Development and Application of a New Technique for ... - OnePetro

SPE 89435 Development and Application of a New Technique for Upscaling Miscible Processes 1

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Mun-Hong Hui, SPE, Dengen Zhou, SPE, Xian-Huan Wen, SPE and Louis J. Durlofsky, 1 2 Stanford University, ChevronTexaco Energy Technology Co. Copyright 2004, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2004 SPE/DOE Fourteenth Symposium on Improved Oil Recovery held in Tulsa, Oklahoma, U.S.A., 17–21 April 2004. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal 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 a proposal of not more than 300 words; illustrations may not be copied. The proposal 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 Miscible gas injection offers both high local displacement efficiencies and a means for handling produced gas. As fine scale simulations of miscible processes can be very timeconsuming, there is a clear need for an accurate and robust coarse scale miscible simulation capability. In this paper, we present a new upscaling technique for the accurate coarse scale simulation of first-contact miscible displacements. Our method contains two key components: Effective Flux Boundary Conditions (EFBCs) and the Extended Todd & Longstaff with Pseudo relative permeabilities (ETLP) formulation. EFBCs, which were applied previously within the context of immiscible displacements, account approximately for the effect of the global flow field on the local upscaling problems and as a result mitigate inaccuracies introduced by standard procedures. The ETLP formulation modifies the way effective fluid properties and upscaled relative permeabilities are computed in a limited compositional framework such that residual oil that is immobile and unavailable for mixing is efficiently represented. The accuracy of the new technique is demonstrated for several example problems. Using synthetic models with differing permeability correlation lengths, we show that the new method provides results superior to those obtained using existing upscaling procedures (standard upscaled relative permeabilities, upscaled absolute permeabilities only, and nonuniform coarsening) for partially layered cases. The breakthrough time, oil production curve and saturation profiles are predicted accurately. The method is also shown to perform well over a wide range of coarsening factors (4 – 2500) for several fields with different permeability heterogeneity structures. Finally, we apply the method to a real field case with encouraging results, indicating the practical applicability of the new technique.

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Introduction In many oil fields with significant amounts of associated gas, miscible gas injection is a potentially attractive recovery method because it can yield high local displacement efficiencies and may also offer a solution for gas-handling. In a miscible gas displacement, channeling and viscous fingering occur due to heterogeneities and the adverse mobility ratio between the injected gas and oil.1 These phenomena need to be properly captured for an accurate estimation of displacement efficiency but the grid resolution required is extremely high. This fact, coupled with the high number of components2 and complex phase behavior, render fine scale simulations of miscible processes prohibitively timeconsuming. For this reason, most field scale simulations of miscible processes are not performed on very fine grids. Approaches to modeling miscible/near-miscible displacements can be broadly divided into two categories. For multi-contact miscible processes, fully compositional (FC) simulations are generally required. When first-contact miscibility is applicable, the limited compositional (LC) formulation3-5 may be preferable due to its computational efficiency. The LC methods allow the simulator to model miscibility within a black-oil framework and empirically account for viscous fingering by modifying the fluid properties of the pseudocomponents. Recently, compositional streamline techniques6-9 have been proposed to efficiently perform FC simulations, and these may eventually allow for fine scale simulations in cases of practical interest. However, issues associated with capillary pressure, gravity, compressibility, streamline updating and nonuniform initial conditions have yet to be fully resolved. Zhang and Sorbie10 proposed an LC approach that employs upscaled relative permeabilities to better capture the effects of permeability heterogeneity. However, they did not account for the fact that, within reservoir simulation length scales, there exists an amount of oil that is practically immobile and not available for mixing (Sorb). The computation of effective fluid properties and upscaled relative permeabilities therefore should not include this Sorb. Moreover, they were mainly concerned with matching experimental results and therefore considered relatively homogeneous cases and a small coarsening factor. Fully compositional approaches involving pseudoization11-14 have also been investigated, but again they did not include the Sorb concept.

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In this work, we present a novel upscaling method for the fast and accurate coarse scale simulations of first-contact miscible processes. The method has two key components: the use of Effective Flux Boundary Conditions (EFBCs)15 for the calculation of upscaled (pseudo) relative permeabilities and the Extended Todd & Longstaff with Pseudo relative permeabilities (ETLP) formulation. EFBCs have been previously applied for the local computation of upscaled relative permeabilities in immiscible cases.16 They act to incorporate approximate global flow information into the upscaling calculation. In particular, EFBCs suppress the flux through a high permeability streak if the streak is not continuous throughout the domain. This property rectifies the problem of premature breakthrough, which can occur due to the over-estimation of flux through cells when using standard boundary conditions. We note that none of the previous approaches entailing upscaled relative permeabilities mentioned above employed specialized boundary conditions such as EFBCs. Our ETLP formulation is a limited compositional approach that is based on the Todd and Longstaff3 method. By adopting the Sorb concept, we will show that the behavior of upscaled relative permeabilities is improved. We investigate the performance of the method for a large number of cases, evaluating the impact of varying correlation structure and coarsening factor. Using synthetic models of dimensionless permeability correlation lengths (lx) ranging from 0.1 to 5.0, we systematically compare the use of standard boundary conditions against EFBCs and also explore two different saturation averaging schemes (volume and outlet averaging). We conclude that EFBCs, when used in conjunction with the outlet averaging of saturations, provide the most effective method for partially layered systems (lx ≤ ~1.0). For these cases, our method yields coarse scale simulation results that match the fine scale solutions closely in terms of breakthrough time, oil production curve and saturation profile. The observation that EFBCs are inappropriate for systems with very high degrees of layering is consistent with the assumptions of the model underlying EFBCs. In fact, their applicability can be quantified via a parameter (Nmf) defined by Wallstrom et al.,16 which we discuss below. In our evaluation of the performance of the method for varying coarsening factor, we quantitatively compare the fractional oil cut and breakthrough errors given by the proposed method against other commonly used techniques such as standard upscaled relative permeabilities, single-phase upscaling and nonuniform coarsening. We test the methods on two fields with different degrees of heterogeneity over a wide range of coarsening factors (4 – 2500). The upscaling errors given by the proposed method are generally lower compared to the other methods. We observe that the errors do not necessarily decrease with increasing refinement and this observation can be attributed to the interplay of numerical dispersion, heterogeneity and displacement physics. Finally, we apply the method to an actual field model and consider a range of coarsening factors (114 – 432). Although the Nmf value of the field suggests that EFBCs may not be fully applicable, our method still outperforms the other upscaling techniques.

