KR102357108B1 - Method of calculating permeability variation of porous material using pore-scale simulation - Google Patents

Abstract

The present invention provides a method for calculating the change in permeability of a porous medium using pore-scale simulation. The method for calculating the change in permeability of a porous medium of the present invention comprises the steps of: providing a porous medium model having porosity by an arrangement of particles; deriving geometric parameters including crack height for the porous medium model; setting an inlet flow rate of the fluid to the porous medium model; and deriving permeability by performing pore-scale simulations on the crack height and the inlet flow rate.

Description

Method of calculating permeability variation of porous material using pore-scale simulation

The technical idea of the present invention relates to a flow analysis method in a porous medium, and more particularly, to a method for calculating a change in permeability of a porous medium using a pore-scale simulation.

The present invention refers to the research tasks of the Energy Resources Convergence Source Technology Development Project Serial Nos. 20132510100060 and 20172510102150 No.

Porous fluids are widely used in various industrial fields such as heat transfer, refining, static pressure, decompression, artificial organs, groundwater, nuclear and geothermal power generation, greenhouse gas storage technology, and oil and gas production. In particular, the reliable prediction of the change in permeability correlated with the decrease in the thickness of hydraulic fracturing cracks in the hydraulic fracturing formation is known as a very important factor in shale gas and tight gas storage areas, geothermal power plants, and the greenhouse gas storage industry. have. For example, it is known that the reduction of the pore pressure due to the continuous production of fluid continuously reduces the gap of individual hydraulic fracturing cracks in the shale gas storage area. These changes are found in various types of storage areas ranging from tight gas to traditional oil and gas fields, as well as hydraulic fracturing cracks for actual shale gas. Therefore, it has been found that these are the most important factors for optimal oil and gas production regardless of the type of storage area. Accordingly, the change in transmittance according to the change of geometrical conditions is systematically analyzed and quantified, and thus is essential for reliable production prediction or economical management of minefields. In such a porous flow analysis, Darcy's equation or Forchheimer's equation is generally used, including the case of numerical analysis.

However, these equations provide a simple relationship to the flow rate and pressure intensification, providing only the permeability to the Darcy flow region or the permeability to the Darcy flow region and the non-Darcy flow region, respectively, and the detailed correlation of the permeability change, the geometric change There is a limitation in not being able to present a correlation for the detailed transmittance change starting from . Therefore, in order to predict the change in transmittance, many experiments using actual core specimens or production logging experiments in the field must be conducted for all predicted geometrical change conditions, and thus cost and time and require great effort. Furthermore, they have significant limitations in predicting changes in transmittance experimentally or theoretically.

First, no rock in nature has the same pore structure. Even when rocks extracted from the same strata have the same rock properties, for example, the same porosity, specific surface area, and particle size, shape, and rock properties are the same, the actual size, shape, and types of internal particles are They are completely identical so that the paths and structures inside each pore are not identical. Therefore, the transmittance values for these same types of family rock core specimens are not the same. Of course, this is because the original pore structure is different in the individual rocks. This limitation becomes even more severe when geometric conditions change. In fact, they may be more problematic due to the sensitivity of the measuring instrument, the method, or damage such as breakage of the specimen or deformation contamination that may occur during the course of the experiment. Ideally, hundreds of identical test rock core specimens would be needed to measure all of them rigorously. The reason is that a series of experiments are required for various stress conditions, pressure conditions, and fluid components. However, these are practically impossible.

Second, even if these are possible, it is also impossible to directly measure the flow characteristics within the pore at the pore-scale, that is, the micrometer level of the pore level, that is, the flow velocity structure of the actual supply flow path. This is because it is virtually impossible to fully characterize the extremely complex pore structure and the voids that form between the astronomical number of individual particles.

As a result, the actual pore flow path, called torsion degree, can never be characterized or elucidated through such experimental methods or production tests. Furthermore, these pore structures are not only very small, but because they are actually surrounded by rocks, it is generally impossible to measure practically with devices such as lasers and hot wires that we use.

The technical problem to be achieved by the technical idea of the present invention is to provide a method for calculating the change in permeability of a porous medium using a pore-scale simulation.

However, these tasks are exemplary, and the technical spirit of the present invention is not limited thereto.

According to the technical idea of the present invention for achieving the above technical problem, a method for calculating a change in permeability of a porous medium includes: providing a porous medium model, having porosity by an arrangement of particles; deriving geometric parameters including crack height for the porous medium model; setting an inlet flow rate of the fluid to the porous medium model; and deriving permeability by performing pore-scale simulations on the crack height and the inlet flow rate.

In some embodiments of the present invention, comparing the transmittance with the transmittance calculated from the equation correlated with the porosity and the total volume; may further include.

In some embodiments of the present invention, comparing the transmittance with the transmittance calculated by the following formula; may further include.

Here, k is the transmittance, V _f is the total volume of the medium, Φ is the porosity, and m is the exponential coefficient.

In some embodiments of the present invention, the exponential coefficient (m) may be in the range of 1.5 to 1.9.

In some embodiments of the present invention, the index coefficient (m) may be in the range of 3.6 to 4.0.

In some embodiments of the present invention, comparing the transmittance with the transmittance calculated from the equation correlated with the specific surface area and the total volume; may further include.

In some embodiments of the present invention, comparing the transmittance with the transmittance calculated by the following formula; may further include.

Here, k is the transmittance, V _f is the total volume of the medium, S _S is the specific surface area, and m is the exponential coefficient.

