A New Method to Quantify Carbonate Rock Weathering

Abstract

The structure and composition of carbonate rocks are modified greatly when they are subjected to phenomena that lead to their weathering. These processes result in the production of residual alterite whose petrophysical, mechanical, and hydrological properties differ completely from those of the unweathered rock. From a geotechnical perspective, it is important that such changes are fully understood as they affect reservoir behavior and rock mass stability. This paper presents a quantitative method of calculating a weathering index for carbonate rock samples based on a range of petrophysical models. In total, four models are proposed, each of which incorporates one or more of the processes involved in carbonate rock weathering (calcite dissolution, gravitational compaction, and the incorporation of inputs). The selected weathering processes are defined for each model along with theoretical laws that describe the development of the rock properties. Based on these laws, common properties such as rock density, porosity, and calcite carbonate content are estimated from the specific carbonate rock weathering index of the model. The propagation of measurement uncertainties through the calculations has been computed for each model in order to estimate their effects on the calculated weathering index. A new methodology is then proposed to determine the weathering index for carbonate rock samples taken from across a weathered feature and to constrain the most probable weathering scenario. This protocol is applied to a field dataset to illustrate how these petrophysical models can be used to quantify the weathering and to better understand the underlying weathering processes.

Appendix A: Mathematical Demonstrations

1.1 A.1 Model 1

1.1.1 A.1.1 Computation of CRWI$_{1}^{i}$

From Eq. (11)

$$\begin{aligned} \hbox {CRWI}_1 ^i=1-\frac{M_1^i }{M_1^f}. \end{aligned}$$

By multiplying the entire fraction by $V_\mathrm{tot}/V_\mathrm{tot}$, the numerator by $M_\mathrm{tot}^{i}{/}M_\mathrm{tot}^{i}$, and the denominator by $M_\mathrm{tot}^{f}{/}M_\mathrm{tot}^{f}$

$$\begin{aligned} \hbox {CRWI}_1 ^i=1-\frac{\frac{M_1^i}{M_\mathrm{tot}^i }\cdot \frac{M_\mathrm{tot}^i }{V_\mathrm{tot} }}{\frac{M_1^f }{M_\mathrm{tot}^f }\cdot \frac{M_\mathrm{tot}^f }{V_\mathrm{tot} }}. \end{aligned}$$

From the definitions of the rock properties [Eqs. (1), (2), and (3)]

$$\begin{aligned} \hbox {CRWI}_1 ^i=1-\frac{m_1^i \cdot \gamma _\mathrm{r}^i }{m_1^f \cdot \gamma _\mathrm{r}^f }=1-\frac{\gamma _1 \cdot v_1^i }{\gamma _1 \cdot v_1^f }=1-\frac{v_1^i }{v_1^f }. \end{aligned}$$

1.1.2 A.1.2 Computation of $\gamma _\mathrm{r}^{i}=f(\hbox {CRWI}_{1}^{i})$

From the definition of the weathering index [Eq. (13)]

$$\begin{aligned} v_1^i =(1-\hbox {CRWI}_1 ^i)\cdot v_1^f. \end{aligned}$$

(37)

Thanks to Eq. (9)

$$\begin{aligned} \gamma _\mathrm{r}^i =\gamma _1 \cdot v_1^i +\gamma _2 \cdot v_2 =\gamma _1 \cdot ( {1-\hbox {CRWI}_1 ^i})\cdot v_1^f +\gamma _2 \cdot v_2. \end{aligned}$$

Following the definition of the bulk density [Eq. (6)]

$$\begin{aligned} \gamma _\mathrm{r}^i =( {1-\hbox {CRWI}_1 ^i})\cdot {\Gamma }_1^f +{\Gamma }_2. \end{aligned}$$

1.1.3 A.1.3 Computation of $v_{0}^{i}=f(\hbox {CRWI}_{1}^{i})$

By combining Eqs. (7) and (37)

$$\begin{aligned} v_0^i =1-v_1^i -v_2 =1-( {1-\hbox {CRWI}_1 ^i})\cdot v_1^f -v_2. \end{aligned}$$

