A Novel Method for Gas–Water Relative Permeability Measurement of Coal Using NMR Relaxation

Abstract

Using the conventional volumetric method in unsteady-state relative permeability measurements for unconventional gas reservoirs, such as coal and gas shale, is a significant challenge because the movable water volume in coal or shale is too small to be detected. Moreover, the dead volume in the measurement system adds extra inaccuracy to the displaced water determination. In this study, a low-field nuclear magnetic resonance (NMR) spectrometer was introduced into a custom-built relative permeability measurement apparatus, and a new method was developed to accurately quantify the displaced water, avoiding the drawback of the dead volume. The changes of water in the coal matrix and cleats were monitored during the unsteady-state displacement experiments. Relative permeability curves for two Chinese anthracite and bituminous coals were obtained, matching the existing research results from the Chinese coalbed methane area. Moreover, the influences of confining pressure on the shape of the relative permeability curve were evaluated. Although uncertainties and limits exist, the NMR-based method is a practical and applicable method to evaluate the gas/water relative permeability of ultra-low permeability rocks.

Appendix: The JBN Calculation Procedure

In the JBN calculation method, N_p and W_i can be obtained by the volume of gas injected and water produced from the experimental data. According to Eqs. (5) and (6), the slopes of N_p versus W_i and 1/W_iI_r versus W_i are used to calculate f_w and f_w/k_rw. In this study, the natural logarithmic approximation was used to match the relationships here. Plotting N_p versus W_i and fitting the equation,

$$ N_{\text{p}} = a_{0} + a_{1} \ln W_{\text{i}} + a_{2} (\ln W_{\text{i}} )^{2} $$

(A-1)

Thus, we can derive f_w from Eqs. (A-1) and (5):

$$ \begin{aligned} f_{\text{w}} & = \frac{{{\text{d}}N_{\text{P}} }}{{{\text{d}}W_{\text{i}} }} = \frac{{a_{1} }}{{W_{\text{i}} }} + 2a_{2} (\ln W_{\text{i}} )\frac{1}{{W_{\text{i}} }} \\ & = \frac{{a_{1} + 2a_{2} \ln W_{\text{i}} }}{{W_{\text{i}} }} \\ \end{aligned} $$

(A-2)

From Eq. (6), we derive f_w/k_rw:

$$ \frac{{f_{\text{w}} }}{{k_{\text{rw}} }} = \frac{{{\text{d}}\left( {\frac{1}{{W_{\text{i}} I_{\text{r}} }}} \right)}}{{{\text{d}}\left( {\frac{1}{{W_{\text{i}} }}} \right)}} = \frac{{{\text{d}}\left( {\frac{1}{{W_{\text{i}} I_{\text{r}} }}} \right)}}{{ - \frac{1}{{W_{\text{i}}^{2} }}{\text{d}}W_{\text{i}} }} = - W_{\text{i}}^{2} \frac{{{\text{d}}\left( {\frac{1}{{W_{\text{i}} I_{\text{r}} }}} \right)}}{{{\text{d}}W_{\text{i}} }} $$

(A-3)

Then, plotting 1/W_iI_r versus W_i and fitting the resulting curves by the following equation:

$$ \ln (W_{\text{i}} I_{\text{r}} ) = b_{0} + b_{1} \ln W_{\text{i}} + b_{2} (\ln W_{\text{i}} )^{2} $$

(A-4)

i.e., $\frac{1}{{W_{\text{i}} I_{\text{r}} }} = e^{{ - [b_{0} + b_{1} LnW_{\text{i}} + b_{2} (LnW_{\text{i}} )^{2} ]}}$.

We obtain f_w/k_rw:

$$ \frac{{f_{\text{w}} }}{{k_{\text{rw}} }} = - W_{\text{i}}^{2} \frac{{\text{d}\left(\frac{1}{{W_{\text{i}} I_{\text{r}} }}\right)}}{{\text{d}W_{\text{i}} }} = - W_{\text{i}}^{2} e^{{ - \left[b_{0} + b_{1} LnW_{\text{i}} + b_{2} (LnW_{\text{i}} )^{2}\right] }}\left(0 - \frac{{b_{1} }}{{W_{\text{i}} }} - \frac{{2b_{2} LnW_{\text{i}} }}{{W_{\text{i}} }}\right) = \frac{{b_{1} + 2b_{2} \ln W_{\text{i}} }}{{I_{\text{r}} }} $$

(A-5)

Knowing f_w/k_rw from Eq. (A-5) and f_w from Eq. (A-2), we can obtain k_rw and then calculate k_rg using Eq. (4).

The outlet gas saturation is as follows:

$$ \begin{aligned} S_{\text{g2}} & = S_{\text{gi}} + N_{\text{p}} - f_{\text{g}} W_{\text{i}} \\ &= S_{\text{gi}} + (a_{0} - a_{1} ) + (a_{1} - 2a_{2} )\ln W_{\text{i}} + a_{2} (\ln W)^{2}_{\text{i}} \\ \end{aligned} $$

(A-6)

Note that the gas volume changes with the pressure change between the inlet gas (P_inlet) and the outlet gas (P_outlet) during gas flow through the core pressure. Thus, W_i is the cumulative gas injected under an average pore pressure:

$$ W_{\text{i}} = W_{{{\text{i}} - 1}} + \frac{{2P_{\text{inlet}} }}{{P_{\text{inlet}} + P_{\text{outlet}} }}\Delta W_{\text{ig}} $$

(A-7)

where ∆W_ig is the injection gas volume under the inlet gas pressure at a given interval value.