upscaling - hurras.oilprocessing.net

Upscaling

Many reservoir flow simulators cannot directly and effectively handle the size of grids used in geological models. Such models can easily contain as many as 10 million cells, whereas single CPU simulations will only run in reasonable time with models of the order of 100,000 cells. Furthermore, grids used in geological models are often unsuitable for simulation due to geometric problems such as inside-out cells.

Upscaling is the process of creating a coarser (lower resolution) grid based on the geological grid which is more appropriate for simulation. While this necessitates the omission of much of the geological models fine detail, the result is intended to preserve representative simulation behavior.

In Petrel, upscaling is split into two steps

Scale up Structure Define the new layering scheme (numbers and shapes of layers) of the simulation grid. See Scale up structure.

Scale up Properties Populate grid properties, such as porosity and permeability, based on those in the fine grid. SeeScale up properties.

The example below shows a geological (left) and an upscaled (right) grid with properties.

� Creating the Pillar Grid for Simulation� Scale up Structure� Scale up Properties

Copyright © 2011 Schlumberger. All rights reserved.Schlumberger Private - Customer Use

Creating the Pillar Grid for Simulation

A simulation grid (a coarser grid that will be used for flow simulation later) must be present in Petrel before the zones can be scaled up. The simulation grid is typically coarser than the geological grid and can be obtained in four different ways:

� Build a new, coarse grid in Petrel using Pillar gridding, Make horizons, Make zones and the Layering process. � Use the Pillar gridding process to create a new pillar grid from the existing fault model with a different cell size. Then

generate the zones directly from the fine grid using the Scale up structure process. The user is free to use different fault pillar geometries and grid layouts between the fine and the coarse grid.

� Build a new, coarse grid in Petrel using Make simple grid and the Scale up structure process. The geometry and faults of an existing fine grid are used as input.

� Import a simulation grid.

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The coarse grid does not need to have the same geological layers or geometry as the fine grid. However, the upscaling process is likely to be quicker if they are similar.

Parent topic: Upscaling


Scale up Structure

Ensure that the coarse grid is active when you open the Scale up Structure process and drop the fine grid in the Input gridbox. The process will sample the zones from the selected fine grid; any existing zonation in the grid will be removed.

Petrel 2005 and older versions used the Scale up Zones/Select Grid process both to upscale grid structure and to attach a fine grid to the coarse grid for later property upscaling. With Petrel 2008.1 and newer versions you can specify a fine grid when doing property upscaling, and you do not need to run Scale up Structure at all if your coarse grid already has the desired zonation.

� Scale up Structure Options� IJK Faults


How to Create a Coarse Grid for Upscaling

1. Go back to the Pillar Gridding process step. 2. Under the Settings tab in the process dialog, rename your 3D grid and define the XY increment of the coarser grid.3. If this grid is made for flow simulation, check the Make zigzag type faults box. 4. Perform the 2D gridding by clicking Apply. You might need to redefine trends and the I- and J-directions of the faults

in order to grid successfully. 5. Press OK to perform the 3D gridding.

Before you start the gridding process, be sure to increase the user-defined number of cells on connections on trends.

How to Scale up Zones

This option allows you to take the layering in your coarse grid directly from the fine scale geological grid.

1. Open the Scale up Structure process dialog. It is important that the coarse-scaled grid (simulation grid) is selected in the Petrel Explorer when the dialog is opened.

2. Define which finer grid you want to upscale from. This is probably the grid from which you created the coarser grid. Highlight (select) this in the Petrel Explorer and insert it by clicking on the blue arrow at the top of the process dialog. All zones and layers from your input grid will be inserted into the spreadsheet automatically.

3. Edit the zones and the layers if needed. 4. Click OK to start the Scale Up process.


Scale up Structure Options

Parent topic: Scale up Structure

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Resample Horizon/Fault Intersection

There are three possible ways to to the output grid.

Do not resample faults No resampling will be done

Use faults from the input The horizon/fault intersections from the input grid are used in the output grid to ensure consistency between the grids.

Use faults from the fault model Use this option if you want to sample the horizon/fault intersections stored on the fault model (horizon lines). This option can be used if you have made manual edits to the horizon lines and want to preserve this relationship in the upscaled grid.

Other Settings

After quality controlling the resulting grid it may be necessary to edit some horizon nodes. This can be performed in the Edit3D Grid process. In case the upscaling processes are rerun for any reason, it is possible to lock the edited nodes by checking the Keep locked nodes unchanged checkbox before pressing Apply

Use IJK faulting This option is only available with Advanced Gridding licenses. When selected, all faults in the model will become stair-stepped. For more information see Creating IJK-faults in an upscaled grid.

Collapse the main zones to zero thickness Use this option to collapse cells less than the minimum thickness specified in project units. Cells are collapsed towards the eroded horizon. If no horizon is set to erosional, the cells are collapsed at the mid-point. See Tool Tip for more details.

Make the Layering Between the Zones

Select how the layering is to be calculated. For vertical or near vertical grids, it is recommended to use Along Pillar. See tool-tip for more information. If there are highly inclined layers in the model, it is advisable to select the Horizons with steep slopes option (not available when Along Pillar is selected).

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Use minimum cell thickness It is possible to collapse layers with a thickness below a specified threshold. This setting is especially useful in erosional or onlap settings where thin cells accumulate at the unconformities. All cells below the threshold thickness are collapsedand set to Undefined. If Proportional or Fractions layering methods are selected the direction must be specified.

The following example shows: Follow Top: no minimum thickness specified.

The following example shows: Follow Top: minimum thickness specified

Settings for Each Zone

Zones can be removed using the "delete row" button once the row has been selected by clicking on the zone icon in the left most column, but note that the Base horizon for one zone must be equal to the Top horizon for the next zone.

To change the top or base horizon for a zone after removing one or more zones, select the horizon in the Petrel Explorer under the original grid and click on the blue arrow in the top or base horizon column

The button changes the colors of the zones to follow a rainbow color scale.

The Reset option removes edits and sets the spreadsheet back to the original spreadsheet from the original grid.

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Note that any surfaces used must be measured in depth or in time as appropriate for the grid (which will itself be measured in depth or time). Similarly, the input grid must be in the same domain as the output grid.


IJK Faults

IJK faults are faults which stair-step through the layers of the grid, zigzagging in three dimensions. They do not need to follow the pillars of the grid, and so can represent faults and support transmissibility adjustments without skewing and distorting the grid in the vicinity.

IJK faults are created by upscaling the layering from a fine grid onto the pillars of another grid using the Scale up Structureprocess. Any faults in the fine grid that are not aligned with the pillars of the target grid are stair-stepped.

The resulting IJK-faulted grid is displayed in the models tree with the icon and has the suffix [IJK][U] added to its name.

The IJK faults are displayed with the icon in the Faults folder of the grid.

