## Recommendation

When simulating hydraulically fractured completions if you intend to have generally a bigger fracture cell size than well bore diameter (or more precisely bigger than the Peaceman radius) then we recommend creating a perm log along the well so that the permeability used in calculating the connection factors is the permeability of the propped fracture and not the upscaled value for the grid-cell.

#### Authors

#### Hadi Parvizi, Raj Malpani, Michael John Williams

Teesside University

In case the fracture cell is small (smaller than Peaceman radius) then we recommend setting the diameter of the wellbore in the global completion to a small value so that the Peaceman radius will not be smaller than the wellbore radius.

If running ECLIPSE Compositional it is possible to use the Fractured Completion, which will give similar results to the approach outlined here. For details of the Fractured Completion consult the ECLIPSE Technical Description.

If using PETREL EXPLICIT MODELING OF HYDRAULIC FRACTURING COMPLETION (with LGR) then as Petrel Online manual says:

“Hydraulic fractures that have the Correlation option switched off can be modeled in the simulator using explicitly calculated **transmissibility multipliers on the faces of the cells the fracture intersects**. In this case the fracture is assumed to lie in a single I or J plane of cells (whichever direction best approximates the true geometric orientation of the fracture). The fracture can be surrounded by a logarithmically spaced LGR using the Grid to Hydraulic Fractures option on the Make local grid process. The LGR is symmetrically placed with the plane of host cells. **For completeness a pore volume multiplier is used to reduce the volume of the central plane of LGR cells nominally intersecting the fracture** (usually, the grid size is far larger than the fracture thickness).

The well connection is calculated using the formula for linear flow into an infinite conducting fracture (the FX/FY parameters in Option 13 of the COMPDAT keyword), rather than the more conventional Peaceman formula for radial inflow. When the well is completed in an LGR the FX/FY options can lead to **unreasonably high connection factors**. The reason for this is that **the formula assumes that the linear flow extends 1/4 of the cell size perpendicular to the fracture**. This is reasonable for the host cell, but not for the innermost LGR cells. In this case the **small LGR cell dimension is replaced by the larger when evaluating the connection factor**. It is recommended that for detailed fracture modeling, the results of the Post Fracture test is used **to history match a multiplier on the well PI**. For Fracture design the results can be compared to a model without LGRs that uses the built in Fracture Correlation – or indeed the conventional negative skin approach if that is your preference.”

## Outline

When modeling fractured completions using explicit high-perm cells to represented the fracture, it is found that the well connection factors generated by Petrel produce simulations that are highly sensitive to the cell-size. The **Problem Statement** section provides an overview of the reasons for this behavior and how these can be largely overcome by following the **Recommendation**. **Upscaling and downscaling well connections** provides a general treatment of the relationship between the connection factors calculated at different cell-sizes when upscaling a heterogeneous reservoir; it is then shown how this approach can provide consistent answers even below the conventional domain of validity for the Peaceman calculation.

## Problem statement

Assume a scenario of evaluation of the performance of a well completed in a cell with fracture.

(Figure1)

Now there are two options:

a) The fracture cell width is less than Peaceman radius:

(Figure 2)

Peaceman radius (r_{o}):

Which is used in transmissibility calculation of the well:

where

And

K_{h} is the effective permeability times net thickness of the connection. For a vertical well the permeability used here is the geometric mean of the x- and y- direction permeabilities, K= (K_{x}K_{y})^{1/2}.

θ is the angle of the segment connecting with the well, in radius. In a Cartesian grid its value is 6.2832 (= 2π), as the connection is assumed to be in the center of the grid block. For wells located on an edge (or a corner) of a Cartesian grid block, use keyword WPIMULT after COMPDAT to scale the resulting connection factors by 0.5 (or 0.25).

