New Relative Permeability Models for North American Sandstone Utilizing Artificial Neural Networks

Introduction

The petrophysical properties of a reservoir rock are the characteristics that depict how these rocks hold or allow the transmissibility of reservoir fluids. These properties include porosity, relative permeability, and water saturation. The accurate determination of these properties helps petroleum engineers make accurate and precise reservoir deliverability predictions, production performance, and recovery factors [1].

Absolute permeability is the ability of a fluid to flow through a permeable rock when only one fluid saturates the pore spaces. For multiphase flow, effective permeability describes the ability of each fluid to flow in the porous medium. Relative permeability is the ratio of the effective to the absolute permeability and is one of the critical parameters in predicting two-phase flow and oil recovery. This multiphase flow in porous media has been a common phenomenon in the oil and gas industry [2]. Petroleum engineers use relative permeability, a non- linear fluid and rock property function, to predict fractional flow in reservoirs [3]. Zhang J, et al. [1] used experimental laboratory procedures and mathematical models to determine relative permeability. Guler B, et al. [3] cited four mathematical models, namely capillary, statistical, empirical, and artificial intelligence (AI).

Nowadays, Artificial Intelligence (AI) is gaining more popularity in developing predictive correlations because the prediction of petrophysical properties requires high accuracy and precision since variations in prediction may lead to loss of capital and time [4]. Most AI methodologies deploy the application of two algorithms: ANN and Adaptive Neuro-Fuzzy Inference System (ANFIS) [5]. In as much as these algorithms can be advantageous, their functionality may be limited since the source of wellbore data is sparse [6], and this, together with missing data, could lead to underfitting or overfitting in most scenarios. An ANN is a computational network that mimics the human brain neurons using similar pattern recognition [7]. The networks contain interconnected neurons composed of input, hidden, and output layers. About 70% of the data set is usually used in a ‘training phase’; the remaining 30% is used for validation and testing purposes to accurately predict the model output [8]. Artificial neural networks have shown promising results in predicting relative permeability [5, 9].

Literature Review

Corey AT, et al. [10] developed a correlation for determining relative permeability based on the works of Purcell and Burdine. Generalized models were later created from Corey’s equations to calculate the relative permeabilities of water and oil. These water and oil relative permeabilities (Equations 1 and 2) are applicable for water-wet sandstone reservoirs and will be used for comparison.

4 $$ k _ {r w} = \left(\frac {S _ {w} - S _ {w c}}{1 - S _ {w c}}\right) ^ {4} $$ w wc rw wc

Table 1 shows a set of equations by Ibrahim, et al. [11] and Kalam, et al. [12] for Oil-Gas and Water-Oil Relative Permeabilities. Wyllie suggested an empirical relative permeability model that was accurate for drainage flow in oolitic limestone, consolidated and unconsolidated sandstone [13] & Honarpour M, et al. [11]. These equations are shown in Table 1 and will be compared to our results.

( ) ( )   − − = − + −   − − − −  

S S S S k S S S S S S S

3.6 0.035388 0.010874 0.56556 1 1

w wc w or rw w w wc wc or wc or

Honarpour M, et al. [14] also developed empirical relative permeability correlations (Equations 3 and 4) that predicted accurate results. The authors used data from North American oilfields and oilfields from Libya, Iran, Argentina, and the United Arab Emirates. These equations were devised for carbonate and non-carbonate formations. Equations 3 and 4 will also be utilized for comparison purposes.

2.9 ( ) ( ) ( )

(3)

1.8 $$ S _ {o} = 0. 7 6 0 6 7 \left[ \frac {\left(\frac {S _ {o}}{1 - S _ {w c}}\right) - S _ {o r}}{1 - S _ {o r}} \right] ^ {1. 8} \left(\frac {S _ {o} - S _ {o r}}{1 - S _ {w c} - S _ {o r}}\right) ^ {2} + 2. 6 3 1 8 \phi \left(1 - S _ {o r}\right) \left(S _ {o} - S _ {o r}\right) $$ S S S S S k S S S S S S φ o or wc o or ro or o or or wc or

2 1 0.76067 2.6318 1 1 1

Where oS : = oil saturation, fraction and or S = residual oil saturation, fraction Ibrahim MNM, et al. [15] developed a comprehensive collection of models for relative permeability, utilizing normalized data that spans across various wettability conditions, as well as sandstone and carbonate reservoirs containing gas-condensate, gas-oil, gas-water, and oil- water systems. They deployed linear regression techniques ( ) ( ) ( ) ( )

