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A disadvantage is the more complex shape of cells at the edge of the grid. These points may be triangulated into a mesh of triangles or tetrahedrons. The unknowns in this equation system are the geo-cell pressures inside the flow cell, whereas known quantities are the geo-cell dimensions and permeabilities, as well as the pressure conditions along the faces of the flow cell. The systems of linear equations generated by unstructured grids are also commonly regarded as more difficult to solve than those produced by structured grids. An obvious advantage with the global upscaling approach is that one avoids using artificial boundary conditions around the upscaling region (i. The inserted grid may be Cartesian (center) or radial (upper left). Because the barrier extends across the entire length of the local upscaling region, the resulting effective permeability (in the z -direction) will be zero. 18) The relative upscaling approach that is presented here utilizes a multistep procedure that applies outer bound methods of increasing accuracy until the best possible upscaling method can be identified. For more complex shape-dominated problems, the unstructured approach looks general and flexible, providing that the data-handling and cell-identification methods can be moved to true x, y, z space preprocessing software. 17. A technique known as preconditioning can improve both the efficiency (speed in a typical problem) and robustness (ability to solve a wide range of problems at least reasonably well) of Orthomin or GMRES. Based on the size of the computational region, the single-phase upscaling process may either be described as local, regional, or global. 29. Although the periodic boundary conditions generally result in an effective permeability that is higher than that computed with the sealed-sides boundary conditions, the effective vertical permeability for the upscaling region illustrated in Fig. Local refinement may be regarded as a form of multiple domain structured grid, in that it consists of a number of linked structured grids. Some coarse grid submittals included sophisticated upscaling and gridding techniques with no pseudoization of relative permeability and grids from 4,810 to 70,224 blocks. Apply a tensor method with semi-open or periodic boundary conditions. This technique is rather rarely used in reservoir simulation. The validation process is carried out by first running a streamline simulation on the geo model to compute the reference solution. For nonisothermal problems, an energy balance is added to the system. As in the coreflood experiment, the local numerical flow simulation is in effect 1D because the cell faces parallel to the main flow direction are sealed. Because the CPU consumption of the applied equation solver is proportional to the number of geo cells, a global solution will use approximately the same amount of computational time as the sum of all the local computations. 12 ). Five participants compared their results for that grid with results from their (areally) locally refined or unstructured grids. Models have been referred to by type, such as black-oil, compositional, thermal, generalized, or IMPES, Implicit, Sequential, Adaptive Implicit, or single-porosity, dual-porosity, and more. This is because the open-sides method consistently overestimates the reservoir flow characteristics by narrowing shale barriers and thickening sand channels. F is a function of the N vector unknowns P i, where the I th scalar element of P i is P iI. In this context, it is important to realize that the two flow-based methods (sealed and open sides) provide an even narrower uncertainty band for the effective permeability, but at the expense of increased CPU time. The VECTRA 3D camera captures a 3D photograph of the patient, and using multiple cameras, the machine takes a number of photographs from several different angles, offering a simulation or 3D virtual model of the patient. , for small buffer values). The 3D approach is most effective when applied to model a local structure such as a branched well. , the geo model). An important design criterion for the artificial boundary conditions is the conservation of flux in and out of the flow cell. of Science and Technology and Pera. Because of the nature of the reservoir simulation equations, only certain preconditioners are effective in solving them. This is discussed in the Stable Step and Switching Criteria sections to follow. A literature search and discussions with numerous developers and users have failed to establish consensus on whether NF or ILU preconditioning is better. This is normally done for extra couplings introduced by dip angles, which are often small. It is usually possible to identify many more layers in the geological model than it is practical to include in reservoir flow simulation, so some upscaling of rock properties will normally be carried out. The matrix A is very sparse because A ij is 0 unless block j is a neighbor of block i. From a practical standpoint, it does not matter which is used. Unless extremely fine, a true regular grid that is orthogonal in all three axes will be unable to assign rock properties accurately to cell volumes. Preconditioning involves transforming the original matrix system into one with the same solution that is easier to solve. As previously mentioned, the method using open-sides boundary conditions is a good estimator of sand continuity and quality, whereas the sealed-sides boundary conditions method is better at detecting the presence and effect of barriers. This is illustrated by Fig. Each of the grid elements will be assigned a single permeability or porosity value. 