Polymer Flood
Polymer Flood
Process Manual
Process Manual
Revision 0I
Revision 0I
By Computer Modelling Group Ltd.
By Computer Modelling Group Ltd.
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Contents
Contents
Introduction
3
Introduction
3
Purpose ... 3 Purpose ... 3 Organization ... 3 Organization ... 3Polymer Flood Process ... 3
Polymer Flood Process ... 3
Polymer
Polymer Flood
Flood
5
5
Introduc Introduction tion ... ... 55 Theoretical Phenomena ... 5Theoretical Phenomena ... 5
Polymer Polymer AdsorptiAdsorption...on... . 66 Permeability Reduction ... 7
Permeability Reduction ... 7
Inaccessible Pore Volume (IPV) ... 10
Inaccessible Pore Volume (IPV) ... 10
Relative Permeability/Wettability Alteration Effects ... 11
Relative Permeability/Wettability Alteration Effects ... 11
Polymer Degradation Polymer Degradation ... ... 1414 Composition-Dependent Viscosity Effects ... 17
Composition-Dependent Viscosity Effects ... 17
Shear-Dependent Viscosity Effects ... 22
Shear-Dependent Viscosity Effects ... 22
Power-Law Expression for Shear-Thinning or Pseudoplastic Fluids ... 26
Power-Law Expression for Shear-Thinning or Pseudoplastic Fluids ... 26
Power-Law Expression for Shear-Thickening or Dilatant Fluids ... 27
Power-Law Expression for Shear-Thickening or Dilatant Fluids ... 27
Tabular Input Option for Velocity/Shear-Rate-Dependent Viscosity ... 30
Tabular Input Option for Velocity/Shear-Rate-Dependent Viscosity ... 30
Velocity Dependent Skin Factor ... 31
Velocity Dependent Skin Factor ... 31
Convert Shear Rates to Velocities ... 32
Convert Shear Rates to Velocities ... 32
Salinity-Depen Salinity-Dependent Viscosity Efdent Viscosity Effects fects ... ... 3333 Lab and Field Information ... 34
Lab and Field Information ... 34
Using the Process Wizard to Model a Polymer Flood ... 35
Using the Process Wizard to Model a Polymer Flood ... 35
Viewing and Adjusting the Process Wizard Results ... 39
Viewing and Adjusting the Process Wizard Results ... 39
Components Generation Components Generation ... ... 3939 Polymer Consumptio Polymer Consumption Reaction n Reaction ... ... 3939 Polymer Polymer AdsorptioAdsorption...n... ... 4040 Polymer Adsorption Table ... 40
Polymer Adsorption Table ... 40
Langmuir Isotherm Option ... Langmuir Isotherm Option ... 4040 Polymer Viscosity ... 41
Polymer Viscosity ... 41
Other Considerations and Troubleshooting Information ... 42
Other Considerations and Troubleshooting Information ... 42
High Molecular Weight (or Low Mole Fractions) ... 42
High Molecular Weight (or Low Mole Fractions) ... 42
Disproportionate Permeability Reduction Effect ... 43
Disproportionate Permeability Reduction Effect ... 43
Interpreting Polymer Flood Model Outputs... 44
Interpreting Polymer Flood Model Outputs... 44
Appendix
Appendix A
A –
– Equations
Equations
45
45
Introduc
Introduction tion ... ... 4545 General
Parts-per-million (ppm) ... 45
Weight Percentage ... 45
Mass Fraction (wt) ... 45
Mole Fraction (dim) ... 45
Polymer Flood Calculations ... 46
Polymer Mass Fraction ... 46
Polymer Mole Fraction ... 46
Volumetric Reaction Rate ... 46
Stoichiometric Coefficients ... 48
Polymer Adsorption Table ... 49
Langmuir Isotherm Option ... 50
Permeability Reduction... 52
Polymer Viscosity ... 53
Appendix B – Conversion of Eclipse Polymer Option to IMEX
64
Example of Eclipse Data ... 64PLYVISC and PLYSHEAR Conversion to IMEX ... 65
PLYROCK Conversion to IMEX ... 66
PLYADS Conversion to IMEX... 66
Appendix C – Conversion of IMEX Polymer Option to STARS
68
Step 1: Builder Convert Simulator Type ... 68Step 2: Process Wizard ... 68
Input Polymer Data ... 68
Input Adsorption Data ... 68
Input Polymer Viscosity... 69
Step 3: Shear Dependent Viscosity ... 69
Appendix D – Example IMEX and STARS Simulations of Polymer
72
Comparisons of IMEX and STARS ... 72Introduction
Purpose
The purpose of this manual is to provide users with the information they need to model polymer flood processes using the CMG STARS simulator.
Organization
The following information is provided:
• Theoretical concepts and how these concepts are represented in the model • Lab and field data required for the simulation
• Procedure for inputting data using the Builder Process Wizard • Procedure for viewing and adjusting the input data
• View the results in Results 3D
• Appendices B, C, and D provide information for converting an Eclipse polymer option to IMEX, and an IMEX polymer option to STARS.
Polymer Flood Process
Polymer flooding, illustrated below, is an EOR (enhanced oil recovery) technique in which water-soluble polymers are added to the injection fluids to increase the viscosity of injected water and/or formation water. This provides mobility control of the fluids, which improves the volumetric sweep efficiency and reduces channeling and water breakthrough, thereby increasing the oil recovery factor.
Figure 1 shows a conceptual comparison of cumulative oil recovery for water, polymer, alkaline/polymer, and ASP flooding techniques.
Cumulative Oil
Pore Volume Injected
ASP Flood
Alkaline/Polymer Flood
Polymer Flood
Continued Water Flood
Figure 1 Comparison of Chemical Flooding Techniques
Figure 2 illustrates the potential improvement from chemical flood EOR processes:
WaterFlood ChemicalFlood EORPotentialto Extendand Enhance Production OilRate Time
Figure 2 Improved Oil Recovery from Chemical Flood Processes1
Polymer Flood
Introduction
Polymer flooding is a process where a thickening agent (polymer) is added to the injected fluid (typically water) to produce a more favorable mobility ratio between the injected fluid and the displaced oil. Polymers are macromolecules composed of repetitive units called monomers. Some common polymers are hydrolyzed polyacrylamide (HPAM), co-polymers of acrylamide (AMPS, NVP) and xanthan gum (biopolymer).
The polymer flood mechanisms that can be modeled in STARS include: • Viscosity and mobility variations of the injected fluid
• Polymer Adsorption • Permeability Reduction
• Inaccessible Pore Volume (IPV)
• Relative Permeability/Wettability Alteration Effects • Polymer Degradation
• Composition-Dependent Viscosity Effects • Shear-Dependent Viscosity Effects
• Salinity-Dependent Viscosity Effects
Details about modeling these mechanisms are outlined in Theoretical Phenomena. Modeling a polymer flood requires, as a minimum, the viscosity of the water-polymer solution at different polymer concentrations. Other data, such as polymer adsorption,
degradation, and rheology, while not required, will yield a more accurate model. Refer to Lab and Field Information on page 34. As well, refer to Other Considerations and
Troubleshooting Information on page 42 for information about specific polymer issues that you may need to address or resolve.
