An assessment of steady-state scale-up for small-scale geological models

An assessment of steady-state scale-up for small-scale geological models Gillian E. Pickup 1 and Karl D. Stephen 1 1 Department of Petroleum Engineering, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK ABSTRACT: The calculation of pseudo-relative permeabilities can be speeded up considerably by using steady-state methods. The capillary equilibrium limit may be assumed at small scales (30 cm or less), when the flood rate is low. At high flow rates and larger distance scales, we may use a viscous-dominated steady-state method which assumes constant fractional flow. Steady-state pseudos may also be calculated at intermediate flow rates using fine-scale simulations, and allowing the flood to come into equilibrium at different fractional flow levels. The aim of this paper is to assess the accuracy of steady-state scale-up for small-scale sedimentary structures. We have tested steady-state scale-up methods using a variety of small-scale geological models. The success of steady-state scale-up depends not only on the flow rate, but also on the nature of the heterogeneity. If high permeability zones are surrounded by low permeability ones (e.g. low permeability laminae or bed boundaries), oil trapping may occur in a water-wet system. In this case pseudo-oilrelative permeabilities are very sensitive to flow rate, and care must be taken to upscale using the correct viscous/capillary ratio. However, in permeability models, where phase trapping may not occur (unconnected low permeability regions), the pseudos are similar, whatever the viscous/capillary ratio. The disadvantage of steady-state scale-up is that it cannot take account of numerical dispersion, in the manner in which dynamic methods can. However, we show examples of coarse-scale simulations with viscous-dominated steady-state pseudos which agree favourably with fine-scale simulations. Provided there are sufficient grid blocks in the coarse-scale model, the smearing of the flood front due to numerical effects is not serious. KEYWORDS: scale up, capillary pressure, sedimentary structure. INTRODUCTION Recent laboratory studies have shown that small-scale sedimentary features, such as lamination, may have a significant effect on hydrocarbon recovery due to capillary effects (Huang et al. 1995; Honarpour et al. 1995). Huang et al. (1995) describe how there are three different contributions to the remaining oil: (a) residual oil which is oil retained within the pores; (b) capillary-heterogeneity trapping of oil between laminae; and (c) bypassed oil at the macroscopic scale. In this paper, we are concerned with capillary-heterogeneity trapping. In laminated systems, it may have a significant effect on the amount of remaining oil, depending on the wettability of the system. Upscaling from small-scale models is therefore required not only to calculate the effective absolute permeability, but also to determine the appropriate relative permeabilities, with the correct end-points, to use in larger-scale models. Upscaling from models with millimetre-sized grid cells to full-field models with grid cells of several tens of metres in length is a daunting task, and must be carried out in stages. One procedure for multi-stage upscaling, called the Geopseudo Method was developed by Corbett et al. (1992). In this method, the effective flow properties (pseudos) are calculated at geologically based length scales, e.g. lamina-scale, bed-scale, etc. Presented at the 6th European Conference on the Mathematics of Oil Recovery (ECMOR), Peebles, September 1998. Petroleum Geoscience, Vol. 6 2000, pp. 203 210 Originally, the Geopseudo calculations were carried out using the Kyte & Berry (1975) upscaling method. In this method, fine-scale two-phase flow simulations are run on representative parts of the reservoir. Unfortunately there are a number of drawbacks with using this method, and other dynamic upscaling methods: (1) they are time consuming because of the need to perform two-phase, dynamic simulations; (2) they are not general (different pseudos are required for different flow rates and different flood directions); (3) they may produce unrealistic results, such as negative permeabilities (Barker & Thibeau 1996). In order to facilitate the upscaling procedure, a number of people have developed methods for scale-up which assume that a flood is in a steady state (e.g. Smith 1991; Dale et al. 1997; Saad et al. 1995; Kumar & Jerauld 1996). This is because such methods avoid many of the problems incurred by unsteadystate methods. However, steady-state methods also have some disadvantages, such as: (1) reservoirs are not actually in steady state, so the steadystate assumption is not strictly valid; (2) steady-state methods cannot compensate for numerical dispersion, unlike methods such as that of Kyte & Berry (1975). 1354-0793/00/$15.00 2000 EAGE/Geological Society, London

