Upscaled modeling of well singularity for simulating ... - Springer Link

Comput Geosci (2008) 12:29–45 DOI 10.1007/s10596-007-9059-5

ORIGINAL PAPER

Upscaled modeling of well singularity for simulating flow in heterogeneous formations Yuguang Chen · Xiao-Hui Wu

Received: 30 March 2007 / Accepted: 19 October 2007 / Published online: 4 January 2008 © Springer Science + Business Media B.V. 2007

Abstract Subsurface flows are affected by geological variability over a range of length scales. The modeling of well singularity in heterogeneous formations is important for simulating flow in aquifers and petroleum reservoirs. In this paper, two approaches in calculating the upscaled well index to capture the effects of fine scale heterogeneity in near-well regions are presented and applied. We first develop a flow-based near-well upscaling procedure for geometrically flexible grids. This approach entails solving local well-driven flows and requires the treatment of geometric effects due to the nonalignment between fine and coarse scale grids. An approximate coarse scale well model based on a well singularity analysis is also proposed. This model, referred to as near-well arithmetic averaging, uses only the fine scale permeabilities at well locations to compute the coarse scale well index; it does not require solving any flow problems. These two methods are systematically tested on three-dimensional models with a variety of permeability distributions. It is shown

Y. Chen (B) Department of Petroleum Engineering (now Energy Resources Engineering), Stanford University, Stanford, CA 94305, USA e-mail: [email protected] Present Address: Y. Chen Chevron Energy Technology Company, P.O. Box 6019, San Ramon, CA 94583, USA X.-H. Wu ExxonMobil Upstream Research Company, P.O. Box 2189, Houston, TX 77252, USA e-mail: [email protected]

that both approaches provide considerable improvement over a simple (arithmetic) averaging approach to compute the coarse scale well index. The flow-based approach shows close agreement to the fine scale reference model, and the near-well arithmetic averaging also offers accuracy for an appropriate range of parameters. The interaction between global flow and near-well upscaling is also investigated through the use of global fine scale solutions in near-well scale-up calculations. Keywords Upscaling · Permeability · Heterogeneity · Well model · Well index · Singularity · Reservoir simulation · Subsurface flow

1 Introduction Subsurface formations often exhibit high degrees of variability in rock properties. Permeability heterogeneity occurring over multiple length scales can strongly affect subsurface fluid flows. However, direct flow simulations using fine scale permeability distributions are impractical due to the high computational cost. Upscaling or averaging procedures are therefore introduced to coarsen the geological descriptions to the scales suitable for flow simulations. A large number of upscaling techniques have been developed, ranging from simple averaging methods to a variety of flow-based approaches. Some comprehensive reviews were presented by [25, 27] and [14]. This paper considers the modeling of wells, which are often represented as source/sink terms in governing flow equations and play an important role in the simulations of groundwater flows and flows in petroleum reservoirs. Due to the singularity introduced by wells, upscaling in the

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near-well region is fundamentally different than standard upscaling away from wells. The flow field in the vicinity of wells changes rapidly within a short distance, in contrast to the “slowly” varying linear flow assumed in the standard upscaling. A number of researchers (e.g., [9, 13, 19]) have studied this issue and developed near-well upscaling procedures. In these methods, welldriven flows are used to compute coarse scale quantities required for well modeling, e.g., well indexes that relate the wellbore flow rates and pressures to the well block pressures. Most existing near-well upscaling work has focused on Cartesian grids. Although the use of geometrically flexible grids has received much attention in recent years, only limited number of studies are about coarse scale well modeling for these grids. Ding [10] discussed near-well upscaling for corner-point grids and pointed out that the difficulty in upscaling on these grids is due in part to a lack of alignment between the coarse grid and Cartesian fine grid. Wolfsteiner [28] presented a preliminary near-well upscaling procedure (two-dimensional) applicable for a multi-block reservoir simulator. Prévost [24] developed gridding and upscaling procedures for three-dimensional fully unstructured grids, but near-well upscaling was not considered. In this work, we develop a flow-based near-well upscaling procedure for three-dimensional prismatic Voronoi grids, also known as perpendicular-bisectional grids as in the literature of petroleum reservoir simulations (e.g., [20]). In addition to the flow-based near-well upscaling procedure, we also study an approximate coarse scale well model, which does not require solving any flow problems. This approach leads to very efficient calculations of coarse scale well index and is readily applied to non-Cartesian grids. We propose this method based on an analysis of the pressure solution for well-driven flows, which can be decomposed into a singular and a regular part. The singular part, characterized by the fine scale permeability at well locations, has the dominant impact on near-well flows. Therefore, the coarse scale well index can be approximated by averaging the fine scale permeability only along the well trajectories. We refer to this approach as near-well arithmetic averaging. The modeling of well singularities has been considered in the context of multi-scale finite element and finite volume methods (e.g., [5, 30]). However, these studies are not directly related to the calculation of coarse scale well index. More relevant to this work is the approximate well modeling in heterogeneous media investigated by previous researchers. Desbarats [7] calculated “effective permeability for radial flow” using a spatial integral of permeability weighted by the

Comput Geosci (2008) 12:29–45

inverse square of distance from the well. Durlofsky [12] presented a well model using an effective background permeability coupled with a near-well “skin.” This skin varies along the well trajectory and is computed from the near-well permeabilities. Both studies provide some type of representation of the permeability for near-well radial flow. Our approach differs from these methods in that only the permeability in the well location is considered in the near-well arithmetic averaging, whereas the previous approaches include permeabilities in a region around the well. Moreover, our method directly provides the coarse scale well index that can be readily used in numerical flow simulation models. This was not the case in the previous work. For example, the model of Durlofsky [12] was coupled with a semi-analytical solution to compute coarse scale well index for arbitrary grids [29]. The flow-based near-well upscaling approach and the approximate well model are tested on a variety of permeability distributions, including variogram-based models with different correlation lengths and channelized systems, using vertical and horizontal wells. These two methods are compared with a simple (arithmetic) averaging method to compute the coarse scale well index. Both methods provide considerable improvement in accuracy. We also demonstrate that, for highly heterogeneous cases, the global flow has an impact on the near-well upscaling calculations. Global near-well upscaling approaches are considered to address this issue. This paper proceeds as follows. We first review the governing equations and standard upscaling procedures. In Section 3, we present a flow-based near-well upscaling procedure for unstructured grids (here prismatic Voronoi grids), and discuss the geometrical complexity involved. Both local and global approaches are considered. A near-well arithmetic averaging method is then presented in Section 4. Numerical results obtained on a variety of reservoir heterogeneities are presented in Section 5, which is followed by concluding remarks in Section 6.

