application of novel upscaling approaches-king-upscaling1999

SPE 50643

Application of Novel Upscaling Approaches to the Magnus and Andrew Reservoirs M.J. King, D.G. MacDonald, S.P. Todd, H. Leung, SPE, BP Exploration Operating Co. Ltd.

Copyright 1998, Society of Petroleum Engineers, Inc. This paper was prepared for presentation at the 1998 SPE European Petroleum Conference held in The Hague, The Netherlands, 20-22 October 1998. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract Cases studies from three North Sea turbidite reservoirs will be presented, which together demonstrate our current understanding of permeability and relative permeability upscaling. The three formations, the Magnus, Magnus Sand Member (MSM), the Magnus, Lower Kimmeridge Clay Formation (LKCF), and the Andrew reservoir each provide distinct challenges for reservoir modelling, either because of reservoir complexity, the fluids in place, or the phase of field life. To meet these challenges, several novel upscaling approaches have been developed. Their use will be explored and current best practice delineated. This best practice differs significantly from previous definitions of effective permeability by placing more emphasis on extracting multiple properties from the fine scale geologic models. Distinct upscaling calculations are required to assess (i) the quality of sands, (ii) the quality of barriers, and (iii) the tortuosity of flow around these barriers. Similarly, when constructing upscaled relative permeabilities, the effective curves are distinguished from the pseudo curves. The former describe the physical displacement of fluids, while the latter include the additional numerical dispersion corrections required when implementing the relative permeability functions within a coarsely gridded full field simulator.

Introduction Three dimensional geologic modelling escaped from the laboratory approximately three years ago, taking with it a variety of geostatistical and upscaling tools. Along the way, it acquired 3D visualisation and a graphical user interface, making it widely

accessible to asset based non-specialists. A result of this proliferation has been a change in working practice, where the same asset-based teams which build the geologic models now upscale them and use them for well planning and reservoir simulation purposes. This has highlighted limitations within the standard approach to upscaling [1, 2] and has accelerated the development of new upscaling methodologies [3, 4].

Perhaps we should remind ourselves of why we are building these large geologic models, and why we need to upscale them. The second question is easier to answer than the first. 1 - 20 - 1000 million cell local area and full field three dimensional geologic models are being built, some fairly routinely. Black oil simulation for routine engineering calculations limits us to a maximum of about 100,000 active cells. Upscaling is the process whereby the very detailed geologic model is reduced to the coarser flow simulation model.

Why do we need such detailed models? Sometimes we dont: coarse grid mass balance style calculations are often adequate for many operational decisions. But, when we wish to improve our mechanistic understanding of the reservoir, or to explore the dynamic implications of different geologic concepts, then these detailed models provide insight that otherwise would be difficult to obtain.

Industry experience in the full life cycle of these upscaled geologic models is still extremely limited. We are still learning why and how to build them, what value they provide, and how to use them in combination with more conventional techniques. The literature includes references where these models have worked extremely well [5, 6]. However, there are many other (unpublished) examples in which the performance prediction from the upscaled model was significantly different than the performance of the reservoir and/or to the performance prediction of sector models drawn from portions of the original detailed geologic models.

This paper is organised around four case studies, chosen to demonstrate different combinations of successful and unsuccessful upscaling calculations. Three of these case studies, and most of the material within this paper, emphasise permeability upscaling. Multiphase upscaling (pseudoisation) is

2 M.J. KING, D.G. MACDONALD, S.P. TODD, H. LEUNG SPE 50643

included as an alternative approach for the Magnus MSM. The first three case studies are preceded by a discussion of effective permeability and followed by a list of elements that define best practice, as we currently understand it. The multiphase upscaling example is much more exploratory, and should be considered work in progress. For convenience all of the formal derivations are relegated to the Appendices.

Effective Permeability What is effective permeability? A cynic could describe it in terms of putting incorrect information into the wrong model, to get the right answer. The wrong model is that carried in a typical full field simulator: homogeneous blocks of numerical rock that extend for perhaps a hundred metres or more laterally, and which may be tens of metres thick. The incorrect information is the effective property. But, what the cynic would emphasise is that we can only define this property based on our expectation of the right answer.

Consider the simplified sketch of a simple sand/shale reservoir zone, in Figure 1a, and focus your attention onto the coarse cell in the centre of the figure. What is the vertical permeability of the cell? With the standard flow based computation of effective permeability [1], the upscaling region is treated as if it were a laboratory coreflood, Figure 1b. The sides of the system are sealed, a pressure drop is exerted vertically, and the pressures and flux are determined numerically. The volumetric flux defines the effective permeability, according to Darcys Law.

(Q )A

K PL

EFF= = 1 .................................................(1) In this particular case the vertical permeability is zero. It corresponds to the right answer in which two of the shales stretch off to infinity. Vertical flow is not possible within the reservoir.

This is certainly not the only possible right answer. Another is sketched in Figure 1c, in which the upscaling volume is embedded within a region of uniform (but unknown) permeability. It is not necessary to explicitly solve the coupled equations as one can show that they reduce to a linear pressure drop on the boundary of the upscaling volume. This effective medium approach is discussed in [4]. The calculated vertical permeability is positive. Streamlines can run from bottom to top of the upscaling volume, spreading beyond the volume to avoid the shales.

Periodic boundary conditions are used in the volume averaging literature [7, 8]. Here the upscaling volume is embedded in multiple replicates of itself, stretching off to infinity. As in the effective medium approach, the numerical calculation can be re-stated as a local calculation on the upscaling volume. In this case the calculated vertical

permeability would be zero. Finally, another possible flow picture is sketched in Figure

1d. Following Begg et.al., [9], the computational region is extended far beyond the upscaling volume. The total vertical flux is positive, due to tortuous flow paths which stretch the full width of the shales. The vertical volumetric flux is only summed within the upscaling volume, although the pressure solution is determined on the entire computational domain. This is a more expensive calculation than any of the local calculations described previously.

Which value of effective permeability should be used? The answer is that any of these values may be the right one

to use, depending upon the flow pattern which one considers to be important. Hopefully this will become clear after examining the three case studies. However, a few additional comments before examining these examples.

The standard upscaling approach is a local approximation. Although the upscaling region is physically embedded in a larger region, upscaling is typically performed without access to this extended information. Hence, the sand and shale patterns in Figures 1a-c are identical within the upscaling region. However, within a local approximation, there are some rigorous results. The sealed side calculation will always give a lower bound, and the linear pressure boundary conditions will give an upper bound (over the natural set of boundary conditions). If the results of these two local calculations differ significantly, then the pre-dominate flow direction is to leave the cell. Both periodic boundary conditions and the use of a wide computational region will give intermediate values.

If the upper and lower bounds are significantly different, as they are in this case, then the cynics perspective is correct. However, if these two values are very close, then you have a pleasant surprise: a representative effective permeability. This happens more often for the horizontal permeability than for the vertical. In this particular case, the upscaled horizontal permeability is very close to the net-to-gross of the upscaling volume, times the permeability of the sand, irrespective of the manner of calculation.