In some embodiments of the present invention, the exponential coefficient (m) may be in the range of 1.3 to 1.7.

In some embodiments of the present invention, the exponential coefficient (m) may be in the range of 1.1 to 1.5.

In some embodiments of the present invention, the porous medium model may be composed of a simple regular porous medium model in which particles of the same size are arranged.

In some embodiments of the present invention, the particles may have a radius in the range of 45 μm to 55 μm.

In some embodiments of the present invention, the porous medium model may have a crack height in a range of 0.4 mm to 0.5 mm, and a porosity in a range of 35.9% to 45.2%.

In some embodiments of the present invention, the inlet flow velocity may range from 0.00001 m/s to 0.75 m/s.

In some embodiments of the present invention, the porous medium model may be composed of a complex regular porous medium model in which particles of different sizes are arranged.

In some embodiments of the present invention, the particles of different sizes may be staggered from each other.

In some embodiments of the present invention, the particles may have a radius in the range of 30 μm to 55 μm.

In some embodiments of the present invention, the porous medium model may have a crack height in the range of 0.4 mm to 0.5 mm, and a porosity in the range of 13.3% to 21.8%.

In some embodiments of the present invention, the inlet flow rate may range from 1 μm/s to 0.1 m/s.

In some embodiments of the present invention, the geometrical parameter may include at least one of crack height, pore volume, solid surface area, specific surface area, porosity, Kozny hydraulic diameter, and total number of grids.

In some embodiments of the present invention, the particles may have a smooth surface and isothermal properties.

In some embodiments of the present invention, in the porous medium model, the porosity may increase as the crack height increases.

In some embodiments of the present invention, the fluid may include water.

In some embodiments of the present invention, in the step of performing the pore-scale simulation, the total number of grids may be in the range of 62 million to 113 million.

In some embodiments of the present invention, in the step of performing the pore-scale simulation, the streamline distribution of the fluid may be derived by performing it based on 100,000 to 250,000 streamline points uniformly distributed.

In some embodiments of the present disclosure, as the inlet flow velocity of the fluid increases, the streamline distribution of the fluid may change from a Darcy flow region to a non-Darcy flow region.

The change in the permeability of the strata according to the change of geometric conditions is receiving a lot of attention in various industrial fields. However, a practical and reliable method for estimating the permeability is still being suggested as reliable experimental instruments and methods for the pore path due to the very complex particle shape of the porous medium and micrometer scale are insufficient. Accordingly, in accordance with the technical idea of the present invention, a pore-scale simulation technique was introduced in order to investigate the change in transmittance according to the change of the geometric condition, focusing on the investigation of the correlation of key geometric variables. As two types of pore-scale simulation models, five types of simple structure models and four types of complex structure models were introduced exemplarily and used to investigate changes in transmittance according to geometrical conditions change. As a result, the key correlations of the geometric parameters that correlate with the change in permeability, i.e., porosity, specific surface area, and total volume, were successfully presented. In particular, it was derived that the specific surface area and the porosity were correlated with the change in the total volume and could be expressed as a function of two practical permeability changes, respectively, and verified through pore-scale simulation models. As a result, the reliability of the derived relational expressions for various types of porous flow accompanying geometrical changes was also verified. Among them, it was found that the pattern of change in transmittance according to geometrical condition change through correlation with specific surface area was more effective, and it was confirmed that they could not be expressed only as a function of porosity.

The above-described effects of the present invention have been described by way of example, and the scope of the present invention is not limited by these effects.

1 shows a pore-scale simulation model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
2 shows geometrical parameters for a pore-scale simulation model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
3 to 5 show the flow velocity distribution of the average model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
6 shows the Reynolds number, pressure drop, and permeability for each of the pore-scale simulation models in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
7 shows the permeability change for each of the pore-scale simulation models in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
8 shows the permeability error for each pore-scale simulation model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
9 is a graph showing the relationship of permeability to the Reynolds number for each of the pore-scale simulation models in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
10 and 11 show a pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
12 shows geometric parameters for a pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
13 to 16 show the flow velocity distribution of the average complexity model in the complex normal porous medium in the method for calculating the change in permeability of the porous medium according to an embodiment of the present invention.
17 shows the Reynolds number, pressure drop, and permeability for each of the pore-scale simulation models in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
18 shows the permeability change for each of the pore-scale simulation models in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
19 shows the permeability error for each of the pore-scale simulation models in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
20 is a graph showing the relationship of permeability to the Reynolds number for each pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.
21 is a flowchart illustrating a method for calculating a change in permeability of a porous medium according to an embodiment of the present invention.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings. The embodiments of the present invention are provided to more completely explain the technical idea of the present invention to those of ordinary skill in the art, and the following examples may be modified in various other forms, The scope of the technical idea is not limited to the following examples. Rather, these embodiments are provided so as to more fully and complete the present disclosure, and to fully convey the technical spirit of the present invention to those skilled in the art. In this specification, the same reference numerals refer to the same elements throughout. Furthermore, various elements and regions in the drawings are schematically drawn. Accordingly, the technical spirit of the present invention is not limited by the relative size or spacing drawn in the accompanying drawings.

With the recent development of numerical solutions and computer performance, several studies have recently been reported in which pore scale simulation (PSS), a direct numerical solution to the pore scale model, is applied. The pore-scale simulation can be an alternative to solving the above-mentioned limitations, despite the numerical limitations such as the convergence limit and high cost when analyzing a very complex grid system itself and an astronomically large number of micrometer-scale grid systems. have. While continuously maintaining the geometry of the original medium, key properties such as pore pore flow velocity, pressure drop and streamline distribution can be reliably analyzed.