Theoretical Phenomena
The following sections describe polymer flood process phenomena and, at a high level, how they are modeled in STARS. For further information, refer to the STARS User’s Guide.
Polymer Adsorption
Adsorption is the adhesion of ions or molecules onto the surface of another phase 2. It is a physical and/or chemical process by which a porous solid (at the microscopic level), for
example, is capable of retaining particles of a fluid on its surface after being in contact with it. Polymer adsorption in an EOR process is related to the amount of polymer retained in the smallest porous spaces or on the rock surface where the solution has passed. The adsorption levels depend on fluid type and concentration, molecular weight, flow rate, temperature, brine salinity, brine hardness and rock type (e.g. rock mineralogy and permeability)3.
Low polymer retention in the reservoir is essential for the success of a polymer EOR operation. A substantial loss of polymer may be detrimental because the polymer
concentration reduction could impact its viscosity and cause a loss of mobility control or low displacement efficiency. For this reason, adsorption is usually estimated from laboratory core flood experiments conducted under conditions as close as possible to those prevailing in the field.
STARS allows a description of these phenomena, through the input of a set of constant
temperature adsorption isotherms (adsorption level as a function of fluid composition). These isotherms can be entered either in tabular form or using the Langmuir isotherm correlation:
(
i)
ic
B
1
A
Ad
×
+
×
=
c (1)where ci is the fluid component composition, and A and B are generally temperature
dependent. Note that the maximum adsorption level associated with the formula is A/B. Coefficient B controls the curvature of the isotherm, and the ratio A/B, as mentioned, determines the plateau value for adsorption. This is illustrated in Figure 3:
Figure 3 Typical Langmuir Isotherm Shape s3
Refer to Polymer Adsorption Table and Langmuir Isotherm Option in Appendix A –
Equations of this manual for more information about the equations used by STARS to model a polymer flood.
2
Ali, L. and Barrufet, M.A., “Profile Modification due to Polymer Adsorption in Reservoir Rocks”. Energy & Fuels Vol. 8, No. 6, (1994), pp.1217-1222.
3 Lake, L.W., “Enhanced Oil Recovery”, Prentice-Hall (1989).
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 A d s o r b e d C o n c e n t r a t i o n 0 2 4 6 8 10 Concentration b increasing a/b constant 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 A d s o r b e d C o n c e n t r a t i o n 0 2 4 6 8 10 Concentration b constant a increasing
Important Keywords for Modeling Polymer Adsorption:
Use keyword *ADSCOMP to specify components and fluid phase.
Use keywords *ADSLANG or *ADSTABLE to specify the input option used for adsorption, as follows:
*ADSLANG Denotes that composition dependence is specified by Langmuir isotherm coefficients.
*ADSTABLE Denotes that composition dependence is specified by a table of adsorption versus composition values.
To define the dependence of the adsorption data on rock type (permeability) for the component/phase specified by *ADSCOMP, use the following keywords:
*ADMAXT Specifies the maximum adsorption capacity. *ADRT Specifies the residual adsorption level. *ADSROCK Specifies the current rock type number.
*ADSTYPE Used to assign multiple adsorption rock type numbers to grid blocks.
Permeability Reduction
Many papers indicate a mobility reduction in the porous media after the polymer has flowed through. This phenomenon occurs due to increased water viscosity and reduction of
permeability4,5, caused in part by polymer adsorption, particularly if it is the chemical or mechanical (entrapment)4 type. The variation in rock permeability due to this process is given by:
K
=
(2) where:k Absolute rock permeability prior to polymer flooding Rk Permeability reduction factor
, a function of polymer adsorption and the residual resistance factor RRF , is given by:
= 1 + (
1) ×
(3)4
Jennings, R. R., Rogers, J.H., and West, T. J., 1971. “Factors Influencing Mobility Control By Polymer Solutions”. J. Pet. Technol., 23(3): 391-401. SPE 2867-PA.
5
Bondor, P.L., Hirasaki, G.J., Tham, M.J., 1972. “Mathematical simulation of polymer flooding in complex reservoirs”. SPEJ (October), 369–382. SPE 3524-PA.
where:
AD Cumulative adsorption of polymer per unit volume of reservoir rock ADMAXT Maximum adsorptive capacity of polymer per unit volume of reservoir
rock
Rk varies from 1.0 to a maximum of
, as adsorption level increases.The residual resistance factor can be obtained from core flooding experiments and can be expressed as the water (or brine) mobility ratio before and after a polymer treatmen t4,6,7.
=
,
,
= (
⁄
),
(
⁄
),
(4) where: λw Water mobilityk w Effective water permeability
μw Water viscosity
As the water viscosity does not change before or after the treatment, equation (4) can be reduced to:
=
,
,
(5)
If the core flooding experiment is linear, the effective permeability can be calculated using the linear expression of Darcy’s equation:
=
×
×
×∆
(6)If the core flooding experiment is radial, then the effective permeability should be calculated as follows:
=
×
(
⁄
) ×
2
×ℎ
×∆
(7)If any of the above expressions are used to calculate the effective permeability before and after the polymer flood in a core flooding experiment and if the injection rate is the same in
6
Chang, H. L., “Polymer Flooding Technology – Yesterday, Today, and Tomorrow”, paper SPE 7043 presented at the 1978 SPE Symposium on Improved Methods for Oil Recovery, Tulsa, April 16-19.
7
Singleton, M. A., Sorbie, K. S., Shields, R. A., “Further Development of the Pore Scale Mechanism of Relative Permeability Modification by Partially Hydrolized Polyacrylamide”, paper SPE 75184, presented at the oil Recovery Symposium, Tulsa, Oklahoma, 2002.
both tests, the residual resistance factor, RRF, can be expressed in terms of the pressure drops, as follows8:
=∆
∆
,
,
(8)
It is typically assumed that only single-phase flow paths are altered by polymers; therefore, the permeability reduction factor for each phase α can be expressed as:
= 1 + (
1) ×
(9)which affects the effective permeability of phase α,
�
, as follows:�
=�
×
=
×
=
(10)where:
α Water, oil or gas phase
,�
Absolute permeability before and after the treatment, respectively.Therefore, to account for permeability reduction in polymer flood simulations or in any other EOR process in which the adsorption of components plays an important role, it is necessary to input the residual resistance factor and the phase to which the resistance factor will be applied.
Use the following keywords to model permeability reduction:
*ADMAXT Specifies the maximum adsorption capacity, as outlined in Polymer Adsorption.
*RRFT Specifies the residual resistance factor for the adsorbing component “i” specified via *ADSCOMP. The value of *RRFT must be greater than or equal to 1. The default value is 1.
