formations (10m). Corbett et al have shown that it is feasible to scaleup two-phase flow using such geologically based length scales.

Large-scale Invasion Percolation with Trapping for Upscaling Capillary-Controlled Darcy-scale Flow

“…Nevertheless, it is usually observed that at relatively small scales capillary forces dominate the displacement process, while viscous and gravitational forces may become more important at larger scales (Odsaeter et al 2015;Kumar and Jerauld 1996;Jonoud and Jackson 2008a). Numerous upscaling methods are available in the literature for two-phase flow (Guzman et al 1999;Jacks et al 1973;Kyte and Berry 1975;Stone 1991;Pickup and Sorbie 1996;Lohne et al 2006;Virnovsky et al 2004;King 1989;King et al 1993;Coll et al 2001;Wallstrom et al 2002;Durlofsky 1997) among which flow-based methods are considered the most reliable. Under this category, two fundamental approaches are employed: steady-state and dynamic pseudos.…”

Prediction of hydrocarbon recovery from turbidite sandstones with linked-debrite facies: Numerical flow-simulation studies

“…Appropriate values for these parameters were calculated using the computer software 'Geopseudo Toolkit' (Ma et al, 2001). This program calculates curves as a function of the absolute permeability using typical Corey type exponent functions of water saturation (e.g., Pickup and Sorbie, 1996). Saturation curves were calculated for different absolute permeabilities and assigned (4), horizontal permeability (k h ) and vertical permeability (k v ) are shown for each zone.…”

New criteria for the validity of steady-state upscaling

“…Numerous authors have described steady-state upscaling methods (Lemouzy et al 1993;Pickup and Sorbie 1994;Kumar and Jerauld 1996;Pickup and Carruthers 1996;Pickup and Sorbie 1996;Ekrann and Aasen 2000;Pickup and Stephen 2000;Nomura 2002;Lohne et al 2004) and we provide only a brief overview here. Steady-state upscaling consists of finding the solution of the two-phase immiscible flow equations in the limit of quasi-static variations in fluid saturation with time.…”

“…Several studies have used a dimensionless ratio of capillary to viscous pressure drop to determine whether a system is capillary or viscous dominated. Viscous and capillary forces are assumed to act in the same direction within a homogenized averaging volume (Pickup and Sorbie 1994;Zhou et al 1994;Dale et al 1997;Pickup and Stephen 2000;Stephen et al 2001). Yet it is easy to demonstrate that this dimensionless parameter is not sufficient to specify the conditions under which each upscaling method is valid (Jonoud and Jackson 2004).…”

“…Most of the previous studies have considered only water-wet systems (Pickup and Sorbie 1994;Pickup and Stephen 2000) and those which have studied mixed-or oil-wet systems have not individually investigated the effect of wettability on the validity of the CL and the VL methods (Dale et al 1997;Lohne et al 2004). Wettability variations can be associated with varying initial water saturations (Jackson et al 2005) and control residual oil saturations and the shape and magnitude of relative permeability and capillary pressure curves (Mohanty and Salter 1983;Masalmeh 2002;Valvatne and Blunt 2004).…”