The technical idea of the present invention is to derive a correlation of permeability change based on general geometrical properties such as porosity, specific surface area, and total volume change by introducing a pore-scale simulation. The geometrical properties can be measured in general very easily. And the derived permeability correlation can be applied to two types of pore-scale simulation series models and used to predict the permeability change. At this time, the two kinds of series models are models with five simple structures and four very complex structures, which are accompanied by geometrical changes.

Estimation of permeability change according to geometrical condition change

The mainstream of general prior research has been to theoretically analyze porous flow relations and physics approaches for permeability of rocks from very easily measured rock properties such as porosity. The Kozeny-Carman equation, the Ergun equation, or other correction equations based on them are still widely used to estimate the transmittance. Forchheimer presented a generalized flow equation to explain the nonlinear effect. Sunada presented a one-dimensional nonlinear flow equation similar to the Forchheimer equation through empirical correlation between Reynolds number and friction coefficient. However, a general and practical theoretical analysis method for the case of changing atmospheric pressure conditions has not yet been proposed.

Recently, Shin presented a definition of friction equivalent permeability (FEP) based on the correlation between the Reynolds number (Re) and the redefined coefficient of friction (f), and through this, the generalized Darcy friction A flow relation is presented. These are represented by Equation 1 below by combining existing related theories.

From these relations, it can be seen that the transmittance is basically correlated based on the porosity (Φ), the hydraulic diameter (D _h ), and the torsion (T). At this time, it can also be proved that the characteristic parameter (fRe ^* _Dh ) of the porous medium is an intrinsic constant applied for each medium regardless of the geometrical change. The characteristic variable is defined as an intrinsic constant in the frictional equivalent permeability (FEP) method, and may represent the friction flow characteristics of an individual porous medium. Of course, these frictional flow characteristics are determined by the geometric shape of each medium particle or the structural characteristics of the pore flow path. Therefore, when the same medium is in different geometric conditions, the corresponding intrinsic constants can be excluded from the correlation of the transmittance change. This means that when introducing the frictional equivalent transmittance (FEP) relational expression of Equation 1 as described above, the geometric shape or structural characteristics of individual pores can be removed. Of course, this is limited to the case where the same medium undergoes a geometric change.

To this end, under the same flow rate condition, two states of the same type of porous medium can be assumed in a state in which the geometrical conditions are changed as the pore pressure is changed. At this time, it can be assumed that the two types of diverse media are geometrically identical in the beginning, but change according to the change in the crack thickness that occurs as the fluid is generated. From Equation 1, the change in transmittance between the two mediums in different geometric conditions is as shown in Equation 2 below.

^{At this time, the characteristic parameter (fRe *} _Dh ) representing the frictional flow characteristics of the porous medium is the same constant under the same flow conditions for the same type of medium. Therefore, Equation 2 can be defined as Equation 3 below by Kozny's hydraulic diameter definition. Accordingly, the transmittance change can be defined as a function of three geometric properties: porosity (Φ), specific surface area (S _{s ), and torsion (T).}

In addition, the porosity can be directly considered and correlated as a function of the specific surface area by the definition of the specific surface area and the hydraulic diameter, as shown in Equation 4 below.

As a result, Equation 3 can be summarized as each exponential equation for the porosity and specific surface area, which is correlated with the total volume and torsion degree of each medium, as shown in Equation 5 below.

On the other hand, Labrid presented a relational expression such as Equation 6 below, which states that the change in transmittance is exponentially proportional to the change in porosity only with a constant coefficient.

In fact, the Labrid equation is one of the most widely used equations in general field applications for defining the correlation between porosity and permeability. This is also due to having a very simple form. The above equation can be utilized for comparison and verification of the results presented below. From this point of view, the term related to the torsion degree of Equation (5) needs to be removed because it can hardly be determined in actual application.

Archie arranged the stratum modulus and stratum index based on the resistivity of the porous medium. Cornell and Katz presented a stratum exponential correlation based on torsion and stratum index, sloped capillary, and cube models.

Accordingly, the torsion degree can be expressed only as a function of the porosity, as given in Equation 7 below. By introducing these relational expressions, Equation 5 can be simplified to Equation 7, which is very practical and much easier in practical applications.

Finally, Equation 7 may be arranged as a function of porosity as in Equation 8 below, or as a function of specific surface area as in Equation 9 below. Of course, note that these are cases in which the exponent (n) is defined as 1.

As a result, the change in transmittance according to the geometrical condition change can be calculated through the exponential function relationship between the porosity and the specific surface area, respectively, accompanied by the change in the total volume, and is shown in Equations 8 and 9 above. Here, the exponential coefficient (m) should be determined as a special constant expressing the torsional characteristics of each medium. This requires a difference in pressure or a difference in permeability in the two geometric conditions. Of course, these differences are the case under different geometric conditions under the same flow conditions. However, after the index coefficient is determined as described above, the change in transmittance in any geometric condition may be determined through Equations 8 and 9 above. This can be satisfied in both the Darcy flow region and the non-Darcy flow region.

Pore-scale simulation of simple normal porous media with geometrical changes.

For pore-scale simulation analysis, we introduce five porous media models with micrometer-scale pore structures. They are assumed to be reduced as a model that gradually tapers, similar to the gap reduction of cracks. Thereafter, the change in permeability in each flow region according to the rise in flow rate conditions was analyzed through pore-scale simulation analysis.

