Migration Rates and Formation Injectivity to Determine 

Containment Time Scales of Sequestered Carbon Dioxide

By Lauri A. Burke

Open-File Report 2012–1062

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2012 

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Suggested citation:

Burke, L.A., 2012, Migration rates and formation injectivity to determine containment time scales of sequestered 

carbon dioxide: U.S. Geological Survey Open-File Report 2012–1062, 23 p.

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iii

Contents

Abstract ......................................................................................................................................................................... 3

Introduction.................................................................................................................................................................... 4

Methodology .................................................................................................................................................................. 5

Fluid Flow Modeling....................................................................................................................................................... 7 

Permeability Classifications ......................................................................................................................................... 13

Formation Injectivity..................................................................................................................................................... 14

Maximum Pressure Differential.................................................................................................................................... 18

Conclusions ................................................................................................................................................................. 19

Acknowledgments........................................................................................................................................................ 22

References Cited......................................................................................................................................................... 23
iv

Conversion Factors

SI to Inch/Pound

Multiply By To obtain

Length

centimeter (cm) 0.3937 inch (in.)

meter (m) 3.281 foot (ft) 

kilometer (km) 0.6214 mile (mi)

Area

square centimeter (cm2) 0.1550 square inch (in2) 

Volume

cubic meter (m3) 6.290 barrel (petroleum, 1 barrel = 42 gal)

Mass

kilogram (kg) 2.205 pound avoirdupois (lb)

Pressure

kilopascal (kPa) 0.1450 pound per square inch (lb/in2) 

Density

kilogram per cubic meter (kg/m3) 0.06242 pound per cubic foot (lb/ft3) 

kilogram per cubic meter (kg/m3) 0.008345 pounds per gallon (ppg)

kilogram per cubic meter (kg/m3) 0.000434 pounds per square inch per foot 

(psi/ft)

Permeability

square meter (m2) 1.01325×1012 darcy (D)

darcy (D) 9.869233×10−13 square meter (m2)

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

°F=(1.8×°C)+32.

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F-32)/1.8.
v

Table of Symbols

A Cross-sectional area

D Depth

FD Fracture gradient at depth

HD Hydrostatic gradient at depth

K Bulk modulus 

k Matrix permeability

L Lateral distance

P Pressure

Q Darcy fluid-flow rate

v Interstitial pore velocity

Vp Velocity of compressional-waves in the formation

α Hydraulic diffusivity

β Bulk compressibility of the formation 

β f Fluid compressibility

∆ Gradient differential

∆P Maximum pressure differential

η Fluid viscosity 

φ Fractional porosity 

μ Shear modulus 

ρ Density

τ Time scale
Lauri Burke 

U.S. Geological Survey 

Migration Rates and Formation 

Injectivity to Determine Containment 

Time Scales of Sequestered 

 Carbon Dioxide 
Overview 

• Abstract and Introduction 

• Methodology 

• Fluid Flow Modeling 

– Hydraulic Diffusivity 

– Darcy’s Law of Fluid Flow

• Permeability Classifications 

• Formation Injectivity 

• Maximum Pressure Differential 

• Conclusions 

2
Abstract 

 Supercritical carbon dioxide exhibits highly variable behavior over a range of 

reservoir pressure and temperature conditions. Because geologic sequestration of 

supercritical carbon dioxide is targeted for subsurface injection and containment at depths 

ranging from approximately 3,000 to 13,000 feet, the investigation into the physical 

properties of this fluid can be restricted to the pressure and temperature conditions likely 

encountered in the sedimentary strata within this depth interval. A petrophysical based 

approach was developed to study the widest range of formation properties potentially 

encountered in sedimentary strata. Fractional porosities were varied from 5 to 95 percent, in 

5-percent increments, and permeability values were varied over thirteen orders of 

magnitude, from 10.0 darcys down to 1.0 picodarcy. 