2 Governing equations and near-well upscaling 2.1 Governing equations and numerical discretization We consider steady state, incompressible, single-phase flow for the upscaling calculations; the upscaled parameters are applicable to more general (two-phase or multiphase) flow systems. The governing equation is formed by combining the continuity equation


(∇ · u = −q) and Darcy’s law (u = −k · ∇p) (with all quantities dimensionless and viscosity equal to one): ∇ · k(x) · ∇p = q, (1) where p is pressure, u is the Darcy velocity, k(x) is the permeability tensor, which can be highly variable in space x, and q represents the source/sink term, i.e., the well flow rate (positive for production). In the following, we assume the fine scale models are defined on Cartesian grids with diagonal permeability tensors. A finite volume method with two-point flux approximation (TPFA) is used to solve Eq. 1. In this method, the flux from grid block i to block l depends only on the pressures in these two blocks. Therefore, the discrete divergence of the flux of grid block i can be assembled as: Ti,l ( pl − pi ) = qi , (2) l

where Ti,l is inter-block transmissibility between grid blocks i and l. Note that Ti,l relates the flow from one block to an adjacent block in terms of the pressure difference between the blocks. It depends on permeability and grid geometry. In reservoir simulations, a well index WI is used to relate the well flow rate q to the difference between the wellbore pressure pw and the well block pressure p: q = WI p − pw . (3) The well index WI is often computed using Peaceman’s well model [22]. This model is derived on two-dimensional uniform rectangular grids with TPFA. For a vertical well fully penetrating the model in the z direction, WI is given by: 2π kx k y h , (4) WI = ln(r0 /rw ) where kx and k y designate the x and y components of the permeability tensor, h represents the model thickness, rw is the wellbore radius, and r0 is referred to as the equivalent radius of the well block. The quantity r0 is dependent on the permeability and grid geometry and is computed as [21]: 1/2 (k y /kx )1/2 x2 + (kx /k y )1/2 y2 r0 = 0.28 , (5) (k y /kx )1/4 + (kx /k y )1/4 where x and y designate the grid block dimensions. We note that the derivation of Eqs. 4 and 5 depends on grid geometries, well configurations, and numerical discretization schemes. Two types of well constraints can be employed. If the well is under a well flow rate constraint (i.e., qi

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is specified), Eq. 2 can be directly solved. The wellbore pressure ( piw ) can then be computed using Eq. 3. Alternatively, if the wellbore pressure is specified, substituting Eq. 3 into Eq. 2 gives a linear system for pressures. Following its solution, the well flow rate qi can be determined.

2.2 Motivation for near-well upscaling In many cases, flow simulation models are developed from highly detailed geological descriptions (e.g., generated using geostatistics). The direct simulation of Eq. 1 on the fine scale is very demanding computationally. Thus, upscaling procedures are applied to generate approximate coarse scale models with appropriate parameters. The coarse scale flow equation is often taken as the same form as Eq. 1, but with k replaced by coarse scale permeability k∗ . In the discretized equation (Eq. 2), Ti,l ∗ is replaced by coarse scale transmissibility Ti,l . Standard ∗ upscaling methods can provide either k or T ∗ . In either case, local flow problems are solved subject to local boundary conditions that essentially assume “linear” flow fields (e.g., constant pressure – no flow or periodic boundary conditions, see [11, 23, 31]). As discussed in Section 1, the pressure field in the vicinity of wells cannot be expected to be linear by any means. As a result, standard flow-based upscaling approaches introduce significant errors in modeling well-driven flows. In particular, the use of k∗ , computed from a linear flow field, into the calculation of coarse scale well index (WI∗ ) is not justified. Other options, such as the use of simple averaging of permeability (e.g., arithmetic average) over coarse scale well blocks, are not sufficient to accurately capture the fine scale flow effects either. This has led to the development of flow-based nearwell upscaling procedures (e.g., [9, 13]). Local fine scale well-driven flow problems (rather than linear flows in standard upscaling methods) are solved in the near-well regions, from which the coarse scale well index WI∗ are directly computed. So far, most flow-based nearwell upscaling work has focused on Cartesian grids. In addition to WI∗ , the coarse scale transmissibilities of coarse scale well blocks are often computed from the near-well upscaling procedures. In this paper, we study the modeling of wells in coarse scale flow simulation models, i.e., the calculation of coarse scale well index WI∗ . We first develop a flowbased near-well upscaling procedure for unstructured grids (here, prismatic Voronoi grids). This approach only computes the coarse scale well index, therefore

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3.1 Local near-well upscaling for prismatic Voronoi grids

Fig. 1 Examples of prismatic Voronoi grids to resolve wells

simplifying to some extent the geometrical effects introduced by flexible grids in upscaling calculations.

3 Flow-based near-well upscaling procedures for prismatic Voronoi grids Voronoi grids have the flexibility to resolve important geological/engineering features. Figure 1 illustrates the use of prismatic Voronoi grids to resolve wells (vertical wells in Fig. 1a and horizontal wells in Fig. 1b). As shown in Fig. 1a, Voronoi grids can be easily refined around some features (wells) while retaining large coarsening ratios in other regions. Note that, in Voronoi grids, grid block interfaces are perpendicular to their inter-node connections; as a result, TPFA is appropriate to be applied for the discretization, as long as the permeability field is not strongly anisotropic. This is the case for the upscaled permeability considered here. In this section, we first present a local near-well upscaling approach to modeling the coarse scale well index for prismatic Voronoi grids, in which the coarse grid and the fine grid are not aligned. Some general geometrical issues encountered in the upscaling for this type of flexible grids will then be discussed. Finally, we describe a global near-well upscaling procedure to account for the impact of global flow on the upscaled well index.

