The following keywords are additional required keywords to be included in your data set when you use the rock compaction model option. These keywords must be located in the Reservoir Description section (see the keywords *CROCKTYPE, *CROCKTAB,
*CROCKTABH, *CCPOR, *CCPRPOR, *CIRREVERS, and *CTYPE for information).
It is always necessary to define both *CPOR and *PRPOR when the analytical aquifer model is used. The aquifer compressibility is not defined by any of the options available on the keywords following *CROCKTYPE, only *CPOR defines aquifer compressibility.
When a region of the reservoir is not explicitly assigned a value of *CTYPE while other regions are assigned *CTYPE values, it is assumed the unassigned region’s compressibility and reference pressure are defined by the *CPOR and *CPRPOR keywords.
Please note that the *INT” option used with input tables is not operational with the
“*CROCKTAB” and “*CROCKTABH” tables.
You need:
a) For the case of constant rock compressibility which varies by *CTYPE region
*CROCKTYPE
b) For the case of reversible nonconstant rock compressibility or the case when permeability varies with pressure. (This option can only be applied on single porosity, dual porosity and dual permeability models).
*CROCKTYPE
**pressure porosity multiplier multiplier (hor.) 1000 1.002956 1.014445 2000 1.005956 1.029286 3000 1.008956 1.044309 4000 1.011956 1.059518
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The above porosity multipliers are computed by using the following equation:
(1)
p is pressure on the table.
and permeability multipliers are determined by the modified equation (2) (Espinoza, C.E. “A New Formulation for Numerical Simulation of Compaction, Sensitivity Studies for Steam Injection”, SPE 12246, 1983.):
por.mult = porosity multiplier
por_input = initial porosity which equals 0.3 in the examples
m is an adjustable parameter that depends on the rock type. Assuming its value to be 4 in the examples.
In the example above because separate horizontal and vertical permeability multipliers were not entered, vertical permeability multipliers default to horizontal permeability multipliers.
or
c) For the case of irreversible rock compressibility and permeability multipliers.
(This option can only be applied on single porosity, dual porosity and dual permeability models).
**pressure porosity multiplier multiplier (hor.) 1000 1.002956 1.014445 2000 1.005956 1.029286 3000 1.008956 1.044309 4000 1.011956 1.059518
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or
d) For the case of rock compressibility and permeability multipliers which undergo hysteresis. (This option can only be applied on single porosity, dual porosity and dual permeability models).
**pressure porosity multiplier multiplier (hor.) 1000 1.002956 1.014445
The above porosity multipliers on hysteresis branches are also computed on the basis of equation (3) as below:
Pr is rebound pressure (first pressure on a hysteresis branch) P is pressure on the table
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A graph of hysteresis branches based on the above data is shown below.
1.002 1.004 1.006 1.008 1.01 1.012 1.014
0 1000 2000 3000 4000 5000
pressure (psi) porosity multiplier rebound pressure
and
1.01 1.02 1.03 1.04 1.05 1.06 1.07
0 1000 2000 3000 4000 5000
pressure (psi)
permeability multiplier
Note that the first point in all the hysteresis tables must match exactly a point in the main table. Failure to ensure this is true could cause numerical difficulties.
When the *CTROCKTAB table is quite nonlinear it is important to define a sufficient number of hysteresis branches using *CROCKTABH. If the *CROCKTABH data is sparse it is possible that there will be a porosity discontinuity at the rebound pressure. This can be minimized by increasing the number of *CROCKTABH tables and can be eliminated entirely by defining a *CROCKTABH table for every pressure point in the main table.
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e) When more than one rock type are used
*CTYPE
For example: A 10x10x4 grid blocks reservoir consists of 4 layers and each layer has one rock type except the first layer which has two rock types. Distribution of rock types on grid blocks of the reservoir is given as follows assuming 4 crocktypes:
*CTYPE 1:5 1:10 1 1
*CTYPE 6:10 1:10 1 2
*CTYPE 1:10 1:10 2 2
*CTYPE 1:10 1:10 3 3
*CTYPE 1:10 1:10 4 4
When the rock dilation model option is used, a number of additional keywords should be included in a data set. Those keywords must be located in the RESERVOIR DESCRIPTION section (see the keywords *CROCKTABE, *CROCKTABD, *CROCKTABU,
*CROCKTABC, *CROCKTABR for more details).
It is also noted that those above keywords can not be combined with any other keywords of the rock compaction model (*CROCKTAB, *CROCKTABH and *CIRREVERS) and vice versa for one rock type.
In one reservoir, rock compaction model and rock dilation model can not be used simultaneously. However, if there are more than one *CROCKTYPE (reservoir) in a simulation, each reservoir can use one of the rock models if those *CROCKTYPE's (reservoirs) are not in contact to each other.
Similar to the rock compaction model, it is always necessary to define both *CPOR and
*PRPOR when the analytical aquifer model is used.
*CROCKTYPE and *CTYPE keywords are also used as in the case of the rock compaction model to define rock type numbers and rock type regions.
