Computational Geosciences 7: 61–82, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Calculation of well index for nonconventional wells on arbitrary grids Christian Wolfsteiner a,b , Louis J. Durlofsky a,b and Khalid Aziz a a Department of Petroleum Engineering, Stanford University, Stanford, CA 94305-2220, USA
E-mail:
[email protected],
[email protected] b ChevronTexaco EPTC, P.O. Box 6019, San Ramon, CA 94583-0719, USA
E-mail:
[email protected]
Received 17 December 2001; accepted 3 October 2002
Wells are seldom modeled explicitly in large scale finite difference reservoir simulations. Instead, the well is coupled to the reservoir through the use of a well index, which relates wellbore flow rate and pressure to grid block quantities. The use of an accurate well index is essential for the detailed modeling of nonconventional wells; i.e., wells with an arbitrary trajectory or multiple branches. The determination of a well index for such problems is complicated, particularly when the simulation grid is irregular or unstructured. In this work, a general framework for the calculation of accurate well indices for general nonconventional wells on arbitrary grids is presented and applied. The method entails the use of an accurate semianalytical well model based on Green’s functions as a reference single phase flow solution. This result is coupled with a finite difference calculation to provide an accurate well index for each grid block containing a well segment. The method is demonstrated on a number of homogeneous example cases involving deviated, horizontal and multilateral wells oriented skew to the grid. Both Cartesian and globally unstructured multiblock grids are considered. In all these cases, the method is shown to provide results that are considerably more accurate compared to results using standard procedures. The method is also applied to heterogeneous problems involving horizontal wells, where it is shown to be capable of approximating the effects of subgrid heterogeneity in coarse finite difference models. Keywords: flexible grids, heterogeneity, nonconventional wells, reservoir simulation, semianalytical methods, s–k∗ , well index, well model
1.
Introduction
The problem of representing wells in large scale finite difference reservoir simulation is fundamentally one of resolution and scale. Because grid blocks are typically two to three orders of magnitude larger than the well diameter, wells are seldom modeled explicitly. Thus, the wellbore pressure differs from that of the grid blocks it intersects and a relationship between these two pressures must be introduced into the simulator. This is accomplished through use of the well index, denoted as WI, which captures the
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interaction between the wellbore and the reservoir. Specifically, for each grid block i intersected by a well, the quantity WI i is required to relate the well flow rate into or out of the block qiw to the difference between the grid block pressure pi and the local wellbore pressure piw : qiw =
WI i pi − piw , µ
(1)
where µ is the fluid viscosity. This definition is based on the assumption of single phase flow. For multiphase flow, the same well index is used though equation (1) generalizes to: w = WI i qpi
krp ppi − piw , µp
(2)
where the subscript p designates the phase and krp is the relative permeability of phase p. Because the flow behavior of wells is a key simulation quantity of interest, the well index represents one of the most important input parameters. For simple vertical wells in homogeneous reservoirs, the well index can be determined analytically (as discussed in the next section). Modern production operations increasingly utilize nonconventional wells (NCWs); i.e., wells with arbitrary trajectory or multiple branches (laterals). Wells of this type can intersect simulation grid blocks in arbitrary ways. In addition, advanced simulation models commonly use irregular grids in order to resolve complex geological features. These grids range from structured curvilinear grids to multiblock grids to fully unstructured grids. The combined use of complex well trajectories and advanced grids therefore introduces a significant challenge for the accurate simulation of well performance. In this paper, we present a framework to compute accurate well indices for NCWs intersecting a complex target simulation grid. The method entails the definition of a local well domain and the solution of a reference well-driven flow problem over this region. The reference solution is achieved in this work using a semianalytical procedure based on Green’s functions. Once this computation is performed, an intersection algorithm is used to identify the well blocks in the target grid and to distribute the