1 shows a pore-scale simulation model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 1, geometric hydraulic fracturing for the "Thickest" model in the simple structured porous medium is shown at the top, and geometric fracture cracks for five models are shown at the bottom.

The length, width, and height of the horizontal and parallel microplate are assumed to be 4 mm, 4 mm, and 0.5 mm, respectively. As support particles, a total of 7900 perfectly spherical homogeneous beads each having a diameter of 0.102 mm are arranged. The beads (ie, particles) may have a radius in the range of 45 μm to 55 μm. The original diameter of a bead excludes the portion cut by the plate or overlapping portions of adjacent beads. They are placed between the two plates, mimicking the staggered distribution of supports in the crack.

A 45.2% void model with an initial 0.5 mm crack height is defined as the thickest model, and is referred to as the "Thickest" model. The next thick model, the "Thicker" model, has a crack height of 0.475 mm. Then, the "Base" model, the average model, has a crack height of 0.45 mm. At this time, it is assumed that beads in each medium are reduced to adjacent beads or walls simply and homogeneously as the crack height changes. In addition, the "Thinner" model, which is the thinnest model, has a crack height of 0.425 mm, and the "Thinnest" model, the thinnest model, has a crack height of 0.4 mm and a porosity of 35.9%.

2 shows geometrical parameters for a pore-scale simulation model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 2 , as basic geometric variables, crack height (h), pore volume (V _p ), solid surface area (A _s ), specific surface area (S _s ), porosity (Φ), Kozny hydraulic diameter (D _h ) and The total cell count is shown. Here, the specific surface area (S _s ) may be calculated from the ratio of the total surface area to the total volume of the solid portion in each model.

A total of 13 different inlet flow rates ranging from 0.00001 m/s to 0.75 m/s were applied to all simulation models. The flow velocity (u) was defined perpendicular to each inlet cross-section, and in each case was set parallel to the flow direction, the X-axis. As shown in the upper right of FIG. 1 , a total of 65 pore-scale simulation cases were defined by five different crack heights and a total of 13 inlet flow velocity conditions. Except for these geometric and flow conditions, all other conditions are the same in all cases. The fluid was assumed to be pure water with a standard density of 998.2 kg/m ³ and a viscosity of 0.001003 kg/ms, respectively.

The solid walls or surfaces of all models were assumed to be perfectly smooth and isothermal. Computational Fluid Dynamics (CFD) modeling and simulation were performed under steady-state flow conditions using Ansys-fluent commercial software. At this time, the basic convergence criterion was set to a ^{residual of 10 -8 or less in both the momentum equation and the continuity equation.} However, this criterion was partially relaxed due to convergence issues for the two fastest flow rate conditions. ^{That is, it was relaxed to 10 - 7} at 0.5 m/s ^{and 10 -6} at 0.75 m/s, and it was difficult to converge the numerical analysis when it actually exceeded 1 m/s. The final total cell count of the unstructured tetrahedral grid system was approximately 113 million to 62 million. The total number of grids is presented in Figs. 1 and 2). These values were identified through trial and error methods for resolution at different grid resolutions. The total grid number cases of 50%, 75%, 100%, 125%, 150%, and 200% were based on the initial thickest and thinnest models, taking into account the usable limitations of the computer system. The upwind scheme of the second order and the SIMPLE method were applied to spatial discretization and pressure-velocity coupling.

3 to 5 show the flow velocity distribution of the average model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Here, FIG. 3 is the case of an inlet flow velocity of 0.00001 m/s in the XZ plane, FIG. 4 is the case of an inlet flow velocity of 0.01 m/s in the XZ plane, and FIG. 5 is an inlet flow velocity of 0.75 m/s in the XZ plane. is the case of

3 to 5 , the streamline distribution of the base model is based on 100,000 homogeneously distributed streamline seeds, and the streamline distribution of the average model for three different flow conditions is presented. has been From this, it is possible to observe the change of the overall flow pattern from the Darcy flow region to the non-Darsi flow region.

Referring to FIG. 3 , a smooth and continuous flow is shown over the entire pore volume.

Referring to FIG. 4 , minute changes in the flow are identified. These changes are due to a slight decrease in the flow path volume and local discontinuities in the flow.

Referring to FIG. 5 , it can be seen that the instability and irregularity of the flow, including flow interruptions and vortex structures, continues until a continuous increase in velocity and a complete turbulence region are observed, and thus can be considered as a Vidarsi flow region.

6 shows the Reynolds number, pressure drop, and permeability for each of the pore-scale simulation models in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 6 , the Reynolds number, pressure drop, and permeability calculated for each of the pore scale simulation models are shown. For reference, "u" is the average internal flow velocity in each model, and the unit is m/s. "Re _u " is the Reynolds number _{calculated as ρuD h} /μ. "ΔP" is the pressure difference between the inlet and outlet surfaces in each model, in units of Pa. "k" is the transmittance calculated from the Darcy equation, in units of Darcy.

7 shows the change in permeability for each of the pore-scale simulation models in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 7 , each change in transmittance calculated by Equation 6, Equation 8, and Equation 9 is shown. For reference, k _Eq(6) is the transmittance calculated by Equation 6, k _Eq(8) is the transmittance calculated by Equation 8, and k _Eq(9) is the transmittance calculated by Equation 9 .