Case Studies: Effective Permeability Models of three submarine fan turbidite reservoirs will be examined in this section. Two of these are sector models and one is a full field model. The fine and coarse three dimensional grids and model sizes are listed in Table 1. All three models have been constructed and upscaled using Smedvigs IRAP Reservoir Modeling System (IRMS version 4.0.7) [10]. The basic upscaling approaches within IRMS follow the methods of [1, 8], which will be demonstrated to be inadequate. Instead, we have needed to make extensive use of the IRMS command language to extend the upscaling toolkit and to provide the results described herein.

SPE 50643 APPLICATION OF NOVEL UPSCALING APPROACHES TO THE MAGNUS AND ANDREW RESERVOIRS 3

Magnus Reservoir, Magnus Sand Member. The Magnus MSM is a large Upper Jurassic, sand dominated, turbidite reservoir deposited above a field wide shale bed, the B shale, Figure 2 [11]. It was discovered in 1974, commenced production in August of 1983, and was on plateau until January of 1995. Decline has been managed since then with an active infill drilling programme, which has also been extremely instructive as to the complexity of the reservoir [12].

The MSM has an average net-to-gross of 76% and an average net porosity of 21%. The reservoir has been depleted with a edge-drive waterflood, with a plateau production rate of ~140 mbd. There is no gas cap and limited aquifer support. The oil is undersaturated, with a gravity of 39 degrees API, a viscosity of 0.48 cp and a GOR of 775 scf/stb. The waterflood is dynamically stable with a mobility ratio close to unity, and a frontal mobility ratio of about 0.3.

The continued infill drilling in Magnus has placed increased emphasis on both the areal and vertical sweep patterns behind the waterflood flood front. Surveillance has shown that in some sands 5 - 10 metres of oil are unswept, in immediate proximity to sands which are being effectively drained. It is for the purpose of understanding these remaining oil targets that detailed reservoir modelling has been under-taken, including the three dimensional cross-sectional model described in [4]. The geocellular resolution of this model was 0.5m vertically, and 30m x 30m areally. The model covered the four reservoir zones above the B shale, each of which was eroded by subsequent reservoir zones. (Of the ~750,000 cells, only ~400,000 were active.) The model extended 2970m from below the OOWC to the crest of the field, and had a width of 630m. Details of the sand and mud facies, and their properties, can be found in that reference.

The simulation was performed using the streamline based Frontsim simulator [13] because of its speed and lack of numerical dispersion. Without upscaling, the convergence of the simulator was impaired because of the large number of pinched-out cells in the geologic model. Even after resampling, simulation took several cpu days when in a history match mode, because of the limited time-step sizes.

To reduce the simulation time to several hours, the decision was made to upscale the model. Working within IRMS, the model was first resampled into a shoebox computational domain of absolutely uniform thickness, which was then upscaled. Upscaled properties were subsequently transferred back to a coarse proportional simulation grid. Local flow directionality and the dip of the geologic layers within each reservoir zone were preserved in the resampling steps using the transmissibility construction derived in [4], and the different tensor representations of permeability within a geologic model and a flow simulator.

The resampling step took the model from a 99 x 21 x 352 3D grid with pinch-out, to a 99 x 21 x 375 layer proportional model. As this computational domain was completely featureless, porosity was converted to cell pore volumes, and directional permeabilities to cell transmissibilities. Over-sampling was utilised to preserve geologic continuity, especially of muds, as shown in Figure 3. Warren and Price flow based upscaling was performed using a uniform 3 x 3 x 3 element to provide a coarse 33 x 7 x 125 layer model. The average cell thickness was less than 1.5m, and so it was believed that this was still a high resolution, mechanistic model. Validation of the upscaling was performed using Time of Flight streamline-based flow visualisation [14, 4].

Was this a successful upscaling calculation? Based on the large scale waterflood performance, as validated using Time of Flight, the answer was yes. However, what about the original question: definition of the remaining oil targets? Unfortunately, inspection of the upscaled permeability patterns showed that lateral dimensions of the muds had increased, sometimes leading to multiple muds merging, generating significantly larger trapped oil accumulations in the upscaled model, than in the original geologic model, Figure 4.

In our experience, this is the single most common error introduced when using flow based upscaling. Another example is given in Figure 5, with more emphasis placed on the loss of sand channel continuity upon upscaling. The sealed sides systematically bias the permeability of sand and mud mixtures towards muds. In a high net-to-gross system, we had not expected this to be a significant effect, and in fact, it does not have an impact on the gross performance of the waterflood. However, as our calculations were intended to understand the detailed habitat of the remaining oil, it was realised that these upscaled models were inadequate.

Andrew Reservoir. The Andrew reservoir was discovered in 1974, and came on production in June of 1996 [15]. The reservoir consists of a low relief structure with four way dip closure, and a relatively thin oil column. The oil is overlain by a gas cap, and is in contact with extensive water bearing sands, which may provide an effective aquifer. The oil is contained within distal Palaeocene turbidite sands, composed mainly of fine to medium grained, clean to moderately poorly cemented sandstones. The reservoir has a high net-to-gross (0.8 to 0.99), medium porosities (16 to 22%), and medium permeabilities. The oil is saturated with gas, 40 degrees API gravity, with a GOR of 871 scf/bbl.

Detailed three dimensional geologic modelling is being used on a fieldwide basis as the fundamental static description of the reservoir. Local area models are extracted, screened and simulated without upscaling, to understand individual well performance. The geologic model is also being upscaled to


construct the full field simulation model. Variations in geologic description are being explored as part of the history match process.

The field is being developed with horizontal wells. Production will be gas constrained, so it is extremely important to understand the extent and continuity of shales, as they will be the major control on the vertical flow of gas within the reservoir.

The Andrew reservoir model was constructed and upscaled using IRMS [10]. Each zone of the reservoir was upscaled separately. The upscaling step reduced the model from approximately 3.8 million cells (140 x 161 x 169) to 74,000 (36 x 44 x 47). After some experimentation it was decided to use proportional layering for the geologic model as it provided better behaved upscaled permeabilities. Because of this simple geologic grid structure, there is no need for a resampling step, as there was for the Magnus MSM.

In the first attempt, permeability was upscaled using the flow base full tensor calculation within IRMS, although only the diagonal terms were retained. Water production is shown for a validation run on a sector model in Figure 6. The fine scale model has the slowest increase in watercut, and the upscaled model has the fastest. Inspection of the three dimensional saturation profiles indicates that the upscaling process has smeared out the shales. Although volumetrically insignificant, the shales are extremely important for understanding the RFT response from the field, and for predicting horizontal well performance. Without the shales to act as barriers, water and gas are free to move vertically throughout the reservoir. The intermediate curve was obtained after modifying the vertical transmissibility multipliers by hand to ensure that the largest (deterministic) shales were correctly modelled as barriers to flow.

The upscaling calculation was unsuccessful, essentially because of the difficulty of modelling the shales. On balance, both deterministic and stochastic shales are important in the Andrew reservoir, e.g., some shales can be mapped over inter-well distances, while others cannot. Nonetheless, most of the stochastic shales are wider than single columns of the coarse simulation model, allowing vertical flow in the reservoir, but requiring that the flow be tortuous.

