**3. Theory and Flowchart**

Figure 4 shows an idealization of the tight sandstone of the Xujiahe Formation, which is characterized by a complex pore structure and microcrack network. The total porosity is the sum of intergranular and microcrack porosity.

**Figure 4.** Scheme of tight sandstone pore structure with intergranular pores and microcrack network.

The flowchart for the prediction of the reservoir properties is given in Figure 5. The microcrack porosity is estimated by using the quantitative relation between the elastic attributes and reservoir properties, based on a rock-physics model and aided by geological, log data, and ultrasonic data.

**Figure 5.** Flowchart for the prediction of the reservoir properties.

The main minerals are quartz, feldspar, and clay. The total porosity is less than 12%. The procedure to establish the rock-physics model is as follows.

(1) The Voigt–Reuss–Hill equation [37–39] is used to compute the elastic modulus of mineral mixture *MVRH* according to the mineral composition:

$$M\_V = \sum\_{i=1}^{N} f\_i M\_{i\prime} \tag{1}$$

$$\frac{1}{M\_R} = \sum\_{i=1}^{N} f\_i / M\_{i\nu} \tag{2}$$

$$M\_{VRH} = \frac{M\_V + M\_R}{2},\tag{3}$$

where *fi* and *Mi* denote the volume fraction and elastic modulus, respectively, of the *i*-th component.

(2) According to the pore structure shown in Figure 4, we use the differential equivalent medium (DEM) theory [40] to add spherical pores and oblate cracks, whose aspect ratio are 1 and 0.0005, respectively, to add pores and microcracks into the matrix, and obtain the bulk and shear moduli of the rock skeleton (starred quantities),

$$(1 - y)d / dy [K^\*(y)] = (K\_2 - K^\*(y))P^{(\*2)}(y),\tag{4}$$

$$d(1-y)d\slash d y[\mu^\*(y)] = (\mu\_2 - \mu^\*(y))Q^{(\ast 2)}(y). \tag{5}$$

The initial conditions are *K*∗(0) = *K*<sup>1</sup> and *μ*∗(0) = *μ*1, where *K*<sup>1</sup> and *μ*<sup>1</sup> are the bulk and shear moduli, respectively, of the initial main material (phase 1); *K*<sup>2</sup> and *μ*<sup>2</sup> are the bulk and shear moduli, respectively, of the inclusion that is gradually added (phase 2); *y* is the volume content of phase 2; and *P*(∗2) and *Q*(∗2) [41] are related to the shape of the inclusion.


$$V\_{\rm P} = \left[ \text{Re} \left( v^{-1} \right) \right]^{-1} \text{ .} \tag{6}$$

$$Q = \frac{\text{Re}\left(v^2\right)}{\text{Im}\left(v^2\right)'}\tag{7}$$

where "Re" and "Im" take real and imaginary parts, respectively.

We consider the inclusion radius of 100 μm, and the bulk and shear moduli of the matrix are 33 GPa and 45 GPa, respectively. Gas has a bulk modulus of 0.02 GPa, a density of 0.089 g/cm3, and viscosity of 0.016 × <sup>10</sup>−<sup>3</sup> Pa·s. The crack porosity is set to 0.2%, and the total porosity varies as shown in Figure 6, where the P-wave velocity and dissipation factor are plotted as a function of frequency. Increasing porosity implies increasing velocity dispersion, attenuation, and relaxation frequency.

**Figure 6.** P-wave velocity (**a**) and dissipation factor (**b**) as a function of frequency and different total porosities.

Figure 7 shows the P- and S-wave velocities as a function of the total porosity for various microcrack porosities at 1 MHz. As can be seen, the velocities decrease with increasing total and microcrack porosities, as expected.

**Figure 7.** Effect of the total and microcrack porosities on the P-wave (**a**) and S-wave (**b**) velocities.

#### **4. Ultrasonic Experiments and Sensitivity Analysis**

To investigate the effects of the microcracks, ultrasonic experiments at 1 MHz were performed. We select a sample with a porosity of 4.39% and a grain bulk modulus of 39 MPa. The grain, dry-rock, and wet-rock densities are 2.691, 2.573, and 2.62 g/cm3, respectively, where the sample is saturated with water. The experimental setup proposed by Guo et al. [43] is used to measure the velocities at 20 ◦C, with the ultrasonic pulse method. The sample is sealed with a rubber sleeve and placed in the vessel. The pore pressure and temperature are fixed. Effective pressures of 5, 10, 15, 20, 25, 30, and 35 MPa are applied to the sample, in both the gas-saturated and water-saturated cases, and the velocities are measured.