204 Gillian E. Pickup and Karl D. Stephen It has been argued (Smith 1991) that when the flood is capillary dominated the fluids will be in capillary equilibrium over small distances (30 cm or less). Alternatively, when the flood is viscous dominated, there is a sharp front (providing the mobility ratio is favourable), where steady state obviously does not hold. After the front has passed, however, the fluids will approach a steady state. Therefore, the fact that the reservoir is not in a steady-state condition, may not be too restricting. The aim of this paper is to assess the errors involved in using steady-state methods. THE SCALE-UP METHODS A simulated flood is in steady state when the saturation in each grid block is constant with time. So the two-phase flow equations become: (k o P o )=0 and (k w P w )=0. (1a & b) The phase permeabilities depend on the water saturation which, in turn, depends on the flood rate and direction, i.e. the results are path-dependent. Equations 1a and 1b are not separate equations, because they are linked by the capillary pressure, and they should not, in fact, be solved separately. However, if we assume that they do hold separately, the effective permeability for each phase may be calculated independently, thus turning a two-phase problem into two single-phase ones. As demonstrated in this paper, pseudos calculated using this method can give reasonable results compared to the fine-scale simulation (but see discussion under tests later in the paper and Fig. 7). In this study, three different steady-state scale-up methods were used, depending on the ratio of viscous/capillary forces. (Gravity was ignored here, since only small-scale models were considered.) Capillary equilibrium If the injection rate is zero, the fluids will come into capillary pressure equilibrium and the water saturation will depend on the capillary pressure curves. In cases where the flow rate is very low, we may assume that capillary equilibrium holds approximately. The capillary equilibrium method has been presented in previous papers (e.g. Smith 1991; Pickup & Sorbie 1996). However, we outline the method here for the sake of completeness. It is assumed that the effective absolute permeability has already been calculated. + Select a capillary pressure value. + Using the capillary pressure curves for each zone, determine the water saturation in each region. + Calculate the average water saturation for the coarse block, using porosity weighting. + Use the fine-scale relative permeability tables to determine the relative permeabilities to oil and water for the saturation values calculated in the second stage. Then calculate the phase permeabilities by multiplying by the absolute permeability (for each fine grid cell). + Perform single-phase steady-state simulations separately on the oil and the water phase permeabilities, and then calculate the effective phase permeabilities to oil and to water. + Calculate the effective relative permeabilities by dividing the effective phase permeabilities by the effective absolute permeabilities. + Select another capillary pressure value, and repeat the steps again. In this way, effective relative permeability curves for oil and water may be obtained. Note that a variety of methods may be used to calculate effective permeability from single-phase flow simulations; see, for example, methods discussed in Pickup et al. (1994). In capillary equilibrium, one would expect that simulations only need to be carried out for one phase (because there should be a constant difference between the pressures), but this is not the case in practice. To reach capillary equilibrium, the velocity must tend to zero, in which case there will be no pressure gradient across the model. However, to get an effective permeability, a pressure gradient must be applied across the model (which would, obviously destroy the equilibrium state). The method is therefore approximate, and the oil and water pressures determined from the single-phase flow simulations, will not, in general, have the same gradient. In spite of this, the method may still give reasonable results, as the tests presented later will show. The main disadvantage with the capillary equilibrium method is that when the saturations for one phase are low, the permeabilities to that phase are also very small, so the pressure equations are ill-conditioned and the equations may not converge. In this case, the results should be monitored carefully. Viscous-dominated steady state At steady state, the fractional flow of water, f w is constant with time, and this can be used to determine the water saturation. In the viscous-dominated case (negligible capillary pressure and gravity), there is a simple relationship between fractional flow and the relative permeabilities (which are a function of water saturation): f k /μ. (2) k /μ k /μ. If we assume water is injected into the model with a uniform fractional flow over the boundary and that the relative permeabilities are isotropic, then the fractional flow will be uniform throughout the model. In this case, the water saturations may be determined by inverting the fractional flow function f w (S w ), where S w is water saturation. Once again, this method has been presented by others (e.g. Kumar & Jerauld 1996), but we outline the steps here for completeness. + Select a fractional flow value. + Determine the water saturation in each region. + Calculate the porosity-weighted average water saturation for the coarse block. + Determine the relative permeabilities in each region, and then calculate the total mobility to oil and water in each region: λ k k μ k μ. (3). + Perform a steady-state, single-phase, flow simulation using the total mobilities and then calculate the effective total mobility. + Use the fractional flow of water to determine the effective relative permeabilities: where the over-bar represents an average value.