The upscaling calculation has another feature: volumetrically insignificant shales which form vertical barriers will be modelled as cells with zero permeability, PERMZ = 0. This over-estimates the volume of shale, and also over-estimates its flow impact, as indicated in Figure 7. In a finite difference calculation, we can visualise flow as running along pipes from cell centre to cell centre [16]. With this image in mind, we see that zero permeability not only prevents flow through a cell, but it also prevents flow through the half cells on either side. The spatial resolution can be doubled, and the volumetric impact of shales removed, if instead shale barriers are modelled as

transmissibility barriers, MULTZ = 0. With this construction in mind, a half cell upscaling

approach was developed for the vertical permeabilities: (1) Upscale using the full tensor permeability algorithm, into a

3D grid identical to the full field model in X and Y, but with twice as many layers;

(2) Each cell in the FFM now has two corresponding cells on this grid, and two permeability values. Select the greater of the two PERMZ values for the FFM;

(3) Upscale using the sealed side flow based algorithm, into a 3D grid shifted up one-half cell from the original grid, and centred on the faces of the FFM;

(4) Calibrate the vertical inter-cell transmissibility multiplier from the permeability on the face (from Step 3), and the harmonic average of vertical transmissibilities from the two adjacent cells (from Step 2):

( ) ( )( ) ( )

KZDZ

MULTZKZ DZ KZ DZ

KZ DZ KZ DZkk

k

k k

=

++

+

+1 21

1

2

/

k ......... (2)

The harmonic average is the standard expression for inter-cell transmissibility [16, 17], here simplified since the cross-sectional area is in common throughout the coarse column.

Values of MULTZ typically vary from zero to unity. When sand is juxtaposed again shale, then MULTZ = 0. In regions of uniform properties, MULTZ 1 . Otherwise, MULTZ takes on intermediate values, depending upon the local contrast in vertical permeabilities.

The results of this extra effort have been quite gratifying. In Figure 6, the water-cut prediction of the upscaled model now follows the fine scale prediction extremely closely. In Figure 8, a cross-section through the FFM is shown, with a clearly resolved gas under-run indicated. Finally in Figure 9, the actual gas production of well A04 tracks the most likely prediction.

Magnus Reservoir, Lower Kimmeridge Clay Formation. The Magnus reservoir consists of two units: the MSM described earlier, and a lower unit, the mud-dominated Lower Kimmeridge Clay Formation (LKCF) [18], Figure 2. The LKCF is a geologically complex generally low net-to-gross reservoir, consisting mainly of sequences of thinly bedded sandstones and mudstones inter-bedded on the centimetre to metre scale. The field average net-to-gross is less than 25%, but it varies from ~65% in the crest of the field, to essentially zero on the margins.

The dominant reservoir facies is structureless, fine to coarse grained sandstone, with permeabilities up to 800 mD. These high density turbidites are often deformed by water escape structures, injected sands, and slumping.

As with the MSM, a three dimensional model was constructed to explore different geologic concepts of the LKCF, and to support LKCF development planning. A waterflood pilot in the southern basin of the LKCF has been successful, but its


extension across the field is problematic. Fieldwide RFT response indicates that pressure communication exists within the LKCF, and between the LKCF and the MSM. However, it is not clear what kind of oil recovery efficiencies are likely, nor how the recovery factor will vary with well spacing.

The LKCF sector model was constructed below the crest of the MSM, Figure 2, because of the availability of LKCF core data, and as a compromise between calibration with field performance of the southern basin waterflood, and prediction of crestal recoveries. This model covered a significantly greater gross volume than the MSM model, covering an area of 3.2km x 3.2km, and all six reservoir zones. The geocellular resolution was 50m x 50m areally, and as before, 0.5m vertically. The model consisted of approximately 1.7 million cells, of which about half were active. An upscaled model was required in order to perform well spacing studies in reasonable amounts of simulation time.

A typical geocellular model is shown in Figure 10. The reservoir is mud dominated, and in the model, all sands are in immediate proximity to the muds. As discussed earlier, such sand and mud mixtures are expected to upscale to mud, destroying the reservoir quality. After the success of the half cell upscaling approach for Andrew, it was both simplified and extended for the LKCF.

The extension has to do with the complexity of fluid flow within the LKCF. In Andrew, half cell upscaling was used to improve the vertical resolution of the coarse model. In the LKCF it is as important to retain lateral flow: the sketch of Figure 5 is expected to be typical of the sands within the LKCF. Hence, the approach was extended to all three directions.

It was also possible to simplify the half cell upscaling approach described earlier. Steps (1) and (2) together were used to calculate a cell permeability which was rarely zero, and that only in regions of uniform thick shales. Step (3) independently calculated the inter-cell transmissibility with a second upscaling calculation, which was then encoded as a multiplier with respect to the cell permeabilities in Step (4). However, as long as the cell permeabilities did not vanish, then these inter-cell transmissibilities in no way depended upon the cell permeabilities. Hence, we may avoid the expense of a flow based calculation for the cell permeabilities, and instead replace their determination with a simpler method.

We have chosen to calculate the cell permeabilities using a well productivity based upscaling approach described in the Appendix. In other words, we ask that the productivity of a perforation within each coarse cell of the model, be identical to the average of the productivities within all of the corresponding fine cells. As these hypothetical perforations can have three possible orientations, we obtain three equations for KXEFF, KYEFF, and KZEFF. This may either be viewed as an artifice to calculate non-zero permeabilities, or, one may recognise that

permeabilities enter into the calculation of well connection factors, just as they do the inter-cell transmissibilities [19]. Either equation may be used to calibrate the permeabilities. As discussed earlier, the simulator has higher spatial resolution if transmissibilities are modelled as face properties, instead of cell properties. The advantage of this approach is obvious in Figure 11, where in contrast to Figure 5, the continuity of the sands are preserved.

So, in summary, the half cell upscaling approach for the LKCF consists of: (1) Upscale all three directional cell permeabilities using the

algebraic well productivity algorithm; (2) Upscale across each face of the coarse 3D grid, using the

sealed side flow based algorithm; (3) Calibrate the three inter-cell transmissibility multipliers from

the permeability on the face (from Step 2), and the harmonic average of directional transmissibilities from the two adjacent cells (from Step 1).

There is one additional implementation detail for the LKCF which was not required for Andrew. The LKCF model is constructed of geologic layers which erode at the zone boundaries. It is important that the spatial correlation of sands within the geologic model be preserved when IRMS performs the flow based upscaling calculations, even in the presence of erosion of these layers. This was ensured by reconstructing the eroded portions of each zone so that each coarse cell consisted or apparently non-eroded fine cells. Although each zone was upscaled separately, this reconstruction accessed fine scale permeability information from the adjacent zone before initiating the flow based upscaling calculation.