25 will also be zero. 17. The reservoir may contain faults, at which the strata are displaced. Cells in such a grid may be simply identified using their ( i, j, k ) index values. The physical implication of this is most clearly seen in the case of a bimodal permeability system of sand and shale. This is a simple way of representing variable dip, but is difficult to represent graphically in a consistent way. The stability analyses involved are complex and may be impractically complex when allowing the implicit vs. Because an exact validation of the upscaling process cannot be performed unless a multiphase flow simulation is carried out on the geo model itself, two alternative upscaling approaches for identifying the proper homogenization method are presented here. The CPU expense per Newton iteration of an AIM model lies between those of IMPES and Implicit models, tending toward the former as more blocks are taken implicit in pressure only. Whitson, Norwegian U. Although this corner-point description handles the fault issue, the basic coordinate system remains a regular grid (i. A simple variation of a regular grid, in which the regular grid is rotated to bring the layers of cells into alignment with the bedding planes. Specifically, large relates to the number of active grid cells. However, most of the examples used to study those effects are highly adverse mobility ratio displacements in homogeneous, horizontal reservoirs. Such grids are often called corner-point grids. These techniques range from simple statistical averages to advanced numerical methods. It should be noted that the different upscaling schemes are only relevant when considering flow-based (tensor) methods. The latter effect also has a tendency of disconnecting stacked sand channels. Such inactive cells are usually compressed out of reservoir simulation solution arrays prior to the memory and time-intensive flow solution stage, and enable reservoirs with irregular boundaries to be represented within extended simulation grids. When they are, a single one-layer single-phase run can be made to determine the index correction factor for grid (a) wells located in the corners of their gridblocks. The periodic-based method turns out to give an effective permeability estimate that generally lies in between the two previous methods. Fitting such a basically regular system to the irregular shapes of a reservoir remains a difficulty that may be solved in two basic ways—either by distorting the grid and fitting the cells into the geometry, or by truncating the grid to the reservoir position. Non-Cartesian grids are argued to define irregular reservoir boundaries more efficiently than Cartesian grids. However, it may be more true to say that optimal solution schemes are simpler to find for structured grids, where the row and plane order provides a natural ab initio solution variable-ordering scheme. This chapter concerns the numerical mathematical model requiring a digital computer. Instead of analyzing simulation profiles, a normalized difference parameter may be computed on a cell by cell basis using the formula. Grid refinement occurs naturally in areas where solution points are closely spaced. , three separate single-phase simulations need to be performed). If the performance of any of the two methods in Step 1 is within an acceptable range of the geo model, then terminate the procedure and choose the appropriate method. 17. The calculations for a timestep consist of a number of Newton ( nonlinear or outer ) iterations terminated by satisfaction of specified convergence criteria. e. This yields a discretization scheme which is conservative (each outflow from one cell is an inflow to another) and for which the fluid in place may be obtained straightforwardly. Such regular gridding systems have advantages for upscaling and downscaling—for example, a natural coarsening of a regular grid may be simply defined by grouping sets of coordinates in each direction. It is used in the Newton step to solve the discretized nonlinear partial differential equations. 18 represent different cells, but may have the same i, j indices, so this creates a multiple-domain grid. 7 are scalar multiplications, requiring a small fraction of the work of the matrix-matrix and matrix-vector multiplications of the implicit formulation. The percentage of total CPU time spent in the linear solver for IMPES (Implicit) was 23. Take, for example, the situation illustrated by Fig. The actual cell volumes may have a variety of shapes, depending on the exact placement of the solution points, but are typically hexagonal in two dimensions. Because it used to be too time-consuming to compute the fine-scale pressure field for the complete geo grid in a single operation, the flow-based methods have traditionally been restricted to solving the pressure field locally—that is, for a single flow cell at a time. Using the previously described validation scheme, the modeler may choose to evaluate any upscaling method until one with a satisfactory performance is found. 4) for SPE10. Run the finite-volume simulator for each of the two composite methods (or compute a grid-based difference parameter) and perform the diagnostics. If the performance of any of the methods in Step 2 is within an acceptable range of the reference solution, then terminate the procedure and choose the appropriate method. Consider a long horizontal oil producer in the Troll West Gas Province that is protected against coning from the overlying gas cap by a calcite barrier just above the well. 17. All participants used the Implicit formulation for SPE2, SPE4, SPE6, and SPE9. Real problem nonlinearities and heterogeneity render the analysis approximate and the generalization of limited merit. If the performance gap between the two methods in Step 1 is acceptable (small), then terminate the procedure and choose either of the two. Some flow-based methods may provide a full permeability tensor. 22. Even though the barrier extends across the entire length of the local upscaling region, the resulting effective permeability (in the z -direction) will be significantly larger than zero. 21 for flow in the vertical direction (here in the case of a flow cell containing a barrier). Because the latter error was not carried forward, more Newton iteration (and CPU time) was required to keep the permanent, accumulating mass balance error tolerably low. In compositional models, this two-step method can be much faster than one-step methods. In the adaptive implicit formulation, there is a variable number of unknowns per cell. With local upscaling techniques, the computational region is identical to the upscaling region (i. In this renewed effort, two alternative boundary conditions for solving the local pressure solution in a flow-based method were suggested more or less at the same time. This implies that the influence of neighboring geo cells is taken into account in addition to the geo cells inside the flow cell. (17. Thus, the model CPU time per gridblock per Newton iteration for moderate or large n is much less for the IMPES formulation than for the implicit formulation. Not all the elements in the grid need represent active solution variables in the simulation. Fig. Current model capabilities, recent developments, and trends will then be discussed in relation to this generalized model. 