Steady-state scale-up methods 205 Fig. 1. The ripple models: (a) high permeability bottomset; (b) low permeability bottomset. Three ripple models are shown, each being 3 1 cm (54 cells by 18 cells). White represents 200 md and grey 10 md. Fig. 2. The rock relative permeability and capillary pressure curves. Note that, in this method, we assume that the fractional flow is uniform over the coarse block boundary. In a full-scale simulation, this would probably not be the case. However, tests carried out in this study, and by others (e.g. Kumar & Jerauld 1996) show that the method works adequately. The disadvantage of the viscous-dominated steady-state method is that if the relative permeabilities are anisotropic, the fractional flows are constant along streamlines but not, in general, along the coordinate axes. This problem was surmounted in Pickup et al. (2000), by assuming the fractional flow was constant in the x-direction in order to calculate the x-direction effective relative permeabilities, and so on. Intermediate viscous/capillary ratios There is no simple way of calculating the effective oil and water permeabilities for an arbitrary value of the viscous/capillary ratio, because a simulation must be carried out to determine the saturations. (This problem has been addressed in 1D by Dale et al. (1997), but not in higher dimensions.) We need to carry out a series of simulations, as described by Saad et al. (1995), in order to build up effective phase permeability curves. In each simulation, a different fractional flow of water is injected and the simulation is continued until steady state has been reached, i.e. until the fractional flow in the producing well is within a small value (10 4 in this case) of the fractional flow in the injector. Then the oil and water permeabilities are used to calculate the effective phase permeabilities. Unlike the other two methods described above, this method requires fine-scale two-phase flow simulations and is therefore slower. Currently, a fully implicit method for calculating the steady-state watersaturation distribution is being developed for intermediate viscous/capillary ratios (Stephen & Pickup 2000). TESTS OF THE METHODS These three steady-state methods were initially tested using a ripple model, shown in Fig. 1. This model of a small-scale ripple was chosen because it is a good example of a sedimentary unit in which capillary equilibrium is likely to hold. Similarly shaped structures occur at larger scales and, depending on the flow rate, fluids may also come into capillary equilibrium across them. Two versions of the model were investigated, one with a low permeability bottomset and the other with a high permeability bottomset. In reality, the bottomset of a ripple will probably have a low permeability, due to mud drapes, for example. However, such low permeability features may be eroded, so we have also considered high permeability features between the ripples, which provide an example where the low permeability zones are not continuous throughout the model. References to a high permeability bottomset model may not be geologically realistic, but this model provides a useful test for the upscaling methods. The rock relative permeabilities and capillary pressure curves, shown in Fig. 2, represent a water-wet system. (The effects of different wettabilities are discussed later and in Pickup et al. 2000). As in Huang et al. (1995), the connate water saturation is a function of permeability, but the residual oil saturation is constant. These end-points apply for a homogeneous sample. However, in a heterogeneous sample, the final water saturation achieved may be correlated with permeability, depending on capillary trapping. We have assumed that the residual oil saturation in the rock curves was independent of flood rate. In reality the residual oil saturation will tend to decrease with increasing flood rate. However, our assumption does not change the results of this study. Scale-up may be carried out using a variety of boundary conditions, as discussed in Pickup et al. (1994). Here, we have used no-flow boundary conditions, and calculated a diagonal permeability tensor, because the effective permeabilities were compared with Kyte & Berry (1975) pseudos, using the Eclipse simulation package with no-flow boundaries. Table 1 shows the flow rates which were tested. The viscous/capillary ratio, N vc, given in column 4, is the ratio of the viscous to the capillary pressure drop. This ratio may be calculated in a number of

