Abstract
Wave anelasticity of the fast P wave at mesoscopic scales is due to energy dissipation by conversion to slow P diffusive modes at heterogeneities much smaller than the wavelength and much larger than the pore size. We consider frames composed of two minerals and study the dissipation effects based on a generalized White plane-layered model, where the interfaces satisfy mixed boundary conditions, i.e., open, closed and partially-open pores. We consider three models to obtain the effective properties. Model 1 is based on effective mineral properties, Model 2 is a generalization of Biot theory to the case of two solids and one fluid, and Model 3 is based on a generalization of White model to the case of three layers. A particular case is that of closed pores at the interface between the layers, where no flow occurs and, consequently, there is no anelasticity and the stiffness modulus is a real quantity and does not depend on frequency. The type of boundary condition highly affects the location of the relaxation peak, which moves from high to low frequencies for decreasing interface permeability. The first two models predict similar locations of the peaks and strength (reciprocal of the quality factor). The results of Model 3 differ due to different distribution of the solid phases, since the frames are not mixed at the pore scale as in Models 1 and 2, but at a mesoscopic scale. These solutions are useful to test modeling algorithms to compute the effective P-wave modulus in more general cases.
Article highlights
-
The frames composed of two minerals are considered to analyze the dissipation effects based on a generalized White plane-layered model.
-
The three different models are analyzed to obtain the properties perpendicular to layering.
-
The type of boundary condition highly affects the location of the relaxation peak, which moves from high to low frequencies for decreasing interface permeability.
Similar content being viewed by others
References
Ba J, Zhao J, Carcione JM, Huang X (2016) Compressional wave dispersion due to rock matrix stiffening by clay squirt flow. Geophys Res Lett 43(12):6186–6195
Ba J, Xu WH, Fu LY, Carcione JM, Zhang L (2017) Rock anelasticity due to patchy saturation and fabric heterogeneity: a double double-porosity model of wave propagation. J Geophys Res Solid Earth 122(3):1949–1976
Backus GE (1962) Long-wave elastic anisotropy produced by horizontal layering. J Geophys Res 67:4427–4440
Brown RJS, Korringa J (1975) On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid. Geophysics 40:608–616
Carcione JM (2014) Wave Fields in Real Media: theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media, 3rd edn. Elsevier, Amsterdam
Carcione JM, Picotti S (2006) P-wave seismic attenuation by slow-wave diffusion: effects of inhomogeneous rock properties. Geophysics 71(3):O1–O8
Carcione JM, Gurevich B, Cavallini F (2000) A generalized Biot–Gassmann model for the acoustic properties of shaley sandstones. Geophys Prosp 48:539–557
Carcione JM, Santos JE, Ravazzoli CL, Helle HB (2003) Wave simulation in partially frozen porous media with fractal freezing conditions. J Appl Phys 94:7839–7847
Carcione JM, Helle HB, Santos JE, Ravazzoli CL (2005) A constitutive equations and generalized Gassmann modulus for multi-mineral porous media. Geophysics 70:N17–N26
Carcione JM, Picotti S, Gei D, Rossi G (2006) Physics and seismic modeling for monitoring CO2 storage. Pure Appl Geophys 163(1):175–207
Carcione JM, Morency C, Santos JE (2010) Computational poroelasticity — a review. Geophysics 75(5):75A229–75A243
Cavallini F, Carcione JM, Vidal de Ventós D, Engell-Sørensen L (2017) Low frequency dispersion and attenuation in anisotropic partially saturated rocks. Geophys J Int 209(3):1572–1584
Carcione JM, Poletto F, Farina B, Bellezza C (2018) 3D seismic modeling in geothermal reservoirs with a distribution of steam patch sizes, permeabilities and saturations, including ductility of the rock frame. Phys Earth Planet Inter 279:67–78
Deer WA, Howie RA, Zussman J (2013) An introduction to rock forming minerals, 3rd edition. The Mineralogical Society
Deresiewicz H, Skalak R (1963) On uniqueness in dynamic poroelasticity. Bull Seism Soc Am 53:783–788
Gei D, Carcione JM (2003) Acoustic properties of sediments saturated with gas hydrate, free gas and water. Geophys Prosp 51:141–157
Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140
Krief M, Garat J, Stellingwerff J, Ventre J (1990) A petrophysical interpretation using the velocities of P and S waves (full-waveform sonic). Log Anal 31:355–369
Mavko G, Mukerji T, Dvorkin J (2009) The rock physics handbook. Cambridge Univ, Press
Müller TM, Gurevich B, Lebedev M (2010) Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks: a review. Geophysics 75:75A147-75A164
Norris AN (1993) Low-frequency dispersion and attenuation in partially saturated rocks. J Acoust Soc Am 94:359–370
Pride SR, Harris JM, Johnson DL, Mateeva A, Nihel KT, Nowack RL, Rector JW, Spetzler H, Wu R, Yamomoto T, Berryman JG, Fehler M (2003) Permeability dependence of seismic amplitudes. Lead Edge 22(6):518–525
Qi Q, Müller TM, Gurevich B, Lopes S, Lebedev M, Caspari E (2014) Quantifying the effect of capillarity on attenuation and dispersion in patchy-saturated rocks. Geophysics 79(5):WB35–WB50
Rosenbaum JH (1974) Synthetic microseismograms: logging in porous formations. Geophysics 39:14–32
Santos JE, Ravazzoli CL, Carcione JM (2004) A model for wave propagation in a composite solid matrix saturated by a single-phase fluid. J Acoust Soc Am 115(6):2749–2760
Santos JE, Savioli GB, Carcione JM, Ba J (2019) Effect of capillarity and relative permeability on $Q$ anisotropy of hydrocarbon source rocks. Geophys J Int 218:1199–1209
White JE, Mikhaylova NG, Lyakhovitskiy FM (1975) Low-frequency seismic waves in fluid saturated layered rocks, Izvestija Academy of Sciences USSR. Phys Solid Earth 11:654–659
Zimmerman (1991) Compressibility of sandstones, Developments in Petroleum Science, vol 29. Elsevier, Amsterdam
Acknowledgement
The authors are grateful to the support of the National Natural Science Foundation of China (grant no. 41974123), the Jiangsu Innovation and Entrepreneurship Plan, China, and the Jiangsu Natural Science Fund for Distinguished Young Scholars, China.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: White’s plane-layer theory with partially-open pore boundary conditions
The model is a stack of two thin alternating porous layers of thickness \(d_1\) and \(d_2\), such that the period of the stratification is \(D = d_1 + d_2\) and the proportions of media 1 and 2 are \(p_1 = d_1/D\) and \(p_2 = 1 - p_1 = d_2/D\), respectively. Each layer is a porous medium composed of one solid and one fluid. The complex and frequency dependent P-wave stiffness for partially open boundary conditions is
where \(\bar{\kappa }\) is the hydraulic permeability (per unit length) of the interface [see Eq. (19) below for a demonstration], \(\omega\) is the angular frequency, and omitting the layer subindex j for clarity,
for each single layer (White et al. 1975; Carcione and Picotti 2006) [see also Carcione (2014, Eq. 7.453)], where \(\kappa\) is the permeability, \(\eta\) is the fluid viscosity, \(E_G = (p_1 / E_{G1} + p_2 / E_{G2})^{-1}\), \(E_{Gj} = K_{Gj} + (4/3) \mu _{mj}\) and \(E_{mj} = K_{mj} + (4/3) \mu _{mj}\), \(M_j^{-1} = (\alpha _j - \phi _j)/K_{sj}+ \phi _j/K_{fj}\), \(\alpha _j = 1 - K_{mj}/{K_{sj}}\), \(K_{Gj} = K_{mj} + \alpha _j^2 M_j\), with j = 1 and 2 being the two single layers, respectively. Modulus \(E_G\) is obtained at high frequencies or \(\bar{\kappa }\rightarrow 0\). If \(\bar{\kappa }\rightarrow \infty\), we have the case of open-pore boundary conditions, i.e., complete flow across the interfaces, which is the case of the classical White model.
Equation (17) is based on the boundary condition
Carcione 2014; Eq. 7.404, where \(p_{f1}\) and \(p_{f2}\) are the fluid pressures in layers 1 and 2, respectively, and \(w_3\) is the relative vertical displacement of the fluid with respect to the solid, which is continuous (the dot above a variable denotes time differentiation). It is straightforward to show that the discontinuity (19) in the fluid pressure at the interface leads to Eq. (17), based on Eqs. 7.443–7.447 in Section 7.13 of Carcione (2014).
The meaning of \(\bar{\kappa }\) is explained in Deresiewicz and Skalak (1963) as partially communicating pores between the two media, but can also be related to a thin layer at the interface characterized by the permeability \(\bar{\kappa }\). An example is a mud cake much thinner than the wavelength in a borehole (Rosenbaum 1974). A similar Eq. (17) was obtained by Qi et al. (2014), where \(1/\bar{\kappa }\) is interpreted as an additional interface impedance due to capillary forces, i.e., exclusively related to the fluids. However, the effect of capillary forces must be considered in the whole pore space to obtain realistic results, as in Santos et al. (2019). Biot’s theory does not hold when the rock is saturated by two-phase fluids, since capillary pressure effects and interaction between flows are ignored. Capillary pressure is responsible for the existence of the additional slow wave, where the relative motions between the two fluid phases induce additional energy losses not present in the case of single-phase fluids. These effects induce changes in phase velocities and dissipation factors.
The peak relaxation frequency is approximately given by
where \(\kappa\) and \(K_E\) are obtained harmonic (Reuss) averages, and \(\eta\) as arithmetic average.
The slow P wave is the cause of the attenuation, with a diffusivity constant \(D = \kappa M E_m / (\eta E_G)\) (Carcione 2014), and diffusion length \(L_r\) = \(\sqrt{D / \omega }\). The fluid pressures are equilibrated if \(L_r\) is comparable to the layer period. For small \(L_r\) for instance (high frequencies), there is not enough time for the pressures to equilibrate, causing anelasticity. Since \(\omega = 2 \pi f\) and f = \(D / (2 \pi L_r^2)\), substituting D into this equation, the transition frequency (20) is obtained for a diffusion length \(L_r = L_j/4\).
Appendix 2: Constitutive equations for a three-phase composite medium (two solids and one fluid)
The stress–strain relation has been derived by Carcione et al. (2005). We have
where \(\sigma\) denotes stress components, \(p_f\) is the fluid pressure, \(\theta\) denotes dilatations, \(\zeta = - \phi ( \theta _f - \beta _1 \theta _1 - \beta _2 \theta _2)\) is the variation of fluid content, \(d_{kl}\) are the components of the deviatoric strain tensor and \(\phi\) is the porosity.
Particularly, the relative displacement of the fluid relative to the solids is
where u denotes displacements, such that \(\zeta = - \mathrm{div} \ \mathbf{w}\), and the strain components are
The material properties are
where \(K_f\) is the fluid modulus and \(\eta\) is the fluid viscosity; \(\beta _i\) is the fraction of solid i per unit volume of total solid, and \(g_i\) are consolidation coefficients of the frames.
The total stress is
where \(\delta _{kl}\) is Kronecker’s delta.
The theory predicts two additional slow P waves and a slow S wave. More details can be found in Carcione et al. (2003).
Rights and permissions
About this article
Cite this article
Carcione, J.M., Santos, J.E. & Ba, J. Canonical solutions for wave anelasticity in rocks composed of two frames. Geomech. Geophys. Geo-energ. Geo-resour. 7, 60 (2021). https://doi.org/10.1007/s40948-021-00222-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40948-021-00222-z