The results are shown in Figures 12 - 15. In Figure 12, a three dimensional four well flooding pattern through the 1.7 million cell (64 x 64 x 452) geologic model of Figure 10 is shown. The Time of Flight flow visualisation shows the flood progressing from the two downdip injectors to the two updip producers in a fence diagram. Figure 13 shows the same pattern after a 2 x 2 x 6 half cell upscaling, which reduced the model to approximately 70,000 cells (32 x 32 x 72). Both of these calculations had flow rates specified for the four wells. Additional validation is obtained in Figure 14, where the pressure gradients calculated in Figure 12 are applied to the 70,000 cell, half cell upscaled, model. The flooding pattern is extremely similar. Pressures are presented also, and show that internal permeability and flow is being preserved. In contrast, Figure 15 shows a similar flow and pressure visualisation when conventional sealed side flow based upscaling was used. The total flow rate is now only 5% of the previous figure. The pressures indicate that the model has been upscaled into a fairly uniform, extremely low quality reservoir.


Best Practice: Effective Permeability What have we learned from these three case studies? The most important lesson is that upscaling is not simply a mathematical transformation of one three dimensional model into another. Instead, it is very much as described by the cynic: one must decide upon the important flow solution, and select numerical techniques that extract the appropriate effective properties. This is very different than in the upscaling of porosity or net-to-gross, in which there is a physically conserved property (volume) and an obvious manner of calculating its average (addition). Although the lack of an unique upscaling approach may be viewed as an issue, it is in fact an opportunity: we select an upscaling method to extract the information of choice from the detailed geologic model.

The lack of a unique upscaling approach does not mean that there are no guidelines that can be developed. Here is a checklist of six items that have been explored through these examples.

Layering. It is extremely important to preserve the geologic correlations of the fine scale models while upscaling, especially in cross-section. In the Magnus MSM, this rule was violated, but was compensated for by using over-sampling. Andrew explicitly used proportional layering for the geologic model as this fine grid nestled simply into the coarse grid. In the Magnus LKCF, the eroded geologic layers were reconstructed for each of the active coarse cells before upscaling.

In our experience, if this step is not handled carefully, then the geologic model may be randomised upon upscaling. Effective spatial correlation lengths may be limited to fractions of the coarse cell size. Most importantly, shale continuity may be compromised, and upscaled Kv/Kh ratios may approach unity.

Preserve Sand Quality. The most common error in permeability upscaling is to not distinguish between the different uses of permeability within a model. In particular, the sealed side flow based effective permeability calculation of Warren and Price is used far more frequently than it should be. Mathematically it provides a lower bound on the effective permeability. Physically it reduces the permeability of sand and mud mixtures, thickening shales, and narrowing and disconnecting sand channels.

Typically, the productivity of a well in the field starts high and then diminishes. The early production is due to the fluids in immediate proximity to the well, and the quality of the near-by sands. Sustained production requires that these fluids be replenished. The well productivity based upscaling approach calibrates the cell permeabilities to correctly model the first flush of production. The inter-cell transmissibility controls the sustained production.

Preserve Barriers. An excellent use of the Warren and Price upscaling approach is to determine the continuity of reservoir baffles and barriers, especially at the scale of the coarse grid block. In Andrew, it was important to determine if gas could cone directly down from the OGOC, or whether the path of the gas would be far more tortuous and would under-run the shales.

The half cell upscaling approach takes advantage of the redundancy of permeability and transmissibility multipliers within the flow simulators, to model both reservoir quality and barriers simultaneously. Further, modelling barriers as inter-cell transmissibility modifiers has doubled the effective spatial resolution of the simulation, compared to grid based permeabilities. As this benefit is obtained in all three directions, the improvement in volumetric resolution is a factor of 2**3 = 8.

Preserve Flow Around Barriers. In the Andrew and Magnus LKCF examples, inter-cell transmissibilities were calibrated using the Warren and Price algorithm. Is this always the best approach? In the discussion of [9], vertical permeability is thought of as a global property, although its value is placed in individual coarse cells. As the vertical flow of fluid becomes more tortuous, as in Figure 1d, then the tendency of the sealed side calculation will be to still under-estimate flow. As the coarse cell gets thicker compared to the vertical shales and muds, then the calculated permeability may drop to zero, completely removing pressure support from a large portion of the reservoir.

In such a case it is almost certainly preferable to calibrate the vertical transmissibility using the non-local computational region of Figure 1d. However, by so doing, we are in the realm of the cynic, as this contradicts the recommendation to preserve barriers. Again, decide upon which element of the reservoirs performance is most important, and then extract the appropriate upscaled properties.

Validation and Iterations. In our experience, no results from an upscaling calculation have ever been correct the first time. Sometimes it is the upscaling calculation itself which has performed in a manner other than expected. Other times it is the geologic model. More positively, after performing an upscaling calculation one is in a position to ascertain the dynamic consequences of the static model. For example, for the Magnus LKCF, the forward prediction of the geologic model lead to a re-evaluation of the core data, and a decrease in importance of the injected sands.

Some of the validation is conceptual, as in a review of the underlying geologic concepts. However, other forms of validation are purely numerical, as in the comparison of fine and coarse scale simulation of sector models (Andrew) or in the Time of Flight flow visualisation (Magnus MSM and LKCF). In one way or another, distinct forms of validation should be


utilised, corresponding to the distinct uses of permeability already listed.

Still An Approximation! Finally, it should be remembered that upscaling is still an approximation. There will always be permeability distributions which are difficult to upscale. As an example, examine Figure 16, where sands on a mud background are upscaled onto two coarse cells. Most upscaling approaches will merge the two disconnected sands upon upscaling. This problem is most effectively handled by re-gridding the problem. Options are to shift the coarse grid to prevent merger, or to re-grid to a slightly finer coarse mesh, or sometimes to re-grid to a much coarser mesh in which the sands as recognised as isolated.

This concludes the discussion of single phase upscaling. Of the cases studied, two have been successful and one has not. We return to the Magnus MSM example, and consider its upscaling as a multiphase flow example.

Multiphase Upscaling All of the single phase upscaling case studies had one important unspoken assumption: that the upscaled model was to be mechanistic. In other words, it was important that the detailed heterogeneity of the geologic model be preserved, so that we could rely upon the upscaled model to describe the interaction of the displacement mechanisms with this heterogeneity.

In the Magnus LKCF example, we are very close to violating this assumption. In the reservoir, and in the reservoir model, few sands are individually thicker than 2.0m. With the half cell upscaling, the 3.0m thick cells have an effective vertical resolution of 1.5m, allowing these sands, and their contrast from the muds, to be modelled. However, with even a factor of 2 coarser cells, then there would be no expectation of providing a mechanistic simulation of the LKCF using solely single phase upscaling techniques.

In such a case, there is little choice but to include finer scale fluid by-passing and displacement mechanisms within the upscaled effective relative permeabilities, i.e., to use pseudo relative permeabilities. There is an extensive literature on pseudos, which we will not attempt to summarise. However, we do recommend two recent reviews of dynamic pseudos [20, 21] and the literature summarised therein.

The method being presented is a hybrid, with elements selected from different approaches to minimise known artifacts. It is a total mobility approach, in which the phase mobilities

, , are analysed in terms of a total mobility,

, and a fractional water flow,

( )W WS ( )O WS( ) ( ) ( ) S SW W W O W= + S

)( ) ( ) (F S S SW W W W W= . Effective fractional flows and total mobilities will be determined by analysing a suitably averaged sequence of saturation profiles obtained by direct simulation.

The effective phase relative permeabilities will be derived secondarily.