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Upscaling Methodology Here, we will describe the proposed miscible upscaling method, applicable for the limited compositional formulation. The technique consists of two main components: the use of Effective Flux Boundary Conditions (EFBCs) for the calculation of upscaled relative permeabilities and the Extended Todd & Longstaff with Pseudo relative permeabilities (ETLP) formulation. Effective Flux Boundary Conditions. Wallstrom et al.15 formulated the theory of EFBCs as a means of prescribing better flux boundary conditions for the local calculation of upscaled relative permeabilities ( k rj* ). This scale-up approach is in a sense nonlocal because some global flow information is used to enhance the estimation of the local flow field. EFBCs can be motivated by analyzing an idealized problem. Consider an inclusion of homogeneous isotropic permeability (kI) embedded in an infinite background of homogeneous isotropic permeability (kB). In the context of reservoir simulation, the inclusion and background represent a fine scale cell and the surrounding large scale domain respectively. Imposing a boundary condition such that the velocity tends to a constant value far away from the inclusion (u → u0), the velocity through the inclusion can be determined explicitly and is given by the following expression15: ⎛ ⎞ kI ⎟⎟ u 0i e i , u I = ∑ Ri ⎜⎜ (1) ( ) + − k k R 1 I B i i ⎝ ⎠ where ei is the unit vector along the ith axis of the inclusion and Ri is the analytically computed asymptotic flux ratio. As the inclusion permeability becomes large compared to the background permeability; i.e., kI / kB → ∞, then uIi / u0i → Ri. This indicates that the flux asymptotically approaches a maximum value rather than continuing to scale with kI. Conversely, when the inclusion permeability is small; that is, as kI / kB → 0, then u Ii k I ⎛ Ri ⎞ (2) → ⎟, ⎜ u 0i k B ⎝ Ri − 1 ⎠ and the inclusion flux will scale with kI.

Fig. 1 – Flux attenuation behavior of EFBCs for a circular inclusion

Fig. 1 illustrates the solution to Eq. 1 for a circular inclusion. The analytical solution indicates that velocity scales with kI when kI / kB is small but approaches an asymptotic value of 2 when the ratio gets large.

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EFBCs are designed to capture the phenomenon illustrated in Fig. 1. To see the connection between the analytical result in Fig. 1 and the choice of boundary conditions to use in relative permeability upscaling, consider the local fine scale region corresponding to a single coarse scale grid block in Fig. 2. In order to compute k rj* for this block, we need to specify boundary conditions for pressure (or flux) along both the inlet and outlet edges of the block as well as saturation along the inlet. In the fairly typical case in which we have permeability streaks that span the coarse block but do not span the global system (as in Fig. 2), fine scale simulations display results in qualitative agreement with the analytical result shown in Fig. 1. Specifically, velocities through high permeability regions reach some maximum beyond which they no longer increase with increasing local permeability.17 Flux through this high permeability streak is suppressed

Coarse cell to be upscaled

using EFBCs Low perm

High perm

Fig. 2 – A local upscaling problem illustrating the principle of flux attenuation in EFBCs

The standard boundary conditions that are commonly used in upscaling (fixed pressures on opposite boundaries, say p = 1 and p = 0) do not give the flux attenuation behavior illustrated in Fig. 1. Rather, these boundary conditions generally give local velocities that scale close to linearly with kI over the entire range of kI / kB (as shown by the dashed line in Fig. 1). This leads to a large overestimation of the effects of high permeability regions and the resulting escalation of the upscaled relative permeability of the injected phase. This in turn leads to coarse scale predictions with a clear pessimistic bias (e.g., coarse scale models that predict early breakthrough). In analogy with Eq. 1, EFBCs specify the flux at the inlet and outlet cells of the local region (i.e., on the fine cells corresponding to the coarse block boundaries). To compute k rj* in the x direction for a 2D (x-z) problem, the inlet and outlet dimensionless fluxes are specified as follows16: k x 1,k k x n x ,k , u x n +1 2 ,k = , (3) u x 1 2 ,k = x k x 1,k + k B ,x k x n x ,k + k B ,x uz

i ,1 2

= uz

i ,n z +1 2

= 0,

(4)

where nx and nz are the number of fine cells comprising the coarse block in the x and z direction respectively. The background permeability kB is computed from a global singlephase steady state solution. In Eq. 3, the outlet flux must in general be rescaled such that the sum of the inlet and outlet fluxes is equal. The saturation is specified to be constant (and equal to 1) along the inlet face. We note that the application of EFBCs to compute k rj* has been shown to yield excellent results for a number of cases involving immiscible

