*2.1. Porosity Estimation*

The conventional density-based porosity equation is described in Equation (1):

$$\mathcal{Q}D = \frac{\rho\_{\rm ma} - \rho\_b}{\rho\_{\rm ma} - \rho\_f} \tag{1}$$

where Ø*D* = density porosity (%), ρ*ma* = matrix density (g/cc), ρ*b* = bulk density (g/cc), ρ*f* = fluid density (g/cc). Unlike in conventional reservoirs (sandstone or limestone), the bulk density acquired through density log in organic-rich shale usually overestimates the porosity. Therefore, the kerogen correction is applied to avoid porosity overestimation. The kerogen volume is determined by using Equation (2) [25]:

$$V\_k = \frac{\gamma \times \text{TOC} \times \rho\_b}{100 \times \rho\_k} \tag{2}$$

where, *Vk* is the kerogen volume (fractions); TOC is total organic carbon content (wt %); ρ*b* is the bulk density from the density log (g/cc); γ is the kerogen conversion factor; and ρ*k* is the kerogen density (g/cc). TOC is determined by the rock eval pyrolysis method on powdered shale samples, and the continuous TOC for the whole interval is estimated by Passey method [27]; γ is proposed by [25], and the selected values are shown in Table 1.

**Table 1.** Conversion factors for total organic carbon (TOC) to kerogen, adapted from Tissot and Welte [25].


For this study, based on rock eval pyrolysis results, the kerogen types are 30% type-II and 70% type-III. Therefore, the kerogen conversion factor for the studied formation is calculated as 1.18; and ρ*k* is determined by the relationship of lab-based TOC and reciprocal of lab-based derived grain density on shale samples by the Equation (3). A good relationship between TOC and reciprocal of grain density (ρ*g* read as RHOG) is observed in Figure 2. The Equation (3) is derived based on the relationship between TOC and reciprocal of grain density (Figure 2).

$$\frac{1}{\rho\_{\mathcal{K}}} = A \times TOC + B \tag{3}$$

**Figure 2.** The direct relationship between core-based derived total organic carbon and reciprocal of grain density providing helpful information for estimation of kerogen and matrix densities.

ρ*g* is the matrix density if *TOC* is zero and ρ *g k* is kerogen density if *TOC* is 100%. *A* and *B* are

based on the linear relationship seen in Figure 2. From the relation found in Figure 2, the matrix density for the samples of this study is 2.79 g/cc, and kerogen density is 1.24 g/cc. The well logs are calibrated by eliminating the kerogen effect, and the following equations Equations (4) and (5) are applied for matrix porosity estimation through density log:

$$
\rho\_{bk\_c} = \frac{\rho\_b - \rho\_{k \times V\_k}}{1 - V\_k} \tag{4}
$$

$$\mathcal{Q}kc = \frac{\rho\_{\rm ma} - \rho\_{\rm lk}}{\rho\_{\rm ma} - \rho\_f} \tag{5}$$

where, ρ*bkc* is kerogen corrected bulk density (g/cc); ρ*k* is kerogen density (g/cc); *Vk* is kerogen volume (fractions) and Ø*kc* is kerogen corrected density porosity (%). As the porosity in organic-rich shale is associated with organic matter and inorganic minerals, so it is crucial to estimate the porosity within organic matter (kerogen). An equation for kerogen porosity was proposed by [28] using mass-balance relation Equation (6).

$$\mathcal{O}\_k = ([T \mathcal{O} \mathcal{C}\_\upsilon \times \mathcal{C}\_c] \times \mathcal{V})TR \frac{\rho\_b}{\rho} \,\,\_k\tag{6}$$

where, Ø*k* = kerogen porosity (%), *TOCo* = original total organic carbon, *Cc* = convertible carbon fraction and *TR* = transformation ratio.