 Fluid-flow modeling incorporated two constitutive equations from fluid dynamics: 

hydraulic diffusivity for near-surface applications, and Darcy’s Law for deeper formations 

exhibiting higher pressure gradients. Based on the flow modeling results, first-order 

approximations of carbon dioxide lateral migration rates were determined. These first-order 

approximations enable the establishment of a permeability classification system for dividing 

the subsurface into flow units that provide short, moderate, and long-term containment of 

carbon dioxide. These results enable a probabilistic determination of how fluids will enter 

and be contained in a subsurface storage formation, which is a vital step in the calculation of 

the carbon dioxide storage capacity of a reservoir. 

 Additionally, this research establishes a methodology to calculate the injectivity of a 

target formation. Because injectivity describes the pressure increase due to the introduction 

of fluids into a formation, the relevant application of injectivity is to determine the pressure 

increase, due to an injection volume and flow rate, that will induce fractures in the reservoir 

rocks. This quantity is defined mathematically as the maximum pressure differential 

between the hydrostatic gradient and the fracture gradient of the target formation. 

 Injectivity is mathematically related to the maximum pressure differential 

 of the formation, and can be used to determine the upper limit for the 

 pressure increase that an injection target can withstand before fracturing. 

3
Introduction 

• This study quantifies first-order approximations for the time 

scales of carbon dioxide (CO2) lateral migration through a 1.0-

kilometer (km) representative volume of rock 

• For characterization and classification of subsurface strata into 

subdivisions based on petrophysical criteria, and 

• Incorporated into the U.S. Geological Survey assessment 

methodology for fully probabilistic determination of the 

storage capacity of geologic formations for CO2 sequestration 

(Brennan, et al., 2010; Burruss et al., 2009) 

4
1

 Data from Lemmon and others, (2011) 

using Span and Wagner (1996) equations 

2

 Calculated from Vp=[(4/3μ+K)/ρ]

1⁄2

1.0E-05

8.0E-01

0 50 100 150 200

Pressure (MPa)

Fluid Compressibility (m·s2/kg)

B.

 

0

200

400

600

800

1,000

1,200

0 50 100 150 200

Pressure (MPa)

Compressional-wave Velocity (m/s)

A.

0.0E+00

1.0E-04

2.0E-04

0 50 100 150 200

Pressure (MPa)

Viscosity (Pa.s)

B.

0

200

400

600

800

1,000

1,200

0 50 100 150 200

Pressure (MPa)

Density (kg/m3)

A.Density1 Viscosity1

Velocity1 Compressibility2 ) 

Thermophysical Properties of CO2

5 (Burke, 2011) 
Approach 

• CO2 sequestration is targeted for injection and subsurface 

containment at depths from approximately 3,000 to 13,000 ft 

• Midpoint is 8,000 ft 

– Normally geopressured region with 100,000 parts per million total 

dissolved solids: 

 0.465 psi/ft (Schlumberger, 2012) 

– Generalized geothermal gradient for shallow crustal rocks: 

 1.65 °F/100ft (Sheriff, 1994) 

– Average surface temperature: 68 °F 

• Pressure and temperature conditions of an “average” 

sedimentary formation at 8,000 ft: 25.5 MPa and 200 °F 

6
Fluid Flow Modeling 

Hydraulic Diffusivity time scale, , in years: 





2

2

L

hd 























f

k

 



 





1





2

2

L

hd 

L

kA P

Q







where 

from and 

Darcy’s Law time scale, , in years:

Pk

L

D









L

Pk

v







Pk

L

D





2





7
Flow Modeling Parameters 

Property Variable Value Units 

Viscosity 5.00E-05 kg/m·s

Fluid Density 628.06 kg/m3

Fluid Compressibility 1.66E-02 MPa-1

Compressional-wave 

Velocity 390.28 m/s

Bulk Compressibility 3.10E-02 MPa-1

Lateral Distance 1.00 km 

Fractional Porosity varies dimensionless 

Matrix Permeability varies D 

Darcy Pressure Differential 25.5 MPa

Fluid Properties 

•At 25.5 MPa and 200 °F 

Rock Properties 

•Fractional porosity varies from 0.05 to 0.95 

•Matrix permeability varies from 1.00E+01 to 1.00E-12 D 

CO

2

Properties 

Rock 

Properties 





 f

Vp



L



k

P

8
Hydraulic Diffusivity Results 

10.0 D 

1.0 mD 

1.0 D 

1.0 μD

1.0 nD 

1.0 pD 

Permeability: 1.0 pD

0.0E+00

1.0E+11

2.0E+11

3.0E+11

4.0E+11

5.0E+11

6.0E+11

1.00 0.800.90 0.70 0.60 0.400.50 0.30 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 nD

0.0E+00

1.0E+08

2.0E+08

3.0E+08

4.0E+08

5.0E+08

6.0E+08

1.00 0.800.90 0.600.70 0.50 0.300.40 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 mD

0.0E+00

1.0E+02

2.0E+02

3.0E+02

4.0E+02

5.0E+02

6.0E+02

1.00 0.800.90 0.70 0.60 0.400.50 0.30 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 μD

0.0E+00

1.0E+05

2.0E+05

3.0E+05

4.0E+05

5.0E+05

6.0E+05

1.00 0.800.90 0.70 0.60 0.400.50 0.30 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 D

0.0E+00

1.0E-01

2.0E-01

3.0E-01

4.0E-01

5.0E-01

6.0E-01

1.00 0.800.90 0.70 0.60 0.400.50 0.30 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 10 D

0.0E+00

1.0E-02

2.0E-02

3.0E-02

4.0E-02

5.0E-02

6.0E-02

1.00 0.800.90 0.70 0.60 0.400.50 0.30 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

9 (Burke, 2011) 
Hydraulic Diffusivity Results 

Permeability

(Darcy)

Lower Bound

(Years)

Average 

(Years)

Upper Bound 

(Years)

10.0 D 1.0E-3.70 1.0E-2.50 1.0E-1.63 

1.0 D 1.0E-2.70 1.0E-1.50 1.0E-0.31

1.0 mD 1.0E+0.30 1.0E+1.50 1.0E+2.68 

1.0 μD 1.0E+3.30 1.0E+4.50 1.0E+5.68 

1.0 nD 1.0E+6.30 1.0E+7.50 1.0E+8.68 

1.0 pD 1.0E+9.30 1.0E+10.50 1.0E+11.68 

Several days to several weeks 

Up to six months 

Several hundred years, 500 years 

Several thousand years 

Hundreds of millions of years 

Billions of years 

-6

-4

-2

0

2

4

6

8

10

12

14

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Fractional Porosity (unitless)

Logarithm of 

Time Scale (years) 

10 D 1.0 D 1.0 mD 1.0 μD 1.0 nD 1.0 pD 10 (Burke, 2011) 
Darcy’s Law of Fluid Flow

10.0 D 

1.0 mD 

1.0 D 

1.0 μD

1.0 nD 

1.0 pD 

Permeability: 1.0 μD 

1.0E-01

1.0E+04

2.0E+04

3.0E+04

4.0E+04

5.0E+04

6.0E+04

7.0E+04

1.00 0.800.90 0.70 0.60 0.400.50 0.30 0.100.20 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 mD 

1.0E-01

1.0E+01

2.0E+01

3.0E+01

4.0E+01

5.0E+01

6.0E+01

7.0E+01

1.00 0.90 0.700.80 0.60 0.50 0.40 0.200.30 0.10 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 nD 

1.0E-01

1.0E+07

2.0E+07

3.0E+07

4.0E+07

5.0E+07

6.0E+07

7.0E+07

1.00 0.90 0.700.80 0.60 0.50 0.40 0.200.30 0.10 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 pD 

1.0E-01

1.0E+10

2.0E+10

3.0E+10

4.0E+10

5.0E+10

6.0E+10

7.0E+10

1.00 0.90 0.700.80 0.60 0.50 0.40 0.200.30 0.10 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 1.0 D 