30. Hence, the effective cell permeability is computed separately and independently of the other flow cells, which may or may not be correct depending on how representative the imposed pressure conditions along the faces of the flow cell are. However, it is possible that improving computer power will bring such rasterized grids to a level of refinement at which a sufficiently good representation may be obtained. A frequently stated generalization is that numerical dispersion error is significantly larger for Implicit than for IMPES formulations. Hence, the imposed pressure gradient is still constant, but the flow is allowed to enter and leave the cell at any point along the sides parallel to the main flow direction. SPE disclaims any and all liability for your use of such content. , relative permeabilities and associated rock types) that also need to be upscaled in one way or the other. As previously mentioned, this should result in an intermediate estimate of the effective permeability tensor and hence provide a flow model performance that lies somewhere in between the two methods in Step 2. This decouples saturations from the underlying pressure field and allows each streamline to be treated as being independent from the streamlines next to it. 17. It is less known that the uncertainty range in the effective permeability may be narrowed using the composite averages. Strictly speaking, the sealed-sides boundary conditions are only valid if no wells are present and the flow cells are symmetric in each direction of the grid as illustrated in 2D by Fig. The latter effect also has a tendency of connecting isolated sand channels. , the flow cell). Strictly speaking, the periodic boundary conditions are only valid if no wells are present and the fine-scale medium is periodic on the scale of the flow cells (i. Tensor methods are the most accurate techniques available for computing the effective cell permeability. If the deviation in performance is unsatisfactory because of the complexity of the geo model, then use the upper (open-sides) and lower (sealed-sides) bound diagonal tensor methods to narrow the uncertainty in the flow model performance. 25 for flow in the vertical direction (here in the case of a flow cell containing a barrier). The Norsk Hydro tensor method is based on the conservation of dissipation (mechanical energy per unit weight of fluid), although it turns out that fluxes are conserved as well. Hence, the ultimate upscaling scheme for the absolute permeability might be the one that is coupled with the multiphase flow simulator and automatically updates the absolute effective permeability field for each timestep. Then repeat the upscaling of the outer bounds (in Step 2) and redo the validation to check if the performance of either method has improved. The choice of sophistication in the upscaling method generally depends on one or several of the following factors. This nearly always means that NF should be ordered first in the vertical direction. The result is a halving of the number of unknowns. 4) uses phase mobilities, densities, and mol fractions evaluated at the upstream blocks. It is also important to realize that even though the computational region may vary according to the scheme used, the upscaling region remains unchanged and is of course defined by the flow cell, as in the case of the simple, analytical upscaling techniques. The situation is graphically illustrated in Fig. The multistep procedure of the relative upscaling approach is graphically illustrated in Fig. However, because most multiphase flow simulators can only handle a diagonal permeability tensor because of the use of a seven-point stencil in 3D, diagonal tensor methods are most frequently used whether directly or indirectly (through a diagonalization of a full tensor). Rerun the finite-volume simulator (or recompute the difference parameter) and check the performance gap. 17. (17. 24. 17. Arguably, the trend is or should be toward sole use of the AIM formulation. The degree of upscaling that needs to be performed (i. Pousti of Pousti Plastic Surgery, call us at (619) 466-8851 or CLICK HERE. If this reduces the upscaling uncertainty to within acceptable limits, then either of the two diagonal tensor methods may be used to provide the final permeability field for the flow model. Depending on which factors that affect fluid flow in the region of the cell, this may or may not be a representative value for that particular flow cell. Regional upscaling is applied to reduce the influence of the artificial boundary conditions on the effective permeability estimate by moving the boundary of the computational region away from the flow cell. In other words, a combination of the outer bound techniques within one and the same model may be a fourth alternative. The linear equation solver is an important component in a reservoir simulator. As a corollary, time spent in repeated runs fighting model instabilities or time-stepping is counterproductive. In the past, however, this has been too CPU-intensive to be performed in practice. Typically, 3D geological models contain detailed descriptions of the reservoir that can be hard to capture properly with a significantly coarser model. In a classical finite-difference scheme, the point values of pressures and saturations are used as solution variables, and the differential operators that appear in the fluid-flow equations may be expanded as difference expressions of these point values to some order. Coarse-grid submittals included results using upscaling and local grid refinement. Pousti Plastic Surgery Proudly Offers the VECTRA 3D Breast Enhancement Simulator for Patients. In most formulations, pressure is an unknown for each cell. 17. 17. NF and RBILU(0) are commonly used in implicit models. Compute the upper and lower bounds of the effective permeability using the arithmetic-harmonic (or pure arithmetic) and harmonic-arithmetic (or pure harmonic) average techniques, respectively. This approximation may range from very good to very poor, depending on the complexity of the fine-scale permeability distribution as well as the upscaling method used. With open-sides boundary conditions, however, also the off-diagonal components will generally be nonzero. This is constructed around a set of solution points that need have no particular indexing scheme. 17. e. g. Understanding the Basics of the VECTRA 3D Technology. Thus, in each block, only n primary equations in n unknowns need be considered in discussions of model formulation and the linear solver. With the current computer power and the lack of proper integration between the geological model and the simulation model, this is hardly achievable yet. For a given problem, different models using the same formulation can give widely different CPU times. In other words, regional upscaling represents an expansion of the local computational region outside the volume of the flow cell. 14a ), but transmissibilities are calculated on the basis of interpolated values ( Fig. Strictly speaking, these boundary conditions are therefore only valid if the neighboring flow cells are of a uniform, nonzero permeability. Two themes emerge from current trends in reservoir simulation gridding. The premise that Cartesian grids cannot provide required accuracy efficiently in these respects has come to be accepted as a fact. For a diagonal tensor, only the effective permeability in the principal directions of flow ( x, y, and z ) will be nonzero. Hence, unlike the sealed-sides boundary conditions where the effective permeability is limited to that of a diagonal tensor, the open-sides boundary conditions, as previously mentioned, give a full permeability tensor. 