$$TOC\_o = \frac{TOC}{1 - TR \times \mathbb{C}\_c} \tag{7}$$

$$TR = 1 - \frac{H I\_p \left[ 1200 - H I\_o \left( 1 - P I\_o \right) \right]}{H I\_o \left[ 1200 - H I\_p \left( 1 - P I\_p \right) \right]} \tag{8}$$

where: *HIp* = present hydrogen index (mg/g), *HIo* = original hydrogen index (mg/g), *PIp* = present production index and *PIo* = original production index. The following equations were used to estimate the original hydrogen index and present hydrogen index proposed by [28]:

$$HI\_o = \frac{TypeII}{100} \times 450 + \frac{TypeIII}{100} \times 125\tag{9}$$

For this study:

$$HI\_{\theta} = 225 \text{ mg/g}$$

$$HI\_{p} = 170 \text{ mg/g}$$

S1/S1 + S2 = *PIp* = 0.35

The convertible carbon fraction is determined by using the relationship proposed by [29], such as *Cc* = 0.085 × *HIo*=18.91%.

Although, the transformation ratio (TR) can be determined by Claypool equation as explained in Equation (8) [28]. However, for this study, the TR value is taken as 88% that is adapted from [24,30] based on organic geochemistry and basin modelling of Goldwyer shale. So, the equation for kerogen porosity will be as Equation (10). By eliminating the kerogen effect and adding the kerogen porosity Equation (11), the final Equation (12) is applied to compute total density porosity for shale reservoirs.

$$\mathcal{D}\_k = 0.2 \times \text{TOC} \times \rho\_b \tag{10}$$

$$\mathcal{O}D\_{Total} = \left[ \left( \frac{\rho\_{ma} - \rho\_{bk\_c}}{\rho\_{ma} - \rho\_f} \right) + \mathcal{O}\_k \right] \tag{11}$$

$$\mathcal{Q}D\_{Total} = \left[ \left( \frac{\rho\_{\rm ma} - \left( \frac{\rho\_{\rm b} - \rho\_{\rm kc} \upsilon\_{\rm k}}{1 - V\_k} \right)}{\rho\_{\rm ma} - \rho\_f} \right) + (0.2 \times TOC \times \rho\_{\rm b}) \right] \tag{12}$$

#### *2.2. Calculation of Water Saturation*

The water saturation estimation in shale is mainly dependent on its organic (kerogen) and inorganic components (minerals). Archie equation [17] is mainly popular for water saturation calculation in clean reservoirs. The equation was developed based on a function between formation conductivity and the conductivity of fluids in the pore spaces of a reservoir, such as:

$$\mathbf{C}\_{t} = \frac{S\_{w}^{\eta} \times \mathbf{C}\_{w}}{F} \tag{13}$$

where *Ct* = total conductivity (ohm−<sup>1</sup> m<sup>−</sup>1), *Cw* = formation water conductivity (ohm−<sup>1</sup> m<sup>−</sup>1), *n* = saturation exponent usually equals to 2, *Sw* = water saturation (%). The equation can be written in terms of resistivity as follows:

$$\frac{1}{R\_t} = \frac{\mathcal{O}^m \times S\_w^n}{a \times R\_w} \tag{14}$$

where *Rt* = true resistivity measured by logging tool (ohm-m), Ø = porosity (%), *m* = cementation exponent, *n* = saturation exponent usually equals to 2, *a* = tortuosity factor usually considered as 1 and *Rw* = formation water resistivity (ohm-m). The Equation (14) is known as the Archie equation for clean formations. Later, this equation did not provide acceptable and accurate results for the shaly formations. Therefore, other approaches such as Simandoux considered the shale effect on water saturation and developed an equation Equation (15) by considering the volume of shale in the equation that was further modified by Schlumberger, 1972 and the modified Simandoux equation is [18]:

$$\frac{1}{R\_{l}} = \frac{\mathcal{O}^{\rm m} \times S\_{\rm w}^{\rm n}}{a.R\_{\rm w} \times (1 - V\_{\rm sl})} + \frac{V\_{\rm sl} \times S\_{\rm w}}{R\_{\rm sl}} \tag{15}$$