Fig. 2 Schematic showing the extraction of local fine scale region from global coarse grid: a global coarse grid, b target coarse blocks containing a vertical well, c areal view of local fine scale region, and d local fine scale region

Similar to the near-well upscaling procedure for Cartesian grids, near-well upscaling for Voronoi grids requires the following steps: define a local region; impose local boundary conditions; and post-process flow results to compute coarse scale quantities. In the following, we briefly present the procedure and focus on some issues that are not encountered with Cartesian grids, such as the mapping of local fine scale solutions to coarse grids. For detailed description about flow-based near-well upscaling procedures, see, e.g., [9, 13, 19]. We first consider how to extract a local region from the global domain. This is illustrated in Fig. 2. For a given global coarse grid (Fig. 2a), we first identify the well blocks that a well penetrates (Fig. 2b). We then consider a region including the well blocks and a ring of adjacent coarse grid blocks. This is illustrated by the central hexagonal block and its six neighboring blocks shown in Fig. 2c. The local fine scale region is then defined by the fine scale cells inside a bounding box to the above coarse scale blocks. These fine scale cells are indicated in shade in Fig. 2c. Shown in Fig. 2d is the three-dimensional local region. The definition of the local fine scale region follows the previous near-well upscaling methods for Cartesian grids, where coarse scale transmissibilities between well blocks and adjacent blocks are often computed in addition to the coarse scale well indexes. Therefore, not only are coarse scale well blocks included in the local region, but also their neighboring blocks. In this paper, only the coarse scale well index will be computed from the local flow problem. Although the local region can be reduced, we believe that the extra ring helps reduce the effect of local boundary conditions. Generic well-driven flows for vertical and horizontal wells [13, 18] are imposed in the local region. Specifically, we prescribe the wellbore pressure to be one and the pressure or flux on the outer boundaries to be zero. The finite volume discretization with TPFA is


used to solve the local fine scale flow problem, and the Peaceman [22] well model is used to calculate the fine scale well index (Eqs. 4 and 5). With the local fine scale solution, coarse scale quantities are calculated by averaging (or integrating) appropriate fine scale quantities. Flux and pressure are the two quantities required for this upscaling procedure. The flux can be integrated from the fine scale solution, whereas the pressure is determined by averaging fine scale pressures over the coarse block. The averaging procedure for Voronoi (or other geometrically flexible) grids is not as straightforward as that for rectangular Cartesian grids, as the coarse grid is generally not aligned with the underlying fine grid. In this work, we only compute the upscaled well index. Integrated fine scale well rates, averaged well pressures, and averaged well block pressures are the required quantities (see Eq. 6 below). This simplifies the geometry issue, as the integration of the local fine scale flow rate along the well path is not affected by the coarse grid geometry and no flux calculation at coarse grid boundaries is needed. For the coarse scale well block pressure, the fine scale pressure is averaged over the well block. A twodimensional example is shown in Fig. 3, where the fine scale pressure (from a local fine scale well-driven flow) in Fig. 3a needs to be mapped onto the coarse grid shown in Fig. 3b. In this paper, we approximately determine the mapping between a coarse grid cell and fine scale cells using the fine scale cells whose centers lie inside the coarse block (see Fig. 4b). This inexact sampling may have an impact on the accuracy of upscaling when the fine scale cell size is close to the simulation cell size. In this case, the underlying geostatistical grid can be refined to provide better resolution. On the other hand, when the areal upscaling ratio is as small as 3, we did not observe significant effect from the approximate sampling of pressures. After mapping local fine scale solutions onto the appropriate coarse blocks, the upscaled well index (WI∗ )

Fig. 3 Mapping of flow solution between a local fine scale grid, and b global coarse scale grid

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Fig. 4 Schematic showing a coarse block (represented by the heavier lines) and its underlying fine grid cells: a underlying fine cells, and b fine cells contained in the coarse block

for one coarse scale well block (for which a well is penetrating) is calculated as: WI∗ =

qw , pw − p

(6)

where qw designates the integrated fine scale well flow rate, pw is the averaged wellbore pressure, and p is the averaged fine scale block pressures over the corresponding coarse block. Note that Eq. 6 needs to be applied to each coarse scale well block for a given well. 3.2 Discussion on mapping of fine scale flux from fine to coarse scale grids In mapping local fine scale solutions to arbitrary coarse grids, a challenge is to preserve fine scale flux on the coarse grid. Although this is not encountered here (as described above), we discuss two practical approximations to deal with the geometric effects, each with its own advantages and disadvantages: – Accurate representation of coarse block geometry using subcells. To accurately represent the coarse block geometry, a set of subcells is constructed to ensure that the boundaries of local fine cells and coarse blocks are aligned. Permeabilities are then sampled from the fine scale geologic model to the subcells. This approach has been considered by a number of researchers. For example, Khan and Dawson [16] presented such a procedure for transmissibility upscaling in prismatic Voronoi grids, and Prévost [24] considered this issue in upscaling for fully unstructured systems. The main advantage of this approach is that the boundary alignment of fine and coarse cells will facilitate the calculation of coarse scale fluxes and pressure gradients, which is important in upscaling. The disadvantage lies in that the construction of subcells can significantly complicate the upscaling algorithm. In practice, this

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as well as the resampling of subcell permeability can be time consuming. – Approximate treatment of the geometric effects. An alternative approach is to approximately treat the geometric effects. Coarse scale and fine scale grids are applied without their grid boundaries matching exactly. This approach does not require the construction of additional subcells, but makes the accurate calculation of flux on the coarse block interface much more difficult. One example is shown in Fig. 4a, where the coarse grid block and the fine grid blocks are arbitrarily intersected. Ding [10] discussed several treatments to deal with the complex geometric intersections in coarse scale flux calculations. The near-well upscaling procedure presented here only computes the coarse scale well index, which does not require the calculation of fine scale flux at the coarse grid block intersections. This is a reasonable simplification, as the well pressure and well flow rate are often of the most interest for reservoir well-driven flows. The flow results in Section 5 will show that our approach here is effective in the coarse scale modeling of near-well flows. We note that, however, this treatment may not preserve fine scale fluxes in the near-well region, which may require coarse scale transmissibilities to be computed from the well-driven flow in the near-well region in addition to the coarse scale well index. If this is desired, either of the above approaches has to be employed to deal with the geometrical issue. 3.3 Global near-well upscaling In the local near-well upscaling approach (Section 3.1), generic local well-driven flows are used. It is well known that local boundary conditions can affect the accuracy of upscaled properties. In this section, we describe a global approach, where the global fine scale solution is used to compute the upscaled well index. This is different than the previous global upscaling approaches (e.g., [15, 33]) in that, here, we only use the global fine scale solution to upscale the well index. Coarse scale transmissibilities in other regions are computed locally. This allows us to isolate the impact of global flow on the near-well flow behavior. In addition, we introduce a global iteration procedure to achieve the consistency between WI∗ and global coarse scale flow solutions, which has not been considered before. In global near-well upscaling, the global fine scale flow (Eq. 1) with specified boundary conditions at wells