reference well rates onto the grid. A finite difference simulation is then performed over the same local well domain using the target grid with the distributed source terms determined from the semianalytical solution. The combination of grid block pressures (from the finite difference calculation) and rates and well pressures (from the semianalytical reference solution) are used to obtain accurate well indices for use in general finite difference modeling via equation (1). The generality of the semianalytical solution technique allows the method described here to approximately account for the effects of subgrid heterogeneity on the computed well index. A number of previous researchers have considered the calculation of well indices for wells of varying degrees of complexity. Modifications to the classical well index given by Peaceman [20] for a fully penetrating well in an anisotropic Cartesian grid block that account for the well orientation were given in [1,16]. These rescaling approaches are used in some commercial simulators, though they can lead to considerable
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errors in some cases [13]. Approaches that combine semianalytical solutions and finite difference results over domains larger than a single well block are more accurate. Babu and Odeh [3] used their earlier work on horizontal well productivity [4] as a reference solution for grid-aligned horizontal wells. They combined this solution with an analytical expression for the computed pressure field on a high aspect ratio Cartesian grid cross section. This allowed them to compute well indices slice by slice for a three dimensional problem. Palagi and Aziz [19] defined an exact WI for wells in a homogenous reservoir by comparing analytical to finite difference results. In the heterogenous case, they obtained a reference solution from fine grid simulations on Voronoi grids. Lee and Milliken [14] considered an arbitrary monobore well in a layered system of laterally infinite extent. Their semianalytical solution based on slender body theory was combined with a finite difference pressure solution with lateral pressure boundary conditions determined from the analytical solution. Klausen and Aavatsmark [13] solved the Laplace equation in elliptic coordinates using geometric transforms to account for a slanted well in an infinite reservoir. This result was used together with a numerical simulation to determine appropriate connection factors. Ding [6] used the layer potential function to obtain a steady state pressure distribution in the vicinity of the well. In addition, he adjusted well block transmissibilities to account more accurately for radial flow characteristics. Ding and Jeannin [7] later proposed a multipoint discretization in a curvilinear coordinate system and identified the discretization coefficient of the elliptic equation as the appropriate well index. Accurate solutions using grids that resolve the well trajectory and near well heterogeneity have also been discussed in the literature (e.g., finite element grids [17] and locally elliptical hybrid grids [10]). Our method differs from existing approaches both in its generality and enhanced functionality. The current framework allows for the accurate modeling of very complex (e.g., multilateral) wells or even groups of wells that intersect the grid in an arbitrary way. Any sort of grid can be treated as long as a simulator is available that is capable of performing a single phase pressure solution on the target grid. Further, our well index calculation involves a user-defined local well domain rather than single well blocks, cross sections or infinite domains as in previous implementations. The proposed approach is not restricted to the semianalytical reference solution applied here. However, this reference solution is extremely accurate for homogeneous problems and has been shown to provide reasonable accuracy for heterogeneous permeability fields as well. In this way, the effects of subgrid heterogeneity on the computed well index are approximated. This paper proceeds as follows. We first discuss how well indices are used in reservoir simulators. We then briefly present the semianalytical well model and describe its ability to approximate the effects of reservoir heterogeneity. We then describe in detail our overall framework for computing the well index for an arbitrary well and grid. Next, we study complex wells on multiblock and Cartesian grids with homogenous properties and demonstrate the enhanced accuracy of our method relative to conventional approaches. Finally, we apply the method to heterogeneous systems and demonstrate that it is able to provide coarse grid results in reasonable agreement with fine scale heterogeneous reference solutions.