� Creating IJK Faulted Grids� Limitations of IJK Faults� Quality Control of IJK-Faulted Grids

Parent topic: Scale up Structure


Creating IJK Faulted Grids

IJK faulted grids are created by upscaling the zonation of a fine grid onto the pillars of a coarse grid. These pillars can be created in Petrel in two different ways:

� Make Simple Grid process: This will create a grid with vertical columns, and all faults will become stair-stepped. � Pillar Gridding process: This can create a grid which is aligned to some of the faults in the fault model, and allows the

use of trends. Note that you should not incorporate any truncated faults in this step (leave them for the IJK faulting step below).

Make sure that the rotation of your grid reflects the major directions of flow in the reservoir and that the grid cell increment reflects the number of grid cells you are able to handle in the simulation run.

Once the coarse grid pillars have been created, the IJK faults can be created using the Scale Up Structure process (with the coarse grid active) by ticking the "Use IJK faulting" box.

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There are two different algorithms available:

� Segmentation-based IJK faulting works by upscaling and extrapolating the layers from each fine segment separately. At each fault, the algorithm selects cells from either side so as to stair-step the layers. If any fine grid fault does not act as a segment boundary, the auto-segmentation option can be used to further divide the segments and allow it to be stair-stepped as well. Note that this algorithm will not attempt to stair-step reverse faults; they will instead be aligned to the grid pillars.

� Reverse fault enabled IJK faulting is the recommended and default algorithm. It works by detecting where coarse grid columns cross faults and horizons, and layering each stretch of each column separately. Layers in adjacent columns are then either attached or IJK fault faces are placed between them, depending on whether they are separated by a fine gridfault. This algorithm can handle reverse faults, but the layers in the resulting grid may not correspond to cell k-indices anymore; a layer property is created under the grid's Properties folder to identify them instead.

When IJK faulted grids are created, the number of nodes is hard wired into the resulting grid. This means that rerunning Scale up Structure - for example in the Workflow editor - may fail. It is recommended that you always recreate the coarse skeleton grid in Pillar Gridding or Make Simple Grid beforehand.

How to Build a Grid with a Combination of Sloping and IJK Faults

1. Make a coarse grid using the Pillar Gridding process. Use the same Fault Model that was used to build the fine grid, but display in the 2D view only those faults which you wish to incorporate as sloping faults (do not include any truncated faults in this step).

2. Activate the resulting 3D grid, and open the Scale up Structure process. Drop in the fine 3D grid.3. Under Other settings, enable IJK-faulting. For segmentation-based IJK faulting, the remaining faults in the fine grid that

were not incorporated in the Pillar Gridding process for the coarse grid will now be incorporated as IJK-faults. For reverse enabled IJK faulting, all faults will be IJK faulted to ensure that a properly connected set of faults is produced.

The following are examples of the difference between using sloping and vertical pillars in the IJK grid.

Parent topic: IJK Faults

Use IJK Faulting Auto Segment

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For the segmentation-based algorithm to build correct layering at the end of a fault, all faults in the original 3D grid need to be on a segment boundary. To avoid having to go back to the Pillar Gridding process for the original grid, this option has been added to create extra segments so that all faults in the upscaled grid will define a segment boundary.

If this option is not used, only the faults in the original grid that define a segment boundary will be upscaled, all other faultswill be ignored.

This option does not give you the possibility to choose which faults from the original grid you want to upscale as IJK faults. If there are minor faults that you wish to ignore in the upscaled grid, but some faults you need which are not on a segment boundary, the only option is to go back to the original grid and re-grid it with the important faults on a segment boundary (by manipulating trends to be segment boundaries).

The examples below show on the left the Original 3D grid with few segments and on the right the upscaled IJK faulted grid with every fault on segment boundary


Limitations of IJK Faults

Some limitations arise when using IJK faults:

A fault is considered to be reverse if the pillars of the coarse grid pass through any horizon both above and below the fault. This means that some faults which are not considered reverse when upscaling onto vertical pillars may become reverse when upscaling onto inclined pillars, and vice versa.

� IJK faulted grids remove the Segment division in the Segment Folder. IJK faults do not separate Segments as they are identified in the Segment Folder. As a result, it is not possible to define different fluid contact levels on different sides of an IJK fault in the Make Contacts process. The algorithm will, however, return a Segment property with the segment division from the original grid. This property can be used when defining separate fluid in place regions for simulation. If these regions have different contact levels, these can be entered manually in the Make Fluid Modelprocess.

� IJK faults are only visible in the 3D window. It is not possible to visualize IJK faults with maps in the Map window, or to visualize IJK faults on a General Intersection in the 3D window or the Intersection window.

� The Fault Filter does not work for IJK faults. IJK faults are only visible in the 3D window as Allan diagrams, showing the juxtaposition of the reservoir across faults of interest. The Fault Filter has been disabled for these faults.

Some subsequent functionality in Petrel must be used with care if the layers do not have unique k-indices (i.e. after generating a grid with the Reverse fault enabled algorithm ) :

� Horizons will appear jagged and k-slicing using the property player will not display unique layers; similarly, the zone filter will not show specific zones. Use value filtering on the layer or zones properties instead.

� When performing simulations, be sure to use the generated ACTNUM property to ensure that any superfluous cells are deactivated.

Segmentation-based algorithm Reverse fault enabled algorithm

Limitations on types of fault that can be stair-stepped

Reverse faults are made vertical. Faults which do not divide segments require auto-segmentation enabled.

No restrictions, but layers will not always correspond to cell k-indices.

Limitations on fault transmissibility calculations

IJK faults have restricted options in the Fault Analysis process.

All faults are converted to IJK faults, even ifpreviously incorporated with Pillar Gridding, so have restricted options in Fault Analysis.

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� The generated "Stairstepped layers" property should be used when defining the zone mapping in the Scale up properties process.



Quality Control of IJK-Faulted Grids

To ensure that the IJK-faulted grid created is a grid with geometries suitable for simulation, the same type of quality control should be performed on this grid as other 3D grids created in Petrel.


Useful QC Techniques

You should assess the quality of your model before and after performing the IJK fault structure upscaling process. Here are some suggestions for improving the quality of the IJK upscaled model:

Before Performing the IJK Fault Structure Upscaling

The smoother the fine model horizons and faults surface are before upscaling, the better the quality of the upscaled model. It is good practice to assess the quality of the fine model objects, which constrain the upscaling process.

1. Check all the fault and horizon surface geometries. They should be smooth, without pinched areas or sharp angles. 2. Display the contour lines to highlight any potential problems with the horizon geometry.3. Check the fine grid geometry to identify cells with bad or poor geometry. To do this:

a. Create cell angle, cell inside out and cell volume properties in Geometrical Modeling. b. Check each property's statistical information (minimum / maximum) and histogram, looking for high cell angle or

cell inside out values, or negative cell volume values. c. Use value filtering and I-, J- or K- 3D slicing all through the fine grid to isolate areas with values outside the

normal range.� Look at the shape of these cells, searching for hanged or twisted cells. � For each of these areas, also display the closest faults or horizons, and check their geometry. In most cases,

there is a direct link between a fault or horizon surface local pinch or sharp angle, and bad grid cell geometries.