is solved. Given the global fine scale solution, Eq. 6 is used to compute WI∗ for each coarse scale well block. Global near-well upscaling eliminates the assumption of local boundary conditions. The resultant WI∗ may provide a coarse scale solution that is very close to the averaged fine scale solution. In some cases, however, they do not match exactly. For the cases considered here, this discrepancy is due to two facts: (1) Only WI∗ is computed using the global flow – Coarse scale properties in other regions (T ∗ ) are still computed using local methods; (2) there is approximation (due to the geometric effects) in computing the averaged well block pressure p, thus p does not necessarily represent the actual pressure pc from the global coarse scale solution. A global iteration procedure can be introduced to achieve the consistency between the averaged fine scale well block pressure p and the coarse scale pressure pc . It proceeds as follows: ν 1. Apply global near-well upscaling to obtain WI∗ . Set iteration number ν = 0. ν 2. Solve global coarse scale flow with WI∗ (and T ∗ from local upscaling method) to obtain coarse scale pressure ( pc )ν . 3. Compute coarse scale well index as

WI∗

ν+1

=

qw , pw − ( pc )ν

(7)

where qw and pw are computed from the global fine scale flow solution (step 1). 4. If not converged, set ν = ν + 1 and iterate on steps 2–3. Note that, during the iteration, WI∗ is computed to preserve the reference well flow rate qw (obtained from global fine scale solution) by adjusting pressures. The convergence criteria can be based on the difference between the global coarse scale solution and the reference solution (e.g., well flow rate). As pointed out earlier, here, the only coarse scale property that we compute from the global reference solution is the well index WI∗ . In a recent work, Chen et al. [4] studied global (and global iterative) upscaling in a more general setting in that all the coarse scale quantities (T ∗ ) are computed from the global flow. However, wells were not considered by Chen et al. [4]. Also related to the iterative approach is the local– global upscaling presented by Chen and Durlofsky [3]. In that method, global flow was incorporated into local upscaling calculations through local boundary conditions, and global coarse scale and local fine scale flow problems were solved iteratively to achieve the consistency between the global and local flow. We


35

note that the global iterative near-well upscaling presented here solved global fine scale flow, whereas the local–global upscaling approach used only global coarse scale solution. In addition, in the local–global upscaling, adaptivity (a thresholding criterion) was introduced to compute coarse scale transmissibility T ∗ in high flow regions as well as well index WI∗ .

4 Approximate coarse scale well modeling The flow-based near-well upscaling can be viewed as a “rigorous” procedure, as flow information is used to derive the coarse scale property. In this section, we present an alternative approach, which approximately accounts for the effects of fine scale heterogeneity in the near-well region. This approximate well model averages the fine scale permeabilities along well trajectories to compute the coarse scale well index and does not require solving any fine scale flow problems. This approach is proposed based on the following analysis of the pressure solution for two-dimensional well-driven flows. Consider a well-driven flow in a homogeneous porous medium. It can be modeled using the following equation: ∇ · (k · ∇p) =

n

qjδ xwj ,

(8)

j=1

where δ is the Dirac delta function, n is the total number j of wells, and qjδ(xw ) is a singular source representing j the jth well with flow rate qj located at xw . By the principle of linear superposition, the solution for Eq. 8 can be decomposed into two parts: p(x) = pr (x) +

n

psj(x),

(9)

j=1 j ps

is the singular solution due to the presence where of the jth well and pr represents the regular solution j without singularity. Note that ps satisfies: ∇ · k · ∇psj = qjδ xwj , (10) for which the analytical solution can be easily obtained, and pr satisfies ∇ · (k · ∇pr ) = 0 subject to appropriate boundary conditions. This decomposition suggests a solution method for p. The solution pr can be obtained using numerical methods on relatively coarse grids, whereas ps can be obtained analytically and then added to pr . This means that there is no need for grid refinement near singularities (around wells). This approach is also referred to as singularity programming and has been applied

in petroleum engineering (e.g., [1, 2]) to improve the pressure solution in the presence of wells. For heterogeneous media, note that the well-driven flow is dominated by the permeability at the singularity point, i.e., the well location. To see this, consider a single well located at xw with the permeability at the well location designated by k(xw ) = k0 . The pressure solution p = pr + ps satisfies: (11) ∇ · k(x) · ∇p = qδ(xw ). The governing equation for ps is similar to Eq. 10, but with k replaced by k0 : ∇ · (k0 · ∇ps ) = qδ(xw ).

(12)

Substituting p = pr + ps in Eq. 11, the equation for pr can be derived as follows: ∇ · k(x) · ∇( pr + ps ) = qδ(xw ), ∇ · k(x) · ∇pr = qδ(xw ) − ∇ · k(x) · ∇ps (13) Noting that qδ(xw ) = ∇ · (k0 · ∇ps ), the above equation can be written as: (14) ∇ · k(x) · ∇pr = ∇ · (k0 − k(x)) · ∇ps . The solution structure of pr in Eq. 14 is not as simple as its counterpart in the homogeneous system. The following shows that the source term in Eq. 14 is still singular, but the singularity is much reduced. For simplicity, consider an isotropic permeability field (i.e., k = kI, k0 = k0 I), the solution ( ps ) for Eq. 12 can be readily obtained as: q ps = ln |x − xw |. (15) 2π k0 Therefore, the source term in Eq. 14 can be written as (k − k0 )∇ps =

q x − xw (k − k0 ) , 2π k0 r2

(16)

where r = |x − xw |. We see that if k is differentiable at xw , then as x → xw , x − xw ∂k |x − xw | . (17) → 2 r ∂r r Therefore, the singularity of the source term in Eq. 14 is of r−1 type, which is much weaker compared to the delta function in the original equation for p (Eq. 11). The above derivations indicate that the upscaling of well-driven flow actually includes two components and that it is the scale-up of the singular solution ( ps ) that has the dominant impact on the coarse scale near-well flow. It also suggests that the coarse scale well index in heterogeneous formations can be approximately calculated using the fine scale permeability at the well (k − k0 )

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Durlofsky [12] presented a model to represent well-driven flow using an effective background permeability (representing a linear flow) and an effective near-well skin (computed from near-well permeabilities, to capture the radial flow). This can be viewed as a decomposition of the well-driven flow into two parts, as discussed above. It would be of interest to further study and compare these two methods.