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Well modeling with simulators
Complex wells typically intersect many grid blocks in a finite difference simulation model. The effects of the well enter the finite difference formulation via component source or sink terms for these grid blocks. The well is controlled, however, by specifying either the flow rate of one or more phases at specified conditions or pressure at some location (e.g., bottomhole). A well constraint equation, involving the particular well specifications, must therefore be solved for each well. This introduces additional unknowns (well pressure or well rate) into the system. We consider a single phase, single component, slightly compressible system. The mass balance for a grid block i containing a well is given by:
Ti,l ( l − i ) −
w
l
mw i =
φcρVi n+1 i − ni , t
(3)
where Ti,l is the transmissibility between block i and neighboring blocks l (connections), designates grid block potential, mw i represents the mass source or sink due to a well, φ is porosity, c is compressibility, ρ is density, Vi is the grid block volume, t is the time step, and the superscript n designates time level. A similar equation describes more complex systems involving multiple phases and components. However, computation of the well index used in finite difference simulators is based on the assumption of single phase flow. Once the well index is determined it is also applied to multiphase flow, as discussed in the previous section. As discussed in section 1, the key to modeling wells is the determination of the correct well index for each block through which the well passes. The well index accounts for the geometry of the grid block, location and orientation of the well segment in the grid block, and rock properties. The well equation, which relates the mw i in equation (3) to system variables, is given by (cf. equation (1)): w mw i = qi ρ = WI i
ρ pi − piw = Tiw pi − piw . µ
(4)
From equation (4) we see that the segment well index WI i can also be viewed as a well transmissibility Tiw . Peaceman [20,21] provided expressions for WI i for vertical and horizontal wells. His analytical result for the well index for vertical wells in non-square grid blocks with anisotropic (diagonal) permeability tensors is the default in virtually all commercial simulators. For a vertical drainhole this well index, which we designate as WI di , where the superscript d indicates ‘default’, is given by [20]: WI di =
2π kx ky h , ln(rw /r0 ) i
(5)
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where
(ky /kx )x 2 + (kx /ky )y 2 . r0 = 0.28 4 ky /kx + 4 kx /ky
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(6)
In these equations, kx and ky are permeability components in the x- and y-directions, h is the grid block thickness in the z-direction, rw is the wellbore radius, and x and y are the grid block sizes. These equations are appropriate for an isolated vertical well, fully penetrating and centered in the grid block, away from boundaries, in a homogeneous reservoir. The grid is further assumed to be uniform in both x and y. Modified versions of equations (5) and (6) to account for different orientations have also been developed [1,16]. However, it is important to recognize that the assumptions inherent in equations (5) and (6) will almost always be violated to some degree for complex NCWs intersecting general grids in an arbitrary manner. We now describe our procedure for the calculation of the appropriate well index for cases where the default well index is not applicable. 3.
Semianalytical reference solution
As indicated above, our procedure for the calculation of the well index requires an accurate reference solution over the local well domain. We will describe exactly how this reference solution is used in the calculation of WI i in the next section. Here, we give a brief description of the semianalytical solution methodology applied for the determination of the reference solution. The overall procedure has been presented in detail in previous publications [18,23,24]. The basic semianalytical method solves the single phase pressure equation (in potential notation), ∂ , (7) ∇ · (k∇ ) = φµc ∂t for a slightly compressible fluid over a box shaped domain, with all variables as described previously. Either constant potential or no flow boundary conditions can be imposed on each of the six bounding faces. Because the problem is solved using Green’s functions, only the wells, rather than the reservoir, need be discretized. Each well in the domain is described in a hierarchical fashion (well, lateral and segment) and ultimately discretized into a specified number of line segments. The inner boundary condition is a total rate or specified bottomhole pressure for each welltree. Subsequent integrations of the instantaneous point source/sink solutions [9,18] (time dependent Green’s functions in real space) over space and time yield expressions for pressure drawdown for each well segment. These drawdown contributions enter a system of linear equations that is solved to determine the individual segment flow rates and drawdowns. The method can also handle finite conductivity effects; i.e., wellbore hydraulics resulting in pressure losses due to frictional, gravitational and accerational effects. This semianalytical method is very efficient, as computation time scales only with the number of total well segments.
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The basic method has been extended to include an approximate treatment of permeability heterogeneity [24]. This treatment entails a representation of the permeability field in terms of a constant, large scale, background permeability k∗ and a locally varying skin s along the well trajectory. This overall permeability representation is called the s–k∗ model [8]. Given a fine grid permeability realization for the domain, k∗ is estimated through simple power averaging or through the solution of a large scale linear flow problem. An “altered zone” near-well permeability ka,j is then computed for each well segment j through a weighted integration of the fine scale permeability values around the segment (i.e., over a region of effective radius ra,j ). Application of Hawkins’ skin formula [11], ∗ k ra,j − 1 ln , (8) sj = ka,j rw gives a skin value sj for each segment. This skin value is then introduced into the system of linear equations describing flow into the individual well segments. Comparison studies with reference finite difference results for nonconventional wells in heterogeneous systems demonstrate that this procedure is of reasonable accuracy for a number of cases [23–25]. Use of the s–k∗ model for permeability adds a negligible overhead to the basic semianalytical approach. 4.