*ADSPHBLK With sub-keyword phase_des, overrides the default phase to which the resistance factor calculation is applied, as follows:
*W Water (aqueous) phase *O Oil (oleic) phase *G Gas (gaseous) phase *ALL All phases
8
Zaitoun, A. and Kohler, N., 1988, “Two-phase Flow Through Porous Media: Effect of an Adsorbed Polymer Layer”, paper SPE 18085 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, TX, Oct. 2-5.
DEFAULTS:
*RRFT 1 (no resistance effect)
*ADSPHBLK If not specified, the resistance factor is applied to the phase that is the source of the adsorbing component (specified by *ADSCOMP). Note: Some polymers reduce water permeability more than they reduce hydrocarbon permeability. The result of this phenomenon, referred to as disproportionate permeability
reduction (DPR )7,9,10 is that the residual resistance factor is neither the same nor constant for all phases, so it should be calculated in core flooding experiments for each phase, as shown below:
From water-oil experiments:
=
;
;
(11)
=
;
;
(12) From water-gas experiments:
=
;
;
(13)
=
;
;
(14) where:k α Effective permeability to the phase α, where α = w, o or g
S or Residual oil saturation (water-oil experiment)
S gr Residual gas saturation (water-gas experiment)
S wirr Irreducible water saturation (water-oil and water-gas experiments)
In the current version of STARS, it is only possible to assign a single value of the residual resistance factor. For suggestions on how to simulate the DPR effect, refer to the workaround outlined in Disproportionate Permeability Reduction Effect on page 43.
Inaccessible Pore Volume (IPV)
When the flow of polymer molecules through the porous media is restricted in small pore throats, only the passage of water or brine is possible. When these pores cannot be contacted by flowing polymer molecules, they are referred to as inaccessible pore volume (IPV). This
9
Botermans, C.W., van Batenburg, D.W., and Bruining, J., “Relative Permeability Modifiers: Myth or Reality?” in SPE European Formation Damage Conference. 2001, The Hague, Netherlands.
10
Elmkies, Ph., et al., “Polymer effect on gas/water flow in porous media”, SPE/DOE IOR Symposium, April 2002, SPE 75160.
phenomenon was first reported by Dawson and Lautz (1972)11, who showed that some pore spaces may not be accessible to polymer molecules and that this allows polymer solutions to advance and displace oil at a faster rate than predicted on the basis of total porosity. They concluded that about 30% of the total pore volume may not be accessible to polymer
molecules, and this has been corroborated in recent experiments by Pancharoen, Thiele and Kovscek (2010)12. As a result, the effective porosity for a polymer solution is less than the actual reservoir porosity. A reduced polymer porosity,
, can be used to represent the available pore volume to polymer solutions as follows:
= (1
) ×
(15)where
is the original porosity, adjusted for pressure and temperature of the block. IPV can have beneficial effects on field performance. The rock surface in contact with the polymer solution will be less than the total pore volume, thus decreasing polymer adsorption.More importantly, if connate water is present in the smaller pores that are inaccessible to the polymer, the bank of connate water and polymer-depleted injection water that precedes the polymer bank is reduced by the amount of the inaccessible pore volume. One drawback,
however, is that movable oil located in the smaller pores will not be contacted by the polymer and therefore may not be displaced.
In STARS, the difference (1
) is requested directly, to update the porosity that will be used for the adsorbing component “i” and the adsorption rock type. In the simulator, it is denoted as the accessible pore volume or fraction of available pore volume to polymers or any similar component, and it should be specified by keyword *PORFT. The default value of *PORFT is 1, which means that there is no inaccessible pore volume.Relative Permeability/Wettability Alteration Effects
The effect of the modification of relative permeability by polymer adsorption has been intensively studied by many authors in the past and although they have found evidence of a selective reduction of the relative permeability to water with respect to relative permeability to oil13,14,15, the conventional belief is that polymer flooding does not reduce residual oil saturation on a micro-scale; rather, it allows the undisplaced oil to approach this low level
11
Dawson, R. and Lautz, R., “Inaccessible Pore Volume in Polymer Flooding”, SPE Journal, October 1972.
12
Pancharoen, M., Thiele, M.R., and Kovscek, A.R., “Inaccessible Pore Volume of Associative Polymer Floods”, paper SPE 129910, SPE Improved Oil Recovery Symposium, Tulsa, Oklahoma, 2010.
13 Schneider, N. and Owens, W.W., 1982, “Steady-state measurements of relative permeability for
polymer/oil systems”, paper SPE 9408-PA, Societ y of Petroleum Engineers Journal, 79-86.
14
Barrufet, A., and Ali, L., “Modification of Relative Permeability Curves by Polymer Adsorption”, paper SPE 27015 presented at the 1994 Latin American/Caribbean Petrol eum Engineering Conference,
Buenos Aires, Argentina, April 27-29.
15
Zheng, C. G., Gall, B. L., Gao, H. W., Miller, A. E., and Bryant, R. S., “Effects of Polymer
Adsorption and Flow Behavior on Two-Phase Flow in Porous Media”, paper SPE 39632 presented at the 1998 SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, U.S.A, 19-22 April 1998.
more quickly, while producing less water in the process. In more recent studies16,17,18, however, it can be seen that all types of micro-scale residual oil were reduced after flooding with viscous-elastic polymers, increasing micro-scale displacement efficiency in the cores. Figure 4 and Figure 5 show typical relative permeability curves for water flooding and polymer flooding obtained from previous studies (Wang et al):
0 20 40 60 80 100 WaterSaturation% 0 20 40 60 80 100 R e l a t i v e P e r m e a b i l i t y % Kro Krw Krop Krp fw fp
Figure 4 Comparison of kr-curves of Polymer/Oil and Water/Oil 16
R e l a t i v e P e r m e a b i l i t y % 100 80 60 40 20 0 WaterSaturation% 0 20 40 60 80 100 500mg/L 1500mg/L 2000mg/L
Figure 5 Influence of Different Polymer Concentration on kr-curve s17
16 Wang, D., Cheng, J., Yang, Q., Gong, W., and Li, Q., “Viscous-Elastic Polymer Can Increase
Microscale Displacement Efficiency in Cores”, paper SPE 63227-MS presented at the 2000 SPE Annual technical and Exhibition held in Dallas, Texas, U.S.A, 1-4 October 2000.
17
Wang, D., Wang, G., Wu, W., Xia, H., and Yin, H., “The Influence of Viscoelasticity on
Displacement Efficiency – From Micro- to Macroscale”, paper SPE 109016-MS presented at the SPE Annual Technical Conference and Exhibition, Anaheim, California, U.S.A, 11-14 November 2007.
18
Xia, H., Wang, D., Ma, W., and Liu, J., “Mechanism of the Effect of Micro-Forces on Residual Oil In Chemical Flooding”, paper SPE 114335 presented at the 2008 SPE/DPE Improved Oil Symposium Held in Tulsa, Oklahoma, U.S.A, 19-23 April 2008.
