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Approximate Solutions for Pressure Buildup During CO2 Injection in Brine Aquifers

Abstract

If geo-sequestration of CO 2 is to be employed as a key emissions reduction method in the global effort to mitigate against climate change, simple yet robust screening of the risks of disposal in brine aquifers will be needed. There has been significant development of simple analytical and semi-analytical techniques to support screening analysis and performance assessment for potential carbon sequestration sites. These techniques have generally been used to estimate the size of CO 2 plumes for the purpose of leakage rate estimation. A common assumption is that both the fluids and the geological formation are incompressible. Consequently, calculation of pressure distribution requires the specification of an arbitrary radius of influence. In this article, a new similarity solution is derived using the method of matched asymptotic expansions. A large time approximation of this solution is then extended to account for inertial effects using the Forchheimer equation. By allowing for slight compressibility in the fluids and formation, the solutions improve on previous work by not requiring the specification of an arbitrary radius of influence. The validity of both solutions is explored by comparison with equivalent finite difference solutions, revealing that the new method can provide robust and mathematically rigorous solutions for screening level analysis, where numerical simulations may not be justified or cost effective.

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Abbreviations

b :

Forchheimer parameter [L−1].

c o :

Compressibility of CO2 [M−1LT2].

c r :

Compressibility of geological formation [M−1LT2].

c w :

Compressibility of brine [M−1LT2].

g :

Gravitational acceleration [LT−2].

h :

CO2 brine interface elevation [L].

h D = h/H :

Dimensionless interface elevation [–].

H :

Formation thickness [L].

k :

Permeability [L2].

M 0 :

Mass injection rate [MT−1].

p :

Fluid pressure [ML−1T−2].

p D = 2π H ρ o kp/M 0 μ o :

Dimensionless pressure [–].

q o :

CO2 flux [LT−1].

q oD = 2π Hr w ρ o q o/M 0 :

Dimensionless CO2 flux [–].

q w :

Brine flux [LT−1].

q wD = 2π Hr w ρ o q w/M 0 :

Dimensionless brine flux [–].

r :

Radial distance [L].

r D = r/r w :

Dimensionless radius [–].

r w :

Well radius [L].

S = S s H :

Storativity [–].

\({S_{\rm s}=\rho_{\rm w}g\phi(c_{\rm w}+c_{\rm r})}\) :

Specific storage [L−1].

t :

Time [T].

\({t_{\rm D}=M_0t/2\pi\phi Hr_{\rm w}^2\rho_{\rm o}}\) :

Dimensionless time [–].

T = ρ w g kHw :

Transmissivity [L2T−1].

α = M 0 μ o(c r + c w)/2π H ρ o k :

Dimensionless compressibility [–].

β = M 0 kb/2π H r w μ o :

Dimensionless Forchheimer parameter [–].

γ = μ o/μ w :

Viscosity ratio [–].

\({\epsilon=(c_{\rm o}-c_{\rm w})/(c_{\rm r}+c_{\rm w})}\) :

Normalized fluid compressibility difference [–].

κ≈ 0.5772:

Euler-Mascheroni constant [–].

μ o :

Viscosity of CO2 [ML−1T−1].

μ w :

Viscosity of brine [ML−1T−1].

ρ o :

Density of CO2 [ML−3].

ρ w :

Density of brine [ML−3].

σ = ρ o/ρ w :

Density ratio [–].

\({\phi}\) :

Porosity [–].

References

  1. Bachu S.: CO 2 storage in geological media: role, means, status and barriers to deployment. Prog. Energy Combust. Sci. 34, 254–273 (2008)

  2. Bear J.: Hydraulics of Groundwater. McGraw-Hill, New York (1979)

  3. Benson S., Cook P.: Underground geological storage. In: Metz, B., Davidson, O., de Coninck, H., Loos, M., Meyer, L. (eds) IPCC Special Report on Carbon Dioxide Capture and Storage, pp. 195–276. Cambridge University Press, Cambridge (2005)

  4. Bickle M., Chadwick A., Huppert H.E., Hallworth M., Lyle S.: Modelling carbon dioxide accumulation at Sleipner: implications for underground carbon storage. Earth Planet Sci. Lett. 255, 164–176 (2007)