206 Gillian E. Pickup and Karl D. Stephen Table 1. Flow rates used for calculating the pseudos, and the corresponding viscous/ capillary ratios Model number Injection rate (cm 3 per hour) Frontal advance rate (m per day) log(visc/p c ratio) 1 500 700 0.5 2 50 70 1.5 3 5 7 2.5 4 0.5 0.7 3.5 different ways. In this paper it is evaluated at the start of the simulation (where all the quantities are known). N ( P ) ( P ) q μ x y zk P, (5). where q inj is the injection rate (cm 3 s 1 ), µ o is the oil viscosity (cp), Δx is the width of a grid block in the x-direction (cm), Δy and Δz are the width and height of the model (cm), k o is the harmonic average of the oil phase permeabilities in the two laminae (D) and ΔP c is the difference in capillary pressures (atms) between the high and low permeability laminae, evaluated at connate water saturation. (There was φ/k scaling between the two capillary pressure curves.) The frontal advance rates in Table 1 were chosen to give a range of viscous/capillary ratios. Since Δx is small in this model, q inj (and therefore the frontal advance rate) is unrealistically large for rates 1 and 2, but is required to ensure a viscous-dominated flood. In a larger-scale model (in the absence of scale-up), a viscousdominated flood would occur for a lower flux. The fastest flow rate (rate 1) approaches the viscous-dominated case where capillary pressure may be ignored, and the slowest rate (rate 4) approaches the capillary-dominated case. Figure 3 shows the variation of the x-direction (horizontal) effective oil relative permeabilities for each injection rate, and for the viscous-dominated steady-state and the capillary equilibrium cases. (Many of the curves are superimposed, though.) In the high permeability bottomset case, the effective permeabilities do not vary significantly with rate, because the high permeability regions are connected to the outlet, so there is no trapping of oil. However, in the low permeability bottomset case, the oil relative permeability falls to zero at lower water saturations as the flow rate decreases. This is due to oil being trapped in the isolated high permeability regions. (See, for example, Van Duijn et al. (1995) for a discussion of phase trapping.) The water (wetting-phase) relative permeabilities are shown in Fig. 4. They were similar in all cases, except the end-points saturations were lower in cases where there was oil trapping. Pseudo-relative permeabilities were also calculated using the Eclipse simulator and the Kyte & Berry (1975) method. The results for the different flow rates are shown in Figs 5 and 6. The pseudo-relative permeabilities from the Kyte & Berry method always show a dependence on rate, partly because they are calculated to compensate for numerical dispersion, but also Fig. 3. Pseudo-oil-relative permeability curves, calculated using steady-state methods. Fig. 4. Pseudo-water-relative permeability curves, calculated using steady-state methods.

Steady-state scale-up methods 207 Fig. 5. Pseudo-oil-relative permeability curves, calculated using the Kyte & Berry (1975) method. Fig. 6. Pseudo-water-relative permeability curves, calculated using the Kyte & Berry (1975) method. because they take account of the changing balance of viscous/ capillary forces. At faster flow rates, when there is a sharp front between the oil and water, the curves are moved to the right. Note that the pseudo-relative permeability curves for water are always extrapolated to the maximum water saturation (0.7 in this case) by the Eclipse pseudo package although, in practice, the water saturation will not reach this level in cases where oil trapping occurs. Eclipse simulations for fine-grid models, consisting of 20 ripples arranged in the x-direction, were performed. These were compared with coarse-scale simulations using pseudos obtained from the steady-state methods and the Kyte & Berry (1975) method. Figure 7 summarizes the errors in the coarse-scale simulations. The rms errors were calculated from the difference in pore volume injected between the coarse- and fine-scale models, at different water cut levels (0.05, 0.10,... 