5 Numerical results

a Arithmetic averaging

b Near-well arithmetic averaging

Fig. 5 Schematic showing arithmetic averaging and near-well arithmetic averaging for coarse scale well block permeability

location (k0 ). A more rigorous analysis can be found in [5]. The above analysis is based on two-dimensional radial flow. In this paper, we propose an approximate coarse scale well model for three-dimensional problems. Specifically, the three-dimensional problem is approximated by a series of two-dimensional problems. Thus, the coarse scale well block permeability that is used in Peaceman’s well index calculations is the arithmetically averaged fine scale permeability along the well trajectory within that coarse block. We call this approach “near-well arithmetic averaging,” as opposed to the standard arithmetic average, where all of the fine scale permeabilities within the coarse well block are averaged. Figure 5 illustrates the fine scale permeabilities involved in these two approaches. For the regular arithmetic averaging, the coarse scale well block permeabil w kijk Vijk / Vijk , where kijk ity (kw eff ) is given by keff = and Vijk represent the fine cell permeability and volume within the entire coarse scale well block. For the kw Vw / Vw , near-well arithmetic averaging, kw eff = where kw and Vw are the fine cell permeability and volume along the well path within the coarse scale well block. Fig. 6 Log-normal permeability distributions with different horizontal correlation lengths (λx = λ y ): a 0.05, b 0.25, and c 0.5. For all the cases, λz = 0.01

Numerical results are presented in this section. We first describe permeability distributions and flow problems, followed by the definition of flow rate errors. We then demonstrate the impact of reservoir heterogeneity on coarse scale near-well flow. Flow results for different coarse scale well models will be presented in Section 5.4. Finally, we discuss the results for global near-well upscaling. 5.1 Permeability distributions and flow problems We consider two types of permeability distributions: variogram-based permeability fields and channelized models. The variogram-based permeability distributions are generated using a sequential Gaussian simulation with a spherical variogram model [8]. The log-normally distributed permeability fields can be characterized by correlation length λ and variance (of log k) σ 2 . We generate fine scale models of dimensions 60 × 60 × 100, with physical dimensions 6,000 × 6,000 × 500 ft3 . Figures 6 and 7 show the fine scale permeability distributions with different correlation lengths. Note that the correlation length is nondimensionalized by the corresponding system length. The permeability distributions in Fig. 6 have the same vertical correlation length (0.01), but varying horizontal correlation length (λx = λ y ) of 0.05, 0.25, and 0.5. Figure 7 displays permeability distributions with λz = 0.01, 0.05, and 0.2, respectively, and the horizontal correlation length fixed at 0.25. In all the cases, σlog k = 2.0, and k y = kx , kz = 0.1kx . This


37

Fig. 7 Log-normal permeability distributions with different vertical correlation lengths λz : a 0.01, b 0.05, and c 0.2. For all the cases, λx = λ y = 0.25

set of permeability distributions allows us to systematically study the effects of reservoir heterogeneity on upscaling. The other type of permeability field corresponds to channelized models, which are usually more difficult for upscaling. We will describe these models when presenting flow results in Section 5.4. Both vertical and horizontal wells are considered. We compare fine and coarse scale well rates for prescribed wellbore pressures. The coarse scale well models include coarse scale well index computed using arithmetically averaged fine scale permeabilities over the entire coarse scale well block (this is one of the commonly used practical procedures), the local flowbased near-well scale-up approach, and the approximate well model (near-well arithmetic averaging). The use and potential errors of standard flow-based k∗ in the near-well region have been discussed by numerous researchers (e.g., [9, 13, 19]) and will not be considered here. For the inter-well regions, a flow-based local transmissibility upscaling method suitable for Voronoi grids [16] is applied. For the variogram-based model shown in Figs. 6 and 7, we consider eight fully penetrating vertical wells (four injectors and four producers) in the reservoir. This well configuration is schematically shown in Fig. 1a, where the fine scale model is uniformly coarsened by a factor of 10 in the vertical direction, and a coarsening factor of 3 is specified for all the well blocks, and for other blocks, a factor of 4 is used. The problem with horizontal wells involves two injectors and two producers, and rectangular grids are used with the fine scale model (60 × 60 × 100) uniformly coarsened to 20 × 20 × 10.

(on the coarse scale). The fine scale results are obtained by integrating the fine scale flow rate onto the corresponding coarse block. The coarse scale well model using arithmetic average k∗ (dashed curve with square) shows considerable deviation from the fine scale results

5.2 Inflow profiles and flow rate errors We now compare well flow rates for the vertical wells illustrated in Fig. 1a. Figure 8 shows inflow profiles for two vertical wells in permeability distributions with different correlation lengths. Each point in the plot represents well flow rate through one well block

Fig. 8 Inflow profiles for vertical wells in log-normally distributed permeability fields with different correlation lengths

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Comput Geosci (2008) 12:29–45 60%

60%

50%

50%

λ x= λy = 0.05 Well Rate Error

Well Rate Error

Fig. 9 Well rate errors for vertical wells for permeability distributions shown in Fig. 6 (λz = 0.01 for all the cases)

40% 30% 20% 10%

inj2

a

inj3

inj4 pro1 pro2 pro3 pro4

n qfine − qcoarse i i

,

(18)

i=1

where i designates a coarse scale well block, n is the total number of coarse scale well blocks for a given well, qicoarse is coarse scale flow rate, and qifine the integrated fine scale flow rate. Note that the well rate error is a normalized L1 error and avoids the cancellation of local errors for a well. The well rate errors for the results shown in Fig. 8a are 34.7% for the arithmetic average and 8.8% for the flow-based upscaling. For Fig. 8b, the corresponding errors are 18.7% and 8.0%, respectively.

