An Initial Approach of Multiple Linear Regression in CO₂-water Relative Permeability Prediction for Carbon Storage Projects

1. Introduction

Class VI well (EPA, 2013) requires site characterization and prediction of the extent of the injected CO₂ plume and associated pressure front, whichever is further defined as Area of Review. The Area of Review is identified via dynamic modeling, which is highly sensitive to the CO₂-water relative permeability of the injection formation. This work discusses the potential of the previously published relative permeability data of the injection formations (Yu et al., 2023) at one well to be applied to another nearby well. This approach might be useful for projects with limited sources of the special core data for dynamic modeling—relative permeability of the aquifer formations between water and CO₂.

2. Method

This work uses multiple linear regression to determine the dependent variable crosspoint saturation ($C{S}_w$) of water and CO₂ relative permeability curves from the irreducible water saturation (${Sw}_i$) and true reference cross-point saturation ($RCS$). The first independent variable ${Sw}_i$ is a decimal that has the range of 0–1. To constrain this range, another variable, true reference cross-point saturation ($RCS$) is also introduced. $RCS$ has a physical meaning and is half of the dynamic space (Mirzaei-Paiaman, 2021):

$$RCS=0.5\times (1-{Sw}_i)$$

(1)

The multiple linear regression is defined as:

$$C{S}_w=w_1·{Sw}_i+w_2·RCS+b\mathrm{}$$

(2)

$w_1$, $w_2$, and $b$ are fitting parameters. This work has limited core data from two CO₂ injection wells, PRB#1 and PRB#2, for three potential storage reservoirs—the Lower Cretaceous Lakota Sandstone, Jurassic Hulett Sandstone, and Pennsylvanian Minnelusa Formation—at the Dry Fork Station site, where PRB#1 data (Yu et al., 2023) is for training, and PRB#2 data is for testing. Both PRB#1 and PRB#2 relative permeability data were acquired with the unsteady-state method (Johnson et al., 1959). Modified Brooks and Corey model (MBC (Behrenbruch & Goda, 2006) is adopted for the relative permeability curve fitting:

$$krw={krw}_{max}·{\left(\frac{S_w-S_{wi}}{S_{w_{max}}-S_{w_i}}\right)}^{n\_water}$$

(3)

$$krg={krg}_{max}·{\left(\frac{S_{w_{max}}-S_w}{S_{w_{max}}-S_{w_i}}\right)}^{n\_gas}$$

(4)

Where $S_{w_{max}}=1$ for the CO₂ injection into the aquifer scenario. For this application, the crosspoint $C{S}_w$ serves to determine the index $n\_water$ and $n\_gas$ for the predicted curves. Thus, only the endpoint $S_{wi}$ needs to be figured out for a certain core sample, which is the purpose and expectation of the work, while the limitations are concluded in the last section. Refer to the Appendix for the coding. Independent and dependent variables are tabulated in Table 1.

3. Result and Discussion

The predicted $C{S}_w$ is listed in Table 2. The mean absolute percentage errors (MAPE) of the samples Lakota 8063, Hulett 8330, and Minnelusa 9487 are 2.94%, 1.39%, and 1.67%, respectively, considered insignificant. Further, the relative permeability curve expression is displayed in Figure 1.

The predicted $C{S}_w$ and relative permeability expression are agreeably similar and feasible for dynamic modeling from a practical perspective. This suggests that the irreducible water saturation serves as a critical indicator for the rock’s wettability, which dominates the fluid flow in porous media and might even neglect the petrophysical barriers of varying formations at the studied site. Yet, as the training and testing datasets used for this work are limited, this approach using irreducible water saturation (${Sw}_i$) and reference crosspoint saturation ($RCS$) to predict the crosspoint saturation ($C{S}_w$) and further relative permeability curves of the injection formations at the nearby wells might only be applicable to this project. More delicate models and approaches should be quested for similar applications to assist the carbon storage projects efficiently and accurately.

4. Conclusion

This work discusses the feasibility of multiple linear regression in predicting water/CO₂ relative permeability using training and testing datasets from two nearby wells, separately, of the Lower Cretaceous Lakota Sandstone, Jurassic Hulett Sandstone, and Pennsylvanian Minnelusa Formation at the Dry Fork Station site. The outcome is encouraging as the predicted relative permeability data is usable for dynamic modeling from a practical standpoint at the study site. Meanwhile, whether this approach could be generally applicable needs more delicate models and larger training datasets to be determined.