4.
EXERCISES
This chapter contains two types of upscaling exercises: single-phase and two-phase.
In both cases, the equations to calculate the pressures have already been solved and
output, along with inter-block flows and additional data, to Excel Spreadsheets. The
reader is required to use the spreadsheets to calculate the effective permeabilities, and
pseudo relative permeabilities.
4.1 Single-Phase Upscaling Exercise
The aim of this exercise is to calculate effective permeability for a small, 2D,
stochastic model using a) no-flow and b) periodic boundary conditions. The model is
show in Figure 1. Information about the model is shown in table 1, and also in the
Excel Spreadsheets called “no-flow.xls” and “periodic.xls”.
k (mD)
10 18
31 55
97 170 300
Figure 1
The stochastic model for the single-phase upscaling exercise.
Variable
No. of grid cells in the x-direction
No. of grid cells in the y-direction
Cell size in the x-direction
Cell size in the y-direction
Cell size in the z-direction
Pressure in centre of 1st column
Pressure in centre of last column
Viscosity
Inter-block flow rates
Permeabilities
Value
8
8
1
1
1
2
1
1
various
various
Units
cm
cm
cm
atm
atm
cP
cm3/s
mD
Table 1
Stochastic model details.
Note that for Darcy’s Law, the permeability should be in D, not mD. However, for
this steady-state flow example, it does not matter.
The pressure equations have already been solved using both no-flow and periodic
boundary conditions, and the results are listed in the spreadsheets, along with the
inter-block flows.
(The permeabilities and the harmonic averages of the
permeabilities have also been output for interest.) The following steps are required to
calculate the effective permeabilities:
a) No-Flow Boundary Conditions
Start by calculating the effective permeability in the x-direction, using the worksheet
“x-dir”.
 Calculate the net pressure gradient in the x-direction. Note that the pressures
have been fixed in centres of the first and last columns of the model, and that
the distance between these points if 7 cm.
 Calculate the total flow between two columns. (It does not matter which two
columns you use because the flow is constant.)
 Calculate the effective permeability using Darcy’s law.
Repeat for the calculation in the y-direction, using worksheet “y-dir”.
b) Periodic Boundary Conditions
Use the worksheet “x-dir” to calculate the kxx and kyx terms, which describe the flow
in the x- and y-directions respectively, due to a pressure gradient in the x-direction.
 Calculate the next pressure gradient in the x-direction. This is actually just the
applied pressure gradient (which equals 1) divided by the length of the model,
which is 8 cm.
 Calculate the total flows in the x- and y-directions.
 Use Darcy’s law to calculate kxx and kyx.
Repeat these calculations for the case with the net pressure gradient in the y-direction,
to obtain the kxy and kyy terms of the effective permeability tensor.
The results are given in spreadsheets no_flow_results.xls and periodic_results.xls.
4.2 Two-Phase Upscaling Exercise
The aim of this exercise is to calculate pseudo relative permeabilities and capillary
pressures using the Kyte and Berry (1975), the Stone (1991) and the PVW (Eclipse
Manual) methods. Use the spreadsheets kyte_berry.xls, stone.xls and pvw.xls. Since
pseudos take a long to calculate “manually”, you are only asked to calculate the
pseudos at one time (one average water saturation value).
Figure 2 shows the model, and Table 2 gives the model details. These details are also
listed in the first worksheet of the spread sheets. The input (“rock”) relative
permeabilities are presented in the second worksheet (for interest).
Calculate pseudos
for this block
k (mD)
70 80
90 100 110 120 130 140
Figure 2
Model for the two-phase upscaling exercise.
Variable
Total no. of cells in the x-direction
Total no. of cells in the z-direction
Cell size in the x-direction
Cell size in the y-direction
Cell size in the z-direction
Permeabilities
Size of coarse block in x-dir. (5 cells)
Size of coarse block in y-dir.
