a037 use of time-dependent flow-based pebi grids in reservoir

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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

Abstract This paper presents a technique applicable for the generation of 2½ dimensional PEBI grids based on streamlines and isopotentials for specific well configurations. The procedure presented involves a hybrid gradient Laplacian grid optimization procedure, global-local transmissibility upscaling, suitable for the resulting partially unstructured grids, and time-dependent handling of flow-based grids using the windowing technique. Examples for grids in complex environments will be shown and numerical simulation results will be presented for a channelized reservoir.

Introduction Modern geological and geostatistical tools provide highly detailed descriptions of the spatial variation of reservoir properties, resulting in grids consisting of 107 or more grid blocks. These grids can not be used in numerical reservoir simulators directly, but need to be coarsened significantly. To preserve as much of the geological information of the fine grid as possible, the coarsening should not be performed uniformly, but more resolution should be put into areas with large impact on the flow. This can be achieved by the means of flow-based gridding. Flow-based grids are constructed by directly or indirectly using streamlines and possibly pressure isopotentials. Various techniques that rely on these principles have been presented in the reservoir simulation literature. These techniques can be classified after the resulting topology of the grid, as either fully structured, partially unstructured or unstructured. To date, most work has been done in the context of structured grids, e.g. for Cartesian grids by Durlofsky et al. [1], Portella and Hewett [2], Castellini et al. [3] or Wen et al. [4]. Advances in linear solvers have made unstructured grids more attractive for reservoir simulation, as they are inherently much more appropriate to capture the complex three dimensional geometry and connectivity of reservoirs. In the recent literature Prevost [5] has introduced a flow-based technique along this line. As the interplay between reservoir heterogeneity and flow pattern is an evolutionary process, it can hardly be captured by a single flow-based grid. A first attempt to update flow-based grids during a simulation run has recently been published by the authors of this work [6]. The paper

focused on the time-dependent gridding using the “windowing technique”. Efficient grid smoothing and upscaling were not considered. In this paper we will further develop this approach by introducing a new gradient based grid optimization technique and transmissibility upscaling for unstructured grids. This work is organized as follows. We will first summarize the steps involved in the generation and parameterization of 2½ dimensional flow-based PEBI grids. Examples for grids and numerical simulation results for a channelized reservoir will be presented in the subsequent section, highlighting the advantages of the new technique.

A037 USE OF TIME-DEPENDENT FLOW-BASED PEBI GRIDS IN RESERVOIR SIMULATION

Martin J. Mlacnik ([email protected])1, Andreas W. Harrer2 and Zoltan E. Heinemann2

1Dept. of Petroleum Engineering, 367 Panama St., Stanford U., Stanford, CA-94305-2220, USA 2Petroleum Engineering Dept., Mining U. of Leoben, Franz-Josef-Str. 27, 8700 Leoben, Austria

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Flow-Based PEBI Grid Generation This work is fully based on using 2½ dimensional PEBI (Voronoi) grids. The general principles of PEBI grid generation and discretization have been discussed in great detail by other authors, e.g. [7], and will therefore not be reviewed. Our grid generation can be structured into three major steps, which will be repeated whenever the grid is updated. The full procedure involves (1) a pressure solve on the fine Cartesian grid and streamlines tracing, (2) grid optimization, and (3) data mapping and upscaling. The single steps will be discussed in the following sections.

Primary Grid Point Generation

The grid generation starts by performing an incompressible two-phase pressure solve on the full fine scale model, using active wells with their flow rates or flowing pressures as boundary conditions, thus yielding a three dimensional pressure and velocity field. To account for changing rates over time, well rates are averaged over the period for which the resulting coarse grid is considered to be valid. Dynamic fine grid data, such as pressures or saturations, are obtained from conventional initialization at the beginning of the run, or by mapping from the coarse grid, at later times. This step is computationally relatively expensive, but only needs to be repeated for a limited number of times, e.g., changes in the well pattern. To generate a 2½ dimensional grid, potential and velocity information must first be reduced from three to two dimensions. This is achieved by computing a block volume averaged pressure potential for each block column in the fine grid, and an average velocity for each side of a block column. Due to the incompressibility assumption for the fine grid pressure solve, the resulting two dimensional velocity field is divergence free. Streamlines can be traced on this velocity field using Pollock’s particle tracking algorithm [8]. A primary grid point distribution for streamline pressure-potential grids is obtained by projecting the isopotential field onto the streamlines. The process entails the selection of streamlines and isopotentials appropriate to generate a grid with a desired grid cell size. To maximize the control over the resulting grid we decompose the grid into single “grid patches”, each of which can have its own independent set of isopotentials. A grid patch is confined by the two outermost streamlines forming the envelope for all other streamlines connecting an injector with a producer, resulting in a grid patch for each streamline connected injector-producer pair. The pressure isopotentials for each grid patch are defined by sampling in constant arc length intervals along the streamline with the largest average particle travel velocity from injector to producer. Using this grid-patch concept, all steps of the grid generation process, including the generation of point distributions and grid optimization, can be performed independently for each patch. To cover areas of the reservoir where no flow information is available, the streamline based grid patches are embedded in a coarse Cartesian “background grid”.