1.1.4 A.1.4 Computation of $m_{1}^{i}=f(\hbox {CRWI}_{1}^{i})$

By combining Eqs. (10), (14), and (37)

$$\begin{aligned} m_1^i&= \frac{\gamma _1 \cdot v_1^i }{\gamma _\mathrm{r}^i }= \frac{\gamma _1 \cdot (1-\hbox {CRWI}_1 ^i)\cdot v_1^f }{( {1-\hbox {CRWI}_1 ^i})\cdot \gamma _1 \cdot v_1^f +\gamma _2 \cdot v_2 }=\frac{1}{1+\frac{\gamma _2 \cdot v_2 }{( {1-\hbox {CRWI}_1 ^i})\cdot \gamma _1 \cdot v_1^f }} \\&= \frac{1}{1+\frac{{\Gamma }_2 }{( {1-\hbox {CRWI}_1^i})~\cdot ~{\Gamma }_1^f }}. \end{aligned}$$

1.2 A.2 Model 2

1.2.1 A.2.1 Computation of CRWI$_{2}^{i}$

From Eq. (11)

$$\begin{aligned} \hbox {CRWI}_2 ^i=1-\frac{M_1^i }{M_1^f }. \end{aligned}$$

By multiplying the entire fraction by $(V_\mathrm{tot}^{i} \cdot V_\mathrm{tot}^{f})/(V_\mathrm{tot}^{i}\cdot V_\mathrm{tot}^{f})$, the numerator by $M_\mathrm{tot}^{i}/M_\mathrm{tot}^{i}$, and the denominator by $M_\mathrm{tot}^{f}/M_\mathrm{tot}^{f}$

$$\begin{aligned} \hbox {CRWI}_2 ^i=1-\frac{\frac{M_1^i }{M_\mathrm{tot}^i }\cdot \frac{M_\mathrm{tot}^i }{V_\mathrm{tot}^i }}{\frac{M_1^f }{M_\mathrm{tot}^f }\cdot \frac{M_\mathrm{tot}^f }{V_\mathrm{tot}^f }}\cdot \frac{V_\mathrm{tot}^f }{V_\mathrm{tot}^f }. \end{aligned}$$

From the generic definition of the compaction [Eq. (12)]

$$\begin{aligned} \frac{V_\mathrm{tot}^f }{V_\mathrm{tot}^f }=1-c^i. \end{aligned}$$

And with the definitions of the rock properties [Eqs. (1), (2), and (3)]

$$\begin{aligned} \hbox {CRWI}_2 ^i=1-\frac{v_1^i }{v_1^f }\cdot (1-c^i). \end{aligned}$$

1.2.2 A.2.2 Computation of $c^{i}$

From Eq. (12)

$$\begin{aligned} c^i=1-\frac{\text{ V }_\mathrm{tot}^i }{\text{ V }_\mathrm{tot}^f }. \end{aligned}$$

By multiplying the fraction by $V_{2}/V_{2}$ and considering the definition of the volume fraction [Eq. (2)]

$$\begin{aligned} c^i=1-\frac{\frac{V_2 }{\text{ V }_\mathrm{tot}^f }}{\frac{V_2}{\text{ V }_\mathrm{tot}^i }}=1-\frac{v_2^f }{v_2^i }. \end{aligned}$$

1.2.3 A.2.3 Computation of $\gamma _\mathrm{r}^{i}=f(\hbox {CRWI}_{2}^{i},c^{i})$

From the definition of the weathering index [Eq. (17)]

$$\begin{aligned} v_1^i =\frac{(1-\hbox {CRWI}_2^i)}{(1-c^i)}\cdot v_1^f. \end{aligned}$$

(38)

From the definition of compaction [Eq. (18)]

$$\begin{aligned} v_2^i =\frac{v_2^f}{1-c^i}. \end{aligned}$$

(39)

By combining Eqs. (9), (38), and (39)

$$\begin{aligned} \gamma _\mathrm{r}^i =\gamma _1 \cdot v_1^i +\gamma _2 \cdot v_2^i =\gamma _1 \cdot \frac{(1-\mathrm{CRWI}_2 ^i)}{(1-c^i)}\cdot v_1^f +\gamma _2 \cdot \frac{1}{(1-c^i)}\cdot v_2^f. \end{aligned}$$

Following the definition of the bulk density [Eq. (6)]

$$\begin{aligned} \gamma _\mathrm{r}^i =\gamma _1 \cdot v_1^i +\gamma _2 \cdot v_2^i =\frac{(1-\mathrm{CRWI}_2 ^i)}{(1-c^i)}\cdot {\Gamma }_1^f +\frac{1}{(1-c^i)}\cdot {\Gamma }_2^f. \end{aligned}$$