Chun and Pope (2008)19reported that while a tertiary polymer flood did not mobilize the water flood residual oil saturation, a secondary polymer flood did cause a displacement of oil saturation below the water flood residual oil saturation in the same core. Also, the authors indicate that the water flood and secondary polymer flood results could not be matched in the simulations conducted if the same residual oil saturation value is used for both floods. The core flood data could be only matched when the residual oil saturation value for the polymer flood was lower than that for the water flood19.
Although it is known that polymer flooding will not be applied as an enhanced secondary recovery, this is precisely what happens when the oil that was bypassed by a previous water flood process and that is trapped in the low-permeable zone is mobilized by the polymer; therefore, if the reduction in oil residual saturation in the simulations is neglected, the oil recovery might be underestimated, resulting in large errors and improperly forecasted results20,21.
To simulate this modification, STARS can optionally interpolate basic relative permeability and capillary pressure data as functions of concentration. With this option, the curvature and endpoints of the curves can be modified based on laboratory experimental data, and used for each grid block depending on the polymer adsorption or concentration level. Enabling this option in STARS requires the following keywords:
*INTCOMP With sub-keywords comp_name and phase, indicates respectively, the name of the component upon whose composition the rock-fluid
interpolation will depend, and the phase from which the component’s composition will be taken:
*WATER Water (aqueous) mole fraction *OIL Oil (oleic) mole fraction
*GAS Gas mole fraction
*GLOBAL Global mole fraction
*MAX Maximum of water, oil and gas mole fractions *ADS Adsorption phase, fraction of maximum
*KRINTRP Indicates the interpolation set number, local to the current rock-fluid rock type. Values start at 1 for each new rock type and increase by 1 for each additional interpolation set. For example, rock type #1 might have local set numbers 1 and 2 while rock type #2 might have local set numbers 1, 2 and 3.
19
Chun, H., and Pope, G. A., “Residual Oil Saturation From Polymer Floods: Laboratory
Measurements and Theoretical Interpretation”, paper SPE 113417 presented at the 2008 SPE/DPE Improved Oil Symposium Held in Tulsa, Oklahoma, U.S.A, 19-23 April 2008.
20 Kamaraj, K., Zhang, G., and Seright, R., “Effect of Residual Oil Saturation on Recovery Efficiency
during Polymer Flooding of Viscous Oils”, paper OTC 22040 presented at the Arctic Technology Conference Held in Houston, Texas, U.S.A, 7-9 February 2011.
21
Chen, G., Han, P., Shao, Z., Zhang, X., Ma, M., Lu, K., and Wei, C., “History Matching Method for High Concentration Viscoelasticity Polymer Flood Pilot in Daqing Oilfield”, paper SPE 144538 presented at the SPE Enhanced Oil Recovery Conference held in Kuala Lumpur, Mala ysia, 19-21 July
*DTRAPW Indicates the value of the wetting phase interpolation parameter (mole fraction) for the current rock-fluid data set.
*DTRAPN Indicates the value of the non-wetting phase interpolation parameter (mole fraction) for the current rock-fluid data set.
*WCRV Indicates the curvature change parameter for water relative permeability.
*OCRV Indicates the curvature change parameter for oil relative permeability. *GCRV Indicates the curvature change parameter for gas relative permeability. *SCRV Indicates the curvature change parameter for liquid relative
permeability. DEFAULTS:
• If *INTCOMP is absent, interpolation will not be enabled.
• For a rock type, if *KRINTRP is absent then there is no rock-fluid interpolation. • At least one of *DTRAPW and *DTRAPN must be present to enable interpolation.
If only one is present, its value is applied to the absent keyword. • Each of *WCRV, *OCRV, *GCRV and *SCRV default to 1 if absent. For more detail, refer to the STARS User’s Guide.
Polymer Degradation
Polymer degradation refers to any process that breaks down the molecular structure of polymer macromolecules. The main degradation mechanisms that may be of concern in an
EOR process are chemical, thermal, biological and mechanical 22:
• Chemical degradation of polymer, or polymer chemical stability, is mainly controlled by oxidation-reduction reactions and hydrolysis, which are due to the presence of divalent cations such as Ca2+, Mg2+, Fe2+ in the water, and oxygen,
which breaks down the polymer molecular chains23,24.
• Thermal degradation of polymers is commonly associated with chemical
degradation and is defined as molecular deterioration resulting from overheating. At high temperatures, the components of the long-chain backbone of the polymer can separate (molecular scission) and react with one another to change the
properties of the polymer rheology and phase behavior. Thermal degradation generally involves changes to the molecular weight and it can occur at
temperatures which are much lower than those at which mechanical failure is likely to occur.
• Biological degradation is more prevalent for biopolymers than it is for synthetic polymers, however it can occur in both. Biological degradation, the microbial breakdown of polymer macromolecules in the presence of bacteria in the reservoir,
occurs more often at lower temperatures and salinities23,24.
22
Chang, H. L., “Polymer Flooding Technology – Yesterday, Today, and Tomorrow”, paper SPE 7043 presented at the 1978 SPE Symposium on Improved Methods for Oil Recovery, Tulsa, April 16-19.
23
Littmann, W., “Polymer Flooding”, Developments in Petroleum Science, Vol. 24, Elsevier, Amsterdam (1988), pp 32-34.
• Mechanical degradation occurs when a polymer solution is exposed to high shear conditions, which fragment/break the polymer molecular chains, resulting in loss of viscosity and mobility control. This may happen when a polymer solution is forced at high flow rates through a porous medium or is i n the vicinity of an injection well, where it is sheared by the high velocities4,25. Mechanical degradation of polymer is more severe at higher flow r ates and longer flow
distances, and with lower permeability porous media. The behavior of the polymer as a non-Newtonian fluid in the presence of shear conditions and the effects of shear-thinning and shear-thickening can be modeled by STARS as explained in Shear-Dependent Viscosity Effects.
All of the above degradation mechanisms, if applicable, should be considered in an EOR process. To be effective, polymer solutions must remain stable for a long time at reservoir
conditions. These degradations are often neglected at the simulation level because the processes are quite complex. A simple approximation for predicting these phenomena is to
use a first-order reaction where the polymer is degraded to water. In this case, the polymer concentration is reduced in each grid cell over time, which results in a reduced viscosity that can be calculated with the same nonlinear mixing rule used to account for the behavior of the aqueous phase viscosity with polymer concentration.
For the simple case of polymer consumption, the first-order reaction can be expressed as follows: (w) Sto2Water (w) Polymer 1 Sto
→
(16) where:1
Sto
Stoichiometric coefficient of reacting component2
Sto
Stoichiometric coefficient of produced componentTypical values of polymer viscosity stability over time are shown in Table 1 and Figure 6 :
Table 1 Polymer Viscosity Specifications
Time (Days) Viscosity (cps)
Polymer 1 Polymer 2 Polymer 3
0 6.17 7.15 8.02 5 2.84 3.65 5.00 10 1.91 2.97 3.50 15 1.47 2.42 3.00 30 0.92 1.65 2.30 45 0.91 1.47 2.00 60 0.84 1.25 1.80 25
Seright, R.S., “The Effects of Mechanical Degradation and Viscoelastic Behavior on Injectivity of Polyacrylamide Solutions”, paper SPE 9297 presented at the 55th Annual Fall Technical Conference, Society Petroleum Engineers, Dallas, Sept. 1980.