Having a look at the ln(r_{o}/r_{w}) function shows:

Now if Skin=0 and ro/rw would be less than or equal to 1, then the denominator of the transmissibility formula will be equal to zero or a negative value which in both of the cases; the pre-processor (Petrel) cannot export this connection factor-CF and the connection will be ignored. Note that that the Peaceman approach defines ro as the distance at which the reservoir pressure, for radial inflow, matches the well BHP – and so it makes no physical sense to have r_{o}<r_{w} and Petrel is correct in not exporting the negative values. The specific problem here is to allow a ‘reasonable’ calculation of a well connection factor to be made even when the Peaceman approach does not apply.

In order to overcome this issue, one can overwrite the diameter of wellbore in the settings of the global completion to make sure that the CF would be exported. If the well is cased, this value would be used for diameter of the perforations and has no effect on the casing diameter.

Note: If the user is using well deliverability feature of the Petrel (available on Petrel 2012.2 onwards), this might cause problem if the well has openhole intervals.

QC: a very useful QC and validation is to plot WBHP-well bottom hole pressure versus WBP-well block pressure. There should be a minimal difference for tiny frac’ed cells. You may use the PI mulitipliers to make sure the tiny completed cells are not controlling the flow into the wellbore rather than the immediate adjacent cells to the completed cells. On the other hand using very multiplier value might affect the performance of the simulator.

If there is a difference between WBHP and WBP, you are probably under-estimating the production.

b) The fracture cell width is greater than Peaceman radius:

(Figure 3)

If there is a high contrast between the matrix permeability and the fracture (which is likely) extra care should be taken about the Kh term that simulator is using by default in the transmissibility calculation for well connection factor.

The Kh should represent the conductivity at sandface (immediately adjacent to the wellbore) rather than the average of the area around the wellbore. By default, if the user does not provide any perm log, Petrel is using the completion cell perm to calculate Kh and consequently CF. So have a look at the completed fracture cell and the perm reduction due to the upscaling in the below table in which the background (matrix) perm is 0.0001 md and the fracture perm is 100 md.

The result of this option again would be underestimation of CF unless a well perm log would be provided for CF calculation.

The below is a table which shows some sensitivity analysis on fracture cell with for transmissibility calculation of Structured and Unstructured grids:

## Upscaling and downscaling well connections

### Skin for equivalence of well connections

ECLIPSE will calculate a well connection factor based on radial inflow from the grid-block containing the connection. For successively coarser grids this radius will be larger, the permeability and cell dimensions will differ – the aim is to find an adjustment to the connection factor to accommodate this coarsening.

Consider a simplified radial inflow (single phase to a vertical well), the pressure equation will be:

ECLIPSE calculates the well connectivity at the equivalent (Peaceman) radius for the cell, r_{0}.

In the next size up, represent the flow by the radial inflow equation for the (bigger) cell, including a modified skin term, S_{1}, to represent the flow through the (possibly different) permeability of the cell in the higher resolution.

Equate these at the Peaceman radius for the cell in the higher resolution (r_{0}) to define a value for S_{1}:

So, we have, for radial flow from the larger cell into the well,

Consider the next resolution coarser. The radial inflow will be:

This can be equated at the Peaceman radius for the cell in the higher resolution (r_{1}), giving:

Repeating for successively coarser cells gives us the following general relation for this effective skin:

Where n represents the resolution of the outer grid. This skin represents the successively finer grids below as concentric circles of different dimensions and permeability that adjust the radial inflow. As this radial inflow is always on a scale less than the Peaceman effective radius for the grid cell under consideration, it makes sense to include it as a skin term.

Finally, define r_{0} at the smallest grid resolution such that it is the smallest DX and DY for which

This method is also a method for upscaling the skin calculated via a welltest to the larger grid-cells used in a full-field model.

The example below comes from simulation on the 9^{th} SPE comparison case.

**Figure 4** Areal resolution 240×240 ft, with (red) and without (green) the skin calculation. The thick line is a simulation on 7.5×7.5 ft cells.

### Connection factors below the Peaceman validity

The Peaceman relation is not valid if r_{0} extends outside the cell boundaries or if r_{0}/r_{w} ≤ 1.

We can construct an estimate of well inflow for very fine grids, by assuming that the upscaling approximation above is reasonable, and then attempt to estimate ln(r_{0}/r_{w}) directly.