displacements.16 The application of EFBCs for miscible displacements has not been considered previously. Wallstrom et al.15 generalized the EFBC formulation for anisotropic permeability fields. In this case there are two different asymptotic flux ratios ( R'x , R'z - primed to indicate anisotropy). Wallstrom et al.15 set each of these to be 2. This approximation was explained and justified, though technically these values should depend on the statistics of the permeability field (but this dependency may be complex). In the present study, we also use R'x = R'z = 2. Since the theory of EFBCs is an approximate way of incorporating nonlocal information in subgrid simulations, it will become less applicable as its underlying assumptions are violated. There are a few features existing in actual flows in geological media that can lead to violations, but the most important for our purposes is the presence of very large length scale permeability features. By considering how the disturbance in the pressure field caused by the inclusion decays with distance, Wallstrom et al. obtained a parameter Nmf for gauging the applicability of EFBCs: N mf = λ max Lmin , (5) where, in two dimensions, λmax = max{λx, λz}, Lmin = min{Lx, Lz} and λi and Li are the dimensional correlation and domain lengths in the ith direction respectively. EFBCs are expected to be applicable when Nmf ≤ ~1. When Nmf is much greater than one, the system acts as though it is layered, and the use of standard local boundary conditions (rather than EFBCs) may be appropriate. Extended Todd and Longstaff with Pseudo relative permeabilities formulation. In miscible flooding, we can theoretically recover all of the oil within the rock by continuously injecting solvent, thus achieving zero residual oil saturation (Sorb = 0). In practice, due to the cost of solvent, injection is stopped once the oil cut is too low to be economic. To illustrate this, consider the fractional flow curve for a core that is initially full of oil (Fig. 3). The flatness of the curve when solvent saturation is beyond about 0.6 implies that the last 40% of the oil is essentially unrecoverable. Thus, the practical limit is reached when the fractional flow (fs) is 0.98 and Sorb = 0.4. The sigmoidal shape of the curve is typical of a miscible displacement in real media, in which the unfavorable mobility ratio, heterogeneities and density contrast lead to viscous fingering, chanelling and gravity segregation.1 These phenomena cause the bypassing of pockets of oil, so full recovery may take a long time. In the context of subgrid simulations in upscaling, the Sorb concept implies that each coarse cell, usually tens or hundreds of feet in x and y, should have a nonzero Sorb, the magnitude of which depends on the subgrid heterogeneity. In a first-contact miscible displacement of oil by gas/solvent, the mixing phenomenon in the hydrocarbon phase can be approximated using several pseudocomponents with different properties: oil and gas. The original Todd and Longstaff (TL) formulation3 proposed a set of mixing rules to modify the effective fluid properties of the hydrocarbon pseudocomponents (oil and gas) such that the effects of mixing between the two are captured. In a miscible

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displacement, the hydrocarbon pseudocomponents do not interfere with each other’s flow. Thus their relative permeabilities are proportional to their respective saturations.

Fig. 3 – Solvent fractional flow curve showing the practical, nonzero Sorb

In the extended formulation (ETLP), we instead compute the miscible upscaled relative permeabilities ( k rj* ) for each coarse cell via a fractional flow formulation employed by Stone18 for immiscible cases: (6) k rs* (S s ,norm ) = f s* λ*T µ se , * (S s ,norm ) = (1 − f s* ) λ*T µ oe . (7) k ro * We note that the k rj are functions of normalized solvent saturation (Ss,norm) to incorporate the Sorb concept: Ss . (8) S s ,norm = 1 − S orb The fractional flow ( f s* ) and total mobility ( λ*T ) as a function of Ss curves are obtained through local fine grid simulations with prescribed boundary conditions (e.g., EFBCs) on each coarse cell. Note that f s* , λ*T and Ss are all averaged quantities; f s* and λ*T are computed via weighted sums of the appropriate quantities in the fine cells along the outlet column of the coarse cell. For a 2D (x-z) problem: ∑k ∆z k u s nx +1 2 ,k , (9) f s* = ∑k ∆z k (u s nx +1 2,k + u o nx +1 2 ,k )

λ*T =

∑k k x n ,k ∆z k λT n ,k ∑k k x n ,k ∆z k x

x

,

(10)

x

where u s

nx +1 2 ,k

, uo

nx +1 2 ,k

, ∆zk, k x

nx ,k

and λT

n x ,k

are the

solvent Darcy velocity, oil Darcy velocity, z-dimension, absolute permeability and total mobility in the x-direction of the kth cell at the outlet face. The quantity Ss is computed either via a porosity-weighted volume average or outlet average of the fine scale saturations. Wallstrom et al.16 employed the outlet averaging scheme in their work. The effective viscosities (µse, µoe) are required in the computation of k rj* using Eqs. 6 – 7. The mixture formed has a viscosity given by the quarter-power mixing rule as follows: 1 S s ,norm (1 − S s ,norm ) = + (11) 1 1 1 µm

4

µs

4

µo

4

Here, µs and µo are the pure solvent and oil viscosities respectively. The effective fluid viscosities are then given by: (12) µ se = µ (s 1−ω ) µ mω (13) µ oe = µ o( 1− ω ) µ mω The empirical mixing parameter (ω) can be used to account for the effects of reservoir heterogeneity on the mixing of fluids. When ω = 0, the system is so heterogeneous that channeling essentially prevents any mixing at all, while ω = 1 is applicable to a homogeneous system with perfect mixing. We do not use ω as an upscaling parameter because it does not impact the value of Sorb and thus cannot be used to model the nonzero Sorb concept. In fact, we do not even need to specify ω in the coarse model. This is the case because * (S k rs* (S s ,norm ) µ se and k ro s ,norm ) µ oe are independent of ω (this is evident from Eqs. 6 and 7). For fine scale simulations, our assumption is that the resolution is high enough such that ω = 1. In other words, it is assumed that there are no subgrid phenomena within each fine cell.

