## Appendix A  
The workflow to calculate the frequency-dependent elastic moduli of fractured rocks in the Galvin model is show in Figure A1. The elastic stiffness tensor, \(C_{ij}^{sat}(w)\), given by [PERSON] (2015) is of the following form:  
\[\frac{1}{C_{ij}^{sat}(w)}=\frac{1}{C_{ij,hf}^{sat}(w)}[1+(\frac{C_{ij,hf}^{sat }-C_{ij,hf}^{sat}}{C_{ij,hf}^{sat}})/(1-\varsigma+\varsigma\sqrt{1-\frac{i\vec{ w}\tau}{\varsigma^{2}}})]\] (A1)  
where \(C_{ij,hf}^{sat}\) and \(C_{ij,hf}^{sat}\) are the stiffness tensors in the high- and low-frequency limits, which can be obtained by the linear-slip theory and the [PERSON] (2003) model, respectively, and \(\varsigma\) and \(\tau\) are parameters that shape the dispersion and attenuation curves of the elastic coefficient, respectively:  
\[\tau=(\frac{C_{11,hf}^{sat}-C_{11,hf}^{sat}}{C_{11,hf}^{sat}G})\] (A2)  
\[\varsigma=\frac{(C_{11,hf}^{sat}-C_{11,hf}^{sat})^{3}}{2C_{11,hf}^{sat}C_{11,hf }^{sat}{}^{2}TG^{2}}\] (A3)  
where \(C_{11,hf}^{sat}\) and \(C_{11,hf}^{sat}\) are the P-wave moduli of the fractured rock in the fracture normal direction in the low- and high-frequency limits, respectively. For sparsely distributed penny-shaped cracks, the expressions of \(T\) and \(G\) are of the following form:  
\[T=\frac{2(C_{b}-\alpha_{b}M_{b})^{2}(2-4\alpha_{b}g_{b}+3\alpha_{b }^{2}g_{b}^{b})^{2}\epsilon_{f}\eta}{15\mu_{b}g_{b}(1-g_{b})^{2}C_{b}L_{b}\kappa _{b}}\] (A4) \[T=\frac{2(C_{b}-\alpha_{b}M_{b})^{2}(2-4\alpha_{b}g_{b}+3\alpha_{ b}^{2}g_{b}^{b})^{2}\epsilon_{f}\eta}{15\mu_{b}g_{b}(1-g_{b})^{2}C_{b}L_{b} \kappa_{b}}\] (A5)  
where \(C_{b}\) and \(L_{b}\) are the P-wave moduli of the fluid-saturated and dry background matrices, respectively; \(\kappa_{b}\) is the matrix permeability; and \(\alpha_{b}=1-K_{b}/K_{gb}\) is Biot's coefficient of the matrix, with \(K_{b}\) being the bulk modulus of the dry background matrix and \(K_{gb}\) that of the solid grains composing the matrix. \(M_{b}=K_{gb}/[(1-K_{b}/K_{gb})-\varphi_{bg}(1-K_{gb}/K_{f})]\) is Biot's background matrix modulus, with \(K_{f}\) being the fluid bulk modulus and \(\varphi_{bg}\) the porosity of the background. \(\mu_{b}\) is the shear modulus of the dry background matrix, \(a\) is the fracture radius, and \(\varepsilon_{f}\) is the fracture density:  
\[\varepsilon_{f}=\frac{Na^{3}}{V} \tag{10}\]  
In dry conditions, \(C_{ij,lf}^{sat}\) and \(C_{ij,hf}^{sat}\) are equal and the Galvin model returns to the linear-slip model in which the elastic compliance tensor of the fractured medium, \(S_{ij,i}\) is of the following form:  
\[S_{ij}^{dry}=S_{b}^{dry}+\Delta S_{f}^{dry} \tag{11}\]  
where \(S_{b}^{dry}\) is the dry compliance tensor of the background matrix. \(\Delta S_{f}^{dry}\) is the excess compliance tensor associated with the fractures and can be expressed in the following form:  
\[\Delta S_{f}^{dry}= \begin{bmatrix}Z_{N}&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&Z_{\rm T}&0\\ 0&0&0&0&0&Z_{\rm T}\end{bmatrix}  \tag{12}\]  
where \(Z_{N}\) and \(Z_{\rm T}\) denote the so-called normal and shear excess compliances caused by the presence of fractures, respectively. In fluid-saturated conditions, high-frequency-limit fluid does not have enough time to flow between pore space and fractures [33]; this regime is called 'unrelaxed' by [PERSON] and [PERSON] (1991). In the Galvin model, the compliance tensor of fractured rocks in the high-frequency limit was calculated by the linear-slip model using the following form:  
\[S_{ij,hf}^{sat}=S_{ij,b}^{sat}+\Delta S_{f}^{sat} \tag{13}\]  
where \(S_{ij,b}^{sat}\) is the inverse of the fluid-saturated matrix stiffness tensor, \(C_{b}\), which is given by the Gassmann equation [38] and dry moduli of the background matrix. At the low-frequency range, fluid has enough time to flow between pores and fractures.