Capillary pressure curve data measured through the mercury injection method can accurately reflect the pore throat characteristics of reservoir rock; in this study, a new methodology is proposed to solve the aforementioned problem by virtue of the support vector regression tool and two improved models according to Swanson and capillary parachor parameters. Based on previous research data on the mercury injection capillary pressure (MICP) for two groups of core plugs excised, several permeability prediction models, including Swanson, improved Swanson, capillary parachor, improved capillary parachor, and support vector regression (SVR) models, are established to estimate the permeability. The results show that the SVR models are applicable in both high and relatively low porosity-permeability sandstone reservoirs; it can provide a higher degree of precision, and it is recognized as a helpful tool aimed at estimating the permeability in sandstone formations, particularly in situations where it is crucial to obtain a precise estimation value.
In the process of exploring and developing oil and gas fields, permeability has been recognized as one of the key parameters to understand the characteristics of reservoir percolation and to evaluate the productivity of oil and gas wells [
Recently, according to an increasing number of studies, intelligent systems are superior to practical and statistical methods with regard to the relative problems of geosciences and petroleum [
Mercury injection capillary pressure (MICP) curves are primarily utilized for several purposes, including the classification of rock types, calculation of the oil saturation according to the height above free water level (HAFWL) method, evaluation of the rock quality, and estimation of the relative permeability [
Throughout field applications, it can be challenging to estimate the permeability with equation (
In most cases, semilog coordinates are employed to plot MICP curves; the linear coordinates of the
As shown in Figure
Mercury injection capillary pressure (MICP) curves (a) and corresponding Swanson and capillary parachor parameters for one core sample (b).
The
Vapnik [
Introduced by Vapnik (1995), the loss function employed in the SVR method can be
A dual space can be used to reformulate this problem, which is expressed as follows:
to maximize
The desired weight vector of the regression hyper plane can be provided by:
In nonlinear regression, the input data are mapped onto a higher dimensional feature space through a kernel function, such that a linear regression hyper plane is generated. In the SVR method, some of the common kernel functions include the radial basis function (RBF), polynomial function, and sigmoid function. Under the condition of nonlinear regression, the identical approach is employed in the linear case to formulate the learning problem again; that is, the nonlinear hyperplane regression function is obtained as follows:
In mercury injection capillary pressure (MICP) data, several parameters exist that can represent the pore structure, such as the quality coefficient of the reservoir, displacement pressure
Liu et al.[
Data for the thirty core samples that have been studied in this work (Liu et al. [
Core no. | Porosity (%) | Permeability (mD) | Swanson parameter | Capillary parachor parameter | |||
---|---|---|---|---|---|---|---|
1 | 32 | 81.9 | 105.1 | 352.4 | 0.1512 | 1.004 | 1.799 |
2 | 35.1 | 56.3 | 273.9 | 2039.7 | 0.0523 | 0.332 | 4.425 |
3 | 39.3 | 8846.5 | 1402.8 | 44,158.20 | 0.0127 | 0.038 | 20.009 |
4 | 35.1 | 1285.1 | 430.2 | 6485.6 | 0.0412 | 0.139 | 6.979 |
5 | 36.4 | 3473 | 780.3 | 16,743.70 | 0.0218 | 0.076 | 11.395 |
6 | 35.3 | 6997.7 | 1373.8 | 44,489.80 | 0.0126 | 0.039 | 19.125 |
7 | 34.4 | 240.2 | 224.2 | 1182.3 | 0.1019 | 0.275 | 3.102 |
8 | 30.2 | 43.6 | 70 | 233.7 | 0.1551 | 3.159 | 1.36 |
9 | 37.7 | 1988.8 | 510.7 | 6886.2 | 0.0425 | 0.102 | 7.796 |
10 | 35.9 | 625.7 | 286 | 1940.5 | 0.0521 | 0.208 | 4.484 |
11 | 31.8 | 702 | 257.4 | 2543.5 | 0.042 | 0.286 | 4.935 |
12 | 38.8 | 6639.1 | 1429.1 | 40,714.00 | 0.0126 | 0.037 | 18.811 |
13 | 36.8 | 9567.3 | 1787.5 | 58,607.90 | 0.0077 | 0.03 | 22.27 |
14 | 33.1 | 1887.2 | 585 | 10,048.10 | 0.0323 | 0.15 | 7.833 |