As shown in Figure 8a, the measured P- and S-wave velocities increase with effective pressure. The increase of the confining pressure at a fixed pore pressure will lead to the gradual closure of internal microcracks (especially at low effective pressures), so as to stiffen the rock skeleton and increase its elastic moduli (and the wave velocities).

**Figure 8.** P- and S-wave velocities (**a**) and their ratio (**b**) as a function of effective pressure.

Figure 8b shows the velocity ratio. In the full gas saturation case, *V*P/*V*<sup>S</sup> increases with increasing effective pressure and the opposite behavior occurs for full water saturation case. In addition, we measured the total porosity in the range 5–35 MPa. Microcracks with small aspect ratios tend to close first with increasing effective pressure and the relation between porosity and pressure changes from exponential to linear [44].

Experimental measurements can be performed to predict microcrack porosity based on the relation between porosity and effective pressure [45–47]. Stiff porosity can be obtained through a linear extrapolation of this relation and the microcrack porosity can be estimated as the difference between the total and stiff porosities. Figure 9 shows the different porosities as a function of the effective pressure.

**Figure 9.** Total, stiff, and microcrack porosities as a function of the effective pressure.

The sensitivity analysis method like Sobol's indices can be used to analyze the interactions between the parameters, which can be applied with the given forward modeling equation or method/procedure [48]. In this work, sensitivity analysis to the total and microcrack porosities as inputs is performed based on the actually measured data from the experiments and well logs as puts, without considering the detailed modeling equations (or procedures). Based on the observed experimental or log data, the fluid sensitivity indicator (FSI) [49–52] has been proposed to analyze how the rock elastic properties are affected by the different fluid saturation statuses. Whereas, in this study, a similar method is adopted to analyze the relative variations of rock physics attributes with respect to the changes in total/microcrack porosity, and it is defined as

$$SI = \frac{|\overline{A} - A\_{\text{m}}|}{A\_{\text{m}}},\tag{8}$$

where *A*<sup>m</sup> and *A* represent the value of rock physics attribute at the minimal total (or microcrack) porosity and the average value of rock physics attribute within the considered range, respectively. Eleven attributes are considered, including density and the P- and S-wave velocities. The other eight are *V*P/*V*S, P-wave impedance (*I*P), S-wave impedance (*I*S), Poisson's ratio (*ν*), shear modulus (*μ*), the first Lamé constant (*λ*), *λρ*, and *V*P/*ρ*.

Figure 10 shows the sensitivity indices to microcrack porosity at ultrasonic frequencies, where we observe that *λρ*, *λ*, *ν*, *μ*, and *I*<sup>P</sup> are the most sensitive ones.

**Figure 10.** Sensitivity indices varying with the microcrack porosity.

The sensitivity analysis to total porosity is performed for the sonic log data, where we consider the gas-saturated layers of Well P. Figure 11 shows the results, where *λρ*, *λ*, *ν*, *μ*, and *I*<sup>P</sup> are the most sensitive. In Figure 10, *V*P/*ρ*, *V*P, and *I*<sup>P</sup> have a similar value while that of *I*<sup>P</sup> is slightly higher than the values of *V*P/*ρ* and *V*P. However, it is obvious that in Figure 11 the value of *I*<sup>P</sup> is higher than the values of *V*P/*ρ* and *V*P. According to Figures 10 and 11, *ν* and *I*<sup>P</sup> are sensitive to both porosities and can be considered to build the rock-physics templates.

**Figure 11.** Sensitivity indices varying with the total porosity, based on data from Well P.

#### **5. Rock-Physics Templates**

#### *5.1. Modeling*

Figure 12 shows the RPTs with respect to total and microcrack porosities at ultrasonic and log frequencies, where the black and red curves isolines of total and microcrack porosities, respectively.

**Figure 12.** RPTs at ultrasonic (**a**) and sonic (**b**) frequencies.

#### *5.2. Calibration*

Ultrasonic and log data are used to calibrate the RPTs. Figure 13 displays an RPT at 1 MHz and the inclusion radius of 50 μm, where the color of the scatters indicates microcrack porosity, and the maximum value is 0.54%. The Poisson's ratio gradually increases with decreasing microcrack porosity; the plot shows that the template is in agreement with the ultrasonic data.

Figure 14 shows an RPT at 10 kHz and the inclusion radius of 200 μm, where the color bar indicates total porosity (the data are from the log data of well P). Most of the data have a porosity greater than 4%. With the increasing porosity, Poisson's ratio increases and P-wave impedance decreases, and the agreement between data and theory is good.

**Figure 13.** Ultrasonic data and corresponding RPT. The color bar indicates the microcrack porosity.

**Figure 14.** Log data and corresponding RPT. The color bar indicates the total porosity.