4. If you detect any problems, you should consider modifying and editing your input model data.

If the resulting grid is not suitable, it is possible to use the Edit 3D Grid process in conjunction with the Grid I/J- directions (found under the grid's Intersections folder) to move the pillars of the target grid before running IJK faulting so you can adjust precisely which cell columns interact with a fault. This approach can sometimes be used to remove defects caused by awkward grid geometries. Note that the reverse fault enabled algorithm is sensitive to the location of the center of a column in the target grid, rather than the pillars at the corners.

It is not possible to run the process Edit 3D Grid on the IJK-faulted grids.

After Performing the IJK Fault Structure Upscaling

1. Check the coarse grid geometry to identify cells with bad or poor geometry. This is how you do it:a. Create cell angle, cell inside out and cell volume properties in Geometrical Modeling b. Check each property's statistical information (minimum / maximum) and histogram, looking for high cell angle or

cell inside out values, or negative cell volume values. c. Perform a 3D visual volume exploration of the resulting grid

� Switch on the top view of the grid and perform I-, J- slicing all through the coarse grid, particularly around contacts with IJK-faults.

� If any hole (missing cells) is observed, check if there are missing cells as well in the input fine grid. � If not, check if your structural model contains reverse fault and which IJK faulting upscaling option you

have used in the Scale up structure process. Models with reverse faults should always be IJK upscaled using the Reverse fault enabled IJK faulting option.

2. Check the geometry of the IJK faults :a. Make a 3D display of all IJK faults. b. Look at the geometry of all of them. Their surfaces should be continuous, without any holes or stacks of cell faces

collected together, and the surfaces should fit on one of the structural faults.

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Please note that for an IJK fault to be built, it must have at least two adjacent cells on each side of the fault (as a consequence, model boundary IJK faults might not be created).

c. If you find any of the problems above; simultaneously display the associated and closest fine model structural faults and horizons, and check their surfaces; as explained in Before performing the IJK fault structure upscaling. Finding such problems is often an indicator of a bad input data.

3. Define contacts using Make Contacts and calculate volumes of the IJK-faulted grid and compare with the original grid. Make sure the boundaries are the same for a valid comparison.


Scale up Properties

Properties from one grid can be transferred to another grid of a different resolution or orientation using the Scale up propertiesprocess. This is usually done in the context of building a simulation model from a geological model, where the simulation model has been coarsened and reoriented for flow simulation. However, the Scale up properties process places no restrictions on the grids that are used as input and output, and the process can also be used to transfer properties between identical grids and to downscale.

For most properties (e.g. porosity, net-to-gross), it is appropriate to upscale to a coarse grid using weighted averaging of values from a fine grid. The fine grid cells in the vicinity of each coarse grid cell are found, and their property values are weighted (by intersection volume and/or by other property values such as net-to-gross) and aggregated using a specified averaging method (such as arithmetic or geometric averaging). Discrete properties can be handled in the same way using aggregation methods such as "most-of" (also known as "mode").

Permeability can be upscaled in the same way, but more sophisticated upscaling techniques are also available. Flow-based upscaling methods perform a flow simulation on coinciding fine grid cells to arrive at a representative permeability for each coarse grid cell. For these methods, upscaled permeabilities are created for each grid direction by imposing appropriate pressure gradients on the boundaries of the fine cell set, and solving for the internal pressure gradients to establish an overall flow response.

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Directional averaging methods are also available for permeability upscaling, wherein the permeability values of the group ofcoinciding fine cells are averaged using a combination of arithmetic and harmonic (or power) techniques. Again, multiple permeability properties are created for each of the I, J and K directions.

� General Options� Sampling Options� Zone Mapping� Averaging Upscaling Methods� Permeability Tensor Upscaling


How to: Upscale Grid Properties

How to upscale grid properties:

1. Activate the target grid (the grid to upscale to) in the Models pane. 2. Double-click on the Scale up properties process in the Processes pane. 3. Select those properties (from another grid) that you wish to upscale, and drop them into the Scale up properties dialog

using the blue arrow button.

4. One at a time, select each property in the table and review the settings used.5. Select the source grid in the table, and review the sampling settings in use. 6. When satisfied with the configuration click Apply. Depending on the sizes of the grids, the upscale may take some

minutes, during which time the progress is displayed on the progress bar of the main Petrel window.

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It is more efficient to upscale multiple properties together, and the same source property can be dropped multiple times and upscaled with different settings if necessary. Dropping a source grid explicitly will add an entry for the grid; if this grid entry is selected in the table when a property from that grid is next dropped, the new property will be placed under that grid entry. This allows multiple different grid settings to be used in the same run.

Recommended Settings for Commonly Upscaled Properties

The primary concern in upscaling most properties for simulation is ensuring that the volumes (e.g. hydrocarbon volume, pore volume etc.) are consistent between the fine and upscaled properties.

Porosity Use volume-weighted arithmetic averaging, and choose to weight by the net-to-gross property if one is present.

Net-to-gross Use volume-weighted arithmetic averaging with no weighting properties. Note that this ensures consistency with the upscaling of porosity; deriving a coarse net-to-gross value from an upscaled facies property is not as accurate.

Saturation properties (e.g. oil saturation, water saturation) Use volume-weighted arithmetic averaging, weighted by both porosity and net-to-gross.

Volume properties (e.g. bulk volume, pore volume) Use the "summation" algorithm.

Facies Use volume-weighted "most of" averaging with no weighting properties.

Instead of volumetric equivalence, the upscaling of permeability is directed by the need to replicate flow behavior between the two grids. A discussion of the appropriateness of the various methods is beyond the scope of this manual; indeed the effectiveness of any one approach is strongly determined by the specific nature of the geological model and target (upscaled) grid. As such, it is important to conduct the upscaling within the context of matching the fine and coarse model behaviors asmuch as possible.

Harmonic averaging Provides the exact, analytic solution when upscaling in one direction only (e.g. upscaling vertically, between areally identical grids).

Geometric averaging Provides a simple approximation when upscaling by small scale factors which preserves the log-normal distribution.

Power averaging Generalizes the other averaging methods by providing an exponent that can be adjusted to more accurately match distributions between the models.

Directional averaging These slightly more sophisticated methods take account of the directional nature of permeability and produce three separate output components to better acknowledge the anisotropic nature of the upscaled permeability. They mix arithmetic and harmonic (or power) averaging to approximate parallel flow through multiple lines of cells.

Flow-based upscaling Available with the "Advanced gridding and upscaling" license feature, this method performs numerical experiments on the intersecting source cells.

Quality Checking the Upscale Results

There are several methods available for quality checking the results of an upscale.