23. In the case of global upscaling, the computational region is that of the entire geo model. This implies that the estimated effective permeability will be scalar. Then repeat the upscaling of the outer bounds (in Step 2) and rerun the simulations (or recompute the difference parameter) to check if the performance gap (upscaling uncertainty) has narrowed. If this is the case, one needs to consider applying one of the alternatives A or C (alternative B will, in this particular example, give the same result as the sealed-sides method). This approach acknowledges the fact that an exact validation of the upscaling results cannot be achieved in practice. The block hexahedral system gives rise to multiple ( i, j, k ) indexing systems—( i, j, k, l ), where the l index specifies the grid system. Being of the analytical type, these methods are very fast and will provide a first quantification of the upscaling uncertainty. 20 provides a schematic drawing of how the computational region varies with the different upscaling schemes. Another important benefit is that a poor separation of scales in the upscaling will no longer occur because the size of the computational region is the same as the geo model. Therefore, instead of trying to validate the absolute performance of an upscaling method, the approach diagnoses the relative performance of outer bound methods using the actual multiphase finite-volume simulator. However, the ability to identify cells by a simple set of indices is lost, and items such as wells need to be positioned in true space terms. An ILU preconditioner using red-black ordering with zero infill on the reduced system is referred to as RBILU(0) and is the most frequently-used form of ILU. The streamline method and the underlying mathematics for incompressible multiphase flow are briefly outlined here. Nonvertical faults interrupt the matrix structure that makes red-black orderings attractive. The underlying solution points of a PEBI mesh can be linked together into a Delaunay triangulation. If the project is pressed for time, the simulations may be skipped altogether in favor of a faster, although less robust, way of performing the diagnostics. For example, consider the isothermal, three-phase, compositional case with all components present in all three phases. Being of the analytical type, these methods are very fast and will provide a first quantification of the upscaling uncertainty. As it happens, the sealed-sides method provides a lower bound and the open-sides method an upper bound of the effective permeability. The Merriam-Webster Dictionary defines simulate as assuming the appearance of without the reality. 17, in which the bisectors to the heavy lines joining the solution points enclose the control volume, represented by the shaded area. The validation of upscaling methods is best done under single-phase flow conditions to avoid introducing other model parameters (e. Iterative solution based on projection onto Krylov subspaces is typically used. e. For a 3D system, regular grids yield seven-point schemes, in which the flow equations for a cell involve solution values for just the cell and its six neighbors. Mathematicians performed stability analyses for constant-coefficient difference equations bearing some resemblance to IMPES. 10) The well variables are then obtained by back substitution as. Even after this process, the geology to be represented is rarely homogeneous at the scale of the simulation grid. , the coarsening factor). In the past 15 years, numerous papers have described local grid refinement and various non-Cartesian grids, as discussed in the Gridding section. 16) where cell b is the neighbor to cell a in some direction and K is the cell permeability in that direction. Two related issues are involved in choosing a grid for reservoir simulation: the accuracy with which the geological description of the reservoir is matched, and the discretization of the flow equations. e. The purpose of this section is to show, using a few examples, that Cartesian grids can provide adequate accuracy and reservoir and near-well definition efficiently in some cases, even without local grid refinement. e. Thus the two marked volumes in Fig. This in effect conserves both mass and volume because there is no permanent or accumulating volume error—only that of the given timestep. The block is explicit in S g if the old time level value S g n is used. The model formulation has a large effect on the nature and expense of those multiplies. For volumetric (additive) properties such as porosity and saturation, the effective flow-cell value is simply given by the bulk and pore volume weighted arithmetic average, respectively, of the geo cells inside it. Explicit. To compute all the directional components of the permeability tensor, the discretization and solution of the flow equation must be performed for each of the principal flow directions (i. The number of permeability realizations that need to be upscaled. 17. In general, the flows F cpab may involve the solution values of a number of cells, the number of cells involved defining the stencil of the numerical scheme. As geological descriptions have improved, fewer and fewer model reservoirs are found to fit this simple pattern, and something more flexible is required. Homogenization of absolute permeability does not have an exact analytical solution, except for in a few idealized cases. 17. The choices may change from one timestep to the next. Currently, an average-sized flow simulation model consists of approximately 100,000 active grid cells. In some special cases, a transformed coordinate system may be used, based around an expected flow pattern. That is an expensive operation but is straightforward, involving only Gaussian elimination. 17. 17. This approach assumes that there exists a way to properly validate the absolute performance of an upscaling method without resorting to an extremely time-consuming (if at all feasible) finite-volume simulation of the geo model. The generalized model, which represents most models in use and under development today, will be discussed in this chapter. In other words, a combination of the outer bound techniques within one and the same model may be a fourth alternative. For vertical flow, the result, therefore, is a thickening of the shale in the flow model equal to the thickness of the flow cell. The historical trend, though irregular, has been and is toward the generalized model, which incorporates all the previously mentioned types and more. 3) for SPE9 and 35.