3 * 1.6 2 5 * * 2 * 5 4 * 1.5

S k S S S S S S

w rw w or w or w or

0.22120304 0.24933592 21.370925 83.491972 $$ = 0. 2 2 1 2 0 3 0 4 \left(S _ {w} ^ {*}\right) ^ {1. 6} + 0. 2 4 9 3 3 5 9 2 S _ {o r} ^ {2} \frac {\left(S _ {w}\right)}{\phi} + 2 1. 3 7 0 9 2 5 \left(S _ {w} ^ {*}\right) ^ {2} S _ {o r} ^ {5} + 8 3. 4 9 1 9 7 2 \phi^ {4} \left(S _ {w} ^ {*}\right) ^ {5} S _ {o r} ^ {1. 5} - $$ ( ) ( ) ( )

4 2 2.3 3 * 2 * 3 *

( )( ) (4) to determine goodness of fit using the coefficient of determination (R2) values. R2 exceeded 0.6 for all the developed models. Their relative permeability equations (Equations 5 & 6) for water and oil will be used to compare the accuracy of the developed model.

φ φ (5)

0.4562939 1161.07198 8.7866012 ( ) ( ) ( ) ( ) ( ) ( ) ( )

3 2 10 3 0.4 * * 6

0.00000578 1 12.841061 $$ k _ {r o w} ^ {*} = 1 - 3. 0 9 0 9 9 6 S _ {w} ^ {*} + 2. 8 6 7 0 2 2 9 \left(S _ {w} ^ {*}\right) ^ {1. 6} - 0. 7 6 8 9 5 2 \left(S _ {w} ^ {*}\right) ^ {2} \tag {6} $$ $$ S _ {w} ^ {*} = \frac {S _ {w} - S _ {w c}}{1 - S _ {w c} - S _ {o}} $$ and φ = porosity, fraction where: * 1 w wc w wc orw ak = absolute permeability, mD Zhang J, et al. [1] used experimental laboratory procedures and mathematical models to successfully predict gas-water relative permeability using five core samples from a tight sandstone gas reservoir and observed that the relative permeability of gas decreased with increased pore pressure at constant water saturation. They also established that when the pore pressure remains the same, a decrease in the absolute permeability results in a decrease in gas-water relative permeability.

Model Development and Discussion

In this work, data was divided into three data sets: one for training, one for validation, and another for testing. So, during the process, about 70% of the data was used for training, 15% for validation, and the remaining 15% for testing.

Model Comparison

The developed model was compared to Modified Corey’s

and Ibrahim and Koederitz’s. The following table compares RMSE values for the new model and the correlations mentioned above.

As such, the new model showed better results and can be used to predict relative permeability to water and oil for sandstone formations within the tested data range.

Conclusions

Model testing was first attempted with six variables, namely Swc, Sor, ka, φ, Sw, and So, for data set 1 (18 data points). The degree of fit for kro was high, with an R2 that ranged between 0.93 and 0.95.

However, the degree of fit for krw was low, with an R2 of only 0.63. Eight variables were later used with normalized water and oil saturations to improve the model’s fit. Using the first data set, the overall R2 improved for krw and kro to 0.9939 and 0.9881, respectively. Using 2, 3, 4, and 5 neurons did not affect model’s fit.

Using a different data deck (data set 2 with 64 data points), the overall R2 for water and oil with ten hidden layers and three neurons was high, with values of 0.9749 and 0.9832, respectively.

All data (82 data points) has also been used to globalize the model. These data sets were enough to accurately predict the relative permeability of the North American Sandstone.

R2 values for water and oil showed an excellent fit with values of 0.9776 and 0.9769, respectively. The RMSE values for both models were extremely low and came close to zero for training and validation compared to the Corey Ibrahim and Koederitz models.

Acknowledgements

Not applicable.

Competing Interests

The authors declare that they have no competing interests.

Availability of Data and Materials

The data supporting the conclusions of this paper are included within the paper. Any queries regarding these data may be directed to the corresponding author.

Code Availability (Software Application or Custom Code Used)

N/A

Consent for Publication

Authors have agreed to submit it in its current form for publication in the journal.

Ethics Approval and Consent to Participate

Not applicable. No test, measurements or experiments on animals were performed as part of this work.

Funding

This research was not funded.