Run volume calculations Use the Volume Calculation process to generate volume reports for each zone and segment. By including the porosity, net-to-gross and saturations, you can ensure that the volumes of the fine grid are closely replicated in the coarse grid.

Check property histograms and statistics By double-clicking on a property in the Models tree, you can display the property's histogram and statistics, and filter by zone or by properties. Note that the upscaled properties will usually have a reduced variance due to the averaging, but qualitative shape of the distributions should usually be similar. Remember to enable volume weighting when

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comparing histograms. Visually inspect the properties with the property player

When the Scale up properties process is selected, the property player tools appear on (by default on the right of the main window). These can be used to display slices of the grids one at a time. If the grid lines are very different between the grids, you can use general intersections instead (right click on the grid's Intersections folder and choose insert general intersection)

Generate streamlines to compare flow directionsRun a single-step FrontSim simulation with a simple default fluid model (e.g. oil-water) and with representative well injection and production rates. A good upscale will show similarly directed streamlines for the major parts of the grids. Also see the GEOFLOFS keyword, designed specifically for setting up simple streamlines simulations for upscaling verification.

Performing a Reduced Upscale for Testing

It can be useful to upscale a small part of the grid and examine the results to ensure that they are satisfactory before running a full upscale. This can be done in two ways:

1. Define a closed boundary polygon using the Make/edit polygons process and drop it in to the settings for the source grid.

2. Use the property filter to filter out all but a few cells on the target grid, and tick the Upscale to filtered cells only option on the source grid settings in the property upscaling process.

Use the Upscale to filtered cells only option with care; when enabled and overwriting a target property (as opposed tocreating a new one), those cells that are not being upscaled to will be left as they are (instead of being set undefined). Also turn off the "Defined values only" option in the property filter, as otherwise the currently displayed property will influence the behavior of the upscale.


General Options

A number of upscaling options apply to all input properties from a specific source grid. In particular, those are the Samplingand the Zone mapping settings. To access those, left click the input grid in the process dialog.

Optionally, you can specify a boundary to limit the region of the output grid that you want to populate with upscaled properties. This is done by dropping a polygon into the Boundary drop box. As with the Make Local Grids process, a cell is excluded if the topmost defined cell in that column has its center outside the boundary (considering X and Y coordinates only).

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The coarse grid cells to populate can also be restricted by enabling the Upscale to filtered cells only option. This will restrict upscaling to those output grid cells, which are included in the currently active filter.

To access the settings for each individual property that you want to scale up, left click the name of that property in the list in the process dialog.

The option Calculate can be used to decide which properties that should be scaled up every time you run the process. Once the property is scaled up once, the Calculate check box is deselected automatically.

The option Ensure all cells get values will carry out a post-processing step which searches for undefined cells in the coarse grid and populates them with the average of the surrounding defined cells. This is repeated in successive steps, using previously averaged values to continue the averaging if necessary, thereby extrapolating values over any accessible gaps in theresulting property. The averaging method used in this process is the same as that used in the upscaling when using averaging, is arithmetic when using summation, and is geometric when using directional averaging or flow-based upscaling.

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Parent topic: Scale up Properties


Sampling Options

The choice of sampling method determines which source grid cells contribute to the upscaling of each target grid cell. They are configured on a per-source-grid basis by selecting a source grid in the properties table.

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The following sampling methods are available:

All intersecting cells For each cell in the target grid, this method samples all cells in the source grid that are overlapping it in 3D. When used with volume-weighted averaging, the volume weight used is the volume of overlap. Overlapping cells that are not zone-mapped to the target cell will not be sampled.

This method is appropriate in most cases, but the upscale may take a long time if run on large grids due to the expensive volume-of-overlap calculation. Note that flow-based upscaling and directional averaging both include a source cellentirely or not at all (i.e. no volume weighting is used); this means that cells with a small presence in the target cell can

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have a large effect on the end result. For permeability upscaling between proportionate grids (i.e. where target cell boundaries often line up with source cell boundaries), the "Cells with center inside target cell" sampling method is usually more appropriate.

Source cell centers For each target cell, this method samples those cells in the source grid whose centers lie within the target cell. When used with volume-weighted averaging, the volume weight is the entire source cell volume (as each source cell will generally only contribute to the single target cell containing its center point). Cells that are not zone-mapped to the target cell will not be sampled.

This approach is a faster alternative to "All intersecting cells" sampling, and is particularly appropriate when upscaling by a scale factor larger than roughly 3. In these cases, the value of a target cell is decided largely by source cells lying entirely within its volume; those source cells lying on the boundaries of target cells make a relatively small contribution, and the detailed volume-of-overlap calculation is unnecessary.

This sampling method is also more appropriate for upscaling permeabilities between proportionate grids (i.e. where the target grid cells contain an exact division of source grid cells). In these cases, this method selects only those cells with a significant presence in the target cell.

Zone-mapped layers This sampling method finds columns of cells passing through each target cell, and samples all layers in those columns that are mapped to the layer of the target cell. This mapping of layers is specified in the zone mapping tab (see Zone mapping). A column of cells in the source grid is included if the center of a zone-mapped cell falls within the target cell; if no such columns are found, the sampling selects any cells that contain the center of the target cell when projected onto the x/y plane.

This method is only appropriate if the grids are closely aligned, and if the same layering styles have been used for each zone. In these cases, it can be used effectively with the Geometric Overlap zone mapping algorithm, which maps eachsource layer to most overlapped target layer. This combination of settings is appropriate in the specific case of transferring properties between copies of the same grid (or parts of a grid), or to Cartesian refinements or coarseningsthat are represented as separate grids.

Target cell centers This method selects the value of the source property which lies at the center of the target cell. As only one value is involved for each target cell, no aggregation of results is needed (so only the "Sample cell center" upscaling algorithm is offered, and all averaging and tensor upscaling methods are unavailable).

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This method is appropriate for downscaling properties, or transferring properties between copies of a grid (or parts of agrid).

Note: Because the cell center is defined as the average of the eight vertices, cells which are very flat and bent may not contain their centers.

Some upscaling algorithms are not appropriate to use with certain sampling methods, and so are unavailable in the list when the sampling method is chosen. The following table details the constraints:

By default, property upscaling will upscale to all target cells of the global grid and any associated local grids. The target cells can be restricted in two ways:

Upscale to filtered cells only When enabled, only those cells filtered for viewing (i.e. by property value or index filters, zone or segment filters or local grid set filters) will be targeted for upscaling. Note that it is not necessary to upscale to the global grid if only some local grid sets need to be populated - simply enable this option and switch off the visualization of the global grid and any unwanted local grid sets.

Boundary Drop a closed polygon to upscale to columns of cells within that polygon only. A column is included if the center of its topmost cell is within the polygon when projected onto the x/y plane.



Zone Mapping

The zone mapping associates each layer in the target grid with a range of layers in the source grid. This information is used by the sampling method to restrict the search space, and decide which cells will contribute to a particular layer.