Figure 6 Viscosity Stability of Sample Polymers
The STARS reaction modeling capability is quite robust and can be used to model more complex reactions if desired. The STARS simulator allows the user to model kinetic reactions in the formation and/or the breaking of any component as a function of fluid flow velocity, temperature, concentration and type of process.
The mandatory keywords for the chemical reaction data are:
*STOREAC Used to specify the stoichiometric coefficient of reacting component *STOPROD Used to specify the stoichiometric coefficient of produced component *FREQFAC Used to specify the reaction frequency factor
If the reaction is different from the first-order reaction and the components are reacting in a different phase than they were originally part of, *RPHASE and *RORDER must also be specified:
*RPHASE Used to specify a flag defining the phase of the reacting component. The allowed values of this flag are as follows:
0 = non-reacting components,
1 = water phase (fluid components only) 2 = oil phase (fluid components only) 3 = gas phase (fluid components only) 4 = solid phase (solid components only)
Note: An adsorbing component may not react in the adsorbed phase. *RORDER Used to specify the order of the reaction with respect to each reacting
component’s concentration factor. It must be non-negative.
Enter zero for non-reacting components. Normally, you would use a value of one (1); however, if the value is zero (0), the reaction rate will be independent of that component’s concentration.
DEFAULTS:
If *RPHASE is absent, the assumption is: iphas = 0 for non-reacting components
iphas = 1 for aqueous components 1 to numw iphas = 2 for oleic components numw+1 to numx
iphas = 3 for noncondensable components numx+1 to numy iphas = 4 for solid components numy+1 to ncomp
If *RORDER is absent, the assumption is: enrr = 0 for non-reacting components enrr = 1 for reacting components
If the process you are simulating is thermal (i.e., the polymer degradation rate depends on the temperature and activation energy [Ea]), the first-order reaction may be rewritten to depend on the absolute temperature in the grid cells according to the Arrhenius equation. In this case, new keywords *RENTH and *EACT are required:
*RENTH Used to specify the reaction enthalpy (J/gmol | Btu/lbmol). It is positive for exothermic reactions and negative for endothermic
reactions. The default is 0.
*EACT Used to specify the single activation energy (J/gmol | Btu/lbmol). Defines the dependence of reaction rate on grid block temperature. If absent, the reaction is independent of temperature (equivalent to *EACT with E a = 0).
For information about the formulas used in determining the volumetric reaction rate, refer to Volumetric Reaction Rate on page 46. For information about the formulas used to calculate the stoichiometric coefficients, refer to Stoichiometric Coefficients on page 48. For more details refer to the STARS User’s Guide.
Composition-Dependent Viscosity Effects
The main purpose of using a polymer is to achieve a favorable mobility ratio. The efficiency of a waterflood is heavily dependent on the mobility ratio of the displacing and displaced fluids. The mobility ratio is defined as follows:
=ℎ
ℎ
=
/
where:
k w Effective permeability to water
μw Water viscosity
k o Effective permeability to oil
μo Oil viscosity
With a high mobility ratio, the displacing fluid moves much faster than the displaced fluid. As a result, a phenomenon called viscous fingering occurs in which the displacing fluid bypasses the displaced fluid and channels towards the producer. Figure 7 illustrates this
effect. The fingering of the water leaves behind large areas of oil that are unswept by the water. The water also channels itself towards the producer (upper left). Once this
communication has been established, water will go straight from the injector to the producer, bypassing the remaining oil in the reservoir.
Figure 7 Water Fingers through Oil due to Adverse Viscosity Effects26
If polymer is added to the injected water, the mobility ratio is lowered (more favorable). The polymer increases the viscosity of the injected water, thus reducing its mobility. The flood
front is more uniform, and there is very little evidence of viscous fingering. The displacement is more piston-like and leaves very little trapped oil behind. This is illustrated in Figure 8.
Figure 8 Improved Sweep Efficiency and Oil Recovery through Use of Polymer 26
Component viscosity can be entered in two ways. The first way is to enter a viscosity versus temperature table using the keyword *VISCTABLE, which contains a temperature column
followed by columns corresponding to the number and order of the non-solid components listed under the *MODEL keyword. For example, consider a fluid model containing water, oil, gas and polymer. The *VISCTABLE will appear similar to that shown below:
VISCTABLE T1 μw,T1 μo,T1 μg,T1 μP,T1 T2 μw,T2 μo,T2 μg,T2 μP,T2 T3 μw,T3 μo,T3 μg,T3 μP,T3 … Tn μw,Tn μo,T4 μg,Tn μP,Tn
Different VISCTABLEs can be defined at different pressures, allowing temperature and pressure dependency to be modeled. The *ATPRES keyword can be used to specify a
*VISCTABLE at different pressures, as f ollows:
ATPRES pres_1 VISCTABLE ATPRES pres_2 VISCTABLE ATPRES pres_n VISCTABLE
The second way to define component viscosity is to enter the coefficients and use a correlation to calculate the viscosity at different temperatures. The correlation uses the following equation:
=
× exp(
/Tabs
) (18) where:avisc1,bvisc1 Coefficients of the correlation for the temperature-dependence of
a component viscosity in the liquid phase
T abs Absolute temperature
Within STARS, the viscosity of a liquid phase is modeled using either a linear mixing rule or a nonlinear mixing rule. The default method in STARS is the linear mixing rule, which uses the following equation to calculate the viscosity of a multi-component mixture:
(
) =
where:
Viscosity of mixture
phase viscosity in the linear mixing rule.Weighting factor of component “i” in the aqueous (α = w) or oleic (α = o)
Viscosity of component “i” in the aqueous (α = w) or oleic (α = o) phase. nc Number of components in the oleic or aqueous phase.Factors
=
(oil mole fractions) and
=
(water mole fractions) are used for linear mixing. To specify nonlinear mixing (for example, solution gas in the oleic phase or polymer in the aqueous phase), use keywords *VSMIXCOMP, *VSMIXENDP and *VSMIXFUNC (one instance for each key component), where factors
and
are different from the mole (mass) fractions
and
respectively, for the key component.To accomplish nonlinear mixing with alternate weighting factors,
is replaced with the nonlinear mixing function
for each component i ∈ S and with
×
for eachcomponent i ∉ S, where S denotes the set of key components. The nonlinear mixing rule for the liquid viscosity is calculated as:
(
) =
∈
=1
×
+
×
×
∉
=1
(20)where:
Viscosity of mixture aqueous (α = w) or oleic (α = o).