  5. Blunt M., King P.: Relative permeabilities from two- and three-dimensional porescale network modeling. Transp. Porous Media 6, 407–433 (1991)

  6. Damen K., Faaij A., Turkenburg W.: Health, safety and environmental risks of underground CO 2 storage—overview of mechanisms and current knowledge. Clim. Change 74, 289–318 (2006)

  7. DEFRA:. The Scientific Case for Setting a Long-Term Emission Reduction Target, UK (2003)

  8. Doughty C.: Modeling geologic storage of carbon dioxide: comparison of non-hysteretic and hysteretic characteristic curves. Energy Conv. Manag. 48, 1768–1781 (2007)

  9. Doughty C., Pruess K.: Modeling supercritical carbon dioxide injection in heterogeneous porous media. Vadose Zone J. 3, 837–847 (2004)

  10. EPRI: The Power to Reduce CO 2 Emissions. Electric Power Research Institute, Palo Alto, CA (2007)

  11. Forchheimer P.: Wasserbewegung durch Boden. Z. Ver. Deutsch Ing. 45, 1782–1788 (1901)

  12. Gasda S.E., Celia M.A., Nordbotten J.M.: Upslope plume migration and implications for geological CO 2 sequestration in deep, saline aquifers. IES J. Part A: Civ. Struct. Eng. 1(1), 2–16 (2008)

  13. Geertsma, J.: Estimating the coefficient of inertial resistance in fluid flow through porous media. SPE J. pp. 445–450, SPE Paper No. 4706 (1974)

  14. Giorgis T., Carpita M., Battistelli A.: 2D modeling of salt precipitation during the injection of dry CO 2 in a depleted gas reservoir. Energy Conv. Manag. 48, 1816–1826 (2007)

  15. Hesse M.A., Tchelepi H.A., Cantwell B.J., Orr F.M. Jr: Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577, 363–383 (2007)

  16. House K.Z., Schrag D.P., Harvey C.F., Lackner K.S.: Permanent carbon dioxide storage in deep-sea sediments. Proc. Natl. Acad. Sci. 103(33), 12,291–12,295 (2006)

  17. IEA: World Energy Outlook 2007—China and India Insights. International Energy Agency, Paris, France (2007)

  18. IPCC: Climate Change 2007: Mitigation. Contribution of Working Group III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, UK (2007a)

  19. IPCC: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, UK (2007b)

  20. Juanes R., Spiteri E.J., Orr F.M. Jr, Blunt M.J.: Impact of relative permeability hysteresis on geological CO 2 storage. Water Resour. Res. 42, W12,418 (2006)

  21. Kevorkian J.: Partial Differential Equations. Thompson Information/Publishing Group, Pacific Grove, CA (1990)

  22. Korre A., Shi J.Q., Imrie C., Grattoni C., Durucan S.: Coalbed methane reservoir data and simulator parameter uncertainty modelling for CO 2 storage performance assessment. Int. J. Greenhouse Gas Control 1, 492–501 (2007)

  23. Lagneau V., Pipart A., Catalette H.: Reactive transport modelling of CO 2 sequestration in deep saline aquifers. Oil Gas Sci. Tech. Rev. IFP 60(2), 231–247 (2005)

  24. LeNeveu D.M.: CQUESTRA, a risk and performance assessment code for geological sequestration of carbon dioxide. Energy Conv. Manag. 49, 32–46 (2008)

  25. Lyle S., Huppert H.E., Hallworth M., Bickle M., Chadwick A.: Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543, 293–302 (2005)

  26. Mathias S.A., Butler A.P., Zhan H.: Approximate solutions for Forchheimer flow to a well. J. Hydraul. Eng. 134(9), 1318–1325 (2008)

  27. Nordbotten J.M., Celia M.A.: Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307–327 (2006)

  28. Nordbotten J.M., Celia M.A., Bachu S.: Injection and storage of CO 2 in deep saline aquifers: analytic solution for CO 2 plume evolution during injection. Transp. Porous Media 58, 339–360 (2005)