0.0E+00

1.0E-02

2.0E-02

3.0E-02

4.0E-02

5.0E-02

6.0E-02

7.0E-02

1.00 0.90 0.700.80 0.60 0.50 0.40 0.200.30 0.10 0.00

Fractional Porosity (unitless)

Time Scale (years)

Permeability: 10 D 

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

7.0E-03

1.00 0.90 0.700.80 0.60 0.50 0.40 0.200.30 0.10 0.00

Fractional Porosity (unitless)

Time Scale (years)

11 (Burke, 2011) 
Darcy’s Law of Fluid Flow

Permeability

(Darcy)

Lower Bound 

(Years)

Average 

(Years)

Upper Bound 

(Years)

10.0 D 1.0E-3.50 1.0E-2.60 1.0E-2.22 

1.0 D 1.0E-2.50 1.0E-1.60 1.0E-1.22 

1.0 mD 1.0E+0.50 1.0E+1.40 1.0E+1.77 

1.0 μD 1.0E+3.50 1.0E+4.40 1.0E+4.77 

1.0 nD 1.0E+6.50 1.0E+7.40 1.0E+7.77 

1.0 pD 1.0E+9.50 1.0E+10.40 1.0E+10.77 

Several days to weeks 

Several months 

6 months up to 60 years 

Hundreds to several thousands of years 

Tens of millions of years 

Billions of years 

-6

-4

-2

0

2

4

6

8

10

12

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Fractional Porosity (unitless)

Logarithm of 

Time Scale (years) 

10 D 1.0 D 1.0 mD 1.0 μD 1.0 nD 1.0 pD 12 (Burke, 2011) 
Permeability Classifications 

Classification Permeability Range 

(Darcy) 

Class I Class I ≥ 1.0 D 

Class II 1.0 D ≥ Class II ≥1.0 mD 

Class III Class III ≤ 1.0 mD 

13
Formation Injectivity 

• The Oilfield Glossary (Schlumberger, 2012) defines 

an injectivity test as a procedure that is used to 

determine “the rate and pressure at which fluids can 

be pumped into the treatment target without 

fracturing the formation.”

• According to Craft and Hawkins (1991) an injectivity 

index quantifies the pressure increase due to 

pumping a known rate and volume of fluids into the 

formation, and is the ratio of the injection flow rate 

divided by the pressure increase. 

14
Formation Injectivity 

• Injectivity describes the pressure increase due to the 

introduction of fluids into a formation. 

• The most interesting and relevant application of 

injectivity is to determine the pressure increase that 

will fracture the reservoir rocks. This is related to 

the fracture gradient. 

15
Fracture Gradients 

16 (Modified from Eaton, 1969) 
Pressure Gradients 

17 (Burke, 2011) 
Maximum Pressure Differential 

 HFP DDD

fracture gradient as a function of depth, and 

FD

HD

original reservoir pressure gradient 

or hydrostatic gradient as a function of depth 

Maximum Pressure Differential, , evaluated at a specific depth, , 

is defined as: 

PD

D

This relation assumes that the formation is not already fractured due to 

overpressuring, and that will always be a positive value, that is, increasing 

The pressure differential will not close the fractures. 

PD

18 (Burke, 2011) 
Conclusions 

• Quantification of the first-order approximations of the 

time scales involved in the lateral migration of 

sequestered CO2 through a given volume of rock 

enables a general estimation of the containment 

timeframes of the sequestered gas. This study 

investigated these time scales for formations exhibiting 

permeabilities from 10.0 darcy to 1.0 picodarcy and 

porosities from 0.05 to 0.95. 

19
Conclusions (2) 

• Fluid flow modeling for determining fluid migration time scales 

– Calculate generalized time scales of lateral CO2 fluid 

migration, given information about average reservoir 

temperature, pressure, permeability, and porosity. 

– Hydraulic diffusivity time scales exhibit hyperbolic decay 

contours; Darcy fluid flow time scales exhibit decreasing 

linear trends. 