Viscosity of component “i” in the aqueous (α = w) or oleic (α = o) phase.
Weighting factor of the non-key component “i” in the aqueous (α = w) oroleic (α = o) phase viscosity in the nonlinear mixing rule.
Weighting factor of the key component “ i” in the aqueous (α = w) or oleic(α = o) phase viscosity in the nonlinear mixing rule.
∈
Number of key components in the liquid phase.
∉
Number of components in the liquid phase that are not key components. In the determination of the weighting factors for the nonlinear mixing rule, each keycomponent acts independent of other key components, which is reflected in the fact that the nonlinear mixing function
depends only on the mass or mole fraction (
or
). This implies that the function data entries must be generated assuming the absence of other key components.The format for specifying the nonlinear mixing rule for
The format for specifying the nonlinear mixing rule for liquid viscositieliquid viscosities is s is as follows:as follows: VSMIXCOMP
VSMIXCOMP comp_namecomp_name Specifies the component that is exhibiting nonlinearSpecifies the component that is exhibiting nonlinear viscosity mixing
viscosity mixing VSMIXENDP
VSMIXENDP x xlowlow x xhighhigh Specifies the minimum and maximum compositionsSpecifies the minimum and maximum compositions
of the specified
of the specified comp_namecomp_name VSMIXFUNC
VSMIXFUNC f f 11 . . .f . . .f 1111 Nonlinear m Nonlinear mixing function ixing function corresponding corresponding to the 11to the 11
intervals between
intervals between x xlowlow and and x xhighhigh
DEFAULTS: DEFAULTS:
If *VSMIXCOMP is absent, linear mixing is assumed for all components. If *VSMIXCOMP is absent, linear mixing is assumed for all components. If *VSMIXENDP is absent,
If *VSMIXENDP is absent, x xlowlow = 0 and = 0 and x xhighhigh = 1 are assumed. = 1 are assumed.
If *VSMIXFUNC is
If *VSMIXFUNC is absent, entriesabsent, entries
== ((
11))/10,/10, for for ii = 1 = 1 to 11, corresponding toto 11, corresponding to linear spacing from 0 tolinear spacing from 0 to 1.1. CONDITIONS:
CONDITIONS:
The phase to which this data will be assigned depends on which of *LIQPHASE, The phase to which this data will be assigned depends on which of *LIQPHASE, *WATPHASE and *OILPHASE is in force.
*WATPHASE and *OILPHASE is in force.
A nonlinear function may be specified for more than one component in each of the water and A nonlinear function may be specified for more than one component in each of the water and oil phases. At least one component in each liquid phase must not be a key component, since oil phases. At least one component in each liquid phase must not be a key component, since the algorithm involves adjusting the weighting factors of the non-key components.
the algorithm involves adjusting the weighting factors of the non-key components.
Keywords *VSMIXENDP and *VSMIXFUNC are applied to the last key component defined Keywords *VSMIXENDP and *VSMIXFUNC are applied to the last key component defined via *VSMIXCOMP. A key component may not be specified more than once in each liquid via *VSMIXCOMP. A key component may not be specified more than once in each liquid phase.
phase.
An example of how these keywords should be entered in the simulation dataset to model the An example of how these keywords should be entered in the simulation dataset to model the nonlinear mixing rule of the water viscosity with the presence of a polymer in the aqueous nonlinear mixing rule of the water viscosity with the presence of a polymer in the aqueous phase is sh
phase is shown below:own below:
VSMIXCOMP 'Polymer' VSMIXCOMP 'Polymer' VSMIXENDP 0 0.001 VSMIXENDP 0 0.001 VSMIXFUNC 0 0.0 0.0759 0.1598 0.2514 0.3498 0.4534 0.5608 0.6704 0.7808 0.891 1.0 VSMIXFUNC 0 0.0 0.0759 0.1598 0.2514 0.3498 0.4534 0.5608 0.6704 0.7808 0.891 1.0 Internally, STARS will divide the
Internally, STARS will divide the composition interval (composition interval ( x xhighhigh - x - xlowlow) into ) into 11 equal subintervals,11 equal subintervals,
corresponding to the
corresponding to the f f 11... f f 1111 values. Inside STARS, the table will appear as shown in the values. Inside STARS, the table will appear as shown in the
following example: following example:
Composition Composition ((ww p p)) Mixing Function Mixing Function f f ((ww p p)) 0.0000 0.0000 0.0000 0.0000 0.0001 0.0759 0.0001 0.0759 0.0002 0.1598 0.0002 0.1598 0.0003 0.2514 0.0003 0.2514 0.0004 0.3498 0.0004 0.3498 0.0005 0.4534 0.0005 0.4534 0.0006 0.5608 0.0006 0.5608 0.0007 0.6704 0.0007 0.6704 0.0008 0.7808 0.0008 0.7808 0.0009 0.8910 0.0009 0.8910 0.0010 1.0000 0.0010 1.0000 At any composition
At any composition ww p p, a corresponding mixing function, a corresponding mixing function f f ((ww p p) ) will be will be determined eitherdetermined either
directly from the table or through
directly from the table or through interpolatiinterpolation. This function will be on. This function will be used to calculate theused to calculate the viscosity of the
viscosity of the solution. As the solution. As the composition changecomposition changes due s due to injection, decomposition orto injection, decomposition or adsorption, the mixing function will change accordingly, resulting in a change of water adsorption, the mixing function will change accordingly, resulting in a change of water viscosity.
viscosity.
For further information and a calculation example, refer to the
For further information and a calculation example, refer to the Polymer ViscosityPolymer Viscositysection ofsection of Polymer Flood Calculations
Polymer Flood Calculations inin Appendix A – EquationsAppendix A – Equations..
Shear-Dependent Viscosity Effects
Shear-Dependent Viscosity Effects
Flow in aFlow in a porous medium is affected by the morphology of porous medium is affected by the morphology of the medium and the rheology ofthe medium and the rheology of the fluid. Many applications of flow in
the fluid. Many applications of flow in porous media involve Newtonian fluids, in which theporous media involve Newtonian fluids, in which the viscosity is independent of shear rate. Any fluid that does not obey Newton’s law of viscosity viscosity is independent of shear rate. Any fluid that does not obey Newton’s law of viscosity is a
is a non-Newnon-Newtonian fluid.tonian fluid.
