# Study on Seismic Attenuation Based on Wave-Induced Pore Fluid Dissolution and Its Application
## 2 Materials and Methods  
_Rock Physics Model Upscaling by Visco-Elastic Model_  
Pore fluid dissolution, as an attenuation mechanism of seismic waves, will lead to dissolution and dissolution between gas and its surrounding liquid, which will lead to the attenuation of seismic waves. In the sphere area of each bubble, the gas dissolution rate can be expressed as [7]:  
\[dn/dt=-4 ITrD_{w}\bigg{(}C_{w}-\frac{P_{f}+2\gamma/r}{RTK_{H}}\bigg{)} \tag{1}\]  
where \(dn/dt\) is the dissolution rate of gas, \(r\) is the bubble radius, \(D_{w}\) is the diffusion coefficient, \(C_{w}\) is the concentration, \(P_{f}\) is the pressure in the fluid, \(\gamma\) is the surface tension coefficient, \(R\) is the gas constant, \(T\) is the temperature, and \(K_{H}\) is the Henry's Law constant. Rock physics models provide velocity dispersion between the seismic data and logging data for specific rock properties. Attenuation curve \(Q_{rock}(f)\) can be used for generating synthetic seismic records:  
\[u(t,\omega)=u_{0}exp(-i\omega t)exp\bigg{(}-\frac{\omega t}{2Q_{rock}(f)} \bigg{)}exp\bigg{(}\frac{i\omega t}{\pi Q_{rock}(f)}\ln\bigg{|}\frac{\omega}{ \omega_{r}}\bigg{|}\bigg{)} \tag{2}\]  
Considering the explicit \(Q(f)\) expression in rock physics model are unavailable, visco-elastic models are upscaled with non-physical meaning parameters. Attenuation properties can then be well modeled by solving the following problem by using least-squares objective function [25; 26]:  
\[\min\sum_{f}|Q_{rock}(f)\,-\,Q_{visco}(f)|^{2} \tag{3}\]  
SLS model can model most rock physics models [10], and the \(Q_{SLS}(f)\) can be expressed by:  
\[Q_{SLS}(f)=1+\frac{(2\pi f)^{2}\tau_{\sigma}{}^{2}}{1+\omega(2\pi f)^{2}\tau _{\sigma}{}^{2}}\tau/\frac{(2\pi f)\tau_{\sigma}{}^{2}}{1+(2\pi f)^{2}\tau_{ \sigma}{}^{2}}\tau \tag{4}\]  
where \(\tau_{\sigma}\) is stress relaxation time, and \(\tau\) is strain relaxation time. Combining Equations (2) and (3), the upscaled \(Q(f)\) by the SLS model can be derived:  
\[\min_{\tau,\tau_{\sigma},L}\sum_{f}\bigg{|}Q_{rock}(f)-(1+\sum\
olimits_{l=1} ^{L}\frac{(2\pi f)^{2}\tau_{\sigma}(l)^{2}}{1+(2\pi f)^{2}\tau_{\sigma}(l)^{2 }}\tau(l))/(\sum\
olimits_{l=1}^{L}\frac{(2\pi f)\tau_{\sigma}(l)}{1+(2\pi f )^{2}\tau_{\sigma}(l)^{2}}\tau(l))^{2}\bigg{|} \tag{5}\]  
where L is relaxation mechanisms, and \(\tau=\frac{\tau_{\sigma}}{\tau_{\sigma}}-1,\ Q_{\text{rock}}(f)\) is the numerically modeled attenuation curve based on the rock physics model. Here, we use the wave-induced gas exsolution-dissolution (WIGED) model to calculate the \(Q_{\text{rock}}(f)\), and the parameters used are listed in Table 1. The volumetric strain of the pore fluid is calculated, followed by the modulus \(K\), and attenuation can be obtained by \(Q(f)^{-1}=\frac{Im(K(f))}{Re(K(f))}\). 1. Rock physics modeling. Based on the geology, geochemistry and core data, a suitable rock physics model is selected, and the parameters in the model are upgraded. The WIGED model is used here for fluid, and the SCA model is used for rock matrix. Then, the saturated rock is obtained by Gassmann substitution. The key parameters in the WIGED model are the size of the bubble and diffusion length. 2. Attenuation curve calculation. _Q(f)_ curve is then calculated from the rock physics model. The explicit _Q(f)_ expression can be derived through upscaling by visco-elastic model. Here, we use the SLS model for upscaling, which yields best results compared with other visco-elastic models, such as the power-law model. 3. Well-to-seismic data calibration. Based on the reflection coefficient and _Q(f)_, the attenuated synthetic seismic data can be generated. By adjusting the \(Q\) related parameters in visco-elastic model (strain relaxation time and stress relaxation time in the SLS model), the optimal match between synthetic seismic data and field data can be found.