15 | 28.7 | 31.4 | 59 | 172 | 0.1589 | 1.694 | 1.157 |
16 | 36 | 1710.9 | 643.4 | 15,318.50 | 0.0122 | 0.105 | 10.323 |
17 | 35.5 | 1654.2 | 641.5 | 9574 | 0.0081 | 0.098 | 11.533 |
18 | 37.3 | 3363.7 | 914.8 | 17,107.70 | 0.008 | 0.059 | 13.284 |
19 | 37.8 | 1024.5 | 353.1 | 2968.7 | 0.0321 | 0.149 | 5.509 |
20 | 31 | 39.4 | 22.9 | 202 | 0.1577 | 1.694 | 1.225 |
21 | 26.3 | 294 | 397.2 | 4092.3 | 0.0494 | 0.144 | 5.015 |
22 | 27.2 | 455 | 438.7 | 4514.1 | 0.0492 | 0.128 | 5.627 |
23 | 27.5 | 677 | 581.6 | 7925.2 | 0.0354 | 0.093 | 7.29 |
24 | 22 | 48.1 | 144.8 | 1046.7 | 0.0767 | 1.671 | 2.007 |
25 | 19.4 | 7.3 | 75.9 | 189.3 | 0.2123 | 0.93 | 1.129 |
26 | 25.6 | 122 | 277.7 | 2467 | 0.0628 | 0.465 | 3.379 |
27 | 27.4 | 274 | 355.6 | 3516.5 | 0.0628 | 0.247 | 4.021 |
28 | 25.3 | 216 | 293.9 | 2751.7 | 0.0628 | 0.461 | 3.428 |
29 | 24.6 | 347 | 466.6 | 4975.1 | 0.0492 | 0.114 | 6.056 |
30 | 27 | 470 | 451.9 | 5448.4 | 0.049 | 0.128 | 5.958 |
To evaluate the simple relationships between the quality coefficient of the reservoir and MICP parameters, this paper exploited an analysis technique to highlight the sensitivity of the quality coefficient of the reservoir in the MICP data. This process could also be useful to select input variables in that many unrelated MICP parameters could be eliminated. In addition, the number of input parameters of the model is rather large, which may overwhelm the model and potentially generate a certain amount of noise rather than the signal. Furthermore, the simple linear regression test (cross plotting) was used, of which the correlation coefficient (
Cross plots of the MICP parameters of the 30 core samples are presented in Figure
Cross plots of MICP parameters and permeability of rock samples: (a) porosity-permeability, (b) Swanson parameter permeability, (c) capillary parachor parameter permeability, (d)
By virtue of a back-propagation neural network, a stable method was proposed by Dutta and Gupta [
The contribution of each input variable is given by:
An improved approach was proposed, and the optimal number of inputs was calculated. First, as shown in Table
Related effort of every input in estimating the permeability according to the sensitivity analysis and correlation coefficient notion.
Input variables | Parameters | Correlation coefficient ( | Relative contribution (%) | Rank no. |
---|---|---|---|---|
X1 | Porosity | 0.3314 | 4.5097 | 4 |
X2 | Swanson parameter | 0.9169 | 27.5106 | 2 |
X3 | Capillary parachor parameter | 0.9685 | 39.1604 | 1 |
X4 | 0.2587 | 2.5836 | 6 | |
X5 | 0.1088 | 2.9899 | 5 | |
X6 | 0.9092 | 23.2458 | 3 |
The optimal number of introduced inputs was an important parameter influencing the design of the SVR model. Thus, on the basis of their RC values, MICP parameters were introduced into the SVR model one by one, and the performance of the SVR model was assessed for every group of inputs. As shown in Figure
Comparison of the RMSE and correlation coefficient (
As shown in Figure
Flowchart explaining the input variable selection by the sensitivity analysis method.
In terms of the evaluation and comparison of the performance of the suggested SVR model, certain earlier methods, such as the capillary parachor parameter-based model and Swanson parameter-based model, were employed to estimate permeability values by using the same dataset.
According to Swanson [
As mentioned before, the capillary parachor parameter refers to the maximum of the cross plot of the mercury injection saturation
Furthermore, resorting to the method suggested by Guo et al. [
Recently, Liu et al. [
To construct a model aimed at the estimation of the permeability from the MICP data, an epsilon support vector regression (
Model construction primarily relied on data from the group of 30 core samples (Table
The predictive performance of the three models was evaluated through the correlation coefficient (
Comparison between the permeability prediction results and assessments of the core specimens: (a) Swanson parameter-based model results; (b) capillary parachor parameter-based model results; (c) SVR model results; (d) relative error of all models. A notable match is indicated by the results between the SVR model predicted and the measured permeability.