Size of coarse block in z-dir. (5 cells)
Porosity (constant)
Datum depth
Tops (constant)
Oil viscosity
Oil density
Water viscosity
Water density
Injection rate (into column 1)
BHP at producer (column 17)
Table 2
Value
17
5
5
5
5
various
25
5
25
0.2
1000
1000
5
0.876
0.44
1.000
31.75
400
Units
cm
cm
cm
mD
cm
cm
cm
cm
cm
cP
gm/cm3
cP
gm/cm3
cm3/hr
atm
Model details for two-phase exercise. Note that this exercise uses Lab units. The fluids and rock were
assumed to be incompressible.
The spreadsheets kyte_berry.xls, stone.xls and pvw.xls contain the results of an
Eclipse fine-scale flow simulation (a water flood), after 10 hours (injection of 0.15
PV). The following properties are listed in work
Water saturation
A fine-scale simulation of a waterflood was performed in Eclipse. The worksheets
labelled, “10 hours” list the properties obtained after 10 hours (after the injection of
0.15 PV), namely:
PERMS – permeabilities (which are, of course, constant)
PRESSURE – oil pressures
SWAT – water saturations
KRO – oil relative permeabilities
KRW – water relative permeabilities
FLOOILK+ – interblock flows of oil in the K+ direction (downwards)
FLOOILI+ – interblock flows of oil in the I+ direction (left to right)
FLOWATK+ – interblock flows of water in the K+ direction (downwards)
FLOOWATI+ – interblock flows of water in the I+ direction (left to right)
OIL-POT – oil potentials
WAT-POT – water potentials
The calculations should be performed in stages, and separate worksheets are provided
for each stage, with the required information. Use the equations in Section 2.6.
The Kyte and Berry Calculation
Worksheet perms
Calculate the effective permeability. Use the Kyte and Berry method here – i.e. take
the arithmetic average in each column and the harmonica average of all the columns.
(This is not as accurate as using a numerical simulation, but is useful for a quick
estimate.)
Worksheet swat
Calculate the average water saturation of the central coarse block. This is a straightforward average.
Worksheet flows
Calculate the total flow of oil and water out of the down-stream side of the coarse
block.
Worksheet poto
Calculate the average oil potential in the centre of coarse blocks 2 and 3 (as indicated
by the shading). Use (relative permeability x absolute permeability) weighting. Note
that we are averaging potential calculated at datum depth, whereas Kyte and Berry
calculate the pressures at the coarse block centre. However, there is a constant
difference between these two values, and this cancels when calculating the potential
difference. For calculating the average capillary pressure, calculate the pore-volume
weighted potential in the whole coarse of the central coarse block (shaded pink and
green).
Also calculate the average oil pressure using pore-volume weighting. This will be
used to calculate the coarse-scale capillary pressure. You need to correct the
potentials (calculated at the datum depth) to the centre of the coarse cell.
Worksheet potw
Calculate the average water potential in coarse blocks 2 and 3, using the same method
as for the oil. Also calculate the pore-volume weighted average for calculating the
capillary pressure.
Worksheet results
Calculate the pseudo rel perms by applying Darcy’s law at the coarse scale. Also
calculate the capillary pressure from average oil and water pressures.
The Stone Calculations (not now covered in the course)
The average water saturation and the total flows are calculated as in the Kyte and
Berry Method.
Worksheet mob
Calculate the transmissibility at the down-stream edge of the central coarse block
(using the harmonic average to obtain the average permeabilities). Calculate the total
mobility in each of these cells in column 11. Then calculate the average mobility,
using transmissibility weighting.
Worksheet results
Calculate the pseudo rel perms directly from the average fractional flow and the
average mobility. The Stone method ignores capillary pressure. However, if using
the Stone option in the Eclipse Pseudo Package, the capillary pressure using the same
method as for the Kyte and Berry option.
The PVW Calculations
These are similar to the Kyte and Berry calculations, except pore-volume weighting is
used in the calculation of the average pressure for calculating the relative
permeabilities.
The answers for these exercises are included in spread sheets kyte_berry_results.xls,
stone_results.xls and pvw_results.xls. The files also show the full pseudo curves
calculated using the Eclipse Pseudo package.