A Method for Evaluating Reservoir Permeability Based on Machine Learning Flow Unit Index

This research area is the H formation in the Y gas field in the X depression. The reservoir of this formation is the underwater distributary channel sand body formed in the lake-braided river delta system of the H formation, which is superimposed in a complex distribution, forming multiple sets of extremely thick reservoirs [1-3]. Owing to the development of high-quality reservoirs locally in this thick sand layer, which is distributed in a separate network, a unique network seepage system was formed in the study area, allowing the thick block sand body to retain good local permeability, even when buried deeper.

From the perspective of stratigraphic sedimentation, the H formation can be divided into upper and lower segments [4, 5]. The lithology of the upper member of the H formation is mainly brownish-gray, gray mudstone; silty mudstone; light-gray argillaceous siltstone; siltstone; silty fine sandstone; and fine sandstone. The upper part is characterized by mud mixed with sand or sand and mud interbedding; the lower part is characterized by sand mixed with mud; and the top is developed with brown-red, purple-red, brown yellow, and other variegated mudstones. The lower section of the H formation is widely distributed throughout the depression. The lithology is mainly gray, dark gray silty mudstone, mudstone, gray white, light gray argillaceous siltstone, siltstone, fine sandstone, pebbly sandstone, and glutenite, with thin coal seams and coal lines.

Overall, the lithology of the H formation is composed mainly of fine sandstone with locally developed medium sandstone, coarse sandstone, and sandy gravel. The reservoir particle minerals were primarily quartz, followed by rock debris particles containing a small amount of feldspar. The particles were mainly rounded into subangular and subcircular shapes, and the sorting was mainly medium and good. At present, the main target layers discovered in the X depression exploration generally have low porosity and permeability and high bound water content. These reservoirs are influenced by various factors such as sedimentation and diagenesis and exhibit strong heterogeneity. There were significant differences in reservoir connectivity, and the relationship between porosity and permeability was complex.

Using more than 200 cores from the H formation of the Y gas field in the X depression, we completed a large number of core mercury injection experiments. According to the mercury injection analysis parameters, mercury injection amount, displacement pressure, pore throat size, and sorting, the pore throat structure of the H formation in the X depression can be divided into three categories. As can be observed from the characteristics of the capillary pressure curve in Figure 1, the low displacement pressure of the reservoir reflects a large average pore throat, good connectivity, and seepage capacity of the pore throat. The curve presents an obvious platform under a low-mercury inlet pressure, and the gentle section is the main mercury inlet section. The longer curve of this section indicates the more concentrated distribution and better sorting of rock sample throats, and the lower position of this section indicates the larger throat radius. The capillary pressure curve of one type of reservoir showed that the rock sample had a thick pore throat, good sorting, and good physical properties. The displacement pressure of the Type II reservoirs was higher than that of the Type I reservoirs, and the curve platform was shorter than that of the Type I reservoirs, indicating that this type of sample had a thicker pore throat, better sorting, and medium petrophysical properties. The displacement pressure of Type III reservoirs was higher than that of Type II reservoirs, and the curve platform was shorter, indicating that this type of sample had a narrow pore throat, poor sorting, and poor petrophysical properties. Overall, the mercury injection curve platforms of Types I, II, and III reservoirs gradually became shorter and steeper, and the displacement pressure gradually increased, reflecting the gradual deterioration of the physical properties and pore structure of Types I to III reservoirs.

Based on the conventional physical properties and capillary pressure data from the mercury injection method, the pore structure characteristic parameters of the different types of reservoirs were determined, as listed in Table 1. The table shows that the pore structure characteristic parameters of the different reservoir types, such as permeability, average pore throat radius, and displacement pressure, are significantly different.

The pore structure of a reservoir mainly refers to the size, distribution, and connectivity of pores and throats in the rock; the geometric form of the pore throat space; and the connectivity between the pore throats. From the relationship between the porosity and permeability K of the H formation core in the X depression (Figure 2), there is an overall trend of K increasing with the increase of ϕ, but the porosity and permeability cannot be represented by a single-fitting relationship. Within the range defined by the black coil in Figure 2, when the porosity of the core is similar, the difference in permeability can reach two to three orders of magnitude, and some cores show extremely strong heterogeneity. When K < 1 × 10⁻³ μm², the pore permeability relationship is more concentrated; K > 1 × 10⁻³ μm², the pore permeability relationship is dispersed. This indicates that permeability is not only affected by porosity but also by other factors.