8 shows the transmittance error for each of the pore-scale simulation models in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 8 , errors calculated by comparing the transmittance of FIG. 6 and the transmittance of FIG. 7 are shown. That is, by comparing the transmittance (shown in FIG. 6) derived by performing the pore-scale simulation with the transmittances (shown in FIG. 7) calculated by Equations 6, 8, and 9, respectively, Errors were calculated. For reference, ε _Eq(6) is the transmittance error according to Equation 6, ε _Eq(8) is the transmittance error according to Equation 8, and ε _Eq(9) is the transmittance error according to Equation 9 . It can be seen that the error of Equation 6 is larger than the error of Equation 8 and Equation 9.

9 is a graph showing the relationship of permeability to Reynolds number for each pore-scale simulation model in a simple normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 9 , the frictional equivalent transmittance (FEP) relational expressions of Equation 6 (indicated by “k_Phi”), Equation 8 (indicated by “k_FEP_Phi”), and Equation 9 (indicated by “k_FEP_Ss”) The relationship between the change in transmittance calculated by .

In order to calculate the transmittance change using Equation 6, Equation 8, and Equation 9, each specific exponential coefficient (m) in Equation 6, Equation 8, and Equation 9 needs to be decided first. These specific exponential coefficients can be determined through simple change analysis for two different geometrical conditions and arbitrarily selected flow rate conditions. The thickest model "Thickest" model and the average model "Base" model use the case where the flow velocity is 0.0001 m/s for pressure drop or permeability change. As the derived coefficients, the exponential coefficient ω in Equation 6 may be about 0.2186. The exponential coefficient (mφ) for the porosity of Equation 8 may be, for example, in the range of 1.5 to 1.9, for example, about 1.7. _{The index coefficient (m ss} ) for the specific surface area of Equation (9) may be, for example, in the range of 1.3 to 1.7, for example, about 1.514. In addition, the change in transmittance in the selected model for the Darcy to non-Darcy flow regime, selected from different geometric conditions, must be measured in order to estimate the values of the other models. Here, the values of the other models are the transmittance changes of the other four models shown in FIG. 9 . Here, the analysis result of the "Thickest" model was assumed as the reference data as the measured transmittance change. As a result, all analysis results, ie, 13 permeability changes and analysis results from the Darcy flow region to the Vidarsi flow region, are all pore-scale simulation analysis results of the “Thickest” model and one-point data, that is, a flow velocity of 0.0001 m/s One-point data of the average model marked with a large green mark in FIG. 9 as a single transmittance value in , was used as reference data for such transmittance calculation. A total of 14 transmittance values were defined as known or measured values. Measured reference data, that is, remaining models and conditions Transmittance changes of the remaining models and conditions of FIG. 9 were assumed as reference data for 51 transmittance changes. As a result, the transmittance change with respect to the geometrical condition change can be calculated from the transmittance change relationship of Equation 6, Equation 8, and Equation 9, and a specific exponential coefficient determined or measured as described above and reference data.

All results calculated based on Equation 6, Equation 8, and Equation 9 show very good agreement with the actual transmittance derived from the pore-scale simulation analysis. As shown in FIG. 9 , it can be seen that the error in the “Thinnest” model, which is the largest reduction in thickness accompanied by a change in porosity of approximately 10% and a change in transmittance of 300%, increases to a certain extent, but FIGS. As shown in Fig. 9, it can be seen that all errors are at a fairly low level. In addition, the change in permeability with increasing flow rate shows very good results in all the calculated models, despite the change from the Darcy flow region to the non-Darsi flow region, which is the flow region. Here, the result of Equation 9 based on the specific surface area and the total volume, as indicated by the gray dotted line, shows the most accurate transmittance prediction. In the case of Equation 8 based on the porosity and the total volume, as indicated by the blue dotted line, it can be seen that the result is relatively excellent, and in particular, the result is much more consistent than that of the Labrid equation. The Labrid equation is defined as a function of the porosity of Equation 6 based on a constant coefficient, and is indicated by a red solid line. As a result, it can be confirmed that the correlation of permeability change according to the technical idea of the present invention can be used as a fairly reliable method in porous flow analysis accompanied by geometrical condition change. In addition, it can be seen that the permeability change relationship based on the relationship between the specific surface area and the total volume shows better results in the problem of the permeability change according to the geometrical condition change, compared to the relationships based on the void relationship.

Porescale Simulation of Complex Normal Porous Media with Geometric Changes

Hereinafter, we will analyze the results of using 13 different inlet velocity conditions and 5 crack height models for a total of 65 pore-scale simulation cases. These are to confirm the change in transmittance according to the geometrical condition change. Through porous models with more complex structures, for example, the size and arrangement of different supports, and a much smaller pore volume, it is a case very similar to the actual porous medium, confirming the consistency and utility of the above-mentioned permeability relation to do For this purpose, another four types of porous media models, i.e., complex models, were additionally introduced, consisting of three different size support particles (i.e. beads) with a staggered lattice structure, and five layers of structure was assumed.

10 and 11 show a pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 10 , a geometric shape in a state in which five layers of beads of three different sizes of a complex hydraulic failure model (a basic pore-scale simulation model) in the complex structured porous medium are excluded. have. The total number of the beads is 28160. The upper right is enlarged to observe the pore path structure in more detail.

Referring to FIG. 11 , it is a five-layer geometrical shape composed of beads of three different sizes containing a complex base model. In the upper right corner, the three marbles arranged alternately are enlarged to observe in more detail. The radius of (A) is 51 μm, the radius of (B) is 30 μm, and the radius of (C) is 55 μm.