In contrast, the previous or historical IMPES procedure reset saturations to preserve volume and iterated out the mass-balance error. Run the finite-volume simulator for each of the two tensor methods (or compute a grid-based difference parameter) and perform the diagnostics. The IMPES model will require 10,000 timesteps over the 100-day period. In models containing them, ILU is the method of choice. The relaxed volume concept relates to the timestep calculation Steps (d) and (e) given previously. Such a description would only suit a reservoir with a single, constant angle of dip. You can now take a sneak peek at yourself to see your body with simulated implants from all 3 FDA approved breast implant companies ( Sientra, Allergan and Mentor ), and even different implant types so patients can see the difference between a moderate profile implant versus a high profile implant. There may be little need to perform (most of) that work in a gridblock when p and composition are changing slowly. A simple 3D grid is the regular Cartesian grid ( Fig. VECTRA and Pousti Plastic Surgery can help you do this. One option would clearly be to use triangular or tetrahedral cells directly and associate cell volumes with these. The resulting control volume is defined by the perpendicular planes—it is the set of points closer to the node than any other. Different types of artificial boundary conditions for the flow cell have been suggested over the years, all with the objective of providing as good an approximation of the real boundary conditions as possible. (17. Hence, the sealed-sides boundary conditions assume that the flow cell is surrounded by mirror images of itself. A simple model in which transmissibility between blocks is calculated on the basis of linear interpolation between the center values of the cells. By the end of the 1980s, 3D geological models had started to appear more regularly on the modeling scene. The VECTRA 3d simulation technology enables patients to better visualize how they would look with different sizes and types of breast implants. (17. One of the key underlying assumptions in streamline simulation is that the system be close to incompressibility. If, on the other hand, the performance of both methods is unsatisfactory, then the following alternatives may be worth considering. In practice, the diagonal permeability tensor is derived by setting up the boundary conditions for x, y, and z directions, respectively, in three independent single-phase simulations. Transmissibilities for such cells may need to be calculated numerically. In fact, it may be mathematically proven that the harmonic-arithmetic average provides a closer lower limit than the pure harmonic mean, whereas the arithmetic-harmonic average provides a closer upper limit than the pure arithmetic mean (truly valid only for regular grids). Depending on which factors affect fluid flow in the region of the cell, this may or may not be a representative value for that particular flow cell. Parallel iterative solution typically uses a domain decomposition approach in which the grid is partitioned into domains that contain approximately the same number of cells. The tools of reservoir simulation range from the intuition and judgment of the engineer to complex mathematical models requiring use of digital computers. The horizons that delimit rock strata are generally not horizontal, but are dipped, curved, or faulted. 17. Hence, the maximum number of directional permeability components that can be obtained with this type of boundary conditions is three, one for each of the principal directions of flow. If necessary, repeat this step using an increasingly larger buffer region until a satisfactory performance of the flow model is obtained. The aim of gridding in reservoir simulation is to turn the geological model of the field into a discrete system on which the fluid-flow equations can be solved. 17. If the vertical grid resolution in the Troll full-field model was such that one could apply the open-sides boundary conditions technique on the cells containing the well, and the sealed-sides boundary conditions technique on the cells containing the barrier, then this would be the optimum local upscaling approach. The deviation in the simulated performance between outer bound methods will then reflect the part of the model uncertainty that originates from the upscaling process itself. Take, for example, the flow illustrated by Fig. 17. As a result, domains normally are groups of columns of cells. Hence, upscaling is a required part of current reservoir modeling workflows. The partitioning should be done such that coupling is strongest between cells within a domain. This yields a set of equations in which the mass conservation conditions for the fluid in the simulation cell volumes are related to the flows through the interfaces between those cell volumes. The simulated performance of the various upscaling methods is then compared to that of the geo model. Still, a more systematic way of identifying the optimum upscaling method is desirable. For a reservoir engineer, on the other hand, effective properties might be all that matter. With the VECTRA 3D simulation technology, patients will also have the opportunity to see the possible benefits of a breast lift procedure in combination with the breast augmentation. , oil recovery) under one or more producing schemes. Generalizations regarding the best formulation have many exceptions. 16 ). 13 ) in which the orthogonal grid provides a rather poor match to the dipping strata represented by the shaded layers. The simulation software displays your physique from a 3D perspective and at a patient consultation, Dr. An ideal is to separate the construction of the flow-simulation grid from the description of the reservoir geometry. 