The discussion will also distinguish between the calculation of effective and pseudo relative permeabilities. The former describe the physical displacement of fluids, while the latter include the additional numerical dispersion corrections required when implementing the relative permeability functions within a coarsely gridded full field simulator.

We take advantage of the JBN method for interpreting a laboratory coreflood as it is guaranteed to provide water and oil relative permeabilities which exactly reproduce the laboratory experiments [22, 23]. This approach is similar to that of [24]. However, the combination of the treatment of effective total mobility (as opposed to fractional flow), and numerical dispersion corrections are new.

Fractional Flow. The derivations follow [22, 23]. The analysis is based upon the one dimensional Buckley-Leverett equations for incompressible fluids in a variable cross-section. After appropriate transformations, the pore volume becomes the equivalent of spatial position, and the volume of water injected is the equivalent of time. The JBN approach calculates both the fractional water flow and the water saturation at the outlet, based on the average water saturation and its time derivative.

( ) ( )

( ) ( ){ }F T PV

d S TdT

S Td

dT T S T

W OUTW

W OUT W

,

,

=

= 1

11

................................... (3)

A cross-plot of the two provides a single fractional flow curve that is guaranteed to reproduce the average waterflood performance, ( )S TW , within an analytic calculation. Such an effective fractional flow curve can also be utilised directly within the Frontsim simulator, as the latter is free from numerical dispersion.

How may this result be extended to finite difference calculations? If one integrates the differential equations over the space of a grid block and the interval of a time step, then one obtains a finite difference form for the Buckley-Leverett equations [25, 26]. The average saturation in a grid block is then exactly determined from the time averaged fluxes which enter and leave that block. The order of the numerical scheme is then related to how this time averaged flux function is calculated from block averaged saturations [27].

Ignoring for the moment the time discretization, then the equivalent of the JBN analysis is obtained by calculating the average saturation within the last (outlet) grid block of the hypothetical homogeneous solution.


( )( ) ( )

PV S T

PV S T PV PV ST PV

PV PV

OUT W OUT

W OUT WOUT

=

....(4)

In other words, the average saturation in a volume adjacent to the system outlet (a pore volume of ), can be obtained from the average saturation in the total system at the same time and at the later time of ( )

PVOUT

( )T PV PV PVOUT . Two limits are easy to understand: the use of one coarse

simulation cell, , and the use of many cells, PV PVOUT =PV PVOUT 0 . The first is the standard pseudoisation

approach in which the fractional flow at the outlet of the system is cross-plotted against the average saturation of the total system, SW OUT W= S [28]. The second limit is that of the analytic result, S SW OUT W OUT , , Eqn (3). Hence Eqn (4) provides an interpolant between pseudos for use in fine grid simulation (analytic result) and in coarse grids. Because of the intrinsic relationship between Eqns (3) and (4), we will call Eqn (4) the extended JBN analysis.

The fractional flow function obtained by a cross-plot of and ( )F TW OUT, ( )SW OUT T will include the effects of spatial

truncation, appropriate for a first order finite difference scheme. This is an exact result for a pseudo fractional flow, in the limit of very small time steps.

Once a finite time step size is selected, then ( )F TW OUT, may be averaged over each time interval, providing a result which is exact for finite time steps. Unfortunately, the simulation time step size is rarely known when constructing a pseudo, especially as it may vary during the course of a simulation run. However, the impact of numerical scheme and time step size are easy to understand. For an IMPES scheme, the (time averaged) fractional flow is represented as a function of the initial upstream block averaged saturation. For a fully implicit representation, the fractional flow depends upon the final upstream block averaged saturation [16].

Consider the sketch of Figure 17. The underlying curve is the effective fractional flow function: a cross-plot of instantaneous outlet fractional flow versus the instantaneous outlet saturation. In both numerical schemes, the block averaged saturation is at a higher water saturation than the instantaneous outlet saturation, moving a point on the curve to the right. The time averaged fractional flow is greater than its initial value, but less than its final value, either moving a point up or down, depending upon the numerical scheme. For an IMPES method, the spatial and temporal truncation errors tend to compensate for each other. Although individual points will shift, as the local

CFL number approaches unity, the overall impact of numerical dispersion may be negligible. (This may explain why the performance of multiphase renormalisation improves when no attempt is made to correct for numerical dispersion [29].) For fully implicit schemes, the two errors augment, driving the pseudo fractional flow curve down and to the right.

Total Mobility. None of this development has included discussion of the effective total mobility. It has relied solely upon knowledge of the water and oil volumes as functions of time, with no mention of distributed or averaged pressures. As no total mobility upscaling technique is rigorous [20, 21], the decision was made to use a simple approach instead: steady state upscaling [30 - 32]. In particular, if the fine scale (rock) curves do not vary with facies or poro-fabric, then the upscaled curve is

simply the rock curve, . ( ) OUT W OUTS ,The rock relative permeability curves are typically

determined by a reservoir condition unsteady state experiment, and are available in tabular form. The only remaining decision is how to interpolate between these points, especially in the saturation interval below the Buckley-Leverett shock saturation. The JBN analysis provides the necessary guidance: for one dimensional incompressible flow, the pressure gradient is proportional to the inverse of the total mobility, which is what we use. Further for saturations below the shock saturation, the ( )1 SW curve is evaluated using quadratic interpolation to insure that these saturations are less mobile than the shock saturation, as is required physically.

Magnus Reservoir, Magnus Main Sand. We return to the Magnus MSM example, but now upscale to a coarse grid comparable to the 1997 Magnus Full Field Model (FFM97). The average grid block size is less than 100m laterally, but there is only a single layer for each of the reservoir zones. The only possibility of accessing mechanistic information is through multiphase pseudoisation. Although in the reservoir the zones communicate, the JBN analysis can only be used for one dimensional floods. Hence, each of the four zones are upscaled independently. A wall of injectors is placed on the downdip face of the geologic model, and a wall of producers, updip. The multiple perforations are pressure controlled, although the total injection and production rates are fixed to give a Darcy velocity of 0.3m/day.

As there is a degradation of reservoir quality downdip, the flood is initiated at the OWC. Because of erosion of one zone by the next, each reservoir zone may not be continuous, in which case an attempt was made to include the largest possible volume. For convenience, the number of grid blocks along the flow direction was chosen to be a multiple of three. Transversely the entire width of the model was included in the simulation.


Three dimensional saturation distributions are determined as a function of time using the Frontsim simulator. (All pinched-out cells are now on the boundary of the simulation domain, and so the convergence difficulties mentioned earlier do not arise.) These saturation profiles are averaged using the Gridsim post-processor [13] to obtain a one dimensional saturation profile, at the resolution of the coarse grid. Pore volumes were upscaled to the same resolution, providing the necessary spatial coordinates to apply the extended JBN analysis.

As an example, consider the top reservoir zone of the Magnus MSM. The simulation was performed on a 39 x 21 x 61 cell 3D grid. Initial time steps were set at 60 days, increasing gradually to 730 days late in the simulation. Simulation continued until a water-cut in excess of 98% was achieved. Using Gridsim, the saturation profiles and pore volumes were coarsened to a 13 x 1 x 1 domain. As in the above discussion, we will first determine the effective and then the pseudo fractional flows, followed by total mobility and pseudo relative permeabilities.