Basically, a model with a porosity of 21.8% of a crack height of 0.5 mm is defined as the thickest model, and is referred to as the "Thickest" model. This model is defined as "base data" as the above reference data, that is, a reference model for providing base data necessary for the permeability prediction of other crack gap models. The next thick model, the "Thicker" model, has a crack height of 0.475 mm. Then, the "Base" model, the average model, has a crack height of 0.45 mm. In this case, it is assumed that each marble is reduced to the adjacent marble and wall in a simple and homogeneous manner in each medium. In addition, the "Thinnest" model, which is the thinnest model, has a crack height of 0.4 mm, and as one-point data for determining the specific exponential coefficients of Equation 6, Equation 8, and Equation 9, respectively, that is, 0.00001 One-point data with a single transmittance value at m/s flow rate is selected.

12 shows geometric parameters for a pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 12 , as basic geometric parameters, crack height (h), pore volume (V _p ), solid surface area (A _s ), specific surface area (S _s ), porosity (Φ), Kozny hydraulic diameter (D _h ) and The total cell count is shown. Here, the specific surface area (S _s ) may be calculated from the ratio of the total surface area to the total volume of the solid portion in each model. ^{The default convergence condition is set to 10 -8} or less in both the momentum equation and the continuity equation.

The flow velocity u is set to be perpendicular to the inlet cross-section, i.e., the Y-Z plane, and parallel to the X axis, the flow direction. This refers to the lower right of FIG. 10 . At this time, a total of 10 inlet flow rate conditions from 1 μm/s to 0.1 m/s were initially imposed and applied to all simulation models except the “Thinnest” model. On the other hand, in the "Thinnest" model, six flow rate conditions are applied, and the reason is that in the "Thinnest" model, very small voids, extreme flow velocity changes, and convergence problems due to flow conditions change rapidly increase. Other analysis conditions such as fluid properties, boundary conditions, numerical analysis conditions and methods, and use of an unordered triangular pyramid grid system are the same as in the case of a simple regular porous medium.

13 to 16 show the flow velocity distribution of the average complexity model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Here, FIG. 13 is the case of the X-Y plane under the condition of 1 μm/s. Fig. 14 is the case of an inlet flow rate of 1 μm/s in the XZ plane, Fig. 15 is the case of an inlet flow rate of 0.01 m/s in the XZ plane, and Fig. 16 is the case of an inlet flow rate of 0.1 m/s in the XZ plane .

Referring to FIG. 13 , the streamline distribution of the “Base” model, which is an average model, which is a reference for a complex model, is based on 250,000 streamline seeds uniformly distributed, and streamlines in three different flow rate conditions. distribution is shown. This is to show the overall flow rate change pattern. It can be seen that the flow rate change pattern in the overall complex model is quite similar to that in the above-described simple normal porous medium.

Referring to FIG. 14 , a smooth and continuous flow is shown over the entire pore volume.

Referring to FIG. 15 , the transition characteristics of the flow changing from the Darcy flow region to the non-Darsi flow region are shown.

Referring to FIG. 16 , a strongly distributed vortex structure is shown, which can be considered as a Vidarsi flow region.

17 shows the Reynolds number, pressure drop, and permeability for each pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 17 , the Reynolds number, pressure drop, and permeability calculated for each of the pore scale simulation models are shown. For reference, "u" is the average internal flow velocity in each model, and the unit is m/s. "Re _u " is the Reynolds number _{calculated as ρuD h} /μ. "ΔP" is the pressure difference between the inlet and outlet surfaces in each model, in units of Pa. "k" is the transmittance calculated from the Darcy equation, in units of Darcy.

18 shows the permeability change for each of the pore-scale simulation models in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 18 , each change in transmittance calculated by Equation 6, Equation 8, and Equation 9 is shown. For reference, k _Eq(6) is the transmittance calculated by Equation 6, k _Eq(8) is the transmittance calculated by Equation 8, and k _Eq(9) is the transmittance calculated by Equation 9 .

19 shows the permeability error for each pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 19 , errors calculated by comparing the transmittance of FIG. 17 with the transmittance of FIG. 18 are shown. That is, by comparing the transmittance (shown in FIG. 17) derived by performing the pore-scale simulation with the transmittances (shown in FIG. 18) calculated by Equations 6, 8, and 9, respectively, Errors were calculated. For reference, ε _Eq(6) is the transmittance error according to Equation 6, ε _Eq(8) is the transmittance error according to Equation 8, and ε _Eq(9) is the transmittance error according to Equation 9 . It can be seen that the error of Equation 6 is larger than the error of Equation 8 and Equation 9.

20 is a graph showing the relationship of permeability to the Reynolds number for each pore-scale simulation model in a complex normal porous medium in the method for calculating the change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 20 , the frictional equivalent transmittance (FEP) relational expressions of Equation 6 (indicated by “k_Phi”), Equation 8 (indicated by “k_FEP_Phi”), and Equation 9 (indicated by “k_FEP_Ss”) The relationship between the change in transmittance calculated by .