All intersecting cells Source cellcenters

Zone-mapped layers

Target cell center

Averaging(volume-weighted)

Unavailable (as only one fine cell will contribute to each coarse cell)

Averaging (cellcount)

Unavailable (to preserve fine grid volumes, as fine cells wouldcontribute full volume to multiple coarse cells)


Summation Unavailable (as only one fine cell will contribute to each coarse cell)

Sample cellcenter

Unavailable Unavailable Unavailable

Directionalaveraging

(for permeability properties only) (for permeability properties only)

(for permeability properties only)


Flow-based upscaling

(for permeability properties only) (for permeability properties only)

(for permeability properties only)


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While the zone mapping can be specified explicitly, it is generally more convenient to generate it using one of the available algorithms:

Matching horizon names This method identifies which zones in the source grid correspond to those of the target grid by examining the names of the horizons. For this to work, the horizon names must match, but not all horizons need to be present in the target grid. Once zones have been matched, every layer in a target zone is mapped to every layer in the corresponding source zone.

This method is appropriate for any sampling method except "Zone-mapped layers".

The mapping produced by this method is sometimes referred to as "open", indicating that entire zones are mapped to one another (as opposed to individual layers or small groups of layers).

Previous process settings Zones are matched between the two grids by examining Make zones and Scale up structure process settings. This is a useful alternative to "Matching horizon names" when the horizon names do not correspond, but it relies on both grids having been created (or at least layered) using the standard Petrel processes. Note: A layer property, if specified in the Use layer property drop box, will not be taken into account when attempting to match layers using this option.

The mapping produced by this method is sometimes referred to as "open", indicating that entire zones are mapped to one another.

Geometric overlap Each layer in the source grid is mapped to the layer in the target grid that it overlaps most. The amount of overlap is

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determined by comparing layer depths on corresponding pillars between the two grids. For this to work effectively, the pillars in the two grids should be roughly aligned across most of the grid.

This method generates a tight zone mapping that is appropriate for use by the "Zone-mapped layers" sampling method. Where layering styles are similar between the two grids, it can be used with other sampling methods too.

The mapping produced by this method is referred to as "closed", indicating that individual layers or small groups of layers are mapped to one another.

Mapping all layers to all layers Every layer in the target grid is mapped to every layer in the source grid. This method can be used if no sampling restrictions based on layering are needed (e.g. when using the "All intersecting cells" sampling method).

Use specific layer indices Manually specify each layer mapping in the table.

When using All intersecting cells sampling or Source cell centers sampling, Matching horizon names or Previous process settings are good choices. If neither can be used due to horizon name differences or because one of the grids has been built outside of Petrel, then "Map all layers to all layers" is a safe fallback option.

When using Zone-mapped layers sampling, the Geometric overlap algorithm is more appropriate. Because this sampling method will sample the specified layers regardless of whether they genuinely overlap the target cells, it is important to use a"closed" mapping (i.e. one that maps layers based on geometry).

When upscaling to an IJK faulted grid, it is advisable to use a layer property to specify the layer each target cell belongs to. By default, the cell's k-index is used, but this may not correspond to the layer index near reverse IJK faults. In these cases, drop the "Stairstepped layers" property produced during IJK faulting into the layer property box, and select the zone mapping algorithm as usual.



Averaging Upscaling Methods

Most properties are appropriately upscaled by averaging values from the fine grid onto the coarser cells of the target grid. Depending on the property type, different averaging methods are appropriate, and other properties may need to be taken into account as weights. For example, porosity should be upscaled with arithmetic averaging and weighted by net-to-gross, whereas a facies property should be upscaled using "most of" averaging and no weights.

Averaging can be based on volume or on number of cells. Volume-weighted averaging will weight source property values by the volume contributed by the fine cell, whereas cell count averaging will give equal weight to all source cells involved in the average (regardless of variations in their volumes). Cell count averaging is not available when using the All intersecting cellssampling method because a fine grid cell will contribute to multiple coarse grid cells when using this sampling method, sovolume-weighting must be used to ensure its overall contribution respects fine grid volumes.

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Continuous Averaging Methods

The following averaging methods are available for upscaling continuous properties:

Discrete Averaging Methods

Discrete properties can only assume a finite set of distinct values, unlike continuous properties which take values from a continuum. In spite of this, each possible value of a discrete property has a numerical code which is used by some of the algorithms below.

Arithmetic

Typically used for additive properties such as porosity, saturation and net-to-gross. Volume weighting will produce a more appropriate arithmetic mean when input values have variable presence within the resulting cell. A property such as oil saturation, for which the value is only meaningful in relation other properties, should be weighted by the properties it depends on, i.e. porosity and net-to-gross. This will ensure that the hydrocarbon pore volume remains constant when upscaling.

Harmonic

Gives the exact effective permeability vertically if the reservoir is layered withconstant permeability in each layer. The harmonic mean works well with lognormal distributions. It is used for permeability because it is sensitive to lower values. The method is not defined for negative values.

Geometric

Normally a good estimate for permeability if it has no spatial correlation and is log normally distributed. The geometric mean is sensitive to lower values,which will have a greater influence of results. The method is not defined for negative values.

Root mean squared

Will set a bias towards higher values.

MinimumSelects the lowest value from all the original cells that contribute to an upscaled cell.

MaximumSelects the highest value from the original cells that contribute to an upscaledcell.

Power

This method generalizes several of the above methods by accepting an exponent parameter; a value of 1 is equivalent to arithmetic averaging, a value of 2 is equivalent to RMS, and a value of -1 is equivalent to harmonic. Theexponent can be negative and/or fractional, but not zero. In theory, as the exponent tends to 0, the power average tends to the geometric average.Similarly, as the exponent tends to infinity, the power average tends to the maximum, and to minus infinity to the minimum. In this sense, each of the above methods have an "equivalent exponent" which indicates the methods' relative biases toward higher or lower values, i.e. generally, minimum < harmonic < geometric < arithmetic < RMS < maximum.

Most of Also known as "mode", this method assigns the most strongly represented source value to the target cell. For example, when using volume-weighted averaging to upscale a facies property, a particular coarse cell overlapping

fine cells containing a total of 1000 m 3of sand, 2500 m 3of silt, and 3000 m 3of shale, will be set to shale.

Median Takes the median of source cell property values (ordered by their numerical codes). The median is the value for which half of the source weight (e.g. volume weight) is lower and half is higher, or the closest possible discretevalue.

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Weighting Properties

When using averaging to upscale, other source grid properties can be used as weighting factors for the source property values. For example, porosity should be weighted by net-to-gross, and saturation properties should be weighted by porosity and net-to-gross.

Refer to the Volume Calculation process dialog for information about which properties depend on which.


Permeability Tensor Upscaling

While the priority in upscaling properties such as porosity and net-to-gross is to maintain overall pore volumes, the focus inpermeability upscaling is to preserve flow behavior. As such, different algorithms and techniques are applicable.