The extended JBN analysis is based upon the average saturation. In a traditional laboratory experiment the only volume over which this average is available is the entire system. However, the simulator provides an entire three dimensional saturation profile, and so the average can be selected at will. With 13 coarse blocks, each of which are comparable in length to an FFM97 cell, then we can support 13 different pseudoisation calculations from this one data set. We reference the analysis to each of the 13 outlet faces by using a saturation averaged over the entire upstream profile. This manipulation of the saturation profiles, and the entire pseudoisation calculation, are performed using Excel Visual Basic.

Two features are apparent within these fractional flow curves, Figure 18. At low fractional flows, the foot of the curve continues to steepen as the profile evolves. This is a gravity slump, the importance of which is increasing as the flood proceeds. However, at late times, the rate of recovery appears to be essentially independent of cell, i.e., independent of the viscous to gravity ratio.

Figure 19 shows the spatial numerical dispersion correction, with a cross-plot of the outlet block saturation, versus the outlet saturation. For the first series, there is only one coarse block in the profile, providing a significant difference between these two saturations. For the last series (series 13), the differences are quite small, giving essentially a straight line. Once there are more than six or seven blocks, the numerical dispersion corrections are quite small. At late times, the saturation profiles are more uniform, as the flood approaches the irreducible water saturation. Then, even for the early series, the numerical dispersion corrections are small.

Figure 20 shows the pseudo fractional flow curve: a cross-plot of outlet fractional flow versus outlet block saturation.

There is far more spread between these curves, than the effective fractional flows. Apparently, numerical dispersion is more important than the physical displacement mechanisms. The impact of gravity is still present, but not as obvious as in the effective curves.

Figure 21 is a plot of the inverse mobility function, determined from the rock curves. There is no evidence that the rock relative permeabilities vary significantly with facies or with rock fabric for any of the reservoir facies. With a single set of rock water and oil relative permeability curves, then the steady state upscaling approach reproduces this curve. The parabolic fit is used to simplify the evaluation of the inverse mobility, especially at saturations below the shock saturation.

Figures 22 and 23 show the result of combining the inverse mobility function with the pseudo fractional flow curves. The effects of numerical dispersion and of gravity are both evident in these final curves. At late times, dispersion is less important but the gravity slump has had more opportunity to evolve. For early times dispersion is extremely important but the gravity slump has had little opportunity to act.

The question remains: which of these 13 pairs of pseudo relative permeability curves should be used in the full field model? The answer depends upon the injection - production well spacing. A typical well distance within FFM97 is about 400m, and so the series 4 set of curves were applied. These pseudos differed significantly from previous calculations (Dykstra-Parsons, and simulation based) in predicting that when water first arrived at a producer it would only provide a limited increase in water-cut, followed by an extended period of negligible water-cut increase. This is much more typical of the field response than the predictions based on the previous pseudos.

Multiphase Discussion The Magnus MSM pseudoisation has demonstrated the ability to separately resolve physical mechanisms, e.g., gravity slumping, and numerical dispersion. The resulting pseudo curves appear reasonably monotonic and can be used in simulation with little modification. This approach has extended the classic JBN analysis to include numerical dispersion corrections in the fractional flow.

The upscaling of total mobility has been significantly simplified with the use of steady state upscaling, removing much of the difficulty in its calculation. In a one dimensional model of fluid flow, the total mobility decouples from the prediction of saturation, and so the latter cannot be used to invalidate this simplification. Numerically calculated pressures may potentially be used for this purpose, but their averaging from a three dimensional distributed system to a one dimensional pressure profile has sufficient ambiguity that again may make validation (or invalidation) of this simplification difficult.


One important implementation detail in this approach was the treatment of total mobility below the Buckley-Leverett shock saturation. If simple linear interpolation is used, then the total mobility will be artificially high. This is definitely non-physical, as the two phase total mobility must decrease as the phases compete for the pore space. The pseudos calculated in this manner will be artificially increased in this saturation range, leading to non-monotonic pseudo relative permeabilities, potentially with values greater than their endpoint values.

Many questions remain with this approach. We have separated the reservoir zones into four distinct calculations. Gravity slumping is permitted within a zone during the pseudoisation, and between zones when used in the full field model. Is this a valid separation? What about the requirement for incompressible flow? It is definitely required to preserve the scaling behaviour of the Buckley-Leverett solution. Are other treatments possible when rock or fluid compressibility are not negligible? Finally, what about the interaction between upscaled absolute permeability and the pseudoisation? The current development is suitable for when permeability is smoothly varying, but is likely to over-emphasise the impact of heterogeneity otherwise. Extensive use has been made of the exact nature of the JBN formulation, including the numerical dispersion corrections. However, the impact of time step size is difficult to pre-calculate, as time steps in the simulator are often quite large (fully implicit) and variable in magnitude (IMPES and fully implicit).

After this work was completed, the authors received a preprint from Stanford University, which applied similar numerical dispersion corrections [33]. Their application was to one dimensional waterflood, which allowed a treatment of total mobility upscaling identical to that of JBN for laboratory coreflood. Interestingly, the resulting pseudo curves are much more prone to non-monotonicity that those obtained in the present study, although whether this is due to the one dimensional floods, or to the treatment of total mobility, must be added to the list of open questions.

Conclusions 1. The upscaling of single phase permeability and

transmissibility has been shown to be a critical step in extracting maximum value from detailed geologic models. When done with understanding, permeability upscaling has been demonstrated in several case studies to provide excellent reservoir performance prediction.

2. It is recommended that permeability upscaling be performed for distinct reservoir properties: sand quality, barriers, and flow around barriers.

3. Half cell upscaling has been developed to simultaneously extract as much of this information as possible from the geologic model. It also provides roughly a factor of 8

improved resolution compared to standard approaches. 4. Layering, validation, iterations, and managing expectations

are as important elements of upscaling as those listed above. A checklist of six items has been provided to remind the practitioner of this.

5. A novel approach to multiphase upscaling has also been developed. This treatment extends the laboratory-scale JBN approach to include numerical dispersion effects, and allowed the separate evaluation of physical and numerical components within the pseudo relative permeabilities.

Nomenclature = Phase mobility, L3t/m = Viscosity, m/Lt A = Area, L2 da = Area element, L2 dq = Flux density, L/t dv = Volume element, L3 DV = Cell volume, L3 DZ = Cell thickness, L F = Fractional flow, dimensionless K, K[X|Y|Z], PERMZ = Permeability, L2MULT[X|Y|Z] = Transmissibility modifiers, dimensionless n = Normal vector, L2

P, P = Pressure, Pressure drop, m/Lt2 PV = Pore Volume, L3 Q = Volumetric flux, L3/t r = Pressure drop vector, L S = Saturation, dimensionless T = Volume Injected, L3

= Face Transmissibility, LTFace4t/m

u = Darcy velocity, L/t x = Position vector, L Subscripts & Superscripts , S = Upscaling surfaces = Upscaling domain EFF = Effective value h = Horizontal i, j, k = Cell indices O = Oil OUT = Outlet, Outlet block v = Vertical W = Water Acknowledgements The material in this paper, and its exposition, were developed in the course of extensive discussions with many colleagues within BP, too numerous to list individually. However, special thanks


go to the Magnus Subsurface team, especially Ken Wells and Dave Richards, for providing the foundations for much of this work, to Alistair Jones for his review of the multiphase upscaling literature, and to Paul Bowden and Pat Neeve for their work with the Andrew simulations. External to BP, special thanks to Lou Durlofsky, Chris Farmer, Tom Hewett, Lindsay Kaye, Don Peaceman, Jens Rolfness and Jeb Tyrie.

