SPE Reservoir Evaluation & Engineering
Volume 11, Number 2, April 2008, 405-416

Summary

Steady-state upscaling techniques are attractive because they are quick and simple to implement; unlike dynamic methods, there is no need for fine-grid simulation, and the upscaled properties are not case dependent. They are based on the assumption that either capillary forces (capillary equilibrium limit, CL) or viscous forces (viscous limit, VL) dominate flow. However, the reservoir conditions for which these assumptions are valid have not been clearly defined. It is generally supposed that the CL method is valid at “low” flow rates over “small” lengthscales, while the VL method is valid at “high” flow rates over “large” lengthscales. These qualitative criteria are difficult to properly apply and can be easily violated, yielding significant errors in predicted reservoir performance.

We have identified a comprehensive suite of dimensionless groups which can be used to define the validity of steady-state methods. The groups account for the effect of heterogeneity, as well as the other parameters which control the balance between capillary and viscous forces. Numerical simulations have been used to identify the range of values for these groups over which steady-state methods are valid. Our results yield a practical set of quantitative criteria which can be used to determine the validity of steady-state upscaling methods for a wide range of geological models. They capture the effects of capillary trapping and are valid regardless of fluid mobility, wettability, or endpoint saturation.

We test our criteria against three realistic models of small- to intermediate-scale geological heterogeneity. We find that the criteria do a good job of predicting the range of validity for each method, and are conservative in all cases, suggesting that if they are met, then steady-state upscaling techniques can be applied with confidence and may still be valid for slightly less restrictive conditions. However, in the models investigated, we find that the validity of the CL method is restricted to very low flow rates, which are unlikely to be encountered in most production scenarios. This is because the CL method overestimates the amount of capillary trapping. In general, VL upscaling is valid over a much more reasonable range of reservoir flow rates.

Introduction

Steady-state multiphase upscaling has become increasingly popular because it is fast, robust and computationally cheap [e.g., Kumar and Jerauld (1996); Pickup and Sorbie (1996); Barker and Thibeau (1997); Ekrann and Aasen (2000); Huang et al. (1995); Pickup et al. (2000); Pickup and Stephen (2000); Kløv et al. (2003)]. Unlike their dynamic counterparts, steady-state techniques do not need a full fine-grid simulation prior to generating the pseudo (upscaled) rock properties. However, steady-state upscaling is limited to areas in the reservoir where either capillary (capillary limit, CL) or viscous (viscous limit, VL) forces dominate flow. Using steady-state upscaling methods outside their validity range can yield significant errors in predicted recovery [e.g., Pickup et al. (2000)].

It is generally recognized that CL upscaling is valid for “low” flow rates over “small” lengthscales, while VL upscaling is valid for “high” flow rates over “large” lengthscales (Kumar and Jerauld 1996; Huang et al. 1995; Kløv et al. 2003; Lohne et al. 2006; Smith 1991; Virnovsky et al. 2003). However, it is often not clear whether these simple, qualitative criteria have been met. Quantitative criteria are usually based on a single dimensionless ratio of capillary to viscous pressure drop within a homogeneous averaging volume (Pickup and Sorbie 1996; Pickup et al. 2000; Zhou et al. 1997; Dale et al. 1997; Stephen et al. 2001). Yet it is easy to demonstrate that this is not sufficient to specify the conditions under which each upscaling method is valid. For example, Pickup and Stephen (2000) used the ratio

[equation]…………………………………………(1)

to determine capillary and viscous limits in a variety of laminated and cross-bedded rock types. Here q_inj  is the injection rate (cm³ s^–1); μo is the oil viscosity (cp); ∆x is the width of a gridblock in the x-direction, ∆y and ∆z are the width and height of the model (cm); k_o  is the harmonic average of the oil phase permeabilities in high and low permeability laminae (D); and ∆Pc is the difference in capillary pressures (atm) between the laminae, evaluated at connate water saturation. For values of N_vc < 10^–2, the system is assumed to be capillary dominated. However, CL upscaling failed to properly capture flow in their ripple model for a case with N_vc = 3.16×10^–3, which is within the assumed criterion for capillary dominated flow. It is not possible to use a universal dimensionless parameter of this type to identify flow regimes in heterogeneous systems, because the nature of the heterogeneity must also be included (Virnovsky et al. 2004).

Jonoud and Jackson (2008) presented a new set of dimensionless criteria which can be used to specify the conditions under which CL and VL upscaling are valid. Their criteria were derived from a simple layered geological model, and represent the end members of flow parallel to layering and flow perpendicular to layering. They found that steady-state upscaling techniques may be valid for a wider range of reservoir conditions than previously thought. When fluid flow is parallel to layering, the CL method can be applied to thinly laminated systems up to tens of meters in length for low to typical flow rates, while the VL method is valid for thick laminations of several tens to hundreds of meters in length for high to typical flow rates. However, when fluid flow is perpendicular to layering, capillary trapping restricts the validity range of the CL method to very thinly laminated systems (e.g., a few mm) and very low flow rates (e.g., mm/day). Capillary trapping occurs when the wetting phase tends to imbibe into lower permeability layers which act as barriers to flow ((McDougall and Sorbie 1992; Ringrose et al. 1993; Huang et al. 1995). In a water-wet system, water imbibes into low permeability layers and expels the oil, but does not displace the oil in the high-permeability layers. As soon as the water saturation in low permeability layers reaches its maximum (Sw^max_low = 1 - So_r , _low), the remaining oil in the high permeability layers is trapped. Validity of the VL method is not affected, as capillary trapping is negligible.

The criteria derived by Jonoud and Jackson (2008) for flow parallel and perpendicular to layering yield the upper and lower limits on the validity of upscaled multiphase properties using steady-state techniques. They are analogous to the limits on upscaled single-phase permeability obtained by the arithmetic and harmonic means (Renard and de Marsily 1997). However, more complex geological heterogeneities include dipping and truncated rock layers, so the flow field in any direction incorporates some elements of layer parallel flow, and some elements of layer perpendicular flow. These may not be captured by their criteria.

The aim of this paper is to test the criteria presented by Jonoud and Jackson (2008) against realistic geological models, typical of those which might be upscaled using steady-state techniques. We wish to determine whether their criteria are applicable regardless of heterogeneity type. We also wish to identify the reservoir conditions under which steady-state methods can be applied.

© 2008. Society of Petroleum Engineers

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History

• Original manuscript received: 20 February 2006
• Meeting paper published: 12 June 2006
• Revised manuscript received: 11 June 2007
• Manuscript approved: 5 July 2007
• Version of record: 25 April 2008