Due to the concentrated distribution of permeability below 1 × 10⁻³ μm², pore distribution histograms were established for rock samples with K < 1 × 10⁻³μm², 1×10^-3μm²<K < 5 × 10⁻³μm², and K > 5 × 10⁻³μm². The porosities of the samples ranged from 5% to 9%. When K < 1 × 10⁻³μm², the small pore size is the main component (Figure 3), and as K increases, the larger pore size component slightly increases. When 1 × 10⁻³μm²<K < 5 × 10⁻³μm², as K increases, the increase in large pore size components is more significant. When K > 5 × 10⁻³μm², the distribution is mainly for the large pore size. Therefore, permeability is greatly influenced by the pore size distribution, and pore structure is also one of the factors affecting permeability. The reservoir permeability is influenced by both porosity and pore structure.

To address this problem, previous researchers established a permeability model based on the flow zone index (FZI). The FZI, as a quantitative indicator for identifying and dividing flow units, is widely used by scholars in the field of oil and gas development. After using this indicator, the permeability is clearly distinguished, and the correlation between the porosity and permeability of each flow unit is good, which can improve the accuracy of reservoir parameter prediction. However, the drawbacks of the FZI method are evident. In theory, the FZI combines the structure, mineral geological characteristics, and pore-throat characteristics to determine the geometric phase of pores [6]. However, the FZI calculation method is too idealized, weakening the influence of pore structure heterogeneity, and RQI/ϕ_z has no obvious geological significance. If K and ϕ_e increase or decrease by an appropriate multiple simultaneously, the same FZI value will be obtained. It leads to the erroneous conclusion that high porosity and high permeability reservoirs and low porosity and low permeability reservoirs are of the same type of flow unit, which does not meet the requirement of the maximum difference in reservoir physical properties between different flow units [7, 8]. The accuracy of dividing flow units based on FZI is insufficient, resulting in uneven water drive within the same unit in actual development. Therefore, the method of dividing flow units using only FZI as a parameter cannot meet the needs of fine division of flow units in the later stages of development [9, 10]. To eliminate this hidden danger, this paper proposes an FZI method that considers pore throat parameters (Figure 4).

The flow unit is a reservoir genetic unit formed by various geological processes and is a comprehensive product of sedimentation, diagenesis, and later transformation. C. L. Hearn et al. believe that a flow unit is a continuous reservoir in both horizontal and vertical directions, with similar lithological characteristics and physical properties of rocks that affect fluid flow in various parts of the unit [11]. W.J. Ebanks et al. believed that flow units are further subdivided rock masses based on geological and physical changes that affect fluid flow in rocks [12]. D.C. Barr et al. believe that flow units are layers with similar hydraulic characteristics in each rock [13]. Qiu and Mu proposed that the flow unit is a part of the internal architectural structure of a sand body, and a concept is a reservoir unit with the same seepage and water flooding characteristics caused by boundary constraints, discontinuous barrier layers, various sedimentary microinterfaces, small faults, and permeability differences within an oil sand body [14, 15]. Currently, the subdivision of reservoir layers is mainly focused on the subdivision of single sand bodies. In a narrow sense, the flow unit is a further subdivision of the reservoir based on the rock properties that affect the fluid flow after subdividing a single sand body, and a completely different standard is adopted by subdividing a single sand body [16-24]. Among these, the FZI method is widely used. The results of the flow unit method were finer, which is of great significance for subdividing sandstones with significant differences in pore structure. Therefore, through the study and division of flow units, reservoirs can be reasonably divided and evaluated, and the distribution of reservoirs can be predicted, significantly improving the interpretation accuracy of permeability.

In a homogeneous medium system, Kozeny proposed a permeability calculation formula based on the capillary theory [25]. Carman proved the reliability of the formula and established the Kozeny–Carman equation [26]. The commonly used forms are as follows:

In the formula, K is the permeability, 10⁻³ μm²; $φe$ is effective porosity; a is the regional empirical constant; $Sgv$ is the specific surface area of mineral particles, μm⁻¹.

Amaefule et al. officially proposed the FZI method in 1983 [27]. Based on the theory of the average hydrodynamic unit radius, the reservoir pore size was treated as a series of capillaries, and the Poisson equation was solved using Darcy’s law. The Kozeny–Carman equation was modified to obtain the pore permeability relationship under different flow unit types as follows:

Transformed：

In the formula, F_S is the shape coefficient; τ is the tortuosity of porous media; S_gv is the surface area of particles per unit volume; $φe$ is effective porosity; permeability unit adopted (×10⁻³ μm²), define the following parameters:

Reservoir quality index：

Standardized porosity index, which is the ratio of pore volume to particle volume：

Flow zone index：

Take the logarithm of both sides of the above equation to obtain：

where RQI is the reservoir quality index in micrometers. Characterized by the size of reservoir permeability, the larger the value, and the stronger the reservoir permeability, K is the permeability, μm², $φe$ is effective porosity, $andφZ$ parameter reflects the relative size of pore space. FZI is a comprehensive parameter that reflects the physical properties and pore structure characteristics of reservoirs, in μm. Samples with approximate FZI values fell on a straight line with a slope of 1, whereas samples with different FZI values fell on a set of parallel lines. Samples on the same straight line had similar flow and pore-throat characteristics, thus forming a type of flow unit. Therefore, after calculating the FZI and RQI parameters related to the pore throat based on the core data, different flow unit types can be divided based on the FZI values.