0.90). Note that the capillary equilibrium method was not run for rates 1 and 2, and the intermediate steady-state method was not run for rate 1. It can be seen that for all the rates, the Kyte & Berry pseudos give the best results, because they compensate for numerical dispersion (in addition to capturing capillary trapping where appropriate). The Kyte & Berry method works very well here, because we have a 1D system at the coarse scale. (However, there are many cases where dynamic methods like that of Kyte & Berry may give poor results, as noted by Barker & Thibeau 1996.) The capillary-dominated method gives a poor result for rate 3 for the low permeability bottomset. This is because the capillary equilibrium assumption produces a high level of trapping, which is not present unless the flow rate is very low. Figure 8 (left) presents the water-cut versus pore volumes injected for rate 3 with the low permeability bottomset case, and shows how breakthrough occurs too early when capillary equilibrium is assumed. This early breakthrough gives rise to an error in final recovery of 23% of the initial oil in place (OOIP). In the slowest rate case (rate 4), the capillary equilibrium pseudo gives a more accurate answer, but the viscousdominated method produces a fairly large error due to late breakthrough, corresponding to an error in total recovery of 17% OOIP. The errors in total recovery in the other cases were much smaller: less than 4% OOIP. Unexpectedly, the viscous-dominated steady-state pseudos performed well in most cases (Fig. 7), even with a scale-up factor of 50 in the flood direction. When a system is viscous dominated, there is a shock front (Buckley & Leverett 1942) which passes through the model, so the system is certainly not in a steady state. The shock front will be spread out because of the increase in block size due to the scale-up, and this should adversely affect the results. In a miscible flood, numerical dispersion behaves like physical dispersion, and the front grows with t. This means that the larger the distance between the injector and the producer wells, the more spread out the front becomes. However, we are dealing with immiscible floods and, in this case, the Buckley Leverett (1942) shock front is maintained. The width of the shock front is spread out over the size of a coarse grid block, but it does not increase with time. In fact, the effect of numerical dispersion is more noticeable over short times (Hewett & Behrens 1993). Figure 9 shows an example of a viscous-dominated flood, illustrating that when

208 Gillian E. Pickup and Karl D. Stephen Fig. 7. RMS errors in pore volumes injected at different water-cut levels: (a) high permeability bottomset model; (b) low permeability bottomset model. See text for details. a) b) Fig. 8. Comparison of water-cuts for the low permeability bottomset case: (a) flow rate 3 (slow); (b) flow rate 4 (very slow). the results are expressed in terms of pore volumes injected, the simulation is more accurate for large numbers of blocks between the injector and the producer. ADDITIONAL TESTS OF THE METHODS These scale-up simulations were repeated for a correlated random field model of the same grid size and a similar permeability range. The model, which is shown in Fig. 10, had a correlation length of approximately 0.13 cm in the horizontal Fig. 9. Water-cut versus pore volumes injected, showing the effect of different numbers of blocks between injector and producer. direction, and slightly less in the vertical direction. The permeability distribution was root-normal with a mean of 8 (median value of 64 md) and a standard deviation of 2. The rms errors in the scale-up, presented in Fig. 11, show that the errors in this stochastic model are comparable to those in the high permeability bottomset case and, in general, are lower than those in the low permeability bottomset case. In the stochastic model, there will be very little trapping at low flow rates, because the high permeability regions are unlikely to be surrounded by low permeability regions. This suggests that in models where there is little or no trapping, the pseudos are relatively insensitive to flow rate, so steady-state scale-up methods may be feasible for a range of flow rates. It is only cases where the low permeability regions form a continuous network, such as the low permeability bottomset ripple example, where steady-state scale-up methods may have problems. Steady-state pseudos have also been calculated for a range of other models, shown in Fig. 12. These models are: (1) a high permeability model, which is compartmentalized by continuous low permeability regions. This model could represent dunes with low contrast in the foreset laminae, but with low permeability bed boundaries. It is an example of a model where oil could be trapped; (2) similar to model 1, except there are holes in the boundaries, so that there is no trapping of oil; (3) a modification of the ripple model, which contains isolated high permeability zones surrounded by low