where *Rsh* is the resistivity of shale (ohm-m) and *Vsh* is the volume of shale (fraction). The conventional water saturation models, e.g., Simandoux equation, modified Simandoux, total shale, and modified total shale equations provided better results for shaly formations as these equations are derived based on the conductivities of clays and non-clay matrix. However, these models overestimate the water saturation for organic-rich shales. Therefore, a modified water saturation equation is applied in this study. An equation was proposed by [2,4] for water saturation calculation for shale reservoirs. The derivation details of the equation are explained by [17] simplified equation for water saturation:

$$S\_{\overline{w}} = \sqrt{\frac{\mathcal{R}\_o}{\mathcal{R}\_t}}\tag{16}$$

where, *Ro* is the rock resistivity in lean shale interval where water saturation is deemed 100% (ohm-m) and *Rt* is the rock resistivity in the organic-rich shale reservoir with some degree of oil/gas saturation (ohm-m). Therefore, Ro and Rt are the key parameters for water saturation calculations.

As the organic-rich shale reservoirs have a higher content of total clay and organic matter it is necessary to conduct corrections (total organic carbon and total clay) for the true formation resistivity (Rt). The clay minerals decrease the formation resistivity and the kerogen increases the resistivity. So, the TOC and shale corrections are used for Rt. First, the correlation is developed between true resistivity log and TOC measurements (on powdered shale samples through rock eval pyrolysis) (Equation (17), Figure 3).

**Figure 3.** Direct relationship between true resistivity and measured total organic carbon showing influence of organic matter on resistivity tool.

A negative correlation Equation (18) is found between laboratory-based water saturation measured on shale samples and rock eval pyrolysis-based TOC. This relationship shows that with the increase in TOC, the water saturation reduces that provides an indication of hydrocarbon saturation in the shale interval (Figure 4).

$$TOC = 0.1635 \times R\_t \tag{17}$$

$$Sw\_{core} = -0.0981 \times TOC\_{core} + 0.825$$

**Figure 4.** An inverse relationship between core-based total organic carbon and water saturation showing the fact that the organic matter increases gas saturation.

The true resistivity is corrected in terms of subtracting a factor A Equation (19) due to TOC that can be evaluated by making arrangements, such as:

$$A = V\_k^2 \times \mathcal{R}\_k \tag{19}$$

If TOC is 100% then Rt will be considered as kerogen resistivity *Rk* (based on Equation (17)) so for this study based on Figure 3 *Rk* = 613 ohm-m and Figure 5 *Rsh* = 1.97 ohm-m are used.

**Figure 5.** The shale resistivity estimation based on shale volume and true resistivity relationship.

Based on the correlation, the *TOCmax* is found as 4.91 wt %. Another factor *B* Equation (20) because of clay minerals effect on resistivity is defined by many authors [18,31,32], such as:

$$B = V\_{sh}{}^2 \times R\_o \tag{20}$$

The squared form of the shale volume will be more convincing in the calculation of reduced resistivity as a result of shale volume. It can be due to the nonlinear relationship between *Ro* and *Rw* in shales [18,31]. For this study, the *Ro* is taken as 1.97 ohm-m (Figure 5).

By compensating the shale and organic matter e ffects on the true resistivity, the modified equation is introduced as:

$$S\_w = \sqrt{\frac{R\_0}{R\_t - (V\_{kr}^2 \times R\_k) + (V\_{sl}^2 \times R\_{sl})}}\tag{21}$$

#### **3. Results and Discussion**

In this section, the applications of proposed porosity and water saturation equations are implemented for the Ordovician Goldwyer shale formation drilled in Theia-1, Pictor East-1, and Canopus-1 wells in Canning Basin, Western Australia.