Fig. 4 – Upscaled relative permeabilities have better behavior using nonzero Sorb

In short, our proposed upscaling procedure is as follows. We apply EFBCs on each coarse cell, obtain the f s* and λ*T curves, determine the Sorb value, and then use these quantities in the ETLP formulation to compute k rj* . In the coarse scale simulation, each coarse cell therefore has its own set of k rj* and an Sorb value. The benefit of employing the nonzero Sorb concept can be seen in Fig. 4, where the k rj* for a typical coarse block are shown. We see that when Sorb = 0, the k rj* possess a non-monotonic, beyond-unity character, a behavior that has been observed by other investigators.10,11 This behavior can pose numerical problems for the simulator and is an unnecessary complication since the last 40% of the oil is practically unrecoverable. The curves are seen to be much better-behaved for Sorb = 0.4. Since the oil saturation represented by Sorb is unavailable for mixing, to include it in the computation of the mixture fluid properties is to consider more oil in the mixture than is physical. A consistent treatment of Sorb has been implemented in the simulator that we used for this study (ChevronTexaco’s CHEARS simulator).

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Results and Discussion We now evaluate the proposed miscible upscaling method by assessing its performance for varying correlation structure and coarsening factor. We then demonstrate the effectiveness of the method by applying it to an actual field case. Evaluation for varying correlation structure. We now compare the different boundary conditions and saturation averaging schemes for generating miscible upscaled relative permeabilities ( k rj* ). We accomplish this by testing the different combinations on a set of synthetic fields with varying correlation lengths. Our intent here is to compare the effective flux boundary conditions (EFBCs) against the standard approach (constant pressure on the inlet and outlet). Another aspect investigated is the saturation averaging scheme, which determines the saturation associated with each fractional flow and total mobility value for the coarse cell. The two schemes considered are volume and outlet averaging, as described in the Methodology section. There are therefore four possible combinations in this assessment (two different boundary specifications, each with two averaging techniques). Fig. 5 shows the synthetic permeability cross sections used to test the upscaling method. The physical dimensions of the fields in the horizontal (Lx) and vertical (Lz) directions are 1200 and 1300 feet respectively. The fields are generated using the geostatistical software GSLIB19 and have a log-mean (µln) and log-variance (σ2ln) of 3. These fine scale maps are 400 × 100 in dimension and the permeabilities (in mD) vary over 6 orders of magnitude. The fine scale permeability field is anisotropic, with kz = 0.1 kx. The fields shown all have the same dimensionless correlation length in the vertical direction (lz = λz/Lz = 0.1) but the horizontal value (lx = λx/Lx) ranges from 0.1 (Field A) to 5.0 (Field D). Higher values of lx indicate higher degrees of layering.

A (lx = 0.1)

Field A (lx = 0.1)

Field B (lx = 0.5)

B (lx = 0.5) Field C (lx = 1.0)

C (lx = 1.0) -1.5

D (lx = 5.0) 4.0

Fig. 5 – Natural log permeability of Fields A – D (µln = 3, σ2ln = 3, lz = 0.1)

We simulate a first-contact miscible process where solvent/gas is injected into a model containing oil and immobile water (Swc = 0.3). The wells are completed along the two lateral edges of the models. The oil and solvent viscosities are functions of pressure but are approximately 0.13 and 0.05 cP respectively at the reservoir pressures encountered. The solvent injector is set at a constant pressure of 9000 psi while the producer has a constant gas rate of 100 MCF/day with a bottom-hole pressure constraint of 5000 psi.

Field D (lx = 5.0) Fig. 6 – Effects of boundary conditions and saturation averaging schemes

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We evaluate the effectiveness of using different combinations of boundary conditions and saturation averaging schemes on the fields shown in Fig. 5. Note that we always employ the Extended Todd and Longstaff with Pseudo relative permeabilities (ETLP) formulation to generate the upscaled relative permeabilities ( k rj* ) for the x-direction. We use the same k rj* in the z-direction (for a few cases, we also generated a set of k rj* for the z-direction and did not observe much difference in the simulation results). We upscale the fine grid model to a coarse grid of 20 × 10, so the model is coarsened by a factor of 200. The coarse grid is uniform and the absolute permeabilities are upscaled using a local procedure with standard (fixed pressure) boundary conditions. We compare the oil production rate versus pore volume injected curves (Fig. 6) for the fine and coarse grid models. Since the gas production rate is constant, the oil production rate and oil cut curves will exhibit the same trends. We see that for lx ≤ 1.0 (Fields A – C), the proposed method (EFBCs and outlet saturation averaging scheme) clearly produces the coarse scale solution with the best match to the fine scale solution in terms of both breakthrough time prediction and overall level of agreement. When lx = 5.0 (Field D), the system is almost completely layered. This violates an underlying premise in EFBCs; i.e., Nmf here is equal to 5, which is significantly greater than 1. It is therefore not surprising that EFBCs become inappropriate for these systems, causing breakthrough to occur too late. In fact, as the field gets more layered, the standard approach, which imposes fixed pressure boundary conditions in the direction of flow, becomes applicable. This explains the closer match given by the standard approach for Field D. We could conceivably mitigate the over-suppression of flux by EFBCs in fields for which Nmf > 1 through an appropriate generalization of the EFBCs,16 though such a modification is beyond the scope of the current work.