Combined Gradient and Laplacian Grid Optimization

The PEBI grid constructed for the primary grid point distribution usually is not optimal for reservoir simulation. Although the corresponding grid shows a distinct quadrilateral structure, it contains a large number of neighborhood relationships across extremely small interfaces, that connect blocks in the direction of their diagonals. These connections do not contribute to the simulation quality and have an adverse impact on the CPU performance. Grid optimization should ideally account for the following three metrics, presented in the order of their importance: (1) minimize the number of connections in the direction of block diagonals (2) ensure the connectivity of grid cells lying on the same streamline and (3) optimize the shape of grid blocks to avoid unfavorable aspect ratios. Optimizing PEBI grids is particularly challenging, because the grid topology is only defined indirectly through the grid point distribution. The grid optimization applied in this work

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combines the advantages of gradient and Laplacian grid smoothing, resulting in a hybrid gradient and Laplacian approach. Gradient smoothing is a very effective, but computationally expensive means to optimize grids, as it is an iterative process that proceeds by computing a gradient vector from perturbing the location of a given point and shifting the point in the gradient direction. To keep the cost of the gradient technique low, we assume that connections may only arise between a given point and its eight immediate neighboring points (i.e. two neighbors along the streamline, two on the same isopotential and the four in the directions of the block diagonals), avoiding the reconstruction of the full grid for every perturbation. Although covering a wide range of point distributions, this assumption does not hold for all configurations; therefore, gradient smoothing, as used in this work, is not suitable to handle metrics (2) and (3), because they require the accurate construction of the PEBI block around a given node. A simple, but efficient way to handle this problem is Laplacian smoothing. Laplacian smoothing, e.g. [4], shifts a given point towards the centroid of the polygon formed by its neighbors. As a consequence, it maximizes the angles between the connection lines formed by a grid point and its neighbors, and homogenizes the grid point distribution. These two properties make Laplacian smoothing attractive for our purposes, as they qualitatively improve the node distribution along the line of the two latter metrics. Maximizing the angles is equivalent to making the point distribution “more quadrilateral” for the PEBI algorithm, while homogenizing it improves the cell aspect ratios. The full smoothing algorithm proceeds as follows. We start by applying the gradient technique to our initial point distribution, minimizing the number of connections in the direction of block diagonals. We then enter the iterative loop of the algorithm. In its first step, Laplacian smoothing is invoked and the point distribution is perturbed. Laplacian smoothing does not have to be repeated until full convergence, usually three to five sweeps are sufficient. The perturbed point distribution is then smoothed again using the gradient approach. The iterative loop is repeated as long as it increases the quality of the grid, i.e. the total sum of the lengths of all connections in the grid in the direction of the block diagonals evaluated at the end of an iterative loop is less than the sum at the end of the previous loop.

Grid Update and Global-Local Transmissibility Upscaling

Tying the grid generation to the actual well positions, necessitates the update of the grid over time. The handling of time dependent grids follows the concept of the windowing technique, e.g. [9], allowing the time-dependent replacement of grids and parameters during a simulation run. Following the nomenclature used in the context of the windowing technique, the underlying fine Cartesian grid can be referred to as “basic grid” and the resulting flow-based grid as “window grid”, which in this work covers the entire reservoir domain. Dynamic properties such as phase pressures or saturations are mapped between grids based on the volumetric intersection of fine and coarse grid blocks, e.g., pressure potentials are transferred using the following relationship:

∫Φ=Φ

cVf

cci dV

V1

, (1)

Flow-based grids, as generated by the procedure presented above, quite naturally lend themselves to the application of global transmissibility upscaling by exploiting the flow and pressure potential information obtained from the global pressure solve. The actual flux across a given coarse grid interface between two grid blocks i and j can be evaluated by solving

∫ ⋅=

cAfcij dAq nv, (2)

where the three dimensional fine grid velocity vf, is known from the global pressure solve. Average block potentials in the coarse grid cells i and j are obtained from data mapping between the fine and coarse grid following eqn. 1.