The kerogen corrected total porosity (matrix porosity plus kerogen porosity) was estimated by using Equation (12). The total porosity on crushed shale samples (core porosity) ranges from 2 to 13%, measured through the di fference between the bulk volume of shale samples and the grain volume of the crushed, cleaned, and dried samples. The Goldwyer shale porosity shows the same range of porosity as most of the organic-rich shales [5,7,14,33–36]. The Goldwyer shale consists three types of pores such as organic pores, interparticle and intraparticle pores as shown in Figure 6. The results show that the conventional porosity estimation through density log overestimates the porosity that may a ffect the accurate reserve estimation in shale. Such as, the porosity based on Equation (1) provided the porosity range from 8 to 15% for Goldwyer shale (Figure 7). However, after applying the kerogen corrections, the corrected porosity ranging from 5 to 10% gives more accurate results that can be well-compared with core porosity (Table 2 and Figure 7). Moreover, the clay minerals also a ffect the pore structure of shale that directly a ffects the water saturation [37,38]. The Goldwyer shale also consists interparticle pores influenced by illite that may change the water saturation (Figure 6). The core derived TOC varies from 0.35 to 4.5 wt % in this study. The log derived TOC matches well with core-based *TOC* and the equivalent kerogen volume also validates the results (Figure 7). It can also be observed in Table 2 and Figure 7 that the clusters (e.g., siliceous and argillaceous shales) with higher TOC value have higher porosity (about 8–10%) due to the addition of organic pores (kerogen porosity) in the matrix porosity.

**Figure 6.** Di fferent pore types observed in Goldwyer shale based on scanning electron microscope images, such as (**a**) interparticle pores indicated by white arrows and intraparticle pores indicated by red arrows; (**b**) organic matter pores (OM), mineral components include calcite (cal), quartz (qtz) and illite.

**Figure 7.** Petrophysical evaluation of Goldwyer shale providing accurate estimation of porosity and water saturation through proposed equations as validated by core-based measurements. Track-1: Depth in meters; Track-2: Cluster analysis to identify cluster based facies; Track-3: Gamma ray log; Track-4: Deep resistivity log; Track-5: Density log; Track-4: Sonic (DT) log; Track-4: Kerogen volume; Track-4: Shale volume based on Gamma ray log; Track-4: TOC based on Passey's method and core measurements; Track-4: Kerogen corrected total density porosity (PHIDKc) based on proposed equation in this study, density based porosity (PHID) & Total porosity based on core samples; Track-4: Water saturation (Sw) based on Simandoux equation (overestimated) and modified Archie's equation (by this study) and core derived Sw.

The water saturation was estimated by Equation (21) by considering the kerogen and shale effects on the resistivity. The required kerogen volume and kerogen resistivity were computed by using the data set (well logs) and core information from three wells (Theia-1, Pictor East-1 and Canopus-1) drilled in Canning Basin. The results for Theia-1 well are illustrated in Figure 7. Similarly, the shale resistivity was taken based on the data set for these three wells. It can be observed in Figure 7 that with the increase in shale volume (e.g., at depth 1546.5 m), the deep resistivity is decreased that enhances the water saturation. In conventional reservoirs, shale resistivity is usually determined from the averaged deep resistivity log reading against shale interval having higher gamma-ray log reading. However, in shale reservoirs, the shale resistivity is obtained from the average reading of the deep resistivity log against an organic lean interval. In this study, the shale resistivity in the organic lean interval is determined as 1.97 ohm-m based on the relationship between shale volume and true resistivity developed by this study (Figure 5). It is impractical to determine the fluid-water contact in heterogeneous shale reservoirs; therefore, an organic lean shale is treated to be fully brine saturated rock, S w = 1 [4].


**Table 2.** Comparison of averaged total porosity and water saturation determined by conventional equations (PHID and Sw\_Simandoux) and introduced by this study (PHIDKc and Sw\_modified Archie). The conventional equations overestimated the porosity and water saturation in shale.

In the same way, the zones with higher TOC value and kerogen volume (such as organic-rich siliceous shale–cluster 3 (siliceous shale) at depth 1550 m) have the lowest water saturation. The inverse relationship between core-based TOC and Sw is also confirmed in this study (Figure 4). So, the kerogen resistivity ( *Rkr* = 613 ohm-m) is determined by Equation (17) by putting TOC value as 100%. Therefore, the modified Archie equation applied in this study provides much better results (well correlated with core derived Sw) than the Simandoux equation (Table 2 and Figure 7). It can be observed that the Simandoux method overestimated water saturation as it is impossible to have more than 100% Sw. Another key factor of this overestimation is inaccurate determination of water resistivity and cementation exponent (m) values. Therefore, the modified Archie equation applied in this study is simple and accurate subject to the resistivity corrections for shale and kerogen.