Fig. 7 – Typical upscaled relative permeabilities from different saturation averaging schemes (EFBCs)

In general, the use of standard boundary conditions is seen to cause an earlier breakthrough as compared with the equivalent case employing EFBCs, as expected. The outlet averaging of saturations also leads to an earlier breakthrough as compared with the equivalent case employing volume averaging. This is because in the latter case, the solvent fractional flow curve is shifted to the right, as each k rj* value is associated with a larger solvent saturation value (Fig. 7). In

addition, we also observe a “stair-step” behavior in the solutions using volume averaging. The reason is that solvent saturation needs to build up to a critical level before any flow will occur and this critical saturation is different for each coarse cell. When this occurs in a key region of the model, the sudden increase in oil flow brings about a plateau region on the curve. We conclude that for partially layered fields, the proposed method (ETLP formulation, EFBCs and outlet saturation averaging scheme) performs very well. In addition to oil production rate curves, we also compare the solvent saturation maps for Field B, for which lx = 0.5 (Fig. 8). It is clear that the coarse scale saturations generated using the proposed method display many of the essential features existing in the fine scale solution. The speedups offered by our upscaling procedure are quite substantial. For this case (Field B), the fine scale simulation required about 20 hours of CPU time, while the coarse scale model ran in only 27 seconds. The EFBC upscaling procedure required 40 minutes of CPU time. Thus, the overall coarse solution was accomplished using about 1/30th of the CPU time of the fine scale solution. Also, as the upscaling code has not been optimized, the CPU requirements for the coarse solution could be reduced further.

Fine grid

Coarse grid

0.1

0.7

Fig. 8 – Solvent saturation at breakthrough for Field B; coarse model from proposed method

Evaluation for varying coarsening factor. In this section, we consider a wide range of coarsening factors and quantify our observations by computing the errors in oil cut and breakthrough time for the coarse scale solutions. We compare the proposed method against other commonly used techniques (e.g., standard k rj* , use of upscaled permeabilities only, and nonuniform coarsening) using fields with different degrees of heterogeneity. Although a visual inspection is useful as a gauge of the upscaling effectiveness, quantitative measures of the error are more instructive. Further, since many coarsening factors are being considered, it is difficult to compare all solutions on a single plot. To present the results concisely, we employ two parameters to quantify the mismatch in simulation results. The first parameter is the fractional L1-norm of the error in oil cut, given by: t D ,F

e L1 =

∫

f f (t D ) − f c (t D ) dt D

0 t D ,F

∫ f f (t D ) dt D

,

(14)

0

where f is the dimensionless oil cut and tD is the pore volume injected. The error is thus the area of the absolute difference between the coarse (subscript c) and fine (subscript f) curves

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divided by the total area under the fine curve. The final time, tD,F is chosen to be 1.4 PVI because the curves generally approach f = 0 at about this time (see Fig. 6). The other measure is the fractional error in breakthrough time (eBT) defined as: t BT , f − t BT ,c e BT = . (15) t BT , f The breakthrough time (tBT) is defined as the dimensionless time (in PVI) at which the oil cut drops below 0.99. In the results below, we consider a wide range of coarsening factors. We keep the grid-aspect ratio, defined as the ratio of the number of cells in x to that in y, constant at 4. Similar results to those presented below were observed for a fixed aspect ratio of 2. The seven coarse grid models used are 8 × 2, 16 × 4, 20 × 5, 40 × 10, 80 × 20, 100 × 25 and 200 × 50, hence the range of coarsening factors is quite large (4 – 2500). We simulate a miscible displacement with well conditions as described above. We consider two fine grid (400 × 100) permeability fields of different geologies: Field B from Fig. 5 and a second system with much stronger heterogeneities, Field E (Fig. 9). The permeability statistics for Field E are µln = 3, σ2ln = 5, lx = 0.5 and lz = 0.01. We compare the effectiveness of the proposed method against three other upscaling techniques. Thus, there are four methods being studied in this section: 1. Standard k rj* (with outlet saturation averaging) 2. EFBC k rj* (with outlet saturation averaging) 3. Upscaled permeabilities (k*) only in a uniformly coarsened grid 4. Upscaled permeabilities (k*) in a nonuniformly coarsened grid20

-1.5

(a) EFBC k rj*

(b) Standard k rj*

4.0

Fig. 9 – Natural log permeability of Field E (µln = 3, σ2ln = 5, lx = 0.5, lz = 0.01)

We will not be fully exhaustive in evaluating the methods; i.e., although there are seven coarsening factors under consideration, we have generated seven coarse models for each field only for two of the methods (EFBC k rj* and k* only). We create three coarse models for the two fields using the methods of standard k rj* and nonuniform coarsening. We note that the aspect ratio will not be maintained at 4 for nonuniform coarsening, as this would severely limit the degree of grid optimization possible.

(c) k* only

(d) Nonuniform coarsening Fig. 10 – Oil cut curves from various methods (Field B)

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Numerical results for Field B (σ2ln = 3, lx = 0.5, lz = 0.1). In Fig. 10a, we show the oil cut results for the proposed upscaling method for Field B. The trends of the results and the level of agreement with the fine scale solution are the same when plotted in a dimensionless or a dimensional form. Only three out of the seven coarse solutions (in the intermediate range of coarsening factors) are shown to keep the plots legible. We see that the coarse scale solutions cluster around the fine scale solution and that the matches are excellent. The proposed method is thus seen to be very robust with respect to coarsening factor. It is evident that the results from standard * k rj* (Fig. 10b) and k only (Fig. 10c) are less accurate, being overly pessimistic (early breakthrough) and optimistic (late breakthrough) respectively. The nonuniformly coarsened models predict the breakthrough time accurately but are less accurate than the EFBC k rj* results at later times (Fig. 10d). Quantitatively, the fractional oil cut errors from applying EFBC k rj* are all less than 0.1 and are clearly lower compared to errors from other methods (Fig. 11). The trend of fractional breakthrough errors has a more erratic behavior, possibly because these are point values as opposed to integrated quantities like the oil cut errors. One interesting feature is that the error (particularly the error in breakthrough time for the k* only simulations) does not necessarily decrease with increasing refinement. We will discuss this observation below in terms of the interplay of numerical dispersion, heterogeneity and displacement physics.