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Knowing the flux qij,c across the coarse grid interface and the two point potential difference causing this flux, the upscaled transmissibility Tij

* for a coarse grid interface can be calculated from the following equation:

cjci

cijij

qT

,,

,*

Φ−Φ−= (3)

Unfortunately, this approach is not suitable to compute upscaled cross-transmissibilities for connections between blocks around grid points lying on neighboring streamlines and the same isobar. By construction, no potential drop exists between these blocks and no flow information can be extracted from the global solve. To overcome this shortcoming we apply a numerical technique first presented by Prevost [5], suitable for handling general unstructured grids. Following this approach, we construct a “local problem” for each cross-connection. The size and extension of the local domain is determined by the interface size in the x, y and z directions, and the location of the according coarse grid nodes. The domain is covered by a Cartesian grid (x’, y’, z’) with its y’ and z’-axes aligned with the coarse grid interface. Absolute block permeabilities for the local problem are obtained by re-sampling the values from the underlying fine scale grid. Using the boundary conditions p1 = 1 on ∂D1, p2 = 0 on ∂D2 and v·n = 0 on ∂D3 and ∂D4 for solving an incompressible single-phase flow problem on the local grid, aligns the principal direction of flow in the local problem with its x’-axis. The solution of the local problem yields the flux across the interface and the pressures at the grid nodes, allowing to determine Tij

* from eqn. 3. The procedure is illustrated in Fig. 1. Besides transmissibilities, also the well indices need to be upscaled. This is achieved by solving the following equation

wellci

wellqWI

Φ−Φ−=

,

* (4)

where well flow rate qwell and well potential Φwell are known from the global fine scale solution and the average block potential ‹Φi,c› is obtained from mapping from fine to coarse grid. As grids resulting from the procedure presented above are not ideal, i.e., not perfectly aligned with streamlines and isopotentials, more consideration is necessary for the application of this technique for smoothed and partially unstructured flow-based PEBI grids. The grids obtained after smoothing may still exhibit more than the four connections expected for streamline pressure-potential based grids, and, depending on the degree of grid optimization, points may have been shifted considerably away from their original location. Therefore, criteria have to be established to decide where the local upscaling approach needs to be applied to avoid computing anomalous transmissibilities. The transmissibility of a given connection will be upscaled using the global information, if the recovered coarse grid potential drop and flux have the same direction and satisfy the following condition:

ε>Φ−Φ

Φ−Φ

cij

cjci

cij

cjci

q

q

,

,,

,

,,

max

(7)

It is desirable to keep the tolerance limit ε as small as possible to improve the computational efficiency of the technique, as each solution of a local problem requires additional computation. Following the common classification of upscaling techniques, our approach can be described as “global-local”; “global” because it first exploits the pressure potential and flow information obtained from the global fine grid solve wherever possible, and “local”, because it applies a local technique where reliable transmissibilities cannot be computed from the global solve.

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Grid Examples We first investigate the performance of our grid generation and optimization procedure on two examples derived from the tenth SPE comparative solution project [10]. Grids where generated for five-spot configurations for layer 61 (example 1) and layer 59 (example 2), respectively. The resulting streamlines for layer 61 are shown in Fig. 2a, revealing regions of very high density and several channels with relatively small widths. Generating a primary grid based on the highly tortuous streamlines and isopotentials leads to the mesh shown in Fig. 2b. The resulting grid is fully unstructured, and the quadrilateral structure characteristic for streamline-isopotential grids is almost completely lost. Optimization transforms the primary grid into the grid shown in Fig. 2c, reducing the 5223 neighborhood relationships to a total of 2428 connections. The example demonstrates that the new PEBI grid optimization algorithm is capable of optimizing even strongly distorted grid point distributions. It also illustrates a potential deficiency of the technique, which is that the Laplacian smoothing shifts the grid points away from their original locations, resulting in a loss of resolution of flow based features. The objective of the grid generation for layer 59 was to balance grid optimization and reproduction of the flow based structure by reducing the influence of Laplacian smoothing. The resulting grid, shown in Fig. 3c, is capable of capturing some of the thin channels shown in Fig. 3a that are important for the flow caused by the given well configuration, as indicated by the streamlines (Fig. 3b). Although the grid could be further optimized towards a quadrilateral connectivity, this would lead to a loss of its flow-based structure.