17. This resulted in a new demand for advanced upscaling. This topic relates to the observation that lower run turnaround time can increase benefits from a reservoir study allotted a budgeted time period. A is the area of the cell orthogonal to the direction of flow, and d the dimension of the cell in that direction. In finite difference, the material balance equations are solved between gridblocks, whereas in streamline simulation the material balance equations are solved along streamlines. The basic structure of an oil reservoir is a set of geological horizons representing bedding planes. The grid (b) doubles the grid (a) IMPES run CPU time but contributes no greater accuracy. Because the value of the effective permeability is influenced by changes in the pressure field, the flow-cell permeabilities should strictly be recomputed by the global method for every timestep taken by the multiphase flow simulator. Then a streamline simulation is run on the flow model for each of the upscaling methods that are to be evaluated. 1,. The conditional stability stems from the explicit treatment of nonpressure variables in the interblock flow terms. These papers show that non-Cartesian grids can reduce grid-orientation effects and provide definition and accuracy near wells, faults, highly heterogeneous areas, and so on more efficiently than Cartesian grids. The reservoir matrix problem is then solved iteratively and the well variables are obtained by back substitution. Rock properties such as porosity are assumed constant over the cell or controlled volume. Therefore, this type of boundary conditions is often referred to as the no-flow or sealed-sides boundary conditions. 23 for flow in the vertical direction (here in the case of a flow cell containing a barrier). The interblock flow term in Eq. With the absolute upscaling approach that is presented here, the modeler is offered a multistep procedure that is to be terminated as soon as a satisfactory upscaling method has been identified. Observation of model results that represent different producing conditions aids selection of an optimal set of producing conditions for the reservoir. A control volume is constructed around the nodes of the resulting mesh to define the simulator cell volumes. Instead of setting the flow through the sides of the cell to zero, the pressure along the sides is allowed to vary in a linear fashion that matches the constant pressure on the two cell faces perpendicular to the flow. 2) represents the entire model, where the i th element of the vector F is F i. In this case, the well variables may be directly eliminated, and the iterative solution is on the implicitly defined matrix system. The Step (e) flash then gives phase amounts and densities which in turn give new iterate S w, S o, and S g values. Therefore, this type of boundary conditions is also referred to as the open-sides boundary conditions. For vertical flow, the result is therefore a narrowing of the shale in the flow model equal to the horizontal dimensions of the flow cell. (17. No participants in SPE1 through SPE10 used a Sequential model, and, with few exceptions, none used AIM. An unstructured grid may be defined in two dimensions, and then applied to each layer of a reservoir model, so that a typical cell is a hexagonal prism. If the deviation in the performance is unsatisfactory (large), then the following alternatives may be worth considering. These are further discussed in the subsections that follow. Faults, both vertical and inclined, may be described precisely ( Fig. Basically, these techniques approximate A -1 b by p ( A ) b. Unstructured grids yield an elegant and flexible grid description. Pousti Plastic Surgery is one of the first Southern California plastic surgery practices to offer their patients this new technology. By allowing no flow to pass through the sides of the cell, all fluxes are forced to go in the principal direction of flow. If time allows, refine or coarsen the flow grid (whatever is best) to achieve a better separation of the length scales. Apart from the truncated cells, all the grid cells are hexahedra that are rectangular in plan. Pore volumes are calculated on the basis of a series of flat regular cells with variable depths ( Fig. This is because the sealed-sides method consistently underestimates the reservoir flow characteristics by thickening shale barriers and narrowing sand channels. A simplistic illustration of the periodic boundary conditions is given in Fig. The complexity of the fine-scale permeability distribution (i. The increasing sophistication of data preparation and solver technology indicates a move towards unstructured grids as a general method of solving the flow equations for a given reservoir simulation problem. A conceptual illustration of the upscaling process is shown in Fig. Current simulation studies use all of these formulations. , wells). As previously mentioned, this should result in an intermediate estimate of the effective permeability tensor and therefore provide a flow model performance that lies somewhere in between the two methods in Step 2. As a rule, the robustness of the iterative scheme is far more dependent on the preconditioning than on the specific Krylov subspace accelerator used. Validate the performance of the two composite methods against that of the geo model using a single-phase streamline simulator with the actual well pattern. e. For many years, simulation used orthogonal Cartesian grids. The areal grid is rectangular. No generalizations from the examples used are intended. Validate the performance of the two tensor methods against that of the reference solution. Seen through the eyes of the geologist, the upscaling task may be a painful experience because all the geological details that were put into the model seem to be lost in the process. (17. For the most part, the examples are taken from the literature. The physical implication of this is most clearly seen in the case of a bimodal permeability system of sand and shale. 17. Best results are usually obtained by ordering the cells first along the direction of highest transmissibility and then successively along directions of decreasing transmissibility. In 2D, this creates triangles, and in 3D it creates tetrahedra. On the other hand, reservoir simulation is increasingly seen as part of a decision-making process rather than an isolated activity, so the ability to map easily onto the generally regular data structures used in seismic and geological modeling becomes an important issue. Therefore, it would be preferable if upscaling could be avoided. For regional upscaling, the computational region is expanded beyond that of the flow cell to include a buffer region of neighboring geo cells. 7) times larger for Implicit than for IMPES. However, as discussed in the next subsection, the size of the computational region may not necessarily be limited to that of the upscaling region (i. Such a layer-cake structure can be used, but will generally misalign property values ( Fig. The multistep procedure of the absolute upscaling approach is graphically illustrated in Fig. Nine-point discretizations also cause problems for red-black orderings, but cause no difficulty for NF. e. For the permeability, which is intrinsic (nonadditive) by nature, no such simple averaging method exists. Most participants used IMPES for SPE Comparative Solution Project problems SPE1, SPE3, SPE5, and SPE10. 