Generally, a fluid can be classified as Newtonian or non-Newtonian depending on its flow Generally, a fluid can be classified as Newtonian or non-Newtonian depending on its flow behavior; t
behavior; that is, hohat is, how its viscw its viscosity changosity changes in the es in the presence presence of shear stof shear stress and thress and the rate oe rate off shear applied. If the shear stress (
shear applied. If the shear stress (
) is plotted against shear rate () is plotted against shear rate (̇̇
) at constant temperature) at constant temperature and pressure, the so-called “flow curve” or “rheogram” (the response of a Newtonian fluid) is and pressure, the so-called “flow curve” or “rheogram” (the response of a Newtonian fluid) is a straight line with slopea straight line with slope
, passing through , passing through the origin. The the origin. The constant of proportionality,constant of proportionality,
,, referred to as the Newtonian or dynamic viscosity is, by definition, independent of shear rate referred to as the Newtonian or dynamic viscosity is, by definition, independent of shear rate and shear stress and depends only on the material, its temperature, and its pressure.and shear stress and depends only on the material, its temperature, and its pressure. On the other hand, for
On the other hand, for a non-Newtonian fluid, the curve does not pass through the origina non-Newtonian fluid, the curve does not pass through the origin and/or does not result in
and/or does not result in a linear relationship between the shear stress (a linear relationship between the shear stress (
) and the shear rate) and the shear rate ((̇̇
). The coefficient of viscosity is ). The coefficient of viscosity is not constant and is a not constant and is a function offunction of
and anḋ̇
..The relationship between shear stress
The relationship between shear stress ττand shear rateand shear rate
̇̇
is as f is as follows:ollows:
==
××̇̇
(21)(21)where: where:
Shear stressShear stresṡ̇
Shear rateShear rate
Apparent viscosity of the fluidApparent viscosity of the fluid A typical shear stress versus shear rate plotA typical shear stress versus shear rate plot for a non-Newtonian fluid is shown for a non-Newtonian fluid is shown inin Figure 9: Figure 9:
ShearRate(s ShearRate(s-1-1))
S S h h e e a a r r S S t t r r e e s s s s ( ( γ γ ) ) Newtonian Newtonian Fluid Fluid DilatantFluid DilatantFluid (ShearThickening) (ShearThickening) PseudoplasticFluid PseudoplasticFluid (ShearThinning) (ShearThinning)
Figure 9 Typical Plot
Figure 9 Typical Plot of Shear Stress vs. of Shear Stress vs. Shear Rate for Newtonian and non-NeShear Rate for Newtonian and non-Newtonian Fluidswtonian Fluids27 27
Often the relationship between shear stress (
Often the relationship between shear stress (
) and shear rate () and shear rate (̇̇
) for these fluids is plotted on) for these fluids is plotted on log-log coordinates, and the relationship can be approximated as a straight line over a limited log-log coordinates, and the relationship can be approximated as a straight line over a limited range of shear rate (range of shear rate ( or stress), that is:or stress), that is:
==
××̇̇
(22)(22)where: where:
Shear stressShear stresṡ̇
Shear rateShear rate
Fluid consistency coefficient or index.Fluid consistency coefficient or index.
Flow behavior index or power-law exponent.Flow behavior index or power-law exponent.When viscosity decreases with increasing shear rate, the fluid
When viscosity decreases with increasing shear rate, the fluid is called shear-thinning. In theis called shear-thinning. In the opposite case, where viscosity increases as the fluid is subjected to a higher shear rate, the opposite case, where viscosity increases as the fluid is subjected to a higher shear rate, the fluid is called thickening. Shear-thinning behavior is more common than fluid is called thickening. Shear-thinning behavior is more common than shear-thickening. Shear-th
thickening. Shear-thinning fluids are inning fluids are also referred to also referred to as pseudoplastic fluids, while shear-as pseudoplastic fluids, while shear-thickening fluids are referred to as dilatant fluids. The behavior of
thickening fluids are referred to as dilatant fluids. The behavior of these fluids is illustrated inthese fluids is illustrated in the following figures.
the following figures.
27
Shear-ThickeningFluid A p p a r e n t V i s c o s i t y ShearRate ShearRate Shear-ThinningFluid A p p a r e n t V i s c o s i t y
Figure 10 Apparent Viscosity vs. Shear Rate for Shear-Thinning and Shear-Thickening Fluids28
Many shear-thinning and shear-thickening fluids exhibit Newtonian behavior at extreme shear rates, both low and high. For such fluids, when the apparent viscosity is plotted against log shear rate, the curve appears as follows:
NewtonianRegion NewtonianRegion Power-LawRegion Shear-Thinning Fluid l o g A p p a r e n t V i s c o s i t y logShearRate
Figure 11 Apparent Viscosity vs. Shear Rate for a Shear-Thinning Fluid 28
28
Subramanian, R. Shankar, “Non-Newtonian Flows”, Department of Chemical and Biomolecular Engineering, Clarkson University.
NewtonianRegion NewtonianRegion Power-LawRegion Shear-Thickening Fluid logShearRate l o g A p p a r e n t V i s c o s i t y
Figure 12 Apparent Viscosity vs. Shear Rate for a Shear-Thickening Fluid
The regions where the apparent viscosity is approximately constant are known as Newtonian regions. The behavior between these regions can usually be approximated by a straight line on these axes. In this region, which is known as the power-law region, the behavior can be approximated by the following expression:
=
+
×
(̇
) (23) This can be rewritten as:
=
(
) ×̇
(24)Equation (24) can be replaced by one more commonly used in the literature, which comes from combining equations (21) and (22):
=
×̇
−1
(25)K represents the fluid consistency index and is equal to “
(
)” in equation (24).
represents the power-law exponent, and the expression (
1) is equal to “b” in equation (24). Note the following:• When n < 1, the fluid exhibits shear-thinning properties. • When n = 1, the fluid shows Newtonian behavior.
• When n > 1, the fluid shows shear-thickening behavior.
The above shows that rheology in porous media has an important impact on enhanced oil recovery projects where polymer solutions are used, because if the effective viscosity of the polymer solution is high at the high velocities experienced near an injection well, the polymer
injection rate and/or the oil production rate (at the producer well) may be decreased. On the other hand, if the effective viscosity of the polymer solution is too low at the low velocities experienced away from the injection well or deep into the reservoir, oil displacement may be inefficient.
To simulate the behavior of the pseudoplastic or dilatant fluids used in enhanced oil recovery (foam, high molecular weight liquids which include solutions of polymers, as well as liquids in which fine particles are suspended [suspensions]), STARS uses the power-law or Ostwald de Waele model, as described below.
Power-Law Expression for Shear-Thinning or Pseudoplastic Fluids
The *SHEARTHIN keyword is used to represent the pseudoplastic behavior or the shear-thinning effect as follows:
*SHEARTHIN
,
DEFINITIONS:
Power-law index or exponent in the viscosity shear thinning equation (dimensionless). The allowed range is from 0.1 to 0.99, inclusive. Values below 0.3 can result in unacceptable numerical performance and so are notrecommended. Values close to 1 approximate Newtonian behavior.
,
The meaning of
,
depends on *SHEAREFFEC (*SHV | *SHR).*SHV: Reference Darcy velocity (m/day | ft/day | cm/min) in viscosity shear thinning equation. The allowed range is 10-10 to 1010 m/day (3.28×10-10 ft/day to 3.28×1010 ft/day | 6.94×10-12 to 6.94×108 cm/min). *SHR: Reference shear rate (1/day | 1/day | 1/min) in viscosity shear thinning equation. The allowed range is 10-10 to 1010 1/day (6.94×10-14 to 6.94×106 1/min).