Calculation of well index for NCWs
We now present a general framework for the computation of well indices for an arbitrary coarse grid. This procedure is general and can also be applied to unstructured coarse grids. We are interested in obtaining appropriate well indices for each well block i given an arbitrary target grid and an arbitrary well trajectory. We first describe aspects of the general method applicable to both homogeneous and heterogeneous reservoirs and then discuss some additional issues relevant to heterogeneous problems. In the following discussion, the subscript i denotes a well block quantity (finite difference simulation), while j denotes a quantity for a well segment (semianalytical model). Let us consider a local well driven flow problem as depicted on the top of figure 1. The local well domain is selected by the user and would typically correspond to the drainage volume for a single or multiple wells. Due to the nature of the current semianalytical reference solution, the selected well domain must be box shaped. The boundary conditions can be constant potential or no flow, as appropriate. We found that the computed well index is not very sensitive to these boundary conditions, as long as they are consistent between the reference and target grid solutions. The specification of the total well flow rate and no flow conditions for all boundaries except the bottom, where constant pressure is specified, allows for an efficient solution because the well will reach steady-state (thus avoiding transient calculations). The local problem is solved first, using the semianalytical method described above. For a heterogeneous problem, the semianalytical tool uses the geostatistical fine grid permeability field, represented in terms of sj and k∗ as explained above. In either case
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Figure 1. Flowchart for obtaining accurate well indices.
the solution provides the well pressure pjw and inflow qjw for each segment j . These results are reference quantities that will be used to compute the appropriate well indices for the simulation grid. The same local domain and boundary conditions as used in the semianalytical solution are applied to obtain the numerical solution for block pressures on the target grid. For a heterogeneous problem, the fine grid permeability information must first be upscaled to the target grid. We then solve the single phase steady state pressure equation on the target grid, ∇ · (k · ∇ ) = q,
(9)
where the source term q is nonzero in each block i in which a well is completed, i.e., we prescribe sources qiw for each block intersected by the well (rather than specifying the total well rate as before). As a segment j is generally not contained by exactly one block i (as indicated in figure 1), we need to apply a mapping procedure to compute the intersections of the well with the grid. Our intersection algorithm is quite general [2] in that any polyhedral unstructured grid can be considered as long as the grid cells are convex. With this information, the rates qiw are determined from the qjw based on the fractional length lij of the well segment j in block i: j =jN
qiw
=
lij qjw ,
(10)
j =1
where the summation over all jN segments in the domain accounts for the possibility that multiple wells may intersect block i.
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The wellbore pressure corresponding to block i, piw , which will be needed for the final calculation of the well index, is simply computed from the reference pjw corrected by the hydrostatic head between the level of the segment midpoint and the block midpoint. Specifically, for a downward pointing z-axis, we have piw = pjw + ρg(zi − zj ).
(11)
qiw
act as simple source terms in the simulator. It is important The prescribed block rates to note that the well model of the target simulator does not enter this procedure, as this is the quantity we seek to compute. Following the finite difference solution of equation (9), the pressure of each well block pi is known. The correct well index for block i can now be computed from the semianalytical (reference) well pressure and rate information, interpolated onto the well blocks using equations (10) and (11), and the block pressures obtained from the target simulator: WI ∗i =
qiw µ . pi − piw
(12)
The ∗ superscript is used to distinguish this well index from the default well index WI d computed by equation (5). Use of the WI ∗i computed from equation (12) will provide finite difference results w for pi consistent with the pjw (see equation (11)) from the semianalytical solution for the specified reference problem. Similarly, if the finite difference simulator is now run with piw specified, the inflow profiles of the two solutions (provided the qjw are integrated along the well in a manner consistent with equation (10)) should match. However, there is no guarantee that these WI ∗i are appropriate for use in all flow problems. This is equivalent to noting that the WI ∗i computed from the procedure described above will vary somewhat if the boundary conditions are varied. We have, however, observed this variation to be very slight, indicating that the well indices computed using our procedure can be expected to display a high level of robustness. This point will be illustrated in the examples presented in the next section. In the case of heterogeneous systems, our