For instance, the 60 × 27 nonuniform model (1620 cells) is roughly comparable in size to the 80 × 20 uniform model. We see that in terms of fractional oil cut errors, the nonuniform coarsening method is inferior only to the proposed method and there is marked improvement with greater refinement in y. Breakthrough time is predicted very accurately because the nonuniform model is generated to provide the highest resolution in important high flow regions (Fig. 12). Numerical results for Field E (σ2ln = 5, lx = 0.5, lz = 0.01). Since Field E has a more extreme level of heterogeneity (i.e., higher σ2ln and smaller lz), it is more difficult to upscale. In general, the EFBC k rj* show a higher degree of nonmonotonicity (Fig. 13). It is therefore not surprising that the match given by the proposed method (Fig. 14a) is not as good as that observed in Field B (Fig. 10a). Nevertheless, it is still clearly superior to the other methods (Fig. 14b – d). Standard * k rj* and k only give pessimistic and optimistic results respectively as before, but the mismatch is much more pronounced here. The nonuniform coarsening results are also consistent with previous observations: accurate breakthrough time predictions but subsequent loss of accuracy.

Fig. 13 – Typical upscaled relative permeabilities from proposed method for two different fields

Fig. 11 – Fractional oil cut and breakthrough time errors from various methods (Field B)

Fine grid -1.5

Coarse grid 4.0

Fig. 12 – Natural log permeability of the fine and nonuniform coarse (48 × 40) models of Field B

For the nonuniformly coarsened models, the number of coarse layers in the y-direction more strongly impacts the accuracy; hence it is used as the abscissa in Fig. 11. Due to this treatment, the comparison between the nonuniform coarsening and the other methods is not as straightforward.

In Fig. 15, we can see that the error trends are different from those observed in Field B. Instead of staying roughly constant throughout the range of coarsening factors, the fractional oil cut error for EFBC k rj* is highest for the coarsest model and decreases with increasing refinement before becoming approximately constant for 40 cells or more in x. This implies that a certain level of resolution is required in order to capture the essential features of such a heterogeneous field. We can discern a similar trend from the fractional breakthrough error plots. In contrast with the results for Field B where the breakthrough time predicted by the proposed method was somewhat erratic (Fig. 11), here it generally decreases and thus moves closer to the fine scale solution with increasing refinement. In fact, there appears to be general improvement for all methods with increasing refinement. Further, although the oil cut errors for standard k rj* are much higher compared to the proposed method, their breakthrough errors are similar or, in the 40 × 10 case, noticeably smaller. This is because the fine scale model breaks through quite quickly for this field and the pessimistic bias of the standard k rj* is fortuitously beneficial.

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(a) EFBC k rj*

(b) Standard k rj*

(c) k* only

(d) Nonuniform coarsening Fig. 14 – Oil cut curves from various methods (Field E)

Fig. 15 – Fractional oil cut and breakthrough time errors from various methods (Field E)

Interplay of heterogeneity and numerical dispersion. One notable feature of the results presented above is that the error does not necessarily decrease with increasing refinement. In the cases without relative permeability upscaling (e.g., k* only), we believe that this occurs as a result of the interplay of numerical dispersion, heterogeneity and displacement physics. The first-order, finite-difference solution of the gas saturation equation introduces truncation errors that smear sharp saturation fronts.21 This numerical dispersion enhances spreading beyond what is physical and artificially disperses injected fluid longitudinally and transversely. Numerical dispersion is proportional to the grid spacing; thus for the coarser models with fewer cells in the direction of predominant flow (x in this case), the longitudinal numerical dispersion can become important, leading to an earlier breakthrough. Since the breakthrough time tends to shift the entire solution, the overall match between the coarse and fine scale solution will also be impacted. The effect of numerical dispersion on simulations of miscible displacements has been widely reported.22,23 Heterogeneity complicates the description given above. Heterogeneity is “lost” when important heterogeneous features like high permeability streaks that lead to early breakthrough are homogenized. Thus, an increasingly delayed breakthrough time would be predicted as the model is progressively coarsened (in the absence of numerical dispersion). For Field B, we can see this interplay in the trend of breakthrough errors for k* only (Fig. 11) where the error generally increases (breakthrough occurs later compared to the fine scale solution) as the model is coarsened from 200 cells in x to 20 cells in x. When there are fewer than 20 cells in x, the effects of numerical dispersion start to dominate and the error decreases; i.e., breakthrough occurs earlier and is closer to the fine scale solution. In a more heterogeneous system like Field E (see breakthrough errors for k* only in Fig. 15), the loss of heterogeneity is more important. All coarse models predict a much later breakthrough time than the fine scale solution and numerical dispersion plays a role only in the coarsest model (8 cells in x). This interplay can also be seen in the results from previous work.14 The above description is further complicated when the relative permeabilities are also upscaled. This is because the

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k rj* account, to some extent, for the interaction between numerical dispersion and heterogeneity. Nonetheless, this interplay may in part explain the general trends in breakthrough time error for the EFBC k rj* results shown in Figs. 11 and 15. Application on field models. We now apply the proposed method to a cross-sectional model of a real field. The model is 6720 feet × 1382 feet in size (240 × 432 cells) and the permeability field is shown in Fig. 16. The system has the following statistics: µln = –0.76, σ2ln = 12.7, lx = 0.22, lz = 0.07. The Nmf value (Eq. 5) for this field is 1.49, slightly beyond the range for which the proposed method is theoretically applicable (maximum of 1.0). The variance is also very high, but we will show that the results are nevertheless quite encouraging.