The bounded power-law relation between the apparent fluid viscosity
and the Darcy fluid velocity
is:
=⎩
,
for
≤
,
,
×
,
−1
for
,
<
<
,
,
for
≥
,
(26)The upper velocity boundary of the shear thinning regime,
,
, is defined by the point on the power-law curve where the apparent viscosity,
, equals the phase fluid viscosity in the absence of polymer (
,
). The lower velocity boundary of the shear thinning regime,
,
, is defined by the point on the power-law curve where the apparent viscosity,
,equals the fluid phase viscosity in the absence of thinning (
,
). For further discussion on the calculation of phase viscosities for Newtonian flow, refer to the STARS User’s Guide for information on *AVISC and *BVISC.The bounded power law relation of apparent viscosity versus velocity for shear thinning is depicted in the log/log plot of Figure 13. The shear thinning regime is represented by a linear relation of slope (
1).log(DarcyVelocity)
ul=ul,lower ul=ul,upper
µapp=µl,0 µapp=µl,p
log(Apparent Viscosity)
Figure 13 Shear Thinning Power Law - Apparent Viscosity vs. Darcy Velocity
Power-Law Expression for Shear-Thickening or Dilatant Fluids
The *SHEARTHICK keyword is used to represent the dilatant’s behavior or the shear-thickening effect as follows:
*SHEARTHICK
,
,
DEFINITIONS:
Power-law index or exponent in the viscosity shear thickening equation (dimensionless). The allowed range is from 1.01 to 5, inclusive. Values above 2.5 can result in unacceptable numerical performance and so are not recommended. Values close to 1 approximate Newtonian behavior.
,
The meaning of
,
depends on *SHEAREFFEC (*SHV | *SHR).*SHV: Reference Darcy velocity (m/day | ft/day | cm/min) in viscosity shear thickening equation. The allowed range is 10-10 to 1010 m/day (3.28×10-10 ft/day to 3.28×1010 ft/day | 6.94×10-12 to 6.94×108 cm/min). *SHR: Reference shear rate (1/day | 1/day | 1/min) in viscosity shear
thickening equation. The allowed range is 10-10 to 1010 1/day (6.94×10-14 to 6.94×106 1/min).
,
Maximum viscosity (cp) in viscosity shear thickening equation. TheThe power-law relation between apparent fluid viscosity
and Darcy fluid velocity
is:
=⎩
,
for
≤
,
,
×
,
−1
for
,
<
<
,
,
for
≥
,
(27)The lower velocity boundary of the shear thickening regime,
,
, is defined by the point on the power law curve when the apparent viscosity,
, equals the phase fluid viscosity in the absence of thickening (
,
). For further discussion on the calculation of phase viscosities for Newtonian flow, refer to the STARS User’s Guide for information about *AVISC and *BVISC.The upper velocity boundary of the shear thickening regime,
,
, is defined by the point on the power-law curve where the apparent viscosity,
, equals the user-defined maximum viscosity (
,
).The bounded power-law relation of apparent viscosity versus velocity for shear thickening is depicted in the log/log plot of Figure 14. The shear thickening regime is represented by a linear relation of slope (
1).log(DarcyVelocity) log(Apparent
Viscosity)
µapp=µl,p µapp=µl,max
ul=ul,lower ul=ul,max
Figure 14 Shear Thickening Power Law - Apparent Viscosity vs. Darcy Velocity
In cases where fluids exhibit both behaviors (shear-thinning and shear thickening), the keywords can be used together. The apparent viscosity of the combined effect is the sum of the shear thinning and thickening apparent viscosities defined in the above sections:
The summed power-law relation between apparent fluid viscosity,
, and Darcy fluid velocity,
, is:
=
,
for
≤
,
,
+
,
for
,
<
<
,
,
for
≥
,
(29)The lower velocity boundary of the shear thinning and thickening regime,
,
, is defined by the point on the thinning power-law curve where the apparent viscosity,
, equals thefluid phase viscosity in the absence of thinning (
,
). The upper velocity boundary of the shear thinning and thickening regime,
,
, is defined by the point on the thickeningpower-law curve where the apparent viscosity,
, equals the user-defined maximum viscosity (
,
).The summed power-law relation of apparent viscosity versus velocity is depicted in the log/log plot of Figure 15.
log(DarcyVelocity) log(Apparent
Viscosity)
µapp=µl,max
µapp=µl,p
ul=ul,lower ul=ul,max
Figure 15 Shear Thinning and Thickening Power Laws - Apparent Viscosity vs. Darcy Velocity
The above explanations are for velocity-dependent viscosity, which is the default option implemented in STARS. For the shear-rate-dependent viscosity option, the same logic applies, with the term “shear rate” replacing “velocity” throughout.
If you need to replace the default option (velocity-dependent viscosity) with the shear-rate-dependent viscosity option, the *SHEAREFFEC keyword must be used. The format of this keyword is as follows:
DEFINITIONS:
*SHV Viscosity shear depends on Darcy velocity. *SHR Viscosity shear depends on shear rate. DEFAULTS:
If *SHEAREFFEC is absent, *SHEAREFFEC *SHV is assumed.
Tabular Input Option for Velocity/Shear-Rate-Dependent Viscosity
In addition to the keywords described above, STARS has a tabular input option for velocity-dependent viscosity or shear-rate-velocity-dependent viscosity, which is useful when the viscosity-versus-velocity relation or viscosity-versus-shear-rate relation is specified by laboratory data, or when a simple power-law relation is not sufficient. The format is as follows:
FORMAT: *SHEARTAB { velocity viscosity } or *SHEARTAB { shear-rate viscosity } DEFINITIONS:
*SHEARTAB A viscosity-versus-velocity table follows. The maximum allowed number of table rows is 40. The first column is either velocity or shear rate,
depending on *SHEAREFFEC.
Velocity *SHEAREFFEC 0: The first column is phase velocity (m/day | ft/day | cm/min). The allowed range is 10-10 to 1010 m/day (3.28×∙10-10 to 3.28×1010 ft/day | 6.94×10-12 to 6.94×108 cm/min).
shear-rate *SHEAREFFEC 1: The first column is phase shear-rate (1/day | 1/day | 1/min). The allowed range is 10-10 to 1010 1/day (6.94×10-14 to 6.94×106 1/min).
viscosity Viscosity (cp) of the component at corresponding velocity. The allowed range is 10-5 to 106 cp.
The following conditions apply to these keywords:
a) *SHEARTAB is applied to the component and phase specified by the immediately preceding *VSMIXCOMP, so *VSMIXCOMP must be present before
*SHEARTAB.
b) *SHEARTAB may not be used together with *SHEARTHICK and *SHEARTHIN.
c) The first column is either velocity or shear rate, depending on *SHEAREFFEC. d) For phase velocity/shear-rate outside the velocity/shear-rate table range, the nearest