� Directional Averaging Methods� Flow-Based Tensor Upscaling� Flow-Based Tensor Upscaling (Advanced)


Permeability Tensors

All permeability upscaling algorithms will accept different input properties for each of the I, J and K grid directions, thereby allowing the upscaling of anisotropic permeabilities. When selecting a permeability upscaling algorithm, the entry in the table will expand to show the three source properties being used as the I, J and K components. Selecting one of the components in the table will allow them to be changed to other source properties. If necessary, the same source property can be specified multiple times (i.e. if using an isotropic input permeability, specify the same property for all three components).

Note that the list of possible source properties for each direction contains only properties with appropriate template, e.g. the list for I direction contains properties with templates "Permeability", "Permeability I" and "Permeability X". Properties with templates such as "Permeability IJ", which represent off-diagonal components of permeability tensor, are not used as a source for upscaling and hence, they are not included into the lists of possible source properties.

Minimum Selects the discrete value with the lowest numerical code.

Maximum Selects the discrete value with the highest numerical code.

Arithmetic Performs arithmetic averaging of the source values using their numerical codes, and selects the value with the nearest numerical code to the result.

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Similarly, permeability upscaling algorithms will produce separate upscaled permeability properties for each grid direction. These separate properties will be given names with suffixes based on the grid direction they represent, e.g. "Perm_I","Perm_J" etc.

Cell Sampling for Permeability Upscaling

For each target grid cell, a set of source grid cells is found using the configured cell sampling method (see Sampling options). Unlike the averaging methods which can weight source cell contributions by e.g. intersection volume, permeability tensor upscaling methods will use each source cell in its entirety. This can affect the choice of sampling method when upscaling permeability; it is generally recommended to use the "Source cell centers" sampling method in this case.

The permeability tensor upscaling methods will always use an IJK box of source cells for each target cell. If a target cell intersects a ragged-edged set of source cells, the smallest IJK box containing those cells will be used by the permeabilityupscaling algorithm. The algorithm may also further expand this box depending on its configuration. Note that this behavior can impact the validity of upscaling IJK permeabilities between rotated grids, as the box of cells used in upscaling may not closely correspond to the target cell.


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Directional Averaging Methods

The overall effective permeability along a series of connected cells is given exactly by the harmonic average of the individual cell effective permeabilities, whereas the effective permeability across a 1-cell-thick plane of cells is given exactly by the arithmetic average of those individual cell effective permeabilities.

Combining these two rules yields the arithmetic-harmonic and harmonic-arithmetic permeability upscaling methods. Each method applies the two rules in succession to arrive at an overall permeability for the 3D block of fine cells overlapping a given coarse cell; because the order of application matters, the two approaches are offered separately.

Each directional averaging algorithm is applied once in each of the I, J and K flow directions, producing a separate result property for each which together specify the anisotropic upscaled permeability. These methods will also accept a differentsource permeability property for each of the three flow directions if anisotropic permeabilities are to be used as input; if not, the same permeability property can be specified for each of the three directions.

Arithmetic-harmonic Arithmetically averages the planes of cells perpendicular to the flow direction to arrive at plane permeabilities, and then harmonically averages these to produce the final upscaled permeability for the given flow direction.

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Harmonic-arithmetic Harmonically averages the lines of cells along the direction of flow before arithmetically averaging the results to get the upscaled cell's permeability for the given flow direction.

Arithmetic-power Similar to arithmetic-harmonic, but power averaging is used instead of harmonic averaging. The exponent for the power must be given; if it is set to -1 then this method is equivalent to arithmetic-harmonic.

Power-arithmetic Similar to harmonic-arithmetic, but power averaging is used instead of harmonic averaging. The exponent for the power must be given; if it is set to -1 then this method is equivalent to harmonic-arithmetic.

Cardwell-Parsons Gives the geometric average (square root of the product) of the arithmetic-harmonic and harmonic-arithmetic results.

Parent topic: Permeability Tensor Upscaling


Flow-Based Tensor Upscaling

Flow-based upscaling involves performing a numerical pressure simulation on the block of fine cells coinciding with eachcoarse cell to determine a representative coarse cell permeability.

The process will calculate I, J, and K or X, Y, and Z permeabilities from input as permeability in the I, J and K directions, net-

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to-gross and porosity. If the Output full tensor permeabilities option is used, then off-diagonal terms in the permeability tensor (in the IJ, IK and JK directions) will also be calculated. It is assumed that the system is symmetrical, i.e. IJ = JI. Note that Eclipse 100 and FrontSim cannot utilize full tensor permeability.

Tensor upscaling methods always include a fine grid cell in its entirety (or not at all) in the calculation of each coarse grid cell value. As such, small intersections can cause more cells than expected to be incorporated, particularly when using the All intersecting cells sampling method. When doing tensor upscaling between closely aligned grids, always use the Cells with center inside target cell or Zone-mapped layers sampling methods to avoid unexpected "grazing" intersections.

� Calculation of the Permeability Tensor


The Process Dialog for Flow-Based Tensor Upscaling

Choosing the Flow-based upscaling method displays a different set of options in the dialog.

Settings The following options can be specified here:

� Numerical Method: The pressure field is estimated by a numerical method. In order to do this, one degree of freedom (one unknown pressure variable) is introduced at the cell center and at the center of each cell sides. If too few fine cells are present within the upscaled cell, additional nodes are introduced so that the pressure solutioncan be calculated.

There are two options for the algorithm used for solving the pressure solution:

� Harmonic average: is a finite difference method used to calculate the pressure. The solution proceduretakes into account that the permeability is discontinuous over the cell sides, giving a reliable solution with large variations in permeability from cell to cell.

� Finite difference: is a standard finite difference solver. � Results co-ordinate system: Choose whether the resulting permeabilities are expressed as local IJK or global

XYZ permeabilities (use IJK for ECLIPSE and FRONTSim simulations). � Output full tensor permeabilities: Choose whether to output a full tensor or just a diagonal tensor.� Adjoint method:This is a faster version of the open to flow boundary condition. It only performs one flow-like

simulation rather than the usual three. The method for backing out the upscaled values is modified so that they can be retrieved from a single simulation without approximation.

Properties

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The porosity and net-to-gross properties on both source and target grids are required to perform the flow simulation, and should be specified here. If they are not specified, they are assumed to be constant.

Skin region A skin region can be added to increase the number of cells used to compute the pressure gradients from the specified boundary condition. For each of the three directions over which a flow calculation is performed, the skin is added in the two perpendicular directions, and the boundary condition applied to the edge of the skin. There are several options:

� At least: Specify the minimum number of source cells required in each direction. That is, if the block has fewer cells in any specific direction, then cells are added on alternate sides of the block until this minimum number is reached.