A following sample data for rock dilation model that is based on formulae by Beattle, C.I, Boberg, T.C. and McNab, G.S.: "Reservoir Simulation of Cyclic Steam Stimulation in the Cold Lake Oil Sands", SPE Reservoir Engineering, May 1991, pp. 200-206 and STARS User’s Guide Version 98 is under keywords *CROCKTABE, *CROCKTABD, *CROCKTABU,
*CROCKTABC and *CROCKTABR.
Parameters that were used in the formulae to create the data are:
pbase = 14.7 psi Initial pressure at which elasticity begins pdila = 400.0 psi Pressure at which dilation begins
ppact = 200.0 psi Pressure at which recompaction begins crd = 0.0007 (1/psi) Dilation rock compressibility
cp = 0.0001 (1/psi) Elastic rock compressibility
fr = 0.1 Residual dilation fraction
phi0 = 0.3 Initial porosity
pmin = 0.001 psi Minimum pressure that recompaction ends
m = 1.0 An adjustable parameter used in permeability
multiplier formula
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Please see the below figure for locations of the above corresponding pressures.
Pressure
A Reservoir Rock Dilation Model Based on Formulae
Also noted that when using the formulae to create data for rock dilation model, the elastic path, unloading path and reloading path have the same value of rock compressibility.
*CROCKTABE
** PRESSURE POR_MULT HOR_PERM_MULT VER_PERM_MULT 14.7000 1.0000000 1.0000000 1.0000000 Porosity multipliers in the *CRCOCKTABE table are computed by a formula:
por_mult = exp(cp*(p – pbase)) (1) where:
p (psi) is pressure in the first column of *CROCKTABE table.
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Horizontal permeability multipliers are given by Espinoza, C.E.: "A New Formulation for Numerical Simulation of Compaction, Sensitivity Studies for Steam Injection", SPE 12246, 1983.
hor_perm_mult = [(por_mult)**m]*[((1/phi0 – 1)/
(1/phi0 – por_mult))**2] (2)
*CROCKTABD
** PRESSURE POR_MULT HOR_PERM_MULT VER_PERM_MULT 400.0000 1.0392819 1.0751787 1.0000000 Porosity multipliers in the *CROCKTABD table are given by:
por_mult = exp(cp*(pdila-pbase))*exp(crd*(pd(i) – pdila)) (3) where:
pu(i) is pressure at i row in the *CROCKTABU table.
Horizontal permeability multipliers are computed by using the formulae (2).
*CROCKTABU
** PRESSURE POR_MULT HOR_PERM_MULT VER_PERM_MULT 700.0000 1.2821393 1.6591104 1.0000000 Porosity multipliers in this *CROCKTABU table are computed by:
por_mult = exp(cp*(pdila - pbase))*exp(crd*(p – pdila))*
exp(cp*(pu(i) – pu(1))) (4) where:
pu(i) is pressure at i row in the *CROCKTABU table.
pu(1) is the first pressure in the *CROCKTABU table.
In the equation (4), unloading path has the same rock compressibility as that of the elastic path.
Horizontal permeability multipliers are computed similarly by using the formulae (2).
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*CROCKTABC
** PRESSURE POR_MULT HOR_PERM_MULT VER_PERM_MULT 200.0000 1.2196086 1.4861998 1.0000000 150.0003 1.1671184 1.3541457 1.0000000 100.0005 1.1168873 1.2377947 1.0000000 50.0008 1.0688181 1.1347675 1.0000000 0.0010 1.0228177 1.0431193 1.0000000 Porosity multipliers in the *CROCKTABC table are computed by:
phii = phi0*exp(-cp*pbase) + fr*(phi0*exp(cp*(pdila-pbase)))*
(exp(crd*(pu(1)-pdila)) – 1) (5) cr = (ln(phii/ (phi0*exp(cp*(pdila-pbase))*
(exp(crd*pu(1)-pdila)))-cp*(ppact-pu(1)))/ppact (6) por_mult = phii * exp(-cr*pc(i))/phi0 (7)
where:
pc(i) is recompaction pressure which is determined by dividing equally interval pressure (0.001, ppact) by a number of intervals such as 5 as in this example.
Horizontal permeability multipliers were obtained by using equation (2).
*CROCKTABR
** PRESSURE POR_MULT HOR_PERM_MULT VER_PERM_MULT 0.0010 1.0228177 1.0431193 1.0000000 In the above table, porosity multiplier is computed as follows:
i) Pressure at the intersection point between reloading path and dilation path:
Assuming that the reloading path has the same rock compressibility as that of the elastic path. Pressure at the intersection point between reloading path and dilation path is computed as:
pend = (ln(phi0*exp(cp*(pdila – pbase))/por_cend) + cp*pc_end – crd*pdila)/(cp – crd) (8) where:
pend : pressure at the intersection point
por_cend : porosity at the last pressure pc_end in the
*CROCKTABC table
pc_end : last pressure in the *CROCKTABC table (= 0.001 psi as shown above)
Reloading pressure pl(i) in the CROCKTABR table is determined by dividing the pressure interval (0.001,pend) by a number of intervals such as 6 as in the example.
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ii) Porosity multiplier is obtained as:
por_mult = por_cend*exp(cp*(pl(i) – pc_end)))/phi0 (9)
In the equation (9), reloading path has the same rock compressibility as that of the elastic path.
Horizontal permeability multipliers in the *CROCKTABR table are computed through equation (2).
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