The authors would like to thank the Magnus License partners and the Andrew License partners for their permission to publish this paper.

The views expressed in this paper are those of the authors and do not necessarily represent those of BP Exploration.

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11. Shepherd, M.: "The Magnus Field, Blocks 211/7a, 12a, UK North Sea", from I.L. Abbotts, ed., United Kingdom Oil and Gas Fields, 25 Years Commemorative Volume, Geological Society Memoir No. 14, 153 (1991).

12. Day, S., Griffin, T., and Martins, P.: Redevelopment and Management of the Magnus Field for Post-Plateau Production, SPE 49130, 1998 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, (2730 September 1998).

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17. ECLIPSE 100 Technical Reference Manual, GeoQuest, Abingdon, UK, (1997).

18. Speers, R.: The Magnus LKCF Reservoir, ibid. 19. Peaceman, D.W.: "Interpretation of Well-Block Pressures in

Numerical Reservoir Simulation With Nonsquare Grid Blocks and Anisotropic Permeability," SPEJ 531, (June 1983).

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21. Barker, J.W., and Thibeau, S.: A Critical Review of the Use of Pseudorelative Permeabilities for Upscaling, SPE Reservoir Engineering, 138, (May 1997).

22. Johnson, E.F., Bossler, D.P., and Naumann, V.O.: Calculation of Relative Permeability from Displacement Experiments, Journal of Petroleum Technology, 61, (Jan. 1959).

23. Jones, S.C., and Roszelle, W.O., Graphical Techniques for Determining Relative Permeability From Displacement Experiments, Journal of Petroleum Technology, 807, (May 1978).

24. Muggeridge, A.H.,: Generation of Pseudo Relative Permeabilities from Detailed Simulation of Flow in Heterogeneous Porous Media, Proceedings of the 2nd International Reservoir Characterization Technical Conference, NIPER/DOE, Dallas, TX, (25-28 June 1989).

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26. Ekrann, S., and Mykkeltveit, J.: "Dynamic Pseudos: How Accurate Outside Their Parent Case, SPE 29138, 13th SPE Reservoir Simulation Symposium, San Antonio, TX, (12-15 February 1995).

27. Shubin, G.R., and Bell, J.B.: Higher Order Godonov Methods for Reducing Numerical Dispersion in Numerical Simulation, SPE 13514, 8th SPE Reservoir Simulation Symposium, Dallas, TX, (10-13 February 1985).

28. Kyte, J.R., and Berry, D.W.: New Pseudo Functions to Control Numerical Dispersion, SPEJ, 269, (August 1975).

29. Christie, M.A., Mansfield, M., King, P.R., Barker, J.W., and Culverwell, I.: A Renormalisation-Based Upscaling Technique for WAG Floods in Heterogeneous Reservoirs, SPE 29127, 13th SPE Reservoir Simulation Symposium, San Antonio, TX, (12-15 February 1995).

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31. Smith, E.H.: The Influence of Small-Scale Heterogeneity on Average Relative Permeability, in Reservoir Characterisation II, Acaemic Press, 52, (1991).


32. Saad, N., Cullick, A.S., and Honarpour, M.M.: Effective Relative Permeability in Scale-Up and Simulation, SPE 29592, Joint Rocky Mountain Regional Meeting and Low Permeability Reservoirs Symposium, Denver, CO, (20-22 March 1995).

33. Hewett, T.A., Suzuki, K., and Christie, M.A.: Analytical Calculation of Coarse-Grid Corrections for Use in Pseudofunctions, Proceedings of the 6th European Conference on the Mathematics of Oil Recovery, (Sept. 1998).

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35. Kreyszig, E.: Advanced Engineering Mathematics, John Wiley & Sons, Inc., New York, (1993).

36. Pickup G.E., Jensen, J.L., Ringrose, P.S. and Sorbie, K.S.: "A Method for Calculating Permeability Tensors using Perturbed Boundary Conditions," Proceedings of the 3rd European Conference on the Mathematics of Oil Recovery, Delft, (1992).

37. King, M.J.: "Application and Analysis of a New Method for Calculating Tensor Permeability" in New Developments in Improved Oil Recovery, ed. H.J. de Haan, Geological Society Special Publication 84, Geological Society Publications, Bath, UK, (1995).

38. Avatsmark, I., Barkve, T. and Mannseth, T.: "Control-Volume Discretization Methods for 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs," SPE 38000, 14th SPE Reservoir Simulation Symposium, Dallas, TX, (8-11 June 1997).

Appendix: Upscaling Fundamentals In the text, three styles of single phase upscaling have been described without explicitly describing the algorithms. Two of them (upscaling for permeability, and upscaling for transmissibility) can be formulated without reference to a particular fine scale 3D grid. In contrast, upscaling for well productivity (well connection factors) relies explicitly on the coordinate system of the fine grid, and is most easily used when the fine and coarse 3D grids are in alignment. Upscaling for Permeability. Consider the upscaling of a region of heterogeneous porous media, Figure A-1. Flow at each position within the region is described by Darcys Law

( ) ( ) ( ) ( )u x K x P x= = 1 .................................. (A-1) which we may integrate over the domain, , to define the effective permeability.

u K P K PEFF = ...................... (A-2) This is a similar approach to Bears use of a representative element volume [34] to define a continuum property. In distinction, an effective property is only representative if the scales of heterogeneity are small compared to the extent of the averaging region. Otherwise the results may depend strongly upon the size and detailed placement of the averaging region. The definition of Eqn (A-2) is suitable in either case.

The averaged velocity involves a volume integral over the

domain, while the averaged pressure gradient may be converted to a surface integral [35].

udv K n P daEFF $

........................................... (A-3)

where is the outwardly directed unit normal vector. In general, the volume averaged Darcy velocity and the volume averaged pressure gradient need not be aligned, necessitating the use of a full permeability tensor [8, 36, 37].

$n

The standard upscaling calculation for permeability due to Warren and Price [1] is a special case of this definition, in which the numerical calculation is configured to emulate a laboratory coreflood. In this treatment the domain has a simple rectilinear shape, with sealed sides and a uniform pressure gradient exerted along the axis of the system. For incompressible fluids, the integrals in Eqn (A-3) are simple to evaluate

( )( )udv Q A L A x = $ , and ($ $n P da x P A

= ) , giving an

effective permeability of K Q LXXEFF = P , as expected from

Darcys Law. Upscaling for Transmissibility. The above definition of effective permeability has attempted to construct an intrinsic property, i.e., one which does not depend strongly upon the volume of the upscaling region. Both the velocity and pressure gradient integrals are performed over the same (fine scale) domain, while the ratio, the effective permeability, is reasonably defined as a property on the coarse scale. In contrast, transmissibility is defined as the volumetric flux per unit pressure drop across a cross-sectional interfacial area.