As mentioned above, it is necessary to determine certain exponential coefficients. The case where the thickest model, "Thickest model," and the thinnest model, "Thinnest" model, respectively, have a flow velocity of 0.00001 m/s is used. As the derived coefficients, the exponential coefficient (ω) of Equation 6 is about 0.04741 The index coefficient (mφ) for porosity in Equation 8 may be, for example, in the range of 3.6 to 4.0, for example, about 3.799. Exponent for specific surface area in Equation 9 The coefficient (m _ss ) may be, for example, in the range of 1.1 to 1.5, for example, m _ss is about 1.3047. Here, the analysis result in the pore-scale simulation of the “Thickest” model is the transmittance value as the measured transmittance change. assume that

In summary, the analysis result of the "Thinnest" model marked in large yellow in FIG. 20 and the "Thinnest" model marked in large red in FIG. 20 are selected as "one-point data", and these are used as reference data for calculating transmittance. As a result, the change in transmittance according to the geometrical condition change can be calculated using the relational expression given therefrom.

Data produced from the Labrid equation of Equation 6 indicated by a red dotted line as shown in FIG. 20 from Equation 6, Equation 8, and Equation 9 and the determined specific exponential coefficient and selected base data shows inconsistent results. This is a different result from the original expectation, and it is analyzed that the change in permeability according to the geometrical condition is not only a function of the porosity.

On the other hand, it can be seen from FIGS. 18 to 20 that Equation 8 and Equation 9 continuously grow and show good results. Specifically, even in the case of the "Thinnest" model, a porosity of about 40% and a transmittance change of 1860% were shown.

In particular, Equation 9, which includes the correlation between the specific surface area and the total volume, shows very good results in all models. In conclusion, it can be seen that the permeability change relationship of Equations 8 and 9 can be very useful and reliably utilized in various types of porous flow analysis, particularly, when accompanied by a change in geometric conditions. Furthermore, it is analyzed that it would be appropriate to correlate the change in permeability according to the geometrical condition change as a function of the total volume change and the specific surface area change, rather than a single function of the porosity.

21 is a flowchart illustrating a method ( S100 ) for calculating a change in permeability of a porous medium according to an embodiment of the present invention.

Referring to FIG. 21 , the method for calculating the change in permeability of the porous medium (S100) includes: providing a porous medium model having porosity by an arrangement of particles (S110); deriving geometric parameters including crack height for the porous medium model (S120); setting the inlet flow rate of the fluid to the porous medium model (S130); and deriving permeability by performing a pore-scale simulation on the crack height and the inlet flow rate (S140).

Comparing the transmittance with the transmittance calculated from the equation correlated with the porosity and the total volume; may further include. Comparing the transmittance with the transmittance calculated in the following formula; may further include.

Here, k is the transmittance, V _f is the total volume of the medium, Φ is the porosity, and m is the exponential coefficient. The index coefficient (m) may be in the range of 1.5 to 1.9. The index coefficient (m) may be in the range of 3.6 to 4.0.

Comparing the transmittance with the transmittance calculated from the equation correlated with the specific surface area and the total volume; may further include. Comparing the transmittance with the transmittance calculated in the following formula; may further include.

Here, k is the transmittance, V _f is the total volume of the medium, S _S is the specific surface area, and m is the exponential coefficient. The index coefficient (m) may be in the range of 1.3 to 1.7. The index coefficient (m) may be in the range of 1.1 to 1.5.

The porous medium model may be composed of a simple regular porous medium model in which particles of the same size are arranged. The particles may have a radius in the range of 45 μm to 55 μm. The porous medium model may have a crack height in the range of 0.4 mm to 0.5 mm, and a porosity in the range of 35.9% to 45.2%. The inlet flow rate may range from 0.00001 m/s to 0.75 m/s.

The porous medium model may be composed of a complex regular porous medium model in which particles of different sizes are arranged. The particles of different sizes may be alternately arranged. The particles may have a radius in the range of 30 μm to 55 μm. The porous medium model may have a crack height in the range of 0.4 mm to 0.5 mm, and a porosity in the range of 13.3% to 21.8%. The inlet flow rate may range from 1 μm/s to 0.1 m/s.

The geometrical parameter may include at least one of crack height, pore volume, solid surface area, specific surface area, porosity, Kozny hydraulic diameter, and total number of lattices. The particles may have a smooth surface and isothermal properties. In the porous medium model, the porosity may increase as the crack height increases. The fluid may include water.

In the step of performing the pore-scale simulation, the total number of grids may be in the range of 62 million to 113 million. In the step of performing the pore-scale simulation, the streamline distribution of the fluid may be derived by performing it based on 100,000 to 250,000 streamline points uniformly distributed. As the inlet flow velocity of the fluid increases, the streamline distribution of the fluid may change from a Darcy flow region to a non-Darci flow region.

The technical spirit of the present invention described above is not limited to the above-described embodiments and the accompanying drawings, and it is the technical spirit of the present invention that various substitutions, modifications and changes are possible without departing from the technical spirit of the present invention. It will be apparent to those of ordinary skill in the art to which this belongs.

Symbol Description (NOMENCLATURE)