-3.0

(a) EFBC k rj*

2.0

Fig. 16 – Natural log permeability of a real field (Field F)

The production scenario is similar to that used in the test cases: the injector is set at a constant pressure of 11000 psi while the producer has a constant oil rate of 80 STB/day with a bottom-hole pressure constraint of 3700 psi. We upscale the fine scale model to three coarse models: 20 × 36, 15 × 27 and 10 × 24, which give coarsening factors of 114, 256 and 432 respectively. We again compare the oil cut versus PVI curves given by the coarse and fine scale simulations. From Fig. 17a, we can see that the oil cut results given by EFBC k rj* are very good. Using standard k rj* (Fig. 17b), we once again observe early breakthrough and low predictions for oil cut. The use of k* only yields surprisingly good results (Fig. 17c), though the error results in Fig. 18 indicate that the proposed method still generally provides the most accurate coarse models. We note that an even coarser model (10 × 18), did not provide accurate results using the proposed method. This may be related to the fact that Nmf > 1, which may limit the degree of vertical coarsening allowable. Apart from the dimensionless plots, we also compare the gas and oil production rate versus actual dimensional time (not PVI) in Fig. 19. The overall matches are quite good until the producer switches from rate to BHP control. The coarse models predict the oil production rates accurately even after the switch in well control (Fig. 19b), though the gas production and injection rates display some error. This suggests that perhaps the well injectivities and productivities are not accurate; i.e., the well treatments in the coarse and fine scale models may not be exactly equivalent. The use of a nearwell upscaling procedure24 may resolve this issue. At the time of the switch (more than 1.4 PVI), the oil fractional flow is much less than 0.1, so the mismatch at late time is not of much practical concern. We reiterate that for the test cases shown in previous sections, the EFBC results are equally good whether we use dimensionless or dimensional plots.

(b) Standard k rj*

(c) k* only Fig. 17 – Oil cut curves from various methods (Field F)

Fig. 18 – Fractional oil cut and breakthrough time errors from various methods (Field F)

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indicate that the proposed method is able to provide very useful predictions for a practical miscible displacement study. For this case, the fine scale simulation required about 68 hours of CPU time while the coarse scale simulation ran in only 57 seconds. The EFBC upscaling procedure took 1 hour, 50 minutes, giving an overall speedup factor of 37. Optimizing the simulation parameters (e.g., time step, material balance tolerance) in the fine run gave a speedup factor of about 6. It will, however, be difficult in general to determine these optimal parameters. Even with optimized parameters, our method still gives a speedup of about 6 (which, again, can be improved by optimizing the upscaling code). (a) Gas production rate

(b) Oil production rate Fig. 19 – Dimensional results for Field F (EFBC

Fig. 20 – Oil production curves for Field F (EFBC

Fine grid 0.1

k rj*

k rj* )

and k* only)

Coarse grid 0.7

Fig. 21 – Solvent saturation maps at breakthrough for Field F (15 x 27 model using EFBC k rj* )

In Fig. 20, we show the improvement in oil production rate prediction given by the proposed method over that using k* only for the 15 × 27 model. We also see from Fig. 21 that the coarse scale saturation map given by the proposed method captures the essential features of the fine scale map very well, including the layer that leads to breakthrough. These results

Conclusions In this paper, we presented a new miscible upscaling technique that enables accurate coarse scale simulations of first-contact miscible processes. The method incorporates two key components: Effective Flux Boundary Conditions (EFBCs) and the Extended Todd and Longstaff with Pseudo relative permeabilities (ETLP) formulation. The former leads to a better breakthrough time prediction by incorporating some global flow information in the local upscaling problems while the latter efficiently models immobile and unmixed oil to generate better-behaved upscaled relative permeabilities. Through our study, we arrive at the following conclusions: 1. For permeability fields of varying correlation lengths, the proposed method (with outlet averaging of saturations) yields an excellent match to the fine scale simulation results for breakthrough time, oil production curves and saturation profiles. EFBCs become less applicable compared to standard boundary conditions as the permeability field becomes more layered (Nmf > 1), consistent with the model underlying EFBCs. 2. By quantitatively comparing the oil cut and breakthrough errors, the proposed method is seen to consistently outperform other techniques such as standard k rj* , k* only and nonuniform coarsening, over a wide range of coarsening factors (4 – 2500). 3. Application of the proposed method on a real field was successful for a range of coarsening factors (114 – 432). A mismatch in gas rates at late time was encountered, which is likely due to near-well effects. Nomenclature EFBCs Effective Flux Boundary Conditions ETLP Extended Todd & Longstaff with k rj* formulation e upscaling error upscaled fractional flow of fluid j (j = o,s) f j* fk oil cut of model k (k = c,f) k absolute permeability tensor upscaled (pseudo) relative permeability of fluid j k rj* li dimensionless correlation length in the ith direction Li domain length in the ith direction Nmf parameter to gauge applicability of EFBCs p pressure Ri asymptotic flux ratio (primed for an anisotropic field)

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Sorb Ss Ss,norm tD u

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bypassed oil saturation in miscible flooding solvent saturation solvent saturation normalized with respect to Sorb dimensionless time (pore volume injected) Darcy velocity vector

Greek

λ*T λi µj µje µln ω σ2ln

upscaled total mobility dimensional correlation length in the ith direction viscosity of fluid j effective viscosity of fluid j log-mean of permeability field Todd-Longstaff mixing parameter log-variance of permeability field