1.2.4 A.2.4 Computation of $v_{0}^{i}=f(\mathrm{CRWI}_{2}^{i}, c^{i})$

By combining Eqs. (7), (38), and (39)

$$\begin{aligned} v_0^i =1-v_1^i -v_2^i =1-\frac{(1-\mathrm{CRWI}_2 ^i)}{(1-c^i)}\cdot v_1^f -\frac{1}{1-c^i}\cdot v_2^f. \end{aligned}$$

1.2.5 A.2.5 Computation of $m_{1}^{i}=f(\mathrm{CRWI}_{2}^{i})$

By combining Eqs. (10), (20), and (38)

$$\begin{aligned} m_1^i =\frac{\gamma _1 \cdot v_1^i }{\gamma _\mathrm{r}^i }=\frac{\frac{(1-\mathrm{CRWI}_2 ^i)}{(1-c^i)}\cdot {\Gamma }_1^f }{\frac{(1-\mathrm{CRWI}_2 ^i)}{(1-c^i)}\cdot {\Gamma }_1^f +\frac{1}{(1-c^i)}\cdot {\Gamma }_2^f }=\frac{1}{1+\frac{{\Gamma }_2^f }{( {1-\mathrm{CRWI}_1 ^i})\cdot {\Gamma }_1^f }}. \end{aligned}$$

1.3 A.3 Model 3

1.3.1 A.3.1 Computation of $\mathrm{CRWI}_{3}^{i}$

As the weathering is isovolumetric, $V_\mathrm{tot}^{i}$ is equal to $V_\mathrm{tot}^{f}$, the definition and demonstration of CRWI$_{3}^{i}$ are the same as that for CRWI$_{1}^{i}$ (Appendix A.1).

1.3.2 A.3.2 Computation of $v_{0}^{i}=f(\mathrm{CRWI}_{3}^{i}$, $\gamma _\mathrm{r}^{i}$, $\gamma _{3}^{i}$)

As the material contains a third constituent phase, $v_{0}^{i}$ can no longer be independently expressed from $\gamma _\mathrm{r}^{i}$. From Eq. (9)

$$\begin{aligned} v_3^i =\frac{1}{\gamma _3^i }\cdot (\gamma _\mathrm{r}^i -\gamma _1 \cdot v_1^i -\gamma _2 \cdot v_2 ). \end{aligned}$$

By combining Eqs. (7) and (9)

$$\begin{aligned} v_0^i =1-v_1^i -v_2 -v_3^i =1-\left( {\frac{1}{\gamma _1 }-\frac{1}{\gamma _3^i }}\right) \cdot \gamma _1 \cdot v_1^i -\left( {\frac{1}{\gamma _2 }-\frac{1}{\gamma _3^i }}\right) \cdot \gamma _2 \cdot v_2 -\frac{\gamma _\mathrm{r}^i }{\gamma _3^i }. \end{aligned}$$

(40)

And following the definition of the weathering index [Eq. (37)]

$$\begin{aligned} v_0^i&= 1-\left( {\frac{1}{\gamma _1 }-\frac{1}{\gamma _3^i }}\right) \cdot ( {1-\mathrm{CRWI}_3 ^i})\cdot \gamma _1 \cdot v_1^f -\left( {\frac{1}{\gamma _2 }-\frac{1}{\gamma _3^i }}\right) \cdot \gamma _2 \cdot v_2 -\frac{\gamma _\mathrm{r}^i }{\gamma _3^i } \\&= 1-\left( {\frac{1}{\gamma _1 }-\frac{1}{\gamma _3^i }}\right) \cdot ( {1-\mathrm{CRWI}_3 ^i})\cdot {\Gamma }_1^f -\left( {\frac{1}{\gamma _2 }-\frac{1}{\gamma _3^i}}\right) \cdot {\Gamma }_2 -\frac{\gamma _\mathrm{r}^i }{\gamma _3^i }. \end{aligned}$$

1.3.3 A.3.3 Computation of $m_{1}^{i}=f(\mathrm{CRWI}_{3}^{i},\gamma _\mathrm{r}^{i},\gamma _{3}^{i})$

As the material contains a third constituent phase, $m_{1}^{i}$ can no longer be independently expressed from $\gamma _\mathrm{r}^{i}$. By combing Eqs. (10) and (37)

$$\begin{aligned} m_1^i =\frac{\gamma _1 \cdot v_1^i }{\gamma _\mathrm{r}^i }=\frac{( {1-\mathrm{CRWI}_3^i})\cdot \gamma _1 \cdot v_1^f }{\gamma _\mathrm{r}^i }=( {1-\mathrm{CRWI}_3^i})\cdot {\Gamma }_1^f \cdot \frac{1}{\gamma _\mathrm{r}^i }. \end{aligned}$$

1.4 A.4 Model 4

1.4.1 A.4.1 Computation of $\mathrm{CRWI}_{4}^{i}$

As the weathering is not isovolumetric due to compaction, $V_\mathrm{tot}^{i}$ is no longer equal to $V_\mathrm{tot}^{f}$, the definition and demonstration of CRWI$_{4}^{i}$ are the same as that for CRWI$_{2}^{i}$ (Appendix A.2).