A: Area,

A _s : Solid Surface Area,

D: Diameter,

D _h : Hydraulic Diameter,

f: Friction Factor,

h: Height, Fracture Height (Aperture), Permeability,

k, k _FEP : Friction Equivalent Permeability (FEP),

L: Length,

L _e : Actual Path Length,

M: Proportional Coefficient,

P: Pressure,

ΔP: Pressure gradient,

Re: Reynolds Number,

S: Surface Area,

S _S : Specific Surface Area,

T: Tortuosity,

u: Intrinsic Velocity, Average Flow Velocity through a Medium,

V: Volume,

V _p : Pore Volume,

V _f : Rock (Bulk) Volume,

v: Interstitial Velocity, Average Flow Velocity though Actual Paths,

x, y, z: Position in X-, Y-, and Z-direction (Axis),

μ: Viscosity,

ρ: Density,

σ: Stress,

Δσ: Pressure Gradient,

Φ: Porosity,

Super- & Sub-Script

*: Pseudo, ~ based on the Tortuosity in a Laminar-flow Condition,

h: Hydraulic,

i: Initial Exponential Coefficient,

m, m _Φ : m for Equation (8),

m, m _ss : m for Equation (9),

n: Unique Constant Exponent Value of Hydraulic Diameter of a Medium,

p: pore,

P: Pore Pressure,

S: Specific Surface, Solid,

u: Intrinsic Velocity, Average Flow Velocity through a Medium,

v: Interstitial Velocity, Average Flow Velocity though Paths, Vertical,

w: Exponential Coefficient of Labrids' Equation,

References

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Claims (25)

providing a porous medium model having porosity by an arrangement of particles;
deriving geometric parameters including crack height for the porous medium model;
setting an inlet flow rate of the fluid to the porous medium model;
deriving permeability by performing pore-scale simulations on the crack height and the inlet flow rate; and
Comparing the transmittance derived by performing the pore-scale simulation with the transmittance calculated from the equation correlated with the porosity and the total volume;
The comparing step is performed by comparing the transmittance derived by performing the pore-scale simulation with the transmittance (k _FEP(B) ) calculated in the following formula, the method for calculating the change in permeability of the porous medium.

where k is the permeability, V _f is the total volume of the medium, Φ is the porosity, m is the exponential coefficient,
A and B are initially geometrically identical porous media, where A is the initial state and B is the state changed by the change in crack thickness that occurs as the fluid is generated. delete delete The method of claim 1,
The index coefficient (m) is in the range of 1.5 to 1.9, permeability change calculation method of the porous medium. The method of claim 1,
The index coefficient (m) is in the range of 3.6 to 4.0, the method of calculating the change in permeability of the porous medium. delete providing a porous medium model having porosity by an arrangement of particles;
deriving geometric parameters including crack height for the porous medium model;
setting an inlet flow rate of the fluid to the porous medium model;
deriving permeability by performing pore-scale simulations on the crack height and the inlet flow rate; and
Comparing the transmittance derived by performing the pore-scale simulation with the transmittance calculated from the equation correlated with the specific surface area and total volume;
The comparing step is performed by comparing the transmittance derived by performing the pore-scale simulation with the transmittance (k _FEP(B) ) calculated in the following formula, the method for calculating the change in permeability of the porous medium.

where k is the permeability, V _f is the total volume of the medium, S _S is the specific surface area, m is the exponential coefficient, A and B are initially geometrically completely identical porous media, A is the initial state, B is It is a state changed by the change in crack thickness that occurs as the fluid is generated. 8. The method of claim 7,
The index coefficient (m) is in the range of 1.3 to 1.7, the permeability change calculation method of the porous medium. 8. The method of claim 7,
The index coefficient (m) is in the range of 1.1 to 1.5, permeability change calculation method of the porous medium. The method of claim 1,
The porous medium model is composed of a simple regular porous medium model in which particles of the same size are arranged, a method for calculating the change in permeability of a porous medium. 11. The method of claim 10,
The particles have a radius in the range of 45 μm to 55 μm, a method for calculating the change in permeability of a porous medium. 11. The method of claim 10,
The porous medium model, having the crack height in the range of 0.4 mm to 0.5 mm, and having a porosity in the range of 35.9% to 45.2%, the method for calculating the change in permeability of the porous medium. 11. The method of claim 10,
The inlet flow rate is in the range of 0.00001 m / s to 0.75 m / s, the method for calculating the change in permeability of the porous medium. The method of claim 1,
The porous medium model is composed of a complex regular porous medium model in which particles of different sizes are arranged, a method for calculating the change in permeability of a porous medium. 15. The method of claim 14,
The method for calculating the change in permeability of the porous medium, in which the particles of different sizes are alternately arranged. 15. The method of claim 14,
The particles have a radius in the range of 30 μm to 55 μm, a method for calculating the change in permeability of a porous medium. 15. The method of claim 14,
The porous medium model, having a crack height in the range of 0.4 mm to 0.5 mm, and having a porosity in the range of 13.3% to 21.8%, a method for calculating the change in permeability of the porous medium. 15. The method of claim 14,
The inlet flow rate is in the range of 1 μm / s to 0.1 m / s, the method of calculating the change in permeability of the porous medium. The method of claim 1,
Wherein the geometrical parameters include at least one of crack height, pore volume, solid surface area, specific surface area, porosity, Kozny hydraulic diameter, and total lattice number. The method of claim 1,
The particles have a smooth surface and isothermal properties, a method for calculating the change in permeability of a porous medium. The method of claim 1,
The porous medium model, the porosity is increased as the crack height increases, the method for calculating the change in permeability of the porous medium. The method of claim 1,
The fluid includes water, a method for calculating the change in permeability of a porous medium. The method of claim 1,
In the step of performing the pore-scale simulation, the total number of grids is in the range of 62 million to 113 million, the method for calculating the change in permeability of the porous medium. The method of claim 1,
In the step of performing the pore-scale simulation, the method for calculating the change in permeability of a porous medium to derive the streamline distribution of the fluid by performing it based on 100,000 to 250,000 streamline points uniformly distributed. 25. The method of claim 24,
As the inlet flow velocity of the fluid increases, the streamline distribution of the fluid changes from a Darcy flow region to a non-Darcy flow region.

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