100%

inj2

b

inj3

inj4

pro1 pro2 pro3 pro4

Flow-based near-well upscaling

5.3 Effects of correlation lengths on coarse scale near-well flow We next present results in terms of the well rate errors. The log-normal permeability distributions shown in Figs. 6 and 7 are first considered to illustrate the impact of reservoir heterogeneity (correlation length) on the coarse scale near-well flow. Shown in Fig. 9 are the well rate errors of eight vertical wells for the arithmetic average k∗ and the flow-based near-well upscaling, where different shades designate different horizontal correlation lengths (see the corresponding permeability fields in Fig. 6). In Fig. 9a, large errors (30–40%) are observed for all eight wells for the short correlation length (λx = λ y = 0.05). When the horizontal correlation length increases to 0.5, the overall errors decrease considerably (to about 10–15%). In Fig. 9b, the flow-based near-well upscaling is seen to reduce the error significantly, to less than 10% for all the cases. The results for horizontal wells for different vertical correlation lengths are shown in Fig. 10. For the case with very short vertical correlation length (λz = 0.01), substantial errors exist with the use of arithmetic averaged k∗ (the average error over the four wells is 134%, indicated by the gray bar in Fig. 10a). The average

394%

100% 90%

λz = 0.01 λz = 0.05 λz = 0.2

80%

Well Rate Error

80%

Well Rate Error

Fig. 10 Well rate errors for horizontal wells for permeability distributions shown in Fig. 7 (λx = λ y = 0.25 for all the cases)

inj1

Arithmetic average k

(especially in Fig. 8a), whereas the results from the flow-based near-well upscaling are in very close agreement with the fine scale model. Also evident in Fig. 8a and b are the smoother inflow profile and smaller errors associated with the permeability field with longer correlation length (Fig. 8b). To systematically compare the results for different cases, we now define a well rate error to quantify the mismatch between the fine and coarse scale results:

n qfine i

20%

0%

inj1

i=1

λx= λ y = 0.5

30%

10%

0%

error =

λx= λy = 0.25

40%

60%

40%

70% 60% 50% 40% 30% 20%

20%

10% 0%

0% inj1

inj2

a

pro1

Arithmetic average k

pro2

inj1

b

inj2

pro1

pro2

Flow-based near-well upscaling

Comput Geosci (2008) 12:29–45 Table 1 Average well rate errors for eight vertical wells for log-normal permeability distributions

39 λx (= λ y ) λz

0.05 0.01

0.25 0.01

0.5 0.01

0.25 0.05

0.25 0.2

Arithmetic average (%) Near-well arith. ave. (%) Flow-based nwsu (%)

37.5 18.9 5.5

21.0 3.8 6.4

12.0 2.1 5.3

15.7 3.7 6.4

15.3 2.1 5.0

error is reduced to 23.6% when the vertical correlation length increases to 0.2. The use of flow-based nearwell upscaling again produces accurate coarse scale solutions for the horizontal wells. The errors are in all cases reduced to 15% or less, as shown in Fig. 10b. In this paper, we present only these two cases to demonstrate the errors of coarse scale well models using simple averaging and the significant improvement offered by the flow-based approach. In fact, these errors are closely related to the correlation lengths of the permeability distributions and well configurations. In general, for vertical wells, the use of simple averaging leads to large errors for cases with small λx and λ y . For horizontal wells, the flow is mainly in the vertical direction; therefore, vertical permeability, often displaying a high degree of heterogeneity, can have a strong impact on the flow. As a result, the simple averaging approach gives larger errors for smaller λz . 5.4 Flow results for different coarse scale well models The results described above are focused on lognormally distributed permeability fields and flow-based near-well upscaling. Next, we present flow results using the approximate well model as well as the flow-based approach. Both the variogram-based and channelized models will be considered.

close agreement with the fine scale solution (well rate errors of 8.0 and 17.7%, respectively). The coarse scale model with arithmetic average permeability (dashed line with square), by contrast, shows considerable error (44.1%) relative to the fine scale results. The results for horizontal wells are shown in Table 2. Near-well arithmetic averaging again performs much better than the block arithmetic average, although its overall accuracy is not quite as high as that of the flowbased approach (in contrast to the vertical well cases, Table 1). In accord with the vertical well case, the largest error here with near-well arithmetic averaging is associated with the shortest correlation length. For this case, the inflow profiles for one typical well are displayed in Fig. 12. The near-well arithmetic averaging (dashed–dotted line) captures the overall flow profile (well rate error of 23.1%), showing reasonable accuracy compared to the use of arithmetic average k∗ (error of 105%). The most accurate flow results are again achieved by the flow-based near-well upscaling procedure (8.5% error as indicated by the dotted line in Fig. 12). The above examples illustrate that the approximate coarse scale well model proposed in Section 4 can approximately capture the effects of near-well fine scale heterogeneity and provide considerable improvement

5.4.1 Results for log-normal permeability distributions Table 1 presents the comparison of flow results (averaged well rate errors over eight vertical wells) among the three coarse scale well models. It clearly shows that the approximate well model, near-well arithmetic averaging, provides much better accuracy than that achieved using arithmetic average k∗ . In fact, except for the case with very short correlation length (λx = λ y = 0.05, and λz = 0.01), the accuracy of near-well arithmetic averaging is comparable to that of flowbased approach. For the case with very short correlation length (where near-well arithmetic averaging shows an average error of 18.9% in Table 1), detailed inflow profiles for one well are presented in Fig. 11. Both the flowbased approach and the approximate well model are in

Fig. 11 Inflow profiles for a vertical well (producer 3) in a lognormal permeability field with correlation length λx = λ y = 0.05, and λz = 0.01

40 Table 2 Average well rate errors for four horizontal wells for log-normal permeability distributions

Comput Geosci (2008) 12:29–45 λx (= λ y ) λz

0.05 0.01

0.25 0.01

0.5 0.01

0.25 0.05

0.25 0.2

Arithmetic average (%) Near-well arith. ave. (%) Flow-based nwsu (%)