� Extra: Specifies the number of cells to be added on both sides of the coarse grid block. � Allow skin outside zone mapped layers: Allows skin cells to be added even if the new cells are not mapped to

that coarse cell. � Use skin cells in velocity average: By default, the skin pressure gradients are not included in the calculation of

velocities (and hence resultant permeabilities); by enabling this option, skin pressures are included as well. This causes some blurring of the permeabilities when upscaling.

Boundary condition Here the type of boundary condition to impose, from which the pressure gradients will arise, is specified. Each condition involves a pressure gradient being applied in the direction of flow (I, J or K). Three different boundary conditions are available:

� Open to flow:Flow is permitted through all the cell sides. A pressure drop is applied in one direction at a time tocompute the permeability in this direction. On the remaining boundaries, a linearly varying pressure field is applied.

� Closed to flow: Flow is allowed through the cell in the direction of flow, but not out of the sides. A pressure drop is applied in one direction at a time to compute the permeability in this direction. The remaining boundaries are closed i.e. the pressure gradient across the boundary is kept at zero.

� Closed to flow between layers: This option is identical to the open boundary condition except that the top and base of the cell are closed when applying a pressure drop in a horizontal (i.e. I or J) direction. This is for use when using the Output full tensor permeabilities option without skin and corrects for the fact that in the reservoir, flow is generally forced to be horizontal due to low permeability boundaries, which may not be present in the cell being simulated.

Note: If you wish to experiment with different settings you can upscale the same input permeability properties multiple times in the same run. However ensure that you use different output names so that you create separate upscaled output properties.


Calculation of the Permeability Tensor

The basic algorithm for flow-based upscaling can be broken into the following steps:

1. Find the set of source cells corresponding to the current target cell (according to the selected cell sampling algorithm).2. Expand this set into a clean IJK box. 3. For each of the I, J and K grid directions

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a. Add some skin cells, orthogonal to flow direction, if requested. b. Impose the appropriate boundary condition on the set of cells. c. Solve for the pressures of all cells inside the block, taking into account the permeabilities and pore-volumes of the

cells. d. Derive overall flux from the gradient of the calculated pressures. e. Assign the coarse cell permeability to match the flux given the imposed pressure gradient.

The rest of this section will discuss each of these detailed steps in turn, assuming the corresponding fine cells have beendetermined. We will use the following two example fine cell sets, both 2D (i.e. a single layer seen from above):

The first shows high permeability channels (shown in red) connecting the top and bottom of the set, whereas the second shows a similar example in which the channel has been broken by zero permeability barriers (shown in purple).

Parent topic: Flow-Based Tensor Upscaling

Addition of Skin Cells

A more natural boundary condition than those used in step b can be found by taking the environment of the block of cells into account. This is done by using a skin zone to extend the block of cells in the two grid directions orthogonal to the current flow experiment. The boundary condition is then applied to this extended block, and the original unextended block experiences a boundary condition which is more reasonable given its environment.

Boundary Conditions

All boundary conditions involve applying a high pressure to one side of the block, and a low pressure to the opposing side (be they in the I, J or K grid directions). The remaining decisions concern the other sides of the block. For example, when calculating the I permeability, a high pressure may be applied to the I- side, and a low pressure to the I+ side of the block; the conditions of the J-, J+, K- and K+ sides can be configured in different ways. In this discussion, we call the I grid direction (in this example) the "primary flow" direction and the J and K grid directions are the "orthogonal" directions.

The closed boundary condition intuitively requires that no flow be allowed out of the orthogonal sides. This is enforced by insisting that the pressure gradients at and normal to those sides is zero.

The open boundary condition allows flow through the orthogonal sides. No condition is enforced on the pressure gradients, but the pressure values are set to ensure a linear drop in pressure in accordance with the pressure applied to the primary sides.

The closed to flow between layers uses the open boundary condition for I-, I+, J- and J+ faces, but always ensures the K- and K+ faces are closed when they are the orthogonal sides. Of course, when K is the primary flow direction - i.e. whendetermining the K permeability - the sides are not closed.

Solving for Pressure

The boundary conditions specify pressure values and/or gradients at certain points. What remains is to develop a means of propagating these values to the rest of the cells (those internal to the boundary).

Neglecting the effects of gravity and compressibility, a steady-state pressure p should satisfy

all points given the (spatially varying) diagonal permeability tensor k.

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The solution is found by applying a discretization based on the fine cells i.e. the solution is found at points placed at each fine cell center. If fewer than three cells are present in any grid direction, the cells are split to ensure at least three are present.

In our two example cases, consider the effect of applying a high pressure at the top of the cell sets, and a low pressure at the bottom. In the case of a closed boundary condition, we get the following:

When the channel connects through, the pressure gradually drops from the high pressure boundary to the low pressure boundary. Because the boundary is closed, the side channel builds up to a high pressure which is not allowed to escape. When the channel does not connect, the pressure cannot escape through the bottom or sides, and so simply builds up to the high pressure at the boundary.

By contrast, an open boundary yields the following gradient:

The pressure is now allowed to escape through the left and right sides, so the left channel's previously built up pressure is released. In the latter case, this allows some pressure drop through the set where there was previously none.

Harmonic Average Algorithm

At each such point, the permeability is considered to be spatially constant (it's treated as being piece-wise constant on the fine cells) so we instead solve

This is converted to discrete form by using

where the pressures and cell sizes are defined according to this scheme:

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The pressures on the cell faces are calculated explicitly by insisting that the velocities are the same on both sides:

where the pressures are here sampled as follows:

Together with the chosen boundary condition, these two rules set up a system of linear equations which can be solved for the pressure values at all points.

Finite Difference Algorithm

The finite difference algorithm uses a discretization which associates neighboring cell pressures via transmissibilities instead of cell face pressures. The transmissibility is calculated using a harmonic average of the two cell's permeabilities.

Calculating Overall Flux

The relationship between velocity, pressure and permeability is given by Darcy's law, neglecting gravity and viscosity:

This velocity is calculated over the fine cells. In discrete terms, the pressure gradient is given by the cell face centers:

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The flux is given by the product of velocity, porosity and net-to-gross. These fluxes are calculated and summed over the fine cells excluding any skin region (unless the Use skin cells in velocity average option is enabled).

Calculating Coarse Cell Permeability

The coarse cell permeability must be assigned to ensure identical flux to the fine cell under the same pressure gradient. Because coarse net-to-gross and porosity are constant over the coarse cell, we arrive at

The pressure integral on the left hand side (corresponding to the coarse cell) is transformed using the divergence theorem to integrals over the fine cell faces corresponding to the coarse cell face.


Flow-Based Tensor Upscaling (Advanced)

Choosing Flow-based upscaling (advanced) displays upscaling algorithms from FloGrid. Flow simulation is more realistic and accurate in a target cell. It is available only with the “All Intersecting Cells” sampling method. 'Skin' cells are not supported, and NTG and porosity are not used.