Subscripts B background c coarse f fine I inclusion m mixture o oil s solvent Superscripts * coarse scale parameters BT breakthrough error L1 L1-norm of error in oil cut Acknowledgments We thank ChevronTexaco for their support of this work. We are grateful to S. Hou, T.C. Wallstrom and D.H. Sharp (Los Alamos National Laboratory) and M.A. Christie (Heriot Watt University) for providing the miscible upscaling code and for useful discussions. We also thank S. Jákupsstovù (Technical University of Denmark) for testing the upscaling code. References 1. Tchelepi, H. and Orr, F.M. Jr.: “Interaction of viscous fingering, permeability heterogeneity and gravity segregation in three dimensions,” SPE Reservoir Engineering (1990) 5: 266 – 271. 2. Newley, T.M.J. and Merrill, R.C. Jr.: “Pseudocomponent selection for compositional simulation,” SPE Reservoir Engineering (1991) 6: 490 – 496. 3. Todd, M.R. and Longstaff, W.J.: “The development, testing and application of a numerical simulator for predicting miscible performance,” Journal of Petroleum Technology (1972) 24: 874 – 882. 4. Koval, E.J.: “A method for predicting the performance of unstable miscible displacement in heterogeneous media,” SPE Journal (1963) 3: 145 – 154. 5. Fayers, F.J.: “An approximate model with physically interpretable parameters for representing miscible viscous fingering,” SPE Reservoir Engineering (1988) 3: 551 – 558. 6. Thiele, M.R., Batycky, R.P. and Blunt, M.J.: “A streamline-based 3D field-scale compositional reservoir simulator,” paper SPE 38889, presented at the SPE Annual Technical Conference and Exhibition, San Antonio, TX, October 5 – 8, 1997. 7. Crane, M., Bratvedt, F., Bratvedt, K., Childs, P. and Olufsen, R.: “A fully compositional streamline simulator,” paper SPE 63156, presented at the SPE Annual Technical Conference and Exhibition, Dallas, TX, October 1 – 4, 2000.

8. Peddibhotla, S., Cubillos, H., Datta-Gupta, A. and Wu, C.H.: “Rapid simulation of multiphase flow through fine-scale geostatistical realizations using a new, 3D, streamline model: a field example,” paper SPE 36008, presented at the Petroleum Computer Conference, Dallas, TX, June 5 – 6, 1996. 9. Jessen, K. and Orr, F.M. Jr.: “ Compositional streamline simulation,” paper SPE 77379, presented at the SPE Annual Technical Conference and Exhibition, San Antonio, TX, September 29 – October 2, 2002. 10. Zhang, H.R. and Sorbie. K.S.: “The upscaling of miscible and immiscible displacement processes in porous media,” paper SPE 29931, presented at the SPE International Meeting on Petroleum Engineering, Beijing, P.R. China, November 14 – 17, 1995. 11. Barker, J.W. and Fayers, F.J.: “Transport coefficients for compositional simulation with coarse grids in heterogeneous media,” SPE Advanced Technology Series (1994) 2: 103 – 113. 12. Christie, M.A. and Clifford, P.J.: “Fast procedure for upscaling compositional simulation,” paper SPE 37986, presented at the SPE Symposium on Reservoir Simulation, Dallas, TX, June 8 – 11, 1997. 13. Rubin, B., Barker, J.W., Blunt, M.J., Christie, M.A., Culverwell, I.D. and Mansfield, M.: “Compositional reservoir simulation with a predictive model for viscous fingering,” paper SPE 25234, presented at the SPE Symposium on Reservoir Simulation, New Orleans, LA, February 28 – March 3, 1993. 14. Jerauld, G.R.: “A case study in scaleup for multi-contact hydrocarbon gas injection,” paper SPE 39626, presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 19 – 22, 1998. 15. Wallstrom, T.C., Christie, M.A., Durlofsky, L.J. and Sharp, D.H.: “Effective flux boundary conditions for upscaling porous media equations,” Transport in Porous Media (2002) 46: 139 – 153. 16. Wallstrom, T.C., Hou, S., Christie, M.A., Durlofsky, L.J., Sharp, D.H. and Zou, Q.: “Application of effective flux boundary conditions to two-phase upscaling in porous media,” Transport in Porous Media (2002) 46: 155 – 178. 17. Christie, M.A., Wallstrom, T.C., Durlofsky, L.J., Hou, S., Sharp, D.H. and Zou, Q.: “Effective medium boundary conditions in upscaling,” Proceedings of the 7th European Conference on the Mathematics of Oil Recovery, Baveno, Italy, September 5 – 8, 2000. 18. Stone, H.L.: “Rigorous black oil pseudo functions,” paper SPE 21207, presented at the SPE Symposium on Reservoir Simulation, Anaheim, CA, February 17 – 20, 1991. 19. Deutsch, C.V. and Journel, A.G.: GSLIB: Geostatistical software library and user’s guide, 2nd edition (1998), Oxford University Press, New York. 20. Durlofsky, L.J., Jones, R.C. and Milliken, W.J.: “A nonuniform coarsening approach for the scale up of displacement processes in heterogeneous porous media,” Advances in Water Resources (1997) 20: 335 – 347. 21. Fanchi, J.R.: “Multidimensional numerical dispersion,” SPE Journal (1983) 23: 143 – 151. 22. Stalkup, F.I., Lo, L.L. and Dean, R.H.: “Sensitivity to gridding of miscible flood predictions made with upstream differenced simulators,” paper SPE 20178, presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 22 – 25, 1990. 23. Jessen, K., Stenby, E.H. and Orr, F.M. Jr.: “Interplay of phase behavior and numerical dispersion in finite difference compositional simulation,” paper SPE 75134, presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 13 – 17, 2002. 24. Mascarenhas, O. and Durlofsky, L.J.: “Coarse scale simulation of horizontal wells in heterogeneous reservoirs,” Journal of Petroleum Science and Engineering (2000) 12: 135 – 147.