88.3 28.7 9.1

134.3 13.0 11.0

53.8 18.8 6.4

55.1 7.9 9.1

23.6 8.9 7.1

over the use of arithmetic average block k∗ . Accuracy comparable to that of flow-based near-well upscaling is obtained for cases with long correlation lengths. 5.4.2 Results for Stanford V channelized model We next consider channelized models, which are usually more difficult for upscaling. The first model involves the Stanford V reservoir [17] and horizontal wells. This model represents a moderate level of difficulty for upscaling. The fine scale permeability field, of dimensions 100 × 130 × 30, is coarsened uniformly to 20 × 26 × 6. Four horizontal wells are positioned in different layers of the reservoir, as shown in Fig. 13. For prescribed well pressures, we compare well flow rates computed using the different procedures. For this case, rather than presenting well rate errors, we make detailed comparison between fine and coarse scale flow rates using the cross-plots shown in Fig. 14, where each point represents well flow rate through one coarse scale well block. Fine scale flow rates are computed by summing the fine scale results over corresponding coarse block regions. The dotted line represents 10% error, and the dashed line indicates 40% error. The use of arithmetic average k∗ for the calculation of WI∗ again shows large errors.

Fig. 12 Inflow profiles for a horizontal well (injector 2) in a lognormal permeability field with correlation length λx = λ y = 0.05, and λz = 0.01

Some flow rates are significantly overestimated. These overpredictions are essentially eliminated by the use of near-well arithmetic averaging (Fig. 14b) – The correlation coefficient between the fine and coarse flow rates increases from 0.747 to 0.988. Figure 14c shows the results for flow-based near-well upscaling, which further improves the accuracy (correlation coefficient of 0.995). 5.4.3 Results for SPE 10 channelized model The second channelized model is derived from the Society of Petroleum Engineers (SPE) 10 reservoir [6]. Only the more highly heterogeneous portion of the reservoir (the lower 50 channelized layers) is considered here. The permeability distribution and histogram are displayed in Fig. 15. We see that the permeability varies over seven orders of magnitude. The fine scale model is of dimensions 60 × 220 × 50, and the coarse scale model is 12 × 44 × 10. Eight vertical wells (four injectors and four producers) fully penetrate the reservoir with regular spacing. The results represented by well rate errors are shown in Fig. 16. For this channelized reservoir, larger errors are observed than in log-normal permeability distributions. The near-well arithmetic averaging is not as accurate as in previous cases, but still shows considerable improvement compared to the block arithmetic averaging. The flow-based near-well upscaling provides the best accuracy (errors of around 10%). Shown in Fig. 17 are the inflow profiles for one typical well (injector 4) in the SPE 10 channelized model. The flow-based near-well scale-up procedure still captures the overall fine scale solution. The near-well arithmetic averaging shows inaccurate predictions for some blocks, but it is generally more accurate than use of arithmetic average k∗ . The well rate error of arithmetic average k∗ is 61%, and that for near-well arithmetic averaging is 26.6%. The above observation is consistent with previous findings in upscaling studies – Channelized systems are more challenging for upscaling due to the high permeability contrasts and the complex connectivity in the permeability distributions. In this section, we compared the flow-based approach and the near-well arithmetic averaging to the


41

Fig. 13 Well locations for the four horizontal wells in Stanford V reservoir [17]

fine scale reference solutions. For the cases considered (variogram-based and channelized models), both methods can provide coarse scale models that are in reasonably close agreement with the fine scale solution. For highly heterogeneous reservoirs, the approximate well model is not as accurate as the flow-based approach, but it still offers significant improvement over the simple averaging approach.

Fig. 14 Flow results of various coarse scale well models for four horizontal wells in Stanford V reservoir

5.5 Results for global near-well upscaling There are at least two factors that can affect the accuracy of coarse scale well modeling: the upscaled well index itself and the coarse scale properties in other regions. In this section, we apply global near-well upscaling to illustrate the interaction between global flow and near-well upscaling.

42


Fig. 15 Permeability distribution and histogram of the lower portion of SPE 10 reservoir [6]

5.5.1 Log-normal permeability distributions We again consider both the variogram-based and channelized models. Figure 18a shows the result of global near-well upscaling for a log-normal permeability distribution with very short correlation lengths (λx = λ y = 0.05, λz = 0.01). The fine scale solution and the result from local flow-based near-well scale-up are also presented in Fig. 18a. We see that there is essentially no difference between the local and global near-well upscaling – Both predict the well flow that is very close to the fine scale solution. This indicates that, for the log-normal permeability distributions considered here, near-well flow can be localized, and the upscaling results are not sensitive to local boundary conditions. This is the basic assumption for many existing upscaling methods. Shown in Fig. 18b are the results for global iterative near-well upscaling (Eq. 7 in Section 3.3). We begin with the WI∗ calculated from global near-well

upscaling (the dotted line in Fig. 18b), then iterate the WI∗ calculations based on global coarse scale pressures. The dashed line represents the well rate after one iteration, which improves the results of global near-well upscaling. The converged solution, designated by the dash–dotted line with squares, can match exactly the fine scale solution (these two curves are overlapped in Fig. 18b). This is achieved by adjusting only the coarse scale well index, indicating that the coarse scale transmissibilities in other regions are sufficiently accurate. 5.5.2 SPE 10 channelized model The impact of inter-well coarse scale properties on near-well flow can be further demonstrated by the next example involving three vertical wells in the SPE 10 channelized model (as shown in Fig. 15). A schematic illustration of the well locations (one injector and two producers) is shown in Fig. 19. All the wells are fully

100% Arithmetic average

Well rate error

80%

Near-well arith. ave. Flow-based nwsu

60%

40% 20% 0%

inj1 inj2 inj3 inj4 pro1 pro2 pro3 pro4 Fig. 16 Well rate errors of various coarse scale well models for the eight vertical wells in SPE 10 channelized model

Fig. 17 Inflow profiles for a vertical well (injector 4) in SPE 10 channelized model


