WO2017217565A1

WO2017217565A1 - Method for calculating permeability of porous medium by means of tortuous hydraulic diameter

Info

Publication number: WO2017217565A1
Application number: PCT/KR2016/006298
Authority: WO
Inventors: 신창훈
Original assignee: 한국가스공사
Priority date: 2016-06-14
Filing date: 2016-06-14
Publication date: 2017-12-21

Abstract

The present invention provides a method for calculating the permeability of a porous medium with respect to a laminar flow and a turbulent flow, in view of geometric characteristics and friction loss characteristics. A method for calculating the permeability of a porous medium, according to an embodiment of the present invention, comprises the steps of: providing the porosity of pores of a porous medium; calculating the hydraulic diameter of the pores of the porous medium; providing the friction coefficient of the porous medium; calculating the tortuosity of the porous medium; calculating the tortuous hydraulic diameter, which is a function of the tortuosity, by means of the hydraulic diameter and the tortuosity; and calculating the permeability of the porous medium by means of the porosity, the tortuous hydraulic diameter, the friction coefficient and the tortuosity.

Description

Permeation calculation method of porous media using torsion hydraulic diameter

The technical idea of the present invention relates to a method for calculating permeability, and more particularly, to a method for calculating the permeability of a porous medium using a torsion hydraulic diameter.

Estimation of permeability of porous media has been a major research topic for a long time in the fields of oil and gas development, as well as various academic fields such as nuclear power, biomechanics and civil engineering. Nevertheless, the geometric characteristics of each medium such as various types of rocks, cