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Canonical solutions for wave anelasticity in rocks composed of two frames

Wave anelasticity solutions in composite rocks

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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.

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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.

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Correspondence to Jing Ba.

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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

$$\begin{aligned} E = \left[ \frac{1}{E_G} + \frac{2 (r_2 - r_1)^2}{\mathrm{i}\omega D (I_1 + I_2 + 1/\bar{\kappa })} \right] ^{-1} , \end{aligned}$$
(17)

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,

$$\begin{aligned} r = \frac{\alpha M}{E_G}, \ \ \ I = \frac{\eta }{\kappa a} \mathrm{coth} \left( \frac{a d}{2} \right) , \ \ \ a^2 = \frac{\mathrm{i}\omega \eta }{\kappa K_E} , \ \ \ K_E = \frac{M E_m}{E_G} , \end{aligned}$$
(18)

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

$$\begin{aligned} p_{f1} - p_{f2} = \frac{1}{\bar{\kappa }} \dot{w}_3 \end{aligned}$$
(19)

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

$$\begin{aligned} f_p = \frac{8 \kappa K_{E}}{\pi \eta d^2} , \end{aligned}$$
(20)

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

$$\begin{aligned} \begin{array}{l} \sigma _{kl}^{(1)} = \left[ ( K_{G1} - \phi \alpha _1 \beta _1 M ) \theta _1 + M ( \alpha _1 - \phi \beta _1 ) \left( \alpha _2 \theta _2 - \zeta \right) \right] \\ \qquad \qquad + 2 \bar{\mu }_1 d_{kl}^{(1)} + \mu _{12} d_{kl}^{(2)} , \\ \\ \sigma _{kl}^{(2)} = \left[ ( K_{G2} - \phi \alpha _2 \beta _2 M ) \theta _2 + M ( \alpha _2 - \phi \beta _2 ) \left( \alpha _1 \theta _1 - \zeta \right) \right] \\ \qquad \qquad + 2 \bar{\mu }_2 d_{kl}^{(2)} + \mu _{12} d_{kl}^{(1)} , \\ \\ p_f = M ( \zeta - \alpha _1 \theta _1 - \alpha _2 \theta _2 ) , \end{array} \end{aligned}$$
(21)

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

$$\begin{aligned} w_k = \phi _w [ u_k^{(f)} - ( \beta _1 u_k^{(1)} +\beta _2 u_k^{(2)} ) ] , \end{aligned}$$
(22)

where u denotes displacements, such that \(\zeta = - \mathrm{div} \ \mathbf{w}\), and the strain components are

$$\begin{aligned} 2 \epsilon _{kl}^{(i)} = u_{k,l}^{(i)} + u_{l.k}^{(i)} , \ \ \ \ \ d_{kl}^{(i)} = \epsilon _{kl}^{(i)} - \frac{1}{3} \delta _{kl} \theta _i , \ \ \ \ \ \epsilon _{kk}^{(i)} = \theta _i . \end{aligned}$$
(23)

The material properties are

$$\begin{aligned} \begin{array}{l} K_{G1} = K_{m1} + \alpha _1^2 M , \ \ \ K_{G2} = K_{m2} + \alpha _2^2 M , \\ \\ M = \left( \displaystyle \frac{\alpha _1 - \beta _1 \phi }{K_1} + \displaystyle \frac{\alpha _2 - \beta _2 \phi }{K_2} + \displaystyle \frac{\phi }{K_f} \right) ^{-1} , \\ \\ \bar{\mu }_1 = [(1-g_1) \phi _1]^2 \bar{\mu }+ \mu _{m1} , \ \ \ g_1 = \mu _{m1} / (\phi _1 \mu _1), \\ \\ \bar{\mu }_2 = [(1-g_2) \phi _2]^2 \bar{\mu }+ \mu _{m2} , \ \ \ g_2 = \mu _{m2} / (\phi _2 \mu _2), \\ \\ \mu _{12} = (1-g_1) (1-g_2) \phi _1 \phi _2 \bar{\mu }, \\ \\ \bar{\mu }= [( 1 - g_1) \phi _1 / \mu _1 + \phi / (\mathrm{i}\omega \eta ) + (1 - g_2 ) \phi _2 / \mu _2 ]^{-1} , \\ \\ \alpha _i = \beta _i - \displaystyle \frac{K_{mi}}{K_i} , \ \ \ \beta _i = \displaystyle \frac{\phi _i}{1 - \phi } , \ \ \ \beta _1 + \beta _2 = 1, \end{array} \end{aligned}$$
(24)

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

$$\begin{aligned}&\sigma _{kl} = \sigma _{kl}^{(1)} + \sigma _{kl}^{(2)} - \phi p_f \delta _{kl} = [ ( K_{G1} + \alpha _1 \alpha _2 M ) \theta _1 + ( K_{G2} \nonumber \\&\qquad + \alpha _1 \alpha _2 M ) \theta _2 - M ( \alpha _1 + \alpha _2 ) \zeta ] \delta _{kl} \nonumber \\&\qquad + ( 2 \bar{\mu }_1 + \mu _{12} ) d_{kl}^{(1)} + ( 2 \bar{\mu }_{2} + \mu _{12} ) d_{kl}^{(2)} , \end{aligned}$$
(25)

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).

Table 1 Material properties.
Table 2 Material properties.
Fig. 1
figure 1

Three models of mesoscopic attenuation. Flow through layers generates the Biot slow (diffusive) wave, which is responsible of the attenuation of the fast P wave by conversion to slow P wave. Subindices ij denote solid i in layer j.

Fig. 2
figure 2

Dispersion coefficient and dissipation factor, corresponding to Models 1 and 2 for different values of the interface hydraulic permeability [\(\bar{\kappa }\) = 0 (closed pores), \(\infty\) (open pores) and 10\(^{-14}\) m\(^2\) s/kg (mixed)]. The properties of the media are listed in Table 1 (A= 3). The peaks at low frequencies correspond to the mixed boundary condition

Fig. 3
figure 3

Dispersion coefficient and dissipation factor, corresponding to Model 1 for different values of the interface hydraulic permeability (units are m\(^2\) s/kg). The properties of the media are listed listed in Table 1 (A= 3)

Fig. 4
figure 4

Comparison of the dispersion coefficient and dissipation factor for the three models. The boundary condition at the interfaces of Models 1, 2 and 3.2 is open and that of Model 3.1 is closed. The properties of the media are listed in Table 1, but solid 21 has been replaced with solid 11 (A= 3). The results of Models 1 and 2 are almost identical

Fig. 5
figure 5

Same as Fig. 3, but solid 22 has been replaced with solid 21 (see Table 1)

Fig. 6
figure 6

Comparison of the dispersion coefficient and dissipation factor for the properties given in Table 2. Models 1 and 2 are based on mixed frames, whereas in Model 3.2 the frames are segregated

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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

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