Steady-state scale-up methods 209 Fig. 10. The correlated random permeability field. viscous/capillary ratio are used. In the other models, the pseudos were relatively insensitive to viscous/capillary ratio, so the capillary equilibrium or the viscous-dominated steady-state methods may be applied for a range of ratios. In the case of Model 3, although there was some trapping, the volume of trapped oil was very small so the pseudos for rates 1 and 4 were similar. Fig. 11. RMS errors in pore volumes injected, at different water-cut levels in the correlated random model. permeability background. Capillary trapping may take place in this model. However, because the area of the high permeability regions is small, the amount of oil trapped will be low; (4) vertical barriers with random holes, which prevent trapping. In each case the steady-state pseudos were compared for rates 1 (very fast) and rate 4 (very slow). The oil relative permeabilities (for the horizontal direction) are shown in Fig. 13 (because there is most variation in the pseudos for the non-wetting phase). As expected, there is a large difference in the pseudos for rates 1 and 4 in Model 1, where a lot of oil was trapped in the slow flow rate case. However, in the model with holes (Model 2), there is no trapping, so the pseudos are similar for the two flow rates. In Model 3, there is trapping for rate 4 but, because the area of the high permeability region is small, the amount of trapping is low, so the pseudos are not very different. In Model 4, there was no trapping, so the end-points are the same for both flow rates. From the tests described earlier, we can deduce that scale-up of Model 1, will not be accurate, unless pseudos for the correct CONCLUSIONS AND DISCUSSION At present, the easiest and quickest methods for scale-up are steady-state methods: capillary equilibrium for slow rates and small length-scales, and viscous-dominated steady-state for fast rates, or at large scales where the effect of capillary pressure and gravity are negligible. In this study, these methods were not as accurate as the Kyte & Berry (1975) method. However, steady-state methods are much faster, and are more robust in the sense that they avoid some of the pitfalls of pseudo-relative permeabilities discussed by Barker & Thibeau (1996). The simplicity of the steady-state methods outweighs the disadvantage of small errors. The accuracy of the Kyte & Berry (1975) method results from the fact that this method compensates for numerical dispersion. However, repeated use of such dynamic upscaling methods tends to over-compensate for numerical dispersion (Christie et al. 1995). Therefore, in a multi-stage scale-up process, the best accuracy will probably be achieved using a steady-state method for some of the scale-up stages, and a non steady-state method for the other stages. This work has highlighted the problem of upscaling where capillary trapping is likely to occur. In this case, non steady-state simulation methods, such as the dynamic methods mentioned above, are significantly more accurate. The intermediate steadystate method is also more accurate in cases where trapping may occur. We are currently developing a direct method for calculating the steady-state water saturation distribution, using a fully implicit solver (Stephen & Pickup 2000). In this study, the models were water-wet. However, it is likely that many reservoirs are of mixed wettability. It is important to use the appropriate capillary pressure curves, reflecting the Fig. 12. Additional models used to test steady-state upscaling. In each case white represents 100 md and grey is 10 md. Models (a) and (b) represent dune bounding surfaces and are 300 50 cm. Models (c) and (d) are only 3 1 cm.