– The orders of magnitude can be approximated as linear over 

a wide range of permeability-porosity values. 

– Similar order of magnitude results for diffusivity and Darcy 

flow suggest that these first-order approximations, derived 

from two separate equations with different input values, 

yield a reliable estimation of the CO2 lateral migration time 

scales. 

20
Conclusions (3) 

• Formations categorized by: 

– Class I permeability may not provide adequate, long-term 

containment of sequestered CO2 in the absence of physical 

trapping mechanisms. Fluid migration occurs on the order 

of days to weeks. 

– Class II permeability represents the most favorable scenario 

for injection and containment of CO2. The order of 

magnitude for 1.0-km lateral migration of carbon dioxide 

through a given volume of rock ranges from several years to 

several thousand years. 

– Class III permeability may not represent viable injection 

targets without formation treatments such as hydraulic 

fracturing or permeability enhancement. Lateral fluid 

migration occurs on the order of several hundreds to several 

hundred thousands of years. 

21
Acknowledgments 

 I thank several of my colleagues at the U.S. Geological 

Survey for their collaboration and support in this study, 

especially: 

 S. Brennan, R. Burruss, M. Buursink, R. Charpentier, 

 T. Cook, R. Drake, R. Dubiel, P. Freeman, H. 

Jahediesfanjani, S. Kinney, M. Merrill, P. Nelson, L. 

Ruppert, and P. Warwick. 

22
References Cited 

Brennan, S.T., Burruss, R.C., Merrill, M.D., Freeman, P.A., and Ruppert, L.F., 2010, A probabilistic 

assessment methodology for the evaluation of geologic carbon dioxide storage: U.S. Geological Survey 

Open-File Report 2010–1127, 31 p., accessed December 10, 2010, at 

http://pubs.usgs.gov/of/2010/1127. 

Burke, L., 2011, Carbon dioxide fluid-flow modeling and injectivity calculations: U.S. Geological Survey 

Scientific Investigative Report 2011-5083, 16 p., accessed January 10, 2012 at 

http://pubs.usgs.gov/sir/2011/5083/sir2011-5083.pdf 

Burruss, R.C., Brennan, S.T., Freeman, P.A., Merrill, M.D., Ruppert, L.F., Becker, M.F., Herkelrath, W.N., 

Kharaka, Y.K., Neuzil, C.E., Swanson, S.M., Cook, T.A., Klett, T.R., Nelson, P.H., and Schenk, C.J., 

2009, Development of a probabilistic assessment methodology for evaluation of carbon dioxide 

storage: U.S. Geological Survey Open-File Report 2009–1035, 81 p., accessed January 11, 2011, at 

http://pubs.usgs.gov/of/2009/1035/.

Craft, B.C., and Hawkins, M.F, 1991, Applied petroleum reservoir engineering, 2nd ed., revised by R.E. 

Terry: Englewood Cliffs, N.J., Prentice Hall, 431 p. 

Eaton, B.A., 1969, Fracture gradient prediction and its application in oilfield operations: Journal of 

Petroleum Technology, v. 21, no. 10, p. 1353 – 1360. 

Lemmon, E.W, McLinden, M.O., and Friend, D.G., 2011, Thermophysical properties of fluid systems in 

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Mallard, W.G., eds., National Institute of Standards and Technology: Gaithersburg, MD, accessed 

January 7, 2011, at http://webbook.nist.gov/chemistry/fluids. 

Schlumberger, 2012, Schlumberger oilfield glossary: Normal pressure, accessed January 4, 2012, at 

 http://www.glossary.oilfield.slb.com/Display.cfm?Term=normal%20pressure. 

Sheriff, R.E.,1994, Encyclopedia dictionary of exploration geophysics: Society of Exploration 

Geophysicists, Tulsa, OK, 1994. 

Span, R., and Wagner, W., 1996, A new equation of state for carbon dioxide covering the fluid region from 

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23