Q TFace Face P= ......................................................... (A-4) Transmissibility explicitly depends upon both the specification of an interfacial area and a distance over which the pressure drop is exerted. Hence it is important that both this distance and the interfacial area be defined identically on the fine and the coarse scales.

A construction with these elements in sketched in Figure A-2. Consider first a planar area representing the coarse cell interface. (For a corner point cell, the planar approximation is typically taken from the center of the cell face [16], although generalisations are possible [38].) Define the fine scale interface, S, from the perpendicular projection of this area onto the boundary of the upscaling region, . From the divergence theorem applied to the volume subtended by the cell face and S, it follows that

( )$ $n n da nFaceS

Face = .................................................. (A-5) indicating that the coarse interfacial area may be obtained from an integral over the fine, so long as the fine area is reduced by


its perpendicular projection. The total flux through S may be calculated in an almost identical fashion,

( )Q n nFace FaceS

= $ $ dq ................................................. (A-6) where now dq dq

dada=

is the outwardly directed flux density

on S. To complete the construction for transmissibility, it is

necessary to define the pressure drop across the upscaling region. Figure A-2 includes a reference vector, r , which for the half cell transmissibility, points from the centre of the coarse cell to the centre of the interfacial area. With this vector, the pressure drop may be calculated from the average pressure gradient, defined as in the construction of effective permeability, ( )

P r P r n P da dv = $

. The resulting

expression for effective transmissibility

( ) ( )$ $ $n n dq T r n P da dvFaceS

Face =

Q

........... (A-7)

only includes terms evaluated on the fine scale. With all the apparent complexity of this expression, for the

Warren and Price calculation each of the three integrals are

simple to evaluate: , ( )$ $n n dqFaceS

=( ) ( ) = r n Pda L P A$

, and . The resulting

transmissibility is given by

(dv L A = )

T Q PX = , again as expected. Upscaling for Well Productivity. Following [19], the productivity of a vertical well is proportional to the horizontal permeability, times the gross length of the wellbore.

Well Pi KX KY DZ~ .......................................... (A-8) Consider the stack of fine cells of Figure 3. We can drop a vertical well through each of the nine fine columns, and calculate the total productivity of each. On the possibility that different columns have different cross-sectional areas, we can define the average productivity as the areal weighted sums of the PIs. This in turn defines an average horizontal permeability. Similarly, consideration of well productivities for horizontal wells in the i and j directions, provides two other averages.

{ }{ }{ }

KX KY DV KX KY DV

KX KZ DV KX KZ DV

KY KZ DV KY KZ DV

EFF EFFi j k

i j k i j ki j k

EFF EFFi j k

i j ki j k

i j k

EFF EFFi j k

i j k i j ki j k

=

=

=

, ,, ,

, ,, ,

, ,, ,

, ,, ,

, ,, ,

, ,, ,

.. (A-9)

Once these averages have been taken, they can be converted to three directional permeabilities algebraically. The three directional permeabilities are defined with respect to the fine grid. If the coarse grid has a different orientation, then the elements of the resulting permeability tensor will need to be evaluated in the coarse coordinate system before use.


Figure 1 - (a) A simple sand/shale reservoir zone with three different calculations for effective vertical permeability (b) With sealed sides, the vertical permeability vanishes (c) With linear pressure boundary conditions the vertical permeability is positive (d) With a wide computational region the vertical permeability is also positive as flow diverts around the shales.

Figure 2 - Magnus reservoir stratigraphy showing the MSM and LKCF sector model locations

Figure 3 - Resampling of the Magnus MSM geologic model, 99 x 21 x 352 to 99 x 21 x 375, showing the 3 X 3 fine columns which upscale to one coarse column.

Figure 4 - Detailed examination of the Magnus MSM permeability pattern shows that mud dimensions have increased at the expense of sand channels, leading to an overly optimistic estimate of remaining oil targets.

Figure 5 - Loss of sand permeability and channel continuity when upscaling using the standard flow based (sealed sides) upscaling algorithm. The dot signifies a well, whose performance is significantly degraded within the upscaled model.

Figure 6 - Contrast of the Andrew reservoir performance prediction calculated from a fine scale sector model, that model upscaled using conventional techniques, and after adjusting some of the vertical transmissibility multipliers by hand. The half cell technique gives answer indistinguishable from the fine scale prediction.


Figure 7- Two representations of shale within a flow simulator. In the first, PERMZ=0, and the effective thickness of the shale is 2*DZ. In the second, MULTZ=0, and the shale now has no volume but impacts the flow as if it had the minimum thickness of DZ.

Figure 8 - Performance of the Andrew FFM, showing the resolution of gas under-run, after use of the vertical half cell upscaling approach.

Figure 9 - Successful prediction of GOR in the Andrew A04 well. The most likely case corresponds to the simulation of the previous figure, while the P90 case follows from the use of conventional

upscaling.

Figure 10 - LKCF geocellular model, showing sands (yellow) and regions of injected sands (red) within a mud background.

Figure 11 - Well Productivity based permeability upscaling, preserving the sand channel of Figure 5. Inter-cellular barriers are now modelled as transmissibility multipliers on the face of the cells. Colour coding in this figure is only approximate.

Figure 12 - Time of Flight visualisation of the flooding pattern in the 1.7 million cell (64 x 64 x 452) LKCF geologic model, with two downdip injectors and two updip producers. The flow calculation is in three dimensions but the results are presented as a fence diagram.


Figure 13 - Time of Flight visualisation of the flooding pattern in the 70,000 cell upscaled LKCF model (32 x 32 x 72), using the half cell upscaling approach. As in the previous figure, the flood is in three dimensions.

Figure 14 - Additional validation, showing pressures and time of flight for the 70,000 cell model. In this calculation pressures are specified at the wells, instead of the fluxes, as in the previous two figures.

Figure 15 - Pressure and time of flight information for the 70,000 cell model upscaled using conventional sealed side flow based upscaling. This model has only 5% of the effective connectivity of the model in the previous figure.

Figure 16 - A two cell coarse model whose effective horizontal permeability is very difficult to represent. Most calculations would merge the non-communicating sands into one larger sand.

Figure 17 - Pseudoisation of Effective Fractional Flow for IMPES and Fully Implicit numerical schemes


Figure 18 - Effective Fractional Flows for the 13 coarse cells of the Magnus MSM simulation

Figure 19 - The outlet block water saturation plotted against the outlet water saturation. With few cells the dispersion correction is important. At late times, saturation profiles are more uniform, and dispersion corrections are not important.

Figure 20 - Pseudo fractional flow curves. Dispersion provides more of an impact on the different curves than the physical mechanisms

Figure 21 - Rock curve inverse total mobility function, showing the quadratic fit used, especially below the shock saturation

Figure 22 - Pseudo relative permeability to water

Figure 23 - Pseudo relative permeability to oil


Figure A-1 - Upscaling of effective permeability

Figure A-2 - Upscaling of effective transmissibility

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