1) where M iI is mass of component I in gridblock i, q ijI is the interblock flow rate of component I from neighbor block j to block i, and q iI is a well term. e. Some are strong advocates of one method and others are just as adamantly supportive of the other. For a given problem, the previous four formulations generally give widely different CPU times. 11) If A is large, solution of the matrix equations is impractical using direct methods such as Gaussian elimination because of computer storage or CPU time requirements. The multiplications required in solution of the IMPES pressure Eq. As may be understood from the previous sections, there exists no single upscaling method for absolute permeability that is superior to all other methods in all situations, at least not until it has been fully established that the global upscaling scheme represents the ultimate method of choice. Such a two-point transmissibility assumes a permeability tensor with primary axes aligned along the grid axes. Such a block-center option is suitable for unfaulted reservoirs and is commonly supplied as a simulator option. Compute the upper and lower bounds of the effective permeability using the arithmetic-harmonic (or pure arithmetic) and harmonic-arithmetic (or pure harmonic) average techniques, respectively. In each gridblock, each of the n variables may be chosen explicit or implicit, independent of the choices in other gridblocks. Implicit vs. The I th primary or conservation equation for block i is. The choice of boundary conditions emulates the way core permeability is measured in the lab. For most linear solvers the following preconditioned system is solved: The preconditioned ORTHOMIN(k) algorithm, which retains the last k A -orthogonal direction vectors, is given by. Implementation is an important factor in the relative efficiencies of different formulations. In addition, RBILU( n ) or ILU( n ) have the parameter n (order of allowed infill) which can be increased as needed to solve problems of any difficulty. It is performed for each of the cells in the coarse grid and for each of the grid properties needed in the flow-simulation model. This implies that a full black-oil simulation is run on the flow model for each of the upscaling methods that are to be evaluated. The effect of this variable choice on total CPU time is often small because the affected work is often a small part of total CPU time. A simple rectangular grid is always used in the areal direction, but faults may subdivide the rock volume in a given column. If for some reason none of the previous alternatives are an option, then one needs to apply the method that best satisfies the wanted flow behavior of a given cell or cells in a given region. Reservoirs are typically shaped like pancakes, being much broader than they are thick. Streamline-based flow simulation differentiates itself from cell-based simulation techniques such as finite-differences and finite-elements in that phase saturations and components are transported along a flow-based grid defined by streamlines (or streamtubes) rather than moved from cell-to-cell. The sealed-sides boundary conditions are graphically illustrated in Fig. , the grid is structured). For over 30 years, many models have used active-block coding. The purpose of simulation is estimation of field performance (e. In addition, advances in geophysics have led to geostatistical description of permeability and porosity on a fine scale once unimaginable. As with the sealed-sides boundary conditions, three independent single-phase simulations, with the main flow direction in x, y, and z, respectively, are needed to yield all of the components of the permeability tensor. To ensure a high degree of relevance in the diagnostics, it is important that the test simulations contain a representative description of the actual flow model. e. In the IMPES formulation, there is a single unknown per cell pressure. The question is not whether to simulate, but rather which tool or method to use. This is illustrated in 2D by Fig. , within practical limits). One was based on linear boundary conditions, the other on periodic boundary conditions. A mathematical model is a set of equations that, subject to certain assumptions, describes the physical processes active in the reservoir. 21. 28, showing the behavior of the lower- and upper-bound tensor techniques in the case of increasing buffer size. 4 (73. If necessary, repeat this step using an increasingly larger buffer region until the upscaling uncertainty reaches acceptable limits, at least as far as practically possible. Further improvements in geological modeling threw an emphasis on describing faults, and made it important to distinguish depth displacements due to dip and faulting. The power of streamline simulation lies in its simplicity. The Implicit formulation is generally faster than IMPES for single-well coning studies, and for thermal and naturally fractured reservoir problems. Many models include well-constraint equations that add well pressures to the set of unknowns. In this case, it is possible to obtain the transmissibility value as a harmonic average:. These Krylov subspaces are spaces spanned by vectors of the form p ( A ) v, where p is a polynomial. For SPE9 (SPE10), the average timestep was 67 (9. Therefore, the new global upscaling scheme is just as fast as any local method. The grid is locally orthogonal, and the desirable property of two point flows is obtained. A higher confidence may also be obtained for the validation process if the actual well pattern is used in the streamline simulations. Plastic surgery can be approached with greater confidence when you can visualize the outcome. The near-well y -direction refinement of the specified grid has no effect and is not necessary in this problem. These equations describe mass balances on the individual components treated in the model. While the Cartesian grid extends past boundaries to numerous inactive blocks, those inactive blocks are dropped by the model and require no computer storage or CPU. Each simulation involves the iterative solution of a linear equation system (typically, the linear solver is a conjugate gradient method, preconditioned by incomplete Cholesky or LU factorization). The areal grid is not modified to match the faults. explicit variable choice to include all permutations (in number and identity) of the n variables. The size of the so-called buffer or skin around each flow cell is usually given in number of neighboring geo cells to either side of the flow cell and must be specified by the modeler for each of the three coordinate directions. These primary variables may be chosen as any n independent variables from the many available variables: phase and overall mol fractions, mol numbers, saturations, p, and so on. This difference allows streamlines to be extremely efficient in solving large, heterogeneous models if key assumptions in the formulation are met by the physical system being simulated (see below). 