210 Gillian E. Pickup and Karl D. Stephen Fig. 13. Comparison of the oil relative permeability curves of the extra models, for rates 1 and 4 (very fast and very slow). wettability of the rock, which may be heterogeneous at the fine scale (Huang et al. 1996). In a heterogeneously wet rock, there may or may not be trapping, depending on the form of the capillary pressure curves. However, our conclusions still apply, namely, that in models where trapping may occur, scale-up must be carried out more carefully. When choosing a method of upscaling, one should not just consider the balance of forces, but also the nature of the permeability heterogeneity. This work was part of Phase III of the Heterogeneity Project, and authors would like to thank the following companies for sponsorship: Amerada Hess, BG, Chevron, Conoco, Deminex (now Veba), Elf, Esso, Fina, Mobil, Pan Canadian, Petrobras, Phillips, Shell, Statoil, Talisman and the UK DTI. We should like to thank Schlumberger Geoquest for the use of the Eclipse simulation package and Ken Sorbie and Steve McDougall for useful discussions. We are grateful to Don Best of Mobil, and an anonymous referee for their useful comments. REFERENCES BARKER, J. W. & THIBEAU, S. 1996. A Critical Review of the Use of Pseudo Relative Permeabilities for Upscaling. SPE RE, 12, (2) 138 143. BUCKLEY, S. E. & LEVERETT, M. C. 1942. Mechanism of Fluid Displacement in Sands. Transactions of the American Institute of Mechanical Engineers, 146, 107. CHRISTIE, M. A., MANSFIELD, M., KING, P. R., BARKER, J. W. & CULVERWELL, I. D. 1995. A Renormalisation-Based Upscaling Technique for WAG Floods in Heterogeneous Reservoirs. Paper SPE 29127, presented at the 13th SPE Symposium on Reservoir Simulation, San Antonio, Texas, 12 15 February. CORBETT, P. W. M., RINGROSE, P. S., JENSEN, J. L. & SORBIE, K. S. 1992. Laminated Clastic Reservoirs: The Interplay of Capillary Pressure and Sedimentary Architecture. Paper SPE 24699, presented at the 67th SPE Annual Technical Conference, Washington, 4 7 October. DALE, M., EKRANN, S., MYKKELTVEIT, J. & VIRNOVSKY, G. 1997. Effective Relative Permeabilities and Capillary Pressure for One- Dimensional Heterogeneous Media. Transport in Porous Media, 26, 229 260. HEWETT, T. A. & BEHRENS, R. A. 1993. Considerations Affecting the Scaling of Displacements in Heterogeneous Permeability Distributions. SPE FE, December, 258 266. HONARPOUR, M. M., CULLICK, A. S., SAAD, N. & HUMPHREYS, N. V. 1995. Effect of Rock Heterogeneity on Relative Permeability: Implications for Scale-up. Paper SPE 29311, presented at the SPE Asia Pacific Oil and Gas Conference, Kuala Lumpur, 20 22 March. HUANG, Y., RINGROSE, P. S. & SORBIE, K. S. 1995. Capillary Trapping Mechanisms in Water-Wet Laminated Rocks. SPE RE, November, 287 292.,, & LARTER, S. R. 1996. The Effects of Heterogeneity and Wettability on Oil Recovery from Laminated Sedimentary Structures. SPEJ, 1, (4) 451 461. KUMAR, A. T. & JERAULD, G. R. 1996. Impacts of Scale-up on Fluid Flow from Plug to Gridblock Scale in Reservoir Rock. Paper SPE 35452, presented at the SPE/DOE Tenth Symposium on Improved Oil Recovery, Tulsa, 21 24 April. KYTE, J. R. & BERRY, D. W. 1975. New Pseudo Functions to Control Numerical Dispersion. SPEJ, August 1975, 269 275. PICKUP, G. E. & SORBIE, K. S. 1996. The Scaleup of Two-Phase Flow in Porous Media Using Phase Permeability Tensors. SPEJ, 1, (4) 369 381., RINGROSE, P. S., JENSEN, J. L. & SORBIE, K. S. 1994. Permeability Tensors for Sedimentary Structures. Mathematical Geology, 26, (2) 227 250., & SHARIF, A. 2000. Steady-State Upscaling from Lamina- Scale to Full Field Model. SPEJ, June. SAAD, N., CULLICK, A. S. & HONARPOUR, M. M. 1995. Effective Relative Permeability in Scale-up and Simulation. Paper SPE 29592, presented at the Joint Rocky Mountain Regional Meeting and Low Permeability Reservoirs Symposium, Denver, Colorado, 20 22 March. SMITH, E. H. 1991. The Influence of Small-Scale Heterogeneity on Average Relative Permeability. In: Lake, L. W., Carroll, H. B. & Wesson, T. C. (eds) Reservoir Characterization II. Academic Press, Inc. STEPHEN, K. D. & PICKUP, G. E. 2000. A Fully-Implicit Upscaling Method for Accurate Representation of the Balance of Viscous and Capillary Forces. Paper presented at the 7th European Conference on the Mathematics of Oil Recovery, Baveno, Italy, 5 8 September. VAN DUIJN, C. J., MOLENAAR, J. & DE NEEF, M. J. 1995. The Effect of Capillary Forces on Immiscible Two-Phase Flow in Heterogeneous Porous Media. Transport in Porous Media, 21, 71 93. Received 23 August 1999; revised manuscript accepted 7 February 2000.