43

Fig. 18 Comparison of local near-well upscaling, global near-well upscaling and global iteration for a vertical well (injector 1) in a log-normal permeability distribution (λx = λ y = 0.05, and λz = 0.01)

penetrating the reservoir. This well configuration was studied previously by Wen et al. [26], and large errors were observed for flow-based local near-well upscaling. In this work, we consider a rectangular grid and a Voronoi grid to coarsen the fine scale model. A scale-up ratio of 5 in each dimension gives a coarse scale model containing 5,280 rectangular cells (as shown in Fig. 19a) and a Voronoi grid with 4,790 cells (Fig. 19b). Shown in Fig. 20a are the flow results for the rectangular grid for producer 2. Even the global near-well upscaling presents considerable errors. Global iteration does improve the coarse model, but still showing large errors in some perforations. Figure 20b presents the results for the Voronoi grid. We see that the global near-well upscaling still shows large errors. Of more interest here is the result for global iteration, which is in very close agreement with the fine scale solution. Global iteration applies the pressures from global coarse scale solutions to compute WI∗ . Its intent is to improve the consistency between the pressures used in the calculation of WI∗ and those in the global coarse

Fig. 19 Two grid systems for SPE 10 channelized model with three vertical wells. The well locations are as defined in [26], for which local near-well upscaling shows considerable errors

scale flow. When the accuracy of coarse scale properties in the inter-well region is sufficient, the global iterative near-well upscaling can provide a solution that is close to the fine scale model. Given the fact that the coarse scale well index is iteratively computed in both cases, the essential difference between the rectangular grid and the Voronoi grid lies in the coarse scale transmissibilities in the interwell regions. This implies that coarse scale properties in other regions can have a strong impact on coarse scale flow in the near-well region. Wen at al. [26] applied a local–global upscaling procedure [3] to improve the coarse scale flow prediction for this case. In [32], a global upscaling method is used in conjunction with local near-well upscaling. They show that the use of global upscaling in regions away from wells greatly improves the accuracy for this case. Our results in this section show that, for log-normal permeability distributions considered here, it is sufficient to apply generic local well-driven flows to compute coarse scale well index, whereas for more complex systems (e.g., highly heterogeneous channelized reservoirs), near-well flow may strongly depend on global flow and cannot be adequately localized. As indicated above, the accuracy of modeling near-well flow depends on the accuracy of upscaled well index as well as the coarse scale properties in other regions. Depending on the accuracy of the coarse scale properties in the regions away from wells, the fine scale solution can be matched by adjusting the well index either locally or globally. In some cases, local methods may be unable to provide sufficient accuracy, and global flow needs to be incorporated in some way into the calculation of coarse scale properties. Although it was not studied in this work, changing flow conditions (e.g., well flow rates) may impact the coarse scale properties computed from global flow. In other words, when the flow condition changes, for the cases in which near-well flow cannot be localized, global flow may need to be solved again to compute the

44


Fig. 20 Comparison of global upscaling in a rectangular grid and a Voronoi grid for a vertical well (producer 2) in the SPE 10 channelized model

coarse scale properties using either the global near-well upscaling or iterative approach.

6 Concluding remarks In this work, the coarse scale modeling of flow in near-well regions was studied. We presented several approaches with various levels of accuracy and computational cost. The approximate coarse scale well model (near-well arithmetic averaging) is the most efficient approach, as it does not require solving any type of flow problems. The local near-well upscaling method computes the upscaled well index from the solution of local well-driven flows. It is more rigorous than the approximate model and only requires solving local flow problems. The global approaches use global fine scale solution to compute coarse scale well index. It may provide the highest level of accuracy, although the computational expenses can be prohibitively high for very large scale problems. We note that the appropriate use of the different modeling approaches depends on the type of permeability heterogeneity (and well configurations) encountered in the problem. The near-well upscaling approaches presented here were targeted toward non-Cartesian grids. We considered an approximate treatment to handle the geometrical complexity due to the nonalignment between the fine and coarse scale grids. The methods were applied to vertical and horizontal wells, and it was shown that they provide considerable improvement over cases without near-well treatment. It would also be of interest to consider the approaches presented here for nonconventional wells (e.g., deviated wells and multilateral wells), which would introduce additional complexity and require further development and testing. We point out that this work has focused on coarse scale well modeling under the assumption that the fine scale flow is of sufficient accuracy. For all the calculations at fine scale, Peaceman’s well model was applied. It is known that the Peaceman model has

limitations, especially for general well configurations and heterogeneous reservoirs, and that it depends on the numerical discretization scheme. Some research is needed to understand/improve the fine scale well index calculations, which will in turn benefit the coarse scale modeling procedures described here. The following main conclusions can be drawn from this work: – In the coarse scale modeling of well-driven flows, well index is the most important factor to model the well singularity. The use of simple average permeability can introduce large errors. Permeability distributions with shorter correlation lengths in general present more significant errors and may require more sophisticated near-well upscaling methods. Horizontal wells usually exhibit larger errors due to the high permeability variations in the vertical direction. – A flow-based near-well upscaling procedure for calculating coarse scale well index for prismatic Voronoi grids in three-dimensional reservoir models was developed and applied. This approach approximately deals with the geometrical effects introduced by the use of non-Cartesian grids. It was systematically tested and shown to provide accurate coarse scale well models (with errors within about 10% relative to the fine scale solutions) for the many permeability distributions considered. This suggests that the approximate geometrical treatment is adequate to capture the effects of fine scale heterogeneity on the near-well flow. Such a procedure and the subsequent testing were not reported by previous researchers. – We also presented an approximate coarse scale well model (near-well arithmetic averaging) based on the analysis of well singularities. The basic idea is to decompose the pressure solution into a regular and a singular component. The singular part has the dominant impact on well flow and is dominated by the fine scale permeability at well locations. This


approximate model does not require any local fine scale solutions, but can provide reasonable accuracy over an appropriate parameter range. Thus, it can be used as an efficient estimation tool in practice. – The impact of global flow on the upscaled modeling of wells was investigated, and global and global iterative near-well upscaling procedures were introduced. It was shown that, in many cases, nearwell flow can be localized, thus local near-well upscaling can provide satisfactory results. For highly heterogeneous reservoirs (e.g., channelized models), however, the coarse scale properties are in general dependent on global flow. For those cases, the global flow needs to be incorporated into the upscaling calculations to provide accurate coarse scale models. Acknowledgements This work was performed during a summer internship project in 2004 at ExxonMobil Upstream Research Company. We are grateful to D. Stern, M. Stone, R. Parashkevov, Q. Luo, L. Lee, and S. Lyons for many helpful discussions and suggestions. The first author would also like to thank L. J. Durlofsky (Stanford University) for reviewing part of this work.

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