The following upscaling methods are available:

Linear boundary condition Linear boundary conditions are imposed by choosing a direction vector along the normal to a surface that divides the grid block into two parts. This dividing surface is the arithmetic average of the two opposing faces modeled as bilinear patches. This can be done in three ways, leading to three numerical experiments; one for each component of a diagonalpermeability tensor. The pressure at points on the external faces is given by the scalar product of the direction vector with the coordinates at those points.

The total flux through the system is calculated in the direction joining the mid-points of the two grid block faces in the direction of flow. This is also the direction of the pressure drop. The diagonal upscaled permeability tensor is then defined to be that constant diagonal tensor that would give rise to the same fluxes if we use identical boundaryconditions.

The transmissibilities are calculated in the same way as in the ECLIPSE simulator, using the NEWTRAN method.

Linear boundary condition. Adjoint method The adjoint method very similar to the linear boundary condition method but has the potential to be nearly three times faster. Only one flow-like simulation is performed rather than the usual three.

The flow-based upscaling process needs a boundary condition, a flow solver and a diagnostic function. This last

requirement reduces to the provision of a vector, , and a fixed number, , so that a measure, , of the total flux is given

by where is the vector of pressures obtained from the numerical solution. Normally one would use a

different vector, , and number, , for each flow direction. In the adjoint method we choose the diagnostic function by

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first finding the total flux through each face of the grid block in the direction of increasing x, increasing y and

increasing z and then taking the sum of these fluxes. We use the same vector in the diagnostic functions, while the

constant, , is different in each experiment (depends on the direction of the pressure gradient in each numerical experiment). In the adjoint method we use linear boundary conditions, and the constant.

Thus we can use the adjoint method of solving a sequence of linear equation problems, where only the right hand sides are different, and we only wish to evaluate a fixed linear diagnostic function. We therefore only need to solve the linear system once and we can find the total fluxes by evaluating three dot products.

Linear boundary condition. Full tensor This is very similar to the linear boundary condition method, but allows all six components of the permeability tensor to be extracted. This option cannot be chosen when overwriting an existing tensor, as the number of components is already implied by the tensor selected for overwrite.

No side-flow boundary condition No-side-flow boundary conditions impose a constant pressure on one face of the block, and a different constant pressureon the opposing face. The other four sides of the block are subject to no-flow boundary conditions. There are three different ways of applying these boundary conditions, providing three independent numerical experiments for thecomponents of a diagonal permeability tensor.

We compute the total flux through the block in the direction of the pressure drop, joining the mid-points of the block faces in the direction of flow. The effective permeability is then estimated by solving the same problem with constantpermeabilities chosen to give the same flux. For the no-side-flow boundary conditions, this constant permeability problem is solved using harmonic-arithmetic averaging. An error is introduced by this, unless the grid blocks are rectangular. The use of the linear boundary conditions removes the need for this approximation.

The no-side-flow boundary condition method transforms the problem into an equivalent problem on a cube. The Jacobians and length-scale factors are evaluated at the cell centers. The transmissibilities, on irregular grids, are slightly different between this approach and the NEWTRAN method used in the ECLIPSE simulator.

� Property Sampling for Flow-Based Upscaling (Advanced)



Property Sampling for Flow-Based Upscaling (Advanced)

Parent topic: Flow-Based Tensor Upscaling (Advanced)

Introduction

In this section, we give a more detailed discussion of the steps taken to sample properties from a fine grid to a coarse grid for the flow-based upscaling (advanced) algorithm.

Note on Terminology: For clarity we refer to fine cells on the fine (or input) model, and to grid blocks on the coarse (oroutput) model. To enable the various calculations we subdivide the grid blocks into smaller partswhich we refer to as microcells. Microcells are not generally the same as fine cells, but are approximately the same size as nearby fine cells.

Sampling the properties

The coarse grid construction determines which fine cells are related to a grid block. We note that each grid has two coordinate systems. The first is the usual x, y, z system where the coordinates are the actual values in a reference Cartesian frame. The second are the logical coordinates which can be used to determine the cell index by taking the integer part of the logical coordinates. By subtracting the cell index from the logical coordinates, we determine the fraction of the cell that is traversed by a point placed in the cell.

To sample properties from the fine grid to the coarse grid we go through the following steps:

1. Determine the logical coordinates of each corner of the grid block.

2. Divide the grid block into a set of by by microcells. 3. Interpolate the logical coordinates of the corners of the grid block onto the center of each of the microcells inside the

grid block.

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4. Determine the integer part of the logical value at the center of each microcell. 5. Look up the properties at the calculated integer values.

This procedure has two advantages. One is speed: we only need to do a search for the coarse grid block corners. The microcell properties are fast array lookups. The second advantage is accuracy: layering patterns are often honored, even when the fine model layers of the fine grid are outside the conventional boundary of the coarse grid block. If this seems paradoxical think of the procedure as a definition of the fine model grid cells that are associated with a particular coarse grid block. Note that although the pattern of the fine model properties is honored we may not honor the geometry. This is inevitable as we cannot exactly honor a general grid with a structured grid. Figure 1 and Figure 2 illustrate the property assignment method using logical coordinate interpolation.

Figure 1. Grid block boundaries do not follow the detailed fine grid

Figure 2. Logical coordinate interpolation property assignment honors layers

We now provide further information about steps 1-3 of sampling the properties.

Determining the Logical Coordinates of the Grid Block Corners

Given a particular coarse grid block, each corner of the coarse block is tested against the fine grid to determine the fine grid

cell that is occupied by the corner. Denote this fine cell by . On the unit cube, define a local coordinate system where each of the variables lies between 0 and 1. By assuming a trilinear mapping between the unit cube and the fine cell (see

Trilinear mapping), we determine the value of that is at the same point as the corner of the coarse grid point. We then

add this value to the value to provide the value of the logical index : , , .

Examples

1. If a coarse grid block corner is at the center of fine grid cell (2,4,3) then its value would be (2.5, 4.5, 3.5). 2. If the corner happens to be aerially in the center of fine grid cell (2,1,3) and three-quarters of the way into the fine cell

in the k-direction, then the values are (2.5, 1.5, 3.75).

Trilinear mapping

We define the deformation of the unit cube to an actual simulation block by the trilinear interpolation formula:

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Points 1-4 are on the top, points 5-8 on the base. These labels are as shown in Figure 3.

Figure 3. Point labels on reference unit cube and simulation block

Constructing the Micro Grid

Each grid block is assumed defined by its eight corner points. An by by microgrid is constructed inside the grid

block by using linear interpolation from the corner values. To determine , and we set (refer to Figure 3):

Figure 4. Microcells in reference unit cube (left) and in actual coarse grid block (right)

Interpolating the Logical Coordinates of the Grid Block to the Microcell Center

The corner point logical values of the grid block are interpolated to the center of each microcell using trilinear interpolation:

where are logical coordinates in the grid block and each range from 0 to 1.


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upscaling - hurras.oilprocessing.net

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