The stability of poro-elastic wave equations in saturated porous media

Abstract

Poro-elastic wave equations are one of the fundamental problems in seismic wave exploration and applied mathematics. In the past few decades, elastic wave theory and numerical method of porous media have developed rapidly. However, the mathematical stability of such wave equations have not been fully studied, which may lead to numerical divergence in the wave propagation simulation in complex porous media. In this paper, we focus on the stability of the wave equation derived from Tuncay’s model and volume averaging method. By analyzing the stability of the first-order hyperbolic relaxation system, the mathematical stability of the wave equation is proved for the first time. Compared with existing poro-elastic wave equations (such as Biot’s theory), the advantage of newly derived equations is that it is not necessary to assume uniform distribution of pores. Such wave equations can spontaneously incorporate complex microscale pore/fracture structures into large-scale media, which is critical for unconventional oil and gas exploration. The process of proof and numerical examples shows that the wave equations are mathematically stable. These results can be applied to numerical simulation of wave field in reservoirs with pore/fracture networks, which is of great significance for unconventional oil and gas exploration.

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Source is located at (100 m, 100 m). Receiver is located at (15 m, 100 m)

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References

  1. Anderson TB, Jackson R (1967) A fluid mechanical description of fluidized beds. Ind Eng Chem Fundam 6(4):527

    Article  Google Scholar 

  2. Avseth P, Mukerji T, Mavko G (2010) Quantitative seismic interpretation: applying rock physics tools to reduce interpretation risk. Cambridge University Press, Cambridge

    Google Scholar 

  3. Bernabe Y (2009) Oscillating flow of a compressible fluid through deformable pipes and pipe networks: wave propagation phenomena. Pure Appl Geophys 166(5–7):969–994

    Article  Google Scholar 

  4. Bernabe Y (2009) Propagation of Biot slow waves in heterogeneous pipe networks: effect of the pipe radius distribution on the effective wave velocity and attenuation. J Geophys Res Solid Earth, vol 114

  5. Bernabe Y, Zamora M, Li M, Maineult A, Tang YB (2011) Pore connectivity, permeability, and electrical formation factor: a new model and comparison to experimental data. J Geophys Res Solid Earth, Vol 116

  6. Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164

    Article  Google Scholar 

  7. Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid 1. Low-frequency range. J Acoust Soc Am 28(2):168–178. https://doi.org/10.1121/1.1908239

    Article  Google Scholar 

  8. Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33(4):1482–1498

    Article  Google Scholar 

  9. Brie A, Pampuri F, Marsala A, Meazza O (1995) Shear sonic interpretation in gas-bearing sands. In: Paper read at SPE annual technical conference and exhibition

  10. Burridge R, Keller JB (1981) Poroelasticity equations derived from microstructure. J Acoust Soc Am 70(4):1140–1146

    Article  Google Scholar 

  11. Carcione JM, Cavallini F, Santos JE, Ravazzoli CL, Gauzellino PM (2004) Wave propagation in partially saturated porous media: simulation of a second slow wave. Wave Motion 39(3):227–240

    Article  Google Scholar 

  12. Carcione JM, Picotti S, Gei D, Rossi G (2006) Physics and seismic modeling for monitoring CO2 storage. Pure Appl Geophys 163(1):175–207

    Article  Google Scholar 

  13. Champoux Y, Allard JF (1991) Dynamic tortuosity and bulk modulus in air-saturated porous media. J Appl Phys 70(4):1975–1979

    Article  Google Scholar 

  14. Chapman M, Zatsepin SV, Crampin S (2002) Derivation of a microstructural poroelastic model. Geophys J Int 151(2):427–451

    Article  Google Scholar 

  15. De la Cruz V, Sahay PN, Spanos TJT (1993) Thermodynamics of porous-media. Proc R Soc Math Phys Sci 443(1917):247–255

    Google Scholar 

  16. De la Cruz V, Spanos TJT (1983) Mobilization of Oil Ganglia. Aiche J 29(5):854–858

    Article  Google Scholar 

  17. De la Cruz V, Spanos TJT (1985) Seismic-wave propagation in a porous-medium. Geophysics 50(10):1556–1565

    Article  Google Scholar 

  18. Dutta NC, Ode H (1979) Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (white model). 2. Results. Geophysics 44(11):1789–1805

    Article  Google Scholar 

  19. Dvorkin J, Mavko G, Nur A (1995) Squirt flow in fully saturated rocks. Geophysics 60(1):97–107

    Article  Google Scholar 

  20. Dvorkin J, Nur A (1993) Dynamic Poroelasticity—a Unified Model with the Squirt and the Biot Mechanisms. Geophysics 58(4):524–533

    Article  Google Scholar 

  21. Johnson DL (2001) Theory of frequency dependent acoustics in patchy-saturated porous media. J Acoust Soc Am 110(2):682–694. https://doi.org/10.1121/1.1381021

    Article  Google Scholar 

  22. Johnson DL, Koplik J, Dashen R (2006) Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J Fluid Mech 176:379–402

    Article  Google Scholar 

  23. Kelder O, Smeulders DM (1997) Observation of the Biot slow wave in water-saturated Nivelsteiner sandstone. Geophysics 62(6):1794–1796

    Article  Google Scholar 

  24. Kirchhoff G (1868) Ueber den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung. Ann Phys 210(6):177–193

    Article  Google Scholar 

  25. Liu JW, Yong WA (2016) Stability analysis of the Biot/squirt models for wave propagation in saturated porous media. Geophys J Int 204(1):535–543. https://doi.org/10.1093/gji/ggv463

    Article  Google Scholar 

  26. Mavko GM, Nur A (1979) Wave attenuation in partially saturated rocks. Geophysics 44(2):161–178

    Article  Google Scholar 

  27. Pride SR, Berryman JG (2003) Linear dynamics of double-porosity dual-permeability materials. I. Governing equations and acoustic attenuation. Phys Rev E 68(3):036603

    Article  Google Scholar 

  28. Pride SR, Gangi AF, Morgan FD (1992) Deriving the equations of motion for porous isotropic media. J Acoust Soc Am 92(6):3278–3290

    Article  Google Scholar 

  29. Sahay PN, Spanos TJTT, de la Cruz V (2001) Seismic wave propagation in inhomogeneous and anisotropic porous media. Geophys J Int 145(1):209–222

    Article  Google Scholar 

  30. Shields FD, Lee KP, Wiley WJ (1965) Numerical solution for sound velocity and absorption in cylindrical tubes. J Acoust Soc Am 37(4):724–729

    Article  Google Scholar 

  31. Slattery JC (1967) Flow of viscoelastic fluids through porous media. AIChE J 13(6):1066. https://doi.org/10.1002/aic.690130606

    Article  Google Scholar 

  32. Sun W, Ba J, Carcione JM (2016) Theory of wave propagation in partially saturated double-porosity rocks: a triple-layer patchy model. Geophys J Int 205(1):22–37

    Article  Google Scholar 

  33. Sun W, Xiong F, Ba J, Carcione JM (2018) Effects of ellipsoidal heterogeneities on wave propagation in partially saturated double-porosity rocks. Geophysics 83(3):71–81

    Article  Google Scholar 

  34. Sun W, Yang H (2003) Elastic wavefield calculation for heterogeneous anisotropic porous media using the 3-D irregular-grid finite-difference. Acta Mech Solida Sin 4(16):283–299

    Google Scholar 

  35. Sun W, Yang H (2004) Seismic propagation simulation in complex media with non-rectangular irregular-grid finite-difference. Acta Mech Sin 20(3):299–306

    Article  Google Scholar 

  36. Tuncay K, Corapcioglu MY (1995) Effective stress principle for saturated fractured porous media. Water Resour Res 31(12):3103–3106

    Article  Google Scholar 

  37. Tuncay K, Corapcioglu MY (1996a) Body waves in poroelastic media saturated by two immiscible fluids. J Geophys Res Solid Earth 101(B11):25149–25159. https://doi.org/10.1029/96jb02297

    Article  Google Scholar 

  38. Tuncay K, Corapcioglu MY (1996b) Wave propagation in fractured porous media. Transp Porous Media 23(3):237–258

    Google Scholar 

  39. Tuncay K, Corapcioglu MY (1997) Wave propagation in poroelastic media saturated by two fluids. J Appl Mech Trans Asme 64(2):313–320

    Article  Google Scholar 

  40. Whitaker S (1966) The equations of motion in porous media. Chem Eng Sci 21(3):291–300

    Article  Google Scholar 

  41. White JE (1975) Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics 40(2):224–232. https://doi.org/10.1190/1.1440520

    Article  Google Scholar 

  42. Yong WA (1999) Singular perturbations of first-order hyperbolic systems with stiff source terms. J Differ Equ 155(1):89–132

    Article  Google Scholar 

  43. Zwikker C, Kosten CW (1949) Sound absorbing materials. Elsevier, Amsterdam

    Google Scholar 

Download references

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant No. 41874137, 42074144), and National Key R & D Program of China (2018YFA0702501).

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Correspondence to Weitao Sun.

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Communicated by Michal Malinowski (CO-EDITOR-IN-CHIEF)/Liang Xiao (ASSOCIATE EDITOR).

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Appendix

Appendix

The matrices involved in the proof of the theorem in “Mathematical stability of poro-elastic wave equations” section are listed here.

$${\mathbf{\rm M}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\langle \rho_{s} \rangle } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\langle \rho_{f} \rangle } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\langle \rho_{s} \rangle } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\langle \rho_{f} \rangle } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\langle \rho_{s} \rangle } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\langle \rho_{f} \rangle } \\ \end{array} } \right],$$
(53)
$${\mathbf{J}}_{1} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 \\ { - a_{11}^{*} } & 0 & 0 & 0 & { - a_{12} } & 0 & 0 & 0 & 0 & 0 & 0 \\ { - a_{21} } & 0 & 0 & 0 & { - a_{22} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & G & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],$$
(54)
$${\mathbf{J}}_{2} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 \\ 0 & { - G} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - a_{11}^{*} } & 0 & 0 & 0 & { - a_{12} } & 0 & 0 & 0 & 0 & 0 & 0 \\ { - a_{21} } & 0 & 0 & 0 & { - a_{22} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & G & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],$$
(55)
$${\mathbf{J}}_{3} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} \\ 0 & 0 & { - G} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - G} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - a_{11}^{*} } & 0 & 0 & 0 & { - a_{12} } & 0 & 0 & 0 & 0 & 0 & 0 \\ { - a_{21} } & 0 & 0 & 0 & { - a_{22} } & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],$$
(56)
$${\mathbf{N}} = b\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - 1} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & { - 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & { - 1} \\ \end{array} } \right],$$
(57)
$${\mathbf{W}} = \left[ {\begin{array}{*{20}c} {a_{11}^{*} } & 0 & 0 & 0 & {a_{12} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & G & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & G & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {a_{12} } & 0 & 0 & 0 & {a_{22} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\langle \rho_{s} \rangle }}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\langle \rho_{f} \rangle }}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\langle \rho_{s} \rangle }}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\langle \rho_{f} \rangle }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\langle \rho_{s} \rangle }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\langle \rho_{f} \rangle }}} \\ \end{array} } \right].$$
(58)

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Xiong, F., Sun, W. & Liu, J. The stability of poro-elastic wave equations in saturated porous media. Acta Geophys. (2020). https://doi.org/10.1007/s11600-020-00508-y

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Keywords

  • Wave equations
  • Porous media
  • Volume averaging
  • Stability analysis