As shown in Figure
Analysis was conducted with mercury injection capillary pressure (MICP), porosity, and permeability data of 22 core plugs. In addition, the method mentioned in Figure
MICP curve (a) and related Swanson parameters and capillary parachor parameters (b) for one of the 22 core samples.
Data sets of 22 core samples, including the experimental data and model predicted permeability data.
Core no. | Porosity (%) | Permeability (mD) | Swanson parameter | Capillary parachor parameter | Swanson model results (mD) | Improved Swanson model results (mD) | Capillary parachor model results (mD) | Improved capillary parachor model results (mD) | SVR model results (mD) |
---|---|---|---|---|---|---|---|---|---|
1 | 4.06 | 18.89 | 285.6 | 3171.4 | 30.8 | 27.55 | 32.72 | 23.81 | 18.69 |
2 | 7.89 | 4.56 | 87.6 | 729.2 | 3.66 | 4.25 | 3.9 | 6.49 | 4.26 |
3 | 15.47 | 9.73 | 198.2 | 383.8 | 15.95 | 18.07 | 1.54 | 4.31 | 9.18 |
4 | 9.34 | 7.32 | 170.4 | 1385.5 | 12.12 | 13.09 | 9.87 | 13.84 | 11.37 |
5 | 4.96 | 23.84 | 243.7 | 3270.5 | 23.09 | 21.72 | 34.21 | 26.73 | 24.98 |
6 | 3.86 | 10.73 | 219.5 | 1597.9 | 19.14 | 17.66 | 12.13 | 11.18 | 11.59 |
7 | 17.11 | 5.51 | 92.2 | 442.6 | 4.01 | 5.15 | 1.89 | 5.24 | 5.39 |
8 | 10.45 | 2.84 | 64.5 | 1125.4 | 2.11 | 2.66 | 7.3 | 11.61 | 4.02 |
9 | 3.97 | 27.11 | 245.3 | 3260.4 | 23.44 | 21.36 | 34.06 | 24.29 | 27.61 |
10 | 11.08 | 3.77 | 89.2 | 632.2 | 3.78 | 4.59 | 3.17 | 6.41 | 4.31 |
11 | 7.14 | 25.11 | 198.1 | 1597.9 | 15.95 | 16.25 | 12.13 | 14.43 | 19.22 |
12 | 15.09 | 94.69 | 487.9 | 5260.4 | 81.02 | 80.37 | 68.07 | 70.6 | 76.92 |
13 | 15 | 5.81 | 179.6 | 1201.3 | 13.3 | 15.22 | 8.02 | 14.46 | 7.41 |
14 | 11.8 | 22.84 | 297.2 | 1842.8 | 33.18 | 34.16 | 14.91 | 20.71 | 22.87 |
15 | 12.39 | 30.35 | 255.1 | 2451.2 | 25.19 | 26.69 | 22.53 | 28.69 | 31.65 |
16 | 13.29 | 48.76 | 297.6 | 4242.8 | 33.18 | 34.72 | 49.87 | 53.19 | 48.27 |
17 | 6.44 | 2.58 | 61.5 | 106.3 | 1.93 | 2.3 | 0.24 | 0.76 | 1.39 |
18 | 7.56 | 2.35 | 107.2 | 510.5 | 5.25 | 5.89 | 2.32 | 4.35 | 3.19 |
19 | 5.18 | 6.06 | 243.4 | 1842.8 | 23.09 | 21.85 | 14.91 | 14.72 | 6.36 |
20 | 23.27 | 138.35 | 681.6 | 8544.7 | 148.43 | 148.9 | 137.4 | 142.12 | 140.75 |
21 | 16.43 | 95.69 | 495.8 | 7144.7 | 83.44 | 83.54 | 106.04 | 101.54 | 100.63 |
22 | 8.24 | 5.01 | 113.3 | 2687 | 5.8 | 6.53 | 0.92 | 2.26 | 3.79 |
In the latter stage of this study, the well-known Swanson and capillary parachor models were improved by adding porosity information since preceding studies had indicated that the porosity had a significant influence on predicting the permeability [
Visualization of the final results of the Swanson parameter-based models and capillary parachor parameter-based model: (a) Swanson model; (b) improved Swanson model; (c) capillary parachor model; (d) improved capillary parachor model.