  29. Nordbotten J.M., Celia M.A., Bachu S., Dahle H.K.: Semianalytical solution for CO 2 leakage through an abandoned well. Environ. Sci. Technol. 39, 602–611 (2005)

  30. Oldenburg C.M.: Screening and ranking framework for geologic CO 2 storage site selection on the basis of health, safety, and environmental risk. Environ. Geol. 54(8), 1687–1694 (2007)

  31. Oldenburg C.M., Unger A.J.A.: On leakage and seepage from geologic carbon sequestration sites: unsaturated zone attenuation. Vadose Zone J. 2, 287–296 (2003)

  32. Oldenburg C.M., Unger A.J.A.: Coupled vadose zone and atmospheric surface-layer transport of carbon dioxide from geologic carbon sequestration sites. Vadose Zone J. 3, 848–857 (2004)

  33. Pruess K.: Numerical studies of fluid leakage from a geologic disposal reservoir for CO 2 show self-limiting feedback between fluid flow and heat transfer. Geophys. Res. Lett. 32, L14,404 (2005)

  34. Pruess K., Garcia J.: Multiphase flow dynamics during CO 2 injection into saline aquifers. Environ. Geol. 42, 282–295 (2002)

  35. Pruess K., Spycher N.: ECO 2N—a fluid property module for the TOUGH2 code for studies of CO 2 storage in saline aquifers. Energy Conv. Manag. 48, 1761–1767 (2007)

  36. Pruess K., Garcia J., Kovscek T., Oldenburg C.: Code intercomparison builds confidence in numerical simulation models for geologic disposal of CO 2. Energy 29, 1431–1444 (2004)

  37. Roose T., Fowler A.C., Darrah P.R.: A mathematical model of plant nutrient uptake. J. Math. Biol. 42, 347–360 (2001)

  38. Saripalli P., McGrail P.: Semi-analytical approaches to modeling deep well injection of CO 2 for geological sequestration. Energy Conv. Manag. 43(2), 185–198 (2002)

  39. Shampine L.F., Reichelt M.W.: The MATLAB ODE Suite. SIAM J. Sci. Comp. 18, 1–22 (1997)

  40. Shampine L.F., Reichelt M.W., Kierzenka J.A.: Solving index-1 DAEs in MATLAB and Simulink. SIAM J. Sci. Comp. 41, 538–552 (1999)

  41. Stauffer, P., Viswanathan, H., Guthrie, G., Pawar, R.: CO 2-PENS: a CO 2 sequestration systems model supporting risk-based decisions. In: Proceeding of CMWR XVI—Computational Methods in Water Resources, Copenhagen, Denmark (2006)

  42. Stern N.: Stern Review on the Economics of Climate Change. Cambridge University Press, Cambridge, UK (2006)

  43. Theis C.V.: The relationship between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground water storage. Trans. Amer. Geophys. Union 16, 519–524 (1935)

  44. Woods, E.G., Comer, A.G.: Saturation and injection pressure for a radial gas storage reservoir. J. Petroleum Tech. pp. 1389–1393, SPE Paper No. 401 (1962)

  45. Wu Y.S.: Numerical simulation of single-phase and multiphase non-Darcy flow in porous and fractured reservoirs. Transp. Porous Media 49(2), 1573–1634 (2002)

  46. Zhang Y., Oldenburg C.M., Finsterle S., Bodvarsson G.S.: System-level modeling for economic evaluation of geological CO 2 storage in gas reservoirs. Energy Conv. Manag. 48, 1827–1833 (2007)

  47. Zhou Q., Birkholzer J.T., Tsang C.F., Rutqvist J.: A method for quick assessment of CO 2 storage capacity in closed and semi-closed saline formations. Int. J. Greenhouse Gas Control 2(4), 626–639 (2008)

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Correspondence to Simon A. Mathias.

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Mathias, S.A., Hardisty, P.E., Trudell, M.R. et al. Approximate Solutions for Pressure Buildup During CO2 Injection in Brine Aquifers. Transp Porous Med 79, 265 (2009). https://doi.org/10.1007/s11242-008-9316-7

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Keywords

  • CO 2 injection
  • Forchheimer’s equation
  • Matched asymptotic expansions
  • Pressure buildup