If we use just a single level recursion, to make the argument clear, we had:

We also know the Peaceman relation for the smallest scale cell is (ignoring orientation effects):

We re-arrange the first of these relations so that it gives an estimate of ln(r_{0}/r_{w}):

From this, and the knowledge that we are upscaling from a high permeability fracture cell to a low permeability upscaled cell, which means k_{0}>>k_{1}. From this it is clear that the denominator is always negative for the case of explicit fractures, and the transmissibility is positive only in the case where the numerator is positive.

This is okay, because we can consider the large scale effective skin of the well (i.e. the effect of the high perm fracture as a skin), this is negative provided the well produces better with the fracture than it would produce without (we are only interested in this case!). Conversely the small scale skin (since the high perm is modeled explicitly this small scale skin is inside the fracture and only reflects damage (e.g. fines) within the fracture) is zero at best, or may be positive.

For simple general application we will assume that the skin in the small scale case is zero, the well connection factor transmissibility equation could be replaced by:

Where k_{1} and h_{1} are the upscaled values (upscaled across high perm frac and low perm formation) and S_{1} is the ‘macroscopic’ skin of the fractured well. The values k_{0} and h_{0} correspond to the values in the explicit fracture representation in small grid cells.

This leaves estimation of the macroscopic skin, S1, as an unknown. However, since we are using the Peaceman model, based on radial inflow at the far-field, it seems appropriate to use the equivalent skin at pseudo-radial flow for the fracture to define S1.

The equivalent skin at pseudo-radial flow for the fracture can be estimated as:

S1 = -ln(C_{f}/(4*k*r_{w}))

Here C_{f} = frac. Conductivity in md-ft, k=matrix permeability, r_{w}=well radius. This relation comes from the pseudo-radial inflow model from the transient model of Cino et al., SPE 6014-PA. The approximation is discussed in relation to fracture interpretation in chapter 12 of Reservoir Stimulation 3^{rd} edition (on page 475).

Testing on a simple example

For the purposes of a comparison test, we make the following assumption about the ‘real’ fracture – 1 inch width (0.08333 ft) and 200 md-ft conductivity and 30% porosity. The background reservoir had 1e-4 md permeability and 5% porosity.

We created the same simulation grid (with the Toriad prototype gridder), using the following cell widths (in feet) representing the hydraulic fracture:

0.022, 0.03, 0.04 0.05, 0.06, 0.07, 0.08, 0.1, 0.2, 0.3, 0.5, 1.0, 2.0, 3.0, 4.0.

So there are examples that are both thinner and wider than the assumed ‘real’ fracture. The smallest cell width tried is the limiting case where the fracture porosity goes to 100%.

Figure 5 is the linear-time plot of instantaneous and cumulative rate, Figure 6 is the log-time plot. Cumulative is in black. The reason why there is a bifurcation and an asymptotic approach to both a lower estimate and an upper estimate is because we always set the permeability within the fracture equal to the fracture permeability (i.e. we have (md-ft/cell_width) for permeability in the cross-fracture direction and (md-ft/frac_width) in the other direction for all cases where cell_width>frac_width). This gives a better approximation of the history period (first 2 or 3 years). If you were to simply use homogeneous permeability at (md-ft/cell_width), the ultimate recovery is probably better estimated but your history matching period will have lower rates and so any matching on the coarse model will over-estimate the reservoir parameters.

The log-time plot (Figure 6) shows that if you increase the cell width you will get slightly different flow periods and transitions.

**Figure 5** Gas Production cumulative and rate.

**Figure 6** Gas production cumulative and rate on a log scale.

For timings – these models are 26x15x15 and run in the following CPU times (simply read from the end of the PRT file – there were no multiple tests to get accurate timings). The models were run with 0.5 day maximum timestep in the ‘history period’ which was designated as 2009 to January 2012; and then with a 10 day maximum timestep in the ‘forecast period’. No models failed to converge, the cases with the fracture-cells narrower than 0.3 ft had linear solver convergence difficulties with the ‘forecast period’ timestep target of 10 days and this is reflected in the longer run times for those cases.