Model of Langmuir Volume at Reservoir Temperature
As the methane adsorption in shale is an exothermic process, the methane adsorption capacity is reduced at a higher temperature. It has been observed that the Langmuir volume decreases with increasing temperature [7,17]. A linear negative correlation exists between the Langmuir volume and temperature, which can be written in the following equation:
where VL(T) is the Langmuir volume at reservoir temperature, scf/ton; T is the reservoir temperature, °C; d and e are the fitting coefficients. The value of the trend-line slope, d, is described as the decrease rate of Langmuir volume with increasing temperature, which quantitatively describes the temperature effect on the Langmuir volume.
It has been concluded that the methane adsorbed on the organic matter is more sensitive to the temperature than the methane adsorbed on the clay minerals . The finding is also confirmed in Figure 6, which displays a positive relationship between the TOC and decrease rate of VL, with the R2 of 0.58. With this relationship, the decrease rate of VL (d) can be calculated based on the TOC using Equation (5):
Figure 6. The relationship between the TOC and decrease rate of Langmuir volume with increasing temperature.
Given that the Langmuir volume at experimental temperature discussed in last section, the Langmuir volume at reservoir temperatures can be estimated using the decrease rate of VL or the d value from Equation (5). Thus, the Langmuir volume at reservoir temperature can be written as:
where VL(T) is the Langmuir volume at reservoir temperature, scf/ton; T is the reservoir temperature, °C; T0 is the experimental temperature, °C; TOC is the total organic carbon, wt %; Vsh is the total clay content, %.
As the available data for the temperature dependence of Langmuir volume has a TOC range of 0.23 to 5.15 wt %, the result here may not be reliable for shale with larger TOC. Moreover, the samples with the TOC range of 3.03 to 5.15 wt % have a larger variation on the relationship than low TOC samples. Therefore, more data is required for the shale samples with TOC larger than 3.03 wt % in terms of the temperature dependence of Langmuir volume.
Model of Langmuir Pressure at Experimental Temperature and Reservoir Temperature
Langmuir pressure is also required to calculate the methane adsorption capacity in actual reservoir conditions. The reciprocal of the Langmuir pressure represents the affinity of the gas for sorbent. It has been concluded that the adsorption affinity on the organic matter is stronger than that on clay minerals . Thermal maturity and volume of small pores were also regarded as controlling factors of the Langmuir pressure [5,10].
Herein, the Langmuir pressure shows no trend with the thermal maturity, TOC or clay content, but a logarithmic-law trend exists to the Langmuir volume, with R2 of 0.31 (Figure 7). The low correlation might result from the sensitive and various controlling factors of the Langmuir pressure. The shale sample with a large Langmuir volume has a high Langmuir pressure, which represents a weak adsorption affinity of methane. As reported, the organic matter and small pore have stronger adsorption affinity of methane comparing to the clay and large pore, respectively.
In this case, the adsorption affinity of methane in shale reflects the proportion of adsorbed methane in the small pore and organic matter. Thus, the weak adsorption affinity in the shale sample with a large Langmuir volume might infer that the proportion of adsorbed methane in the small pore and organic matter is low. Since the relationship between the Langmuir volume and Langmuir pressure is obtained from a large amount of data, the relationship can be informative. Therefore, the Langmuir pressure can be predicted using the following equation:
where PL and VL are the Langmuir pressure and Langmuir volume at experimental temperature in psi and scf/ton, respectively.
Figure 7. The relationship between the Langmuir pressure and Langmuir volume at experimental temperature for the studied samples.
The temperature dependence of the Langmuir pressure has been described by the following equation :
where m and n is the fitting coefficient, resulting from the thermal dynamic parameters: the heat of adsorption and the standard entropy of adsorption. These parameters have been compared between the organic matter and clay minerals, concluding that the methane adsorbed on the organic matter releases more heat than the methane adsorbed on clay. A linear relationship between the heat of adsorption and the standard entropy of adsorption has been proposed for different types of kerogen, clay and shale samples at different thermal maturity . It might imply that the thermal dynamic parameters are related to the TOC. For each shale sample, the coefficient m and n are determined using linear fitting on ln(1/pL) and 1/(T+273). In terms of the studied samples, the plot of m with TOC is listed in Figure 8 as the following equation:
Figure 8. The relationship of the TOC to the fitting coefficient m.
Combined with the prediction model for the Langmuir pressure at experimental temperature, the Langmuir pressure at reservoir temperature (PL(T)) can be obtained using the following equations:
where gc(T) is the adsorbed gas content at certain temperature and pressure, scf/ton; VL(T) is the Langmuir volume at reservoir temperature, scf/ton; PL(T)
is the Langmuir pressure at reservoir temperature, psi; P is the reservoir pore pressure, psi.
In this study, we proposed a prediction model for the methane adsorption capacity in shales based on the high-pressure methane adsorption experiment result. The methane adsorption capacity at certain pressure and temperature can be calculated using the Langmuir model with the Langmuir parameters. Herein, the prediction model for methane adsorption in shales was built in 4 steps: a model of the Langmuir volume at experimental temperature, the temperature dependence of the Langmuir volume, a model of the Langmuir pressure at experimental temperature, the temperature dependence of Langmuir pressure.
The model of the Langmuir volume at experimental temperature considers the TOC and clay content without the thermal maturity, which shows no relationship with the TOC-normalized Langmuir volume. The predicted Langmuir volume at experimental temperature was plotted against the measured results, showing a good R-square. However, more data is still required to improve the model, as the shale samples in the TOC range of 10 to 25 wt % is rarely measured.
For the other three steps, the relationships are informative but not precise enough to provide a reliable prediction. A positive relationship exists between the TOC and decrease rate of Langmuir volume with increasing temperature based on the published data, which requires more data for the shale samples with TOC larger than 3 wt %. As the Langmuir pressure is sensitive to too many factors, it is hard to estimate using the TOC and clay content. However, a logarithmic-law trend is observed between the Langmuir volume and Langmuir pressure at experimental temperature; the temperature dependence of Langmuir pressure is related to the TOC. Furthermore, as the high-pressure methane adsorption experiments on shales were measured under different conditions in the references, the amount of samples for the temperature dependence of the Langmuir parameters is insufficient, which constrains the accuracy of the related models.
Moisture was not considered in this study, which is also regarded as an important controlling factor on methane adsorption in shales [22,23,24]. The existing moisture in shales occupies pore volume or blocks pore throat to reduce the methane adsorption capacity. However, the moisture content employed in the references are various, which are not available to compare with each other or collect sufficient data. Moreover, it is very uncertain for the moisture content under in-situ conditions and its variation within a shale reservoir.
The major application of this study is that the well log data can calculate TOC and Vsh without any problem and therefore this study can help to calculate VL and PL at the reservoir condition for volumetric calculation of absorbed gas in shale reservoirs.
Investigation and writing—original draft preparation, J.Z.; writing—review and editing and supervision, R.R.
This research was funded by China Scholarship Council, File No.201506440050.
The authors would like to thank the Unconventional Gas Research Group in Curtin University for providing support for the experimental work.
Conflicts of Interest
The authors declare no conflict of interest.
Table A1. The detailed information about the studied shale samples in this paper.
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