Numerical Simulation Results To investigate the performance of our technique using time-dependent grids, we consider a geostatistic description of a complex channelized reservoir. The permeability field is shown in Fig. 4. Eight wells (four injectors and four producers), operated under pressure control, are active at different times over the period of the run, leading to four different well patterns over time. Two grids for different well patterns are shown in Fig. 5. Results for the fine grid reference solution and the flow based PEBI grid solution are summarized in Fig. 6, showing the field oil and water production. Comparison results have been generated for a 24×24 Cartesian grid model using the local upscaling technique presented above, but are excluded from the discussion. This grid was not at all able to match the reference solution, even at this relatively fine resolution in comparison to the resolution of flow based grids, which ranged from approximately 200 to 400 blocks. The field oil and water production rates obtained for the reference and flow-based PEBI grid show a close match for the three water breakthroughs, but water production tends to be underpredicted at later times. The good agreement in breakthrough times between the streamline grid and reference solution occurs because the grid is based on streamlines that contribute to breakthrough. The error in the prediction of water production can be related to dispersive effects, i.e., water that invades areas gridded by the coarse Cartesian grid. It is anticipated that introducing multiphase upscaling will improve the results.

Conclusions

• A flow-based PEBI gridding technique and a global-local transmissibility upscaling procedure for unstructured grids has been presented. Accurate coarse grid results were obtained with a relatively high degree of grid coarsening.

• A new gradient based grid optimization technique, suitable for point distributed grids, has been introduced and was successfully tested on complex cases. The procedure is not restricted to point distributed grids and may also be applied to optimize other types of grids.

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• Time-dependent gridding based on the windowing technique was used to adapt the grid to changing well patterns over time. Using the windowing technique is not restricted to PEBI grids and may prove beneficial for other flow-based grid generation approaches.

Acknowledgements

M.J. Mlacnik would like to express his gratitude to the Austrian Science Fund FWF for sponsoring this work under the Schrödinger Fellowship grant J2246-N05. The authors would also like to thank Prof. L.J. Durlofsky from Stanford U. for numerous valuable discussions.

Nomenclature T = transmissibility, mD⋅ft, m-1 WI = well index, mD⋅ft, m-1

n = interface unit normal vector p = pressure, psi, Pa q = flow rate, bbl/day, m3/sec v = velocity, ft/sec, m/sec ∂D = domain boundary ε = tolerance limit Φ = pressure potential, psi, Pa

Sub- and Superscripts c = coarse grid property f = fine grid property i, j = cell indices ij = property at cell interface well = well property * = upscaled property

References

[1] Durlofsky, L.J., Behrens, R.A., Jones, R.C., Bernath, A.: “Scale Up of Heterogeneous Three Dimensional Reservoir Descriptions”, SPEJ 1, 313-326, 1996.

[2] Portella, R.C.M., Hewett, T.A.: “Upscaling, Gridding, and Simulation Using Streamtubes”, SPEJ 5, 315-323, 2000.

[3] Castellini, A., Edwards, M.G., Durlofsky, L.J.: “Flow Based Modules for Grid Generation in Two and Three Dimensions”, paper presented at the 7th European Conference on the Mathematics of Oil Recovery, Baveno, Italy, 5-8 Sept., 2000.

[4] Wen, X.H., Durlofsky, L.J., Edwards, M.G.: “Upscaling of Channel Systems in Two Dimensions Using Flow-Based Grids”, Transport in Porous Media 51, 343-366, 2003.

[5] Prevost, M.: “Accurate Coarse Reservoir Modeling Using Unstructured Grids, Flow-Based Upscaling and Streamline Simulation”, Ph.D. thesis, Stanford University, 2003.

[6] Mlacnik, M.J., Harrer, A.W., Heinemann, Z.E.: “Locally Streamline-Pressure-Potential-Based PEBI Grids”, SPE 79684, paper presented at the SPE Symposium on Reservoir Simulation held in Houston, Texas, 3–5 Feb., 2003.

[7] Heinemann, Z.E., Brand, C.W., Munka, M.: “Modeling Reservoir Geometry with Irregular Grids”, SPERE 6, 225-232, 1991.

[8] Pollock, D.W.: “A Semianalytical Method of Path Line Computation for Transient Finite-Difference Models”, Ground Water 26, 743-750, 1988.

[9] Mlacnik, M.J., Heinemann Z.E.: “Using Well Windows in Full Field Reservoir Simulation”, SPERE&E 6, 275-285, 2003.

[10] Christie, M.A., Blunt, M.J.: “Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques”, SPERE&E 4, 308-317, 2001.

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Figures

Fig. 1: Schematic of the local transmissibility upscaling procedure

Fig. 2: (a) Streamlines, (b) primary grid, and (c) optimized grid for gridding example 1

Fig. 3: (a) Permeability field, (b) streamlines, and (c) flow based grid for gridding example 2

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Fig. 4: Permeability field of the dynamic grid example (low permeability areas have been suppressed to emphasize the channelized structure)

Fig. 5: Grids used for the dynamic grid example (0-50 days and 220-300 days)

Fig. 6: Field oil and water production rates for dynamic grid example

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a037 use of time-dependent flow-based pebi grids in reservoir