**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]:

$$dn/dt = -4\Pi r D\_w \left(\mathbb{C}\_w - \frac{P\_f + 2\gamma/r}{RTK\_H}\right) \tag{1}$$

where *dn/dt* is the dissolution rate of gas, *r* is the bubble radius, *Dw* is the diffusion coefficient, *Cw* is the concentration, *Pf* is the pressure in the fluid, *γ* is the surface tension coefficient, *R* is the gas constant, *T* is the temperature, and *KH* 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 *Qrock(f)* can be used for generating synthetic seismic records:

$$u(t,\omega) = u\_0 \exp(-i\omega t) \exp\left(-\frac{\omega t}{2Q\_{\text{rock}}(f)}\right) \exp\left(\frac{i\omega t}{\pi Q\_{\text{rock}}(f)} \ln\left|\frac{\omega}{\omega\_r}\right|\right) \tag{2}$$

Considering the explicit *Q*(*f*) expression in rock physics model are unavailable, viscoelastic 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]:

$$\min \sum\_{f} \left| Q\_{\text{rock}}(f) - Q\_{\text{visco}}(f) \right|^2 \tag{3}$$

SLS model can model most rock physics models [10], and the *QSLS(f)* can be expressed by:

$$Q\_{SLS}(f) = 1 + \frac{(2\pi f)^2 \mathbf{r}\_\sigma^2}{1 + \omega \left(2\pi f\right)^2 \mathbf{r}\_\sigma^2} \mathbf{r} / \frac{(2\pi f) \mathbf{r}\_\sigma^2}{1 + \left(2\pi f\right)^2 \mathbf{r}\_\sigma^2} \mathbf{r} \tag{4}$$

where *τσ* is stress relaxation time, and τ is strain relaxation time. Combining Equations (2) and (3), the upscaled *Q*(*f*) by the SLS model can be derived:

$$\min\_{\pi, \pi\_{\sigma}, L} \sum\_{f} \left| Q\_{\text{rock}}(f) - (1 + \sum\_{l=1}^{L} \frac{(2\pi f)^2 \tau\_{\sigma}(l)^2}{1 + (2\pi f)^2 \tau\_{\sigma}(l)^2} \tau(l)) / (\sum\_{l=1}^{L} \frac{(2\pi f) \tau\_{\sigma}(l)}{1 + (2\pi f)^2 \tau\_{\sigma}(l)^2} \tau(l))^2 \right| \tag{5}$$

where L is relaxation mechanisms, and *τ* = *τε τσ* − 1, *Q*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*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*)−<sup>1</sup> = *Im*(*K*(*f*)) *Re*(*K*(*f*)) .


**Table 1.** Parameters used in three rock physics models: the pore fluid dissolution model, the SLS model and the Pride model.

Well-to-seismic calibration based on rock physics model upscaling. The misfit between the seismic data and the logging data can then be calibrated by implementing the following procedures (Figure 1):


**Figure 1.** Procedures of the proposed well-to-seismic calibration method.


### **3. Synthetic Data Test**