method for computing the well index includes an upscaling component, since the effects of near well, fine scale heterogeneity are included in the coarse grid WI ∗i . Due to the approximate nature of the s–k∗ approach, and slight inconsistencies between the semianalytical and finite difference solution methods, it is possible that the procedure will occasionally result in a block pressure pi that is actually higher than the wellbore pressure piw for an injection well. Application of equation (12) for the calculation of WI ∗ would, in this case, provide a negative well index. When this occurs, we introduce an additional iteration loop into our procedure to force the calculation of positive WI ∗ by iterating on WI ∗ until differences between the semianalytical and coarse scale finite difference flow profiles are minimized. This minimization can be framed as a formal optimization procedure [15]. Our approach is, however, simpler, as it involves modification of the WI ∗ based only on the local mismatch in flow rate. Finally, it should be noted that our overall framework is not restricted to the type of reference solution applied in this work. In fact, any other semianalytical or finite differ-
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ence reference solution is suitable as long as it can provide the required source terms qiw and local wellbore pressures piw . Fine grid finite difference models that honor the well trajectory and share common boundaries with the coarse grid, such as the multiblock method [12] or finite element approaches [22], may be applied to provide a reference solution for complex wells in heterogeneous permeability fields. 5.
Results
We first demonstrate the proposed method on a general three-dimensional grid containing a complex well trajectory. The example illustrates the generality of the method. In subsequent cases we use Cartesian grids to allow for quantitative comparisons between our results and those obtained using conventional models (i.e., Peaceman well index WI d from equation (5)). Homogeneous cases will be used to assess the geometrical effects; studies with heterogeneous reservoirs will demonstrate the ability of this approach to capture the effects of the fine grid permeability field. For all cases presented, we compute the WI ∗ distribution at steady state for a reservoir with a constant pressure boundary at the bottom face. This allows us to use long time semianalytical solutions together with fast steady state finite difference calculations. Our approach does however give well indices that are also suitable for use during the transient period, as we will show. Tables 1 and 2 give properties for most of the runs and the well trajectories used for the Cartesian grid examples. Deviations from these specifications will be noted. The effects of wellbore friction are neglected in all cases, which means that pressure does not vary along the well unless the well is inclined. 5.1. Pinchout multiblock grid For this example we will use a complex grid constructed to model a pinchout (three geologic layers on the right side of the model collapsing to two layers on the left side). The multiblock grid [12] shown in figure 2 is comprised of nine blocks, each of which has 8 × 8 × 8 cells. The physical dimensions of the system are 500 × 250 × 250 m. The permeability field is homogenous and isotropic (k = 100 md). The first step in Table 1 Reservoir, fluid and well properties. Drainage area Thickness Porosity φ Compress. c Density ρ Viscosity µ Form. fac. B Initial potential ini Total rate Q Well radius rw
4500 × 4500 ft2 175 ft 0.25 3.0 × 10−5 psi−1 at pini 60 lbm /ft3 1 cP 1.05 RB/STB 3000 psi 10,000 STB/d 2.4 in
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C. Wolfsteiner et al. / Calculation of well index for NCWs Table 2 Well configurations of the examples presented.
(a) skewed horizontal
(b) multilateral
(c) herringbone
Figure 2. Pinchout multiblock grid with NCW.
our procedure (see figure 1) is to obtain the semianalytical reference solution. In this case, we use a resolution of 15 segments to obtain the reference well influx and pressure profiles, shown in figure 3, for a well specified to produce Q = 10, 000 STB/d. We next perform an intersection of the well trajectory with the grid as illustrated in figure 4. This enables us to map the reference flow rates onto the target grid, where they are used as source terms in the finite difference calculations. The pressure field for the finite difference solution on the multiblock grid is shown in figure 5. The resulting well index distribution WI ∗ , computed from the reference and finite difference calculations via equation (12), is shown in figure 6. We can now use these WI ∗i in our finite difference simulator. Figure 7 compares steady state inflow profiles and well segment pressures along the well trajectory computed using the finite difference method with WI ∗i (points) and the reference semianalytical solution (curves). As expected, the finite difference solution with WI ∗ traces the reference result accurately. This demonstrates the applicability of the method for complex grids and general well trajectories. Note that we do not compare the multiblock grid results to any simulations using a default well index in this case because it is not clear how to compute such well indices for the grid of figure 2 using equations (5) and (6).