1.4.2 A.4.2 Computation of $v_{0}^{i}=f(\mathrm{CRWI}_{3}^{i},\gamma _\mathrm{r}^{i}, c^{i},\gamma _{3}^{i})$

As with CRWI$_{3}^{i}$, the material contains a third constituent phase, so $v_{0}^{i}$ can no longer be independently expressed from $\gamma _\mathrm{r}^{i}$.

By combining Eqs. (38) and (40)

$$\begin{aligned} v_0^i&= 1-\left( {\frac{1}{\gamma _1 }-\frac{1}{\gamma _3^i }}\right) \cdot \gamma _1 \cdot v_1^i -\left( {\frac{1}{\gamma _2 }-\frac{1}{\gamma _3^i }}\right) \cdot \gamma _2\cdot v_2^i -\frac{\gamma _\mathrm{r}^i }{\gamma _3^i } \\&= 1-\left( {\frac{1}{\gamma _1 }-\frac{1}{\gamma _3^i }}\right) \cdot \frac{( {1-\mathrm{CRWI}_4 ^i})}{(1-c^i)}\cdot {\Gamma }_1^f -\left( {\frac{1}{\gamma _2 }-\frac{1}{\gamma _3^i }}\right) \cdot \frac{1}{(1-c^i)}\cdot {\Gamma }_2^f -\frac{\gamma _\mathrm{r}^i }{\gamma _3^i }. \end{aligned}$$

1.4.3 A.4.3 Computation of $m_{1}^{i}=f(\mathrm{CRWI}_{3}^{i},\gamma _\mathrm{r}^{i},c^{i},\gamma _{3}^{i})$

As with CRWI$_{3}^{i}$, the material contains a third constituent phase, so $m_{1}^{i}$ can no longer be independently expressed from $\gamma _\mathrm{r}^{i}$.

By combining Eqs. (10) and (38)

$$\begin{aligned} m_1^i =\frac{\gamma _1 \cdot v_1^i }{\gamma _\mathrm{r}^i }=\frac{( {1-\mathrm{CRWI}_4 ^i})}{(1-c^i)}\cdot \frac{\gamma _1 \cdot v_1^f }{\gamma _\mathrm{r}^i }=\frac{( {1-\mathrm{CRWI}_4 ^i})}{(1-c^i)}\cdot {\Gamma }_1^f \cdot \frac{1}{\gamma _\mathrm{r}^i }. \end{aligned}$$

1.5 A.5 Error on Model Without Gravitational Compaction

For a function $f$ of $n$ variables $x_{i}$, the absolute error on $f$ is given by

$$\begin{aligned} {\Delta }f( {x_i \vert i=1,\ldots ,n})=\sqrt{\mathop \sum \limits _{i=1}^n \left( {\frac{\partial f}{\partial x_i }}\right) ^2\cdot ( {{\Delta }x_i })^2}. \end{aligned}$$

The absolute error on $v_{1}^{i}$ is computed from Eq. (10)

$$\begin{aligned} v_1^i =\frac{\gamma _\mathrm{r}^i \cdot m_1^i }{\gamma _1}. \end{aligned}$$

As $\gamma _{1}$ is an imposed parameter, its error is considered to be zero.

The partial derivatives of $v_{1}^{i}$ are

$$\begin{aligned}&\frac{\partial v_1^i }{\partial m_1^i }=\frac{\gamma _\mathrm{r}^i }{\gamma _1},\\&\frac{\partial v_1^i }{\partial \gamma _\mathrm{r}^i }=\frac{m_1^i }{\gamma _1}. \end{aligned}$$

Thus

$$\begin{aligned} {\Delta }v_1^i =\sqrt{\left( {\frac{\gamma _\mathrm{r}^i }{\gamma _1 }}\right) ^2\cdot \left( {{\Delta }m_1^i }\right) ^2+\left( {\frac{m_1^i }{\gamma _1 }}\right) ^2\cdot ( {{\Delta }\gamma _\mathrm{r}^i })^2}. \end{aligned}$$