In response to the shortcomings of the traditional flow unit index method, pore structure-related parameters were added to analyze the relationship between the FZI and pore structure. This determined whether the FZI can characterize the permeability differences of different internal reservoirs in flow units.

Parameters such as the displacement pressure, saturation median pressure, and pore throat radius were calculated based on the mercury injection capillary pressure curve of the core and could accurately reflect the pore structure of the reservoir. The FZI is a parameter that combines the characteristics of rock minerals and the pore throat structure to determine the pore structure, which is theoretically similar to the capillary pressure curve. Therefore, the correlation between the two can be used to clarify the relationship between the FZI and pore structure.

We extracted the maximum pore throat radius Rmax, displacement pressure Pd, and average throat radius R parameters from the corresponding mercury injection capillary pressure curves and analyzed the correlation between these three pore structure characterization parameters and the FZI. Through analysis, we found that the FZI increases with an increase in the average throat radius R and maximum pore throat radius Rmax and decreases with an increase in displacement pressure Pd, with good correlation (Figure 5). From this, the FZI can effectively reflect the pore structure characteristics of low-permeability tight sandstone reservoirs in the H formation of the Y gas field in the X depression and can serve as a parameter to characterize the permeability differences between different flow units. Therefore, we used this parameter as the basis for the flow unit classification and the corresponding reservoir pore structure evaluation.

We used more than 1100 core analysis data points from the H formation of the Y gas field in X depression to determine the relationship between the FZI and the cumulative frequency of the H formation. From Figure 6, the cumulative frequency curve of the formation FZI in the H Formation has an obvious segmentation, which can be divided into three trend lines with different slopes. Therefore, the sample-point cores can be classified into three types of flow units, and the specific classification criteria are listed in Table 2.

Analyzing the data in Table 2, the low permeability and tight reservoirs in the H formation are mainly composed of Type III flow units with poor physical properties, accounting for 73.3% of the total, whereas Types I and II flow units with good physical properties only accounted for 26.7% of the total. This further confirms the characteristics of the low-permeability tight reservoirs in the study area. Based on the FZI classification standard, the physical property boundaries between each reservoir type tended to be clear. After classification, the correlation between the porosity and permeability of low-permeability tight reservoirs in the H formation was significantly improved (Figure 7), which further demonstrates that the FZI can effectively reflect the pore structure characteristics of H formation reservoirs and accurately classify reservoirs with different pore structures. Therefore, for the low-permeability tight sandstone reservoirs in the study area, it is necessary to first classify the reservoirs and then establish a logging interpretation model.

Establish corresponding permeability calculation models by classifying different flow units：

In the equation, the subscripts I, II, and III represent the types of flow units of types I, II, and III, respectively, that is, different types of reservoirs.

In the actual processing and interpreting of single-well logging data, it is necessary to establish a relationship between the core section FZI and the corresponding depth section logging curve and apply the established relationship to no-core sections to obtain accurate permeability point-by-point curves. To establish the relationship between the FZI of the core section and the logging curve, the first step is to select the logging data that have a clear response to the FZI.

Figure 8 shows the response characteristics of the conventional logging data in the 4320–4345 m well section of Z1 well in the Y gas field. The first track was a shale indicator track, in which the natural gamma value was mainly distributed in the 44–53 API section of the coring well, with almost no change. This indicates that the lithology of the coring well section is stable, and there is no significant correlation between the natural gamma value and the permeability K and FZI. However, the natural gamma value was higher at 4332–43,333 m (the maximum value was close to 80 API). As a result of the increase in shale, it occupies a part of the pore space, making its connecting throats finer and the pore permeability relationship worse; the permeability significantly decreases, and the FZI decreases. The second track is the porosity curve track, which is calculated from the three porosity curves and has a certain correlation with each other; that is, there may be a certain correlation between the three porosity curves and the FZI. The third track was a resistivity track with a resistivity of 4336 m as the boundary. The variation amplitude of the upper resistivity value was relatively small, whereas the variation amplitude of the lower resistivity and the difference between the deep and shallow resistivities increased. The permeability increased, the FZI value increased, and the reservoir properties improved significantly.

The methods for establishing the relationship between the FZI and logging curves include multiple linear regression and support vector regression (SVR) [28-32]. As the most used regression method, multiple linear regression is based on the least-squares method with empirical risk minimization as the criterion, while SVR is based on the linear kernel function and structural risk minimization as the criterion. With the addition of relaxation variables, the support vector was more robust to outliers than the multiple linear regression method.