The following equations show the established model functions.
Swanson model:
Improved Swanson model:
Capillary parachor model:
Improved capillary parachor model:
After model construction, the obtained predicted permeability results for each model are presented in Table
Comparison of the permeability prediction results and measurements of the 22 core samples: (a) Swanson parameter-based model results; (b) improved Swanson parameter-based model results; (c) capillary parachor parameter-based model results; (d) improved capillary parachor parameter-based model results; (e) SVR model results; (f) relative error of all models. The results indicate a notable match between the SVR model predicted and the measured permeability.
Relative error analysis of the predicted permeability of the five types of models.
Core no. | Swanson model relative error (%) | Improved Swanson model relative error (%) | Capillary parachor model relative error (%) | Improved capillary parachor model relative error (%) | SVR model relative error (%) |
---|---|---|---|---|---|
1 | 63.01 | 45.81 | 73.19 | 26.01 | -1.06 |
2 | -19.68 | -6.74 | -14.52 | 42.34 | -6.50 |
3 | 64.00 | 85.77 | -84.19 | -55.71 | -5.59 |
4 | 65.51 | 78.80 | 34.77 | 89.08 | 55.35 |
5 | -3.12 | -8.86 | 43.52 | 12.14 | 4.81 |
6 | 78.40 | 64.58 | 13.04 | 4.17 | 8.07 |
7 | -27.12 | -6.58 | -65.67 | -4.95 | -2.06 |
8 | -25.76 | -6.37 | 157.41 | 309.19 | 41.66 |
9 | -13.54 | -21.21 | 25.64 | -10.41 | 1.85 |
10 | 0.31 | 21.71 | -15.95 | 69.96 | 14.44 |
11 | -36.46 | -35.28 | -51.70 | -42.54 | -23.44 |
12 | -14.43 | -15.12 | -28.10 | -25.44 | -18.76 |
13 | 128.96 | 162.02 | 38.17 | 148.91 | 27.52 |
14 | 45.28 | 49.58 | -34.71 | -9.32 | 0.16 |
15 | -16.99 | -12.05 | -25.75 | -5.46 | 4.29 |
16 | -31.96 | -28.79 | 2.27 | 9.08 | -1.00 |
17 | -25.10 | -11.02 | -90.71 | -70.68 | -46.15 |
18 | 123.83 | 151.04 | -0.94 | 85.29 | 35.98 |
19 | 281.34 | 260.89 | 146.21 | 143.01 | 5.10 |
20 | 7.28 | 7.62 | -0.69 | 2.72 | 1.73 |
21 | -12.80 | -12.69 | 10.82 | 6.12 | 5.16 |
22 | 15.69 | 30.27 | -81.68 | -54.80 | -24.25 |
To assess the models’ performance, two significant concepts were used, including the correlation coefficient (
Once the
Finally, Figure
Error distribution statistics for all models proposed in this study aimed at predicting the permeability. Low values of the mean standard deviation (STD) indicate the exceptional performance of the SVR model: (a) Swanson parameter-based model, (b) improved Swanson parameter-based model, (c) capillary parachor parameter-based model, (d) improved capillary parachor parameter model, and (e) SVR-based model.
As illustrated in Figures
The permeability, as one of the most significant quality parameters of reservoirs, is capable of providing meaningful data for characterizing reservoirs and petro physical studies when used in combination with the porosity. In fact, certain researchers have attempted to determine the formation permeability by virtue of empirical correlations based on related experiments. Numerous studies have estimated the permeability based on mercury injection capillary pressure (MICP) data due to the significance of the called-for permeability knowledge. However, the estimation requires methods with great precision. Throughout the paper, the support vector regression method and two improved models based on the Swanson model and capillary parachor parameter-based model were utilized in response to this requirement. The MICP data and porosity information were utilized in the SVR model, including the Swanson parameter, capillary parachor parameter, mean pore throat radius (
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors are very grateful to the key consulting project of the Chinese Academy of Engineering (2020-XZ-13), the National Key Research and Development Program (2019YFC1904304), and the research on the rheological properties of rock failure in deep mines and its impact on roadway stability (SKLMRDPC19ZZ08) for financial support.