The absolute error on CRWI$^{i\,\mathrm{no}\,\mathrm{compaction}}$ is computed from Eq. (13)

$$\begin{aligned} \hbox {CRWI}^{i\,\mathrm{no}\,\mathrm{compaction}}=1-\frac{v_1^i }{v_1^f}. \end{aligned}$$

The partial derivatives of CRWI$^{i\,\mathrm{no}\,\mathrm{compaction}}$ are

$$\begin{aligned}&\frac{\partial \mathrm{CRWI}^{i\,\mathrm{no}\,\mathrm{compaction}}}{\partial v_1^i}=-\frac{1}{v_1^f },\\&\frac{\partial \mathrm{CRWI}^{i\,\mathrm{no}\,\mathrm{compaction}}}{\partial \gamma _\mathrm{r}^i }=\frac{v_1^i}{v_1^{f2}}. \end{aligned}$$

Thus

$$\begin{aligned} {\Delta }\hbox {CRWI}^{i\,\mathrm{no}\,\mathrm{compaction}}=\sqrt{\left( {\frac{1}{v_1^f }}\right) ^2( {{\Delta }v_1^i })^2+\left( {\frac{v_1^i }{v_1^{f2}}}\right) ^2( {{\Delta }v_1^f })^2}. \end{aligned}$$

1.6 A.6 Error on Model with Gravitational Compaction

The absolute error on $v_{2}^{i}$ is computed from Eq. (7)

$$\begin{aligned} v_2^i =1-v_0^i -v_1^i. \end{aligned}$$

The partial derivatives of $v_{2}^{i}$ are

$$\begin{aligned}&\frac{\partial v_2^i }{\partial v_0^i }=-1,\\&\frac{\partial v_2^i }{\partial \gamma _\mathrm{r}^i }=-1. \end{aligned}$$

Thus

$$\begin{aligned} {\Delta }v_2^i =\sqrt{( {{\Delta }v_1^i })^2+( {{\Delta }v_0^i })^2}. \end{aligned}$$

The absolute error on $c^{i}$ is computed from Eq. (18)

$$\begin{aligned} c^i=1-\frac{v_2^f }{v_2^i }. \end{aligned}$$

The partial derivatives of $c^{i}$ are

$$\begin{aligned}&\frac{\partial c^i}{\partial v_2^f }=-\frac{1}{v_2^i },\\&\frac{\partial c^i}{\partial v_2^i }=\frac{v_2^f }{v_2^{i 2}}. \end{aligned}$$

Thus

$$\begin{aligned} {\Delta }c^i=\sqrt{\left( {\frac{1}{v_2^i }}\right) ^2( {{\Delta }v_2^f })^2+\left( {\frac{v_2^f }{v_2^{i 2}}}\right) ^2( {{\Delta }v_2^i })^2}. \end{aligned}$$

The absolute error on CRWI$^{i\,\mathrm{compaction}}$ is computed from Eq. (17)

$$\begin{aligned} \hbox {CRWI}^{i\,\mathrm{compaction}}=1-\frac{v_1^i }{v_1^f }\cdot ( {1-c^i}). \end{aligned}$$

The partial derivatives of CRWI$^{i\,\mathrm{compaction}}$ are

$$\begin{aligned}&\frac{\partial \mathrm{CRWI}^{i\,\mathrm{compaction}}}{\partial v_1^i }=-\frac{( {1-c^i})}{v_1^f },\\&\frac{\partial \mathrm{CRWI}^{i\,\mathrm{compaction}}}{\partial v_1^f }=\frac{v_1^i }{v_1^{f2}}\cdot ( {1-c^i}),\\&\frac{\partial \mathrm{CRWI}^{i\,\mathrm{compaction}}}{\partial c^i}=\frac{v_1^i }{v_1^f }. \end{aligned}$$

Thus

$$\begin{aligned}&{\Delta }\mathrm{CRWI}^{i\,\mathrm{compaction}}\\&\quad =\sqrt{\left( {\frac{( {1-c^i})}{v_1^f }}\right) ^2( {{\Delta }v_1^i })^2+\left( {\frac{v_1^i }{v_1^{f 2}}\cdot ( {1-c^i})}\right) ^2( {{\Delta }v_1^f })^2+\left( {\frac{v_1^i }{v_1^f }}\right) ^2( {{\Delta }c^i})^2}. \end{aligned}$$