This study uses the SVR method and validates it using multiple linear regression. The basic concept of SVR is to transform the input into a high-dimensional feature space, determine the optimal classification plane in the high-dimensional space, and maximize its classification interval. In the case of nonlinearity, the classification hyperplane is

where w and b represent the normal vector and intercept of the hyperplane, respectively, and g (x) represents a nonlinear mapping function. The objective function is represented as

The constraint conditions are:

where $xi$ is the sample point, $yi$ is the corresponding category, L is the number of samples, $εi$ ≥ 0 is the relaxation variable, and C is the penalty factor.

When processing the regression, the support vector machine will attempt to fit more data into the interval. The width of the interval is controlled by the epsilon hyperparameter, and the data in the interval band do not account for this loss. Finally, the optimal model was obtained by minimizing the total loss and maximizing the interval to achieve better generalization.

SVR also provides kernel functions that can effectively fit both linear and nonlinear data. Common kernel functions include linear, polynomial, Gaussian, Laplacian, and sigmoid functions.

Based on the characteristics of the FZI data, this study conducted logarithmic processing of the FZI values using linear kernel function fitting and correlation analysis to determine the optimal curve for the regression. A grid-optimization algorithm was used to determine the optimal model parameters.

After analyzing the conventional logging response data, the obvious response characteristics are from the natural gamma, density, sound wave, and resistivity curves. Considering that gamma radiation is absorbed by the formation, its absorption capacity is related to the formation density, and the ratio of deep to shallow resistivity reflects the permeability of the reservoir. After comprehensive consideration, the natural gamma, density, sound wave, and deep-to-shallow resistivity have a correlation between seven parameters, including the ratio of natural gamma to density and the ratio of induced resistivity, and FZI (Figure 9).

The correlation between lnFZI and natural gamma rays, density, sound waves, deep shallow resistivity, the ratio of natural gamma rays to density, and the ratio of deep shallow resistivity was −0.67, −0.068, 0.3, −0.042, −0.7, −0.69, and 0.85, respectively. After the combination, from Table 3, the correlation of resistivity increased by approximately 20%, and the correlation of natural gamma increased by 3%, indicating that the combination curve exhibited better performance. The ratio of sound waves, natural gamma rays to density, and the ratio of deep-to-shallow resistivity were used to fit the FZI.

The least squares fitting method yields the FZI response equation as follows:

In support vector machine parameter C = 1, ε = At 0.09, the response equation for FZI is obtained as follows:

This model was used for the feedback. Figures 10 and 11 show a comparison of the FZI values calculated by the least squares and SVR methods, respectively. The mutual validation of the fitting effects between the two methods verified the reliability of the fitting.

Figure 12 shows the logging interpretation results for Well Z1. The sixth and seventh traces were calculated based on the FZI fitting relationship and pore permeability model. Among them, “linear-” represents the least squares method, and “SVR-” represents the support vector machine method. According to the core analysis and statistics, the FZI values of the core in this section were distributed between 1.8 and 6.2, mainly belonging to Types II and III. The FZI curve of the core agreed well with the calculated FZI curve. There were 145 rock cores in this section, of which seven were misjudged, with a coincidence rate of 95.17%. The seventh channel was the permeability channel. Compared with the core overburden permeability, the calculated absolute error of the permeability was 0.054 mD, and the relative error was 16.71%. In the low-porosity and low-permeability sections, the calculated permeability matches the core overburden permeability well, indicating that the calculated permeability after classification meets the requirements for fine reservoir evaluation and has good practical application results.

This study proposes a flow unit index method that considers the pore-throat parameters. Through analysis of the pore permeability intersection, the permeability of the study area is not only affected by the porosity but also by the pore structure. The traditional flow unit index method weakens the influence of the pore structure, leading to misjudgment. The introduction of three pore-throat parameters, namely the maximum pore-throat radius, displacement pressure, and average throat radius, extracted from the mercury injection capillary pressure curve, achieved an accurate division of the flow units. Based on machine learning, a multiparameter fitting equation for the FZI was established that can achieve a continuous evaluation of permeability. This was applied to one well. The calculated absolute error of permeability was 0.054 mD, and the relative error was 16.71% with high precision, which lays the foundation for follow-up development of the study area.

This research was funded by National Natural Science Foundation of China “Physical Parameters Modeling on Carbonate Fracture-Cavity Reservoirs (No. 41402113)” and National Science and Technology major projects of China “Logging identification and comprehensive evaluation technology of low-permeability tight reservoir” (2016ZX05027-002-002).

I hereby declare and guarantee that there is no conflict of interest between all authors.

The data used to support the results of this study can be obtained from the corresponding authors upon request.