15 ). Therefore, the upscaling process is essentially an averaging procedure in which the static and dynamic characteristics of a fine-scale model are to be approximated by that of a coarse-scale model. This is frequently obtained in structured grids by local grid refinement, replacing a set of cells in the original grid by a finer grid ( Fig. Simulation of petroleum reservoir performance refers to the construction and operation of a model whose behavior assumes the appearance of actual reservoir behavior. With the introduction of new and promising solution algorithms such as the Output Least Squares (OSL) method, global upscaling schemes can now be realized. Instead, this chapter attempts to summarize current practices and trends related to development and application of reservoir simulation models. Selecting the proper upscaling method from the many available choices can be quite a challenge. The resulting multiplies in the linear solver are then either matrix-matrix or matrix-vector multiplies, requiring work (number of scalar multiplies) of order n 3 or n 2, respectively. The truncated approach fits in well with the rectangular grids used in geological modeling. These are based on solving a second-order elliptic differential equation describing single-phase, incompressible, steady-state fluid flow in a region without sources and sinks (i. 19. 7 (57. With regard to the outer bounds of effective permeability, it is well known that the harmonic and arithmetic means provide the absolute lower and upper limit of the effective permeability, respectively. The gain in accuracy is largest in the beginning (i. The problems involved in using a regular structured grid to represent reservoir geometry can be avoided by using an unstructured grid. These numerous inactive blocks pose a problem only for models, if any, that do not use active-block coding. If time allows, refine or coarsen the flow grid (whatever is the best) to achieve a better separation of the length scales. 17. Convert the two local tensor methods in Step 2 into regional methods using a buffer region of modest size. , only geo cells inside the flow cell are considered in the upscaling computations). The flow equation is usually discretized with a finite-difference scheme, although finite-element methods are also applied occasionally. In this role, structured grids may have advantages of simplicity and scalability. Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. 17. But many find, like this writer, that the better method is problem-dependent and it is difficult to find a reliable a priori indicator for making an up-front choice. e. To define faulting more precisely it is useful to define the position of grid cell by its corner point locations. If, on the other hand, the performance of both methods is unsatisfactory because of the complexity of the geo model, then use the upper (open-sides) and lower (sealed-sides) bound diagonal tensor methods to narrow the uncertainty in the flow-model performance. Such grids are not well adapted to represent geological data, and have been used less frequently as more detailed reservoir descriptions have become available. Some cells may be inactive, representing volumes of the reservoir with zero porosity. The model itself is either physical (for example, a laboratory sandpack) or mathematical. 14b ). The time available to the project for performing upscaling. , instead of guessing what the boundary conditions for the flow cells might be, the pressure conditions surrounding the cells are explicitly known). The relative performance of the tensor methods that is caused by the various boundary conditions has proven to be of considerable interest. Although the model itself obviously lacks the reality of the reservoir, the behavior of a valid model simulates—assumes the appearance of—the actual reservoir. The resulting full tensor may be either symmetric or nonsymmetric depending on the properties of the method under consideration. A common requirement in reservoir simulation is an increased level of detail around an item of interest such as a well. g. The challenge of computing an accurate effective permeability has resulted in a large number of upscaling techniques. Either volume or flux vice, depending on the type of property that is to be upscaled. e. It iteratively solves for a pressure correction at each iteration, uses the pressure correction to form a new residual, applies an inexpensive implicit preconditioning such as diagonal scaling or line Gauss-Seidel to the new residual, and then uses the sum of the two steps as the approximate solution. Finally, the vector equation. An IMPES model using this internally determined stable step will run stably but may suffer from the weakest-link principle. Whereas the field can be produced only once, at considerable expense, a model can be produced or run many times at low expense over a short period of time. The pressure field is usually solved locally—that is, for one flow cell at a time. This ties in with a further ideal, inherent in many discretization schemes, that the scale of the simulation grid should be below the scale of the problem structure. NF can also be used effectively in cases involving vertical faults and pinchouts because the matrix retains the structure required by NF. Because a typical 3D geological model may consist of approximately 10 million active grid cells, it is obviously infeasible to run fluid-flow simulations directly on the geological model. This is to ensure that the CPU consumption of a simulation run will be reasonable (i. A hexahedral shape with eight corners and bilinear planes as surfaces then describes the cell geometry. 17. (17. , the fine-scale property distribution inside each flow cell must be identical). Others used simple flow-based upscaling to 75- to 2,000-block Cartesian grids with moderate k r changes. This is shown in Fig 17. However, if the grid distortion is mild, it may be possible to ignore some additional couplings and use a low-order transmissibility scheme.
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