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Figure 3. Semianalytical reference solution for the pinchout example.
Figure 4. Intersection of NCW with grid for determination of well blocks.
5.2. Skewed horizontal well The following examples involve Cartesian grids of dimension 45 × 45 × 33 cells. The well in the first example follows configuration (a) as shown in table 2. The well is rotated 30 degrees in the horizontal plane relative to the orientation of the grid lines. The reservoir is homogenous but anisotropic with kxx = 200, kyy = 100, kzz = 50 md and has a constant pressure aquifer (paqu = 3000 psi). We compute WI ∗ for this case as described above. The WI ∗ distribution is compared to the default well index
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Figure 5. Pressure field from finite difference simulation (units are psia).
Figure 6. Variation of WI ∗ along the well trajectory.
in figure 8. Note that the WI ∗ profile, even when it is normalized by the well block segment length as in figure 8, shows significant variation from block to block. The default well index for this geometry and anisotropy has a value of 12.3 STB/d/psi and is constant for all 61 blocks that are intersected by this skewed trajectory. This default well index corresponds to a well of effective well length that is 36% greater than the actual well length. As indicated above, procedures that attempt to rescale the well index based on geometry and anisotropy have been suggested in [1,16]. However, because errors for these types of approaches were reported to be as high as 30%
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Figure 7. Influx and pressure comparisons for pinchout example (curves – semianalytical reference solution; points – finite difference results using WI ∗ ).
Figure 8. WI d and WI ∗ for skewed horizontal well (see table 2(a)).
in some cases [13], we do not attempt to apply them here (though we note that it is certainly possible that some of the default results could be improved using these procedures). The reservoir is then simulated for a period of 100 days (corresponding to early transient and steady state conditions) using both the finite difference and semianalytical techniques. The well pressure as a function of time is shown in figure 9 (pressure does not vary along the well trajectory). Included in the figure are finite difference results
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Figure 9. Drawdown for skewed horizontal well.
using the default well index WI d (dashed curve), finite difference results using the new well index WI ∗ (points), and the semianalytical reference solution (solid curve). Note that the pressure axis is scaled such that its maximum value identifies the initial wellbore pressure (here 2963.5 psia). Hence, the interval between any of the curves and the top of the graph is equal to the drawdown at a given time. Most notable in figure 9 is that simulation results using WI d differ from the reference solution, while those computed using WI ∗ agree closely. The results for wellbore pressure using the default well index are in error by 5.4 psi out of a total wellbore pressure drawdown of 29.9 psi (18% error) at 100 days. Some of this deviation can certainly be attributed to the increased effective well length introduced by the default well index. It is interesting to note that, although the WI ∗i were calculated from the steady state problem, the finite difference result matches the accurate semianalytical solution in the transient region as well. 5.3. Multilateral well We repeat the WI ∗ calculation procedure for a more complex well trajectory (see table 2(b)). The main wellbore is now inclined and has a horizontal lateral extending from it. The reservoir and boundary conditions are the same as in the previous example. We plot the pressure at a point near the heel of the well as a function of time in figure 10. The ordinate is again scaled such that the maximum value identifies the initial static pressure level. In this case use of the default well index results in an even higher error in wellbore pressure than in the previous example (the error in wellbore drawdown here is about 23% at 100 days). We next test if changing the reservoir boundary conditions has an influence on the quality of finite difference results computed using WI ∗ (recall that WI ∗ is computed from a steady state solution). Figure 11 shows the well drawdown for the same well
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Figure 10. Pressure comparison for multilateral well with constant pressure aquifer (see table 2(b)).
Figure 11. Pressure comparison for multilateral well in closed reservoir.
configuration in a closed reservoir (i.e., no aquifer). The reference solution is still closely reproduced. Note that while the absolute deviation of the two finite difference solutions is nearly the same (about 6 psi), the relative error appears smaller for the closed reservoir case than for the previous case. Were we to plot the results in terms of productivity index, however, errors comparable to those in the previous case would be observed for the simulation performed using WI d .
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Figure 12. Pressure comparison for herringbone well (see table 2(c)).
5.4. Herringbone well The next well configuration considered has four symmetric laterals completed from an unperforated horizontal main trunk (see table 2(c)). The grid and permeability anisotropy are the same as in the previous two examples. Each lateral passes through 10 grid blocks and intersects the gridlines at a 45◦ angle. In a conventional simulator, these laterals could be modeled as stair-step wells with well segments oriented in either the x- or y-direction. The decline in well pressure over time for these two types of models (dashed curves) is plotted together with the reference solution in figure 12 (pressure does not vary along the well trajectory). The deviations from the reference are different for the two types of models (i.e., about 6 and 14 psi, respectively) due to the reservoir anisotropy. The errors for both types of conventional models are significant; 16% for the well with segments defined in the x-direction and 33% for the well with segments defined in the y-direction. The WI ∗ computed using our procedure is again seen to be capable of providing much more accurate simulation results. 5.5. Heterogeneous reservoir We now consider the application of our procedure to a heterogeneous reservoir. Recall that in this case our representation of permeability variation via the s–k∗ model is only approximate (i.e., our reference solution is not as rigorous as in the homogeneous cases). We therefore require a different solution to act as the ‘correct’ solution (to be used to gauge the accuracy of our method), though we still use the semianalytical solution as the reference solution for the WI ∗i calculation. For this problem we model a horizontal well aligned with the grid. The fine grid finite difference solution using the default well index can be expected to be of a high level of accuracy since the well is aligned with the
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grid. Therefore, the correct solution will be taken in this case to be the fine grid finite difference solution. The fine grid in this case is again 45 × 45 × 33 while the coarse target grid is 15 × 15 × 11. The coarse grid was generated from the fine grid by upscaling uniformly by a factor of three in each direction using a flow based upscaling procedure [5]. The fine scale permeability field was of mean permeability 200 md, with a coefficient of variation of 1.4. The permeability correlation structure was isotropic with a dimensionless correlation length of 0.3. The horizontal well is fully penetrating and centered on both the fine and coarse grids. The accuracy of the fine grid result was verified by performing a simulation on a downscaled grid of size 135 × 135 × 99 cells, which was obtained by replacing each geostatistical fine grid cell with 3 × 3 × 3 cells of equal permeability. Results obtained on this grid essentially overlay those obtained on the 45 × 45 × 33 grid, indicating that the fine grid finite difference results can be considered to provide the ‘correct’ solution. Well pressure as a function of time is plotted in figure 13. The comparison between the fine (solid curve) and coarse grid (dashed curve) simulations, both generated using default well indices, shows that the coarse grid result does not degrade much with upscaling. The plot also shows that the simulation results for wellbore pressure using WI ∗i (points), computed using the s–k∗ reference solution, are about as accurate as the coarse grid results using WI d . For both cases, the deviation from the fine grid result is about 2 psi. For heterogeneous reservoirs, the inflow profile along the well trajectory can be even more important than the wellbore pressure. In figure 14 we display the various solutions for inflow as a function of position along the wellbore. Here, the coarse grid result with default well indices is in significant error relative to the fine grid result, while
Figure 13. Pressure comparison for horizontal well in a heterogeneous reservoir.
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Figure 14. Influx profiles for horizontal well in a heterogeneous reservoir.
the coarse grid result using WI ∗ tracks the fine grid solution more closely (the fine grid solution is here averaged over the coarse grid blocks to allow for easier comparison). This illustrates the ability of our procedure to provide accurate inflow profiles, even in cases when the wellbore pressure shows some error. 5.6. Channel reservoir We consider a second heterogeneous system, a geostatistical realization of a fluvial channel system [5] on a 50×50×30 fine grid. The overall system dimensions are 5000× 5000 × 150 ft. The permeability distribution is bimodal with a mean of 1516 md for the channel sandstone and a mean of 5 md for the surrounding mudstone. This represents a considerable permeability contrast. A partially penetrating horizontal well was placed such that it intersects as many high permeability channels as possible (figure 15(a)). The well segments are centered and fully penetrate the fine grid blocks through which they pass. This allows us to again view the finite difference result on the fine grid (generated using default well indices) as the ‘correct’ solution for this problem. The fine grid was uniformly upscaled as described above to 10 × 10 × 10 coarse blocks. Figure 15(b) shows that the upscaling has significantly affected both the permeability variability and the clear channel structure evident in the original fine grid. Note that, in contrast to the previous example, the well segments are no longer centered in the resulting coarse well blocks. The WI ∗ we compute account for both the subgrid heterogeneity and the off-centered trajectory. In figure 16, influx profiles for the fine grid solution (averaged onto the coarse grid blocks as above), coarse grid solution using WI d and coarse grid solution using WI ∗ are compared. The coarse grid results using WI ∗ , though not exact, clearly track the fine grid result more closely than do the coarse grid results using WI d . The steady state well pressures are also more accurate for the coarse
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(b)
Figure 15. Horizontal well in a channel system (permeability in md) on (a) fine grid, (b) coarse grid.
Figure 16. Influx profiles for horizontal well in a fluvial channel system.
scale results using WI ∗ . Specifically, the fine grid result for pw is 2927 psi; use of WI d on the coarse grid gives pw = 2911 psi, while use of WI ∗ on the coarse grid gives pw = 2923 psi (the initial well pressure was 2963 psi). These results further demonstrate the ability of our overall procedure to approximate the effects of subgrid heterogeneity. 6.
Discussion
In this paper, we developed a general framework for computing well indices for use in finite difference reservoir simulators. The idea is to replace the well block level
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assessment of well indices by detailed local well domain studies (reference solutions) from which appropriate well indices for large scale simulations can be derived. The method applies a semianalytical reference solution to provide local relationships between wellbore pressure and flow rate. Theses quantities, used in conjunction with a finite difference solution with flow driven by source terms derived from the semianalytical result, provide accurate well indices for use in general finite difference simulations. The method is very precise for homogeneous systems, as the semianalytical solution is virtually exact for an arbitrary well in a homogeneous domain. For heterogeneous cases, the s–k∗ permeability model is used to represent the effects of subgrid heterogeneity. Though only approximate, this approach introduces an upscaling component into the overall methodology. The method was applied to a number of systems involving complex well trajectories (skewed horizontal, inclined and multilateral wells) and Cartesian and multiblock grid systems. For homogeneous models, the method was shown to provide finite difference results that essentially reproduce the accurate semianalytical results. Errors using standard (default) well indices were shown to be significant in many cases. Specifically, for an example involving a complex (herringbone) multilateral well, the default well index provided results that were in error by about 33%. Results for heterogeneous systems were also presented. In this case, although the wellbore pressure was not as accurate as in the homogeneous case, the well indices computed using our methodology were shown to provide more accurate inflow profiles along the well than did the conventional approach. The methodology presented here can be extended in a number of directions. Reference solutions could be generated using finite difference or finite element calculations in which the grid closely follows the complex well trajectory. This could be accomplished, for example, using the multiblock grid approach described in [12] or using an unstructured finite element (or finite volume) approach. Such reference solutions could be used in place of the semianalytical calculations applied here. These more rigorous approaches would provide better accuracy for highly heterogeneous permeability fields, for which the semianalytical solution with the s–k∗ permeability model is only approximate. These more general models could also be used to compute upscaled near-well transmissibilities, which were shown to provide enhanced results in previous procedures for upscaling around horizontal wells [15]. Though these extensions will result in enhanced accuracy, it must be noted that they will render the overall procedure more complex.
7.
Conclusions The following main conclusions can be drawn from this study:
– A practical approach was presented to compute the well index for nonconventional wells intersecting arbitrary grids. The method introduces an approximate treatment for the effects of subgrid heterogeneity.
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– It was shown that in many cases default procedures in commercial simulators can result in significant errors in both wellbore pressure and inflow distribution. – While the well index can change with time and boundary conditions, it was shown that, for practical purposes, it can be computed based on steady state conditions.
Acknowledgements This work was supported in part by the U.S. Department of Energy under contract number DE-AC26-99BC15213 and by a consortium of companies under the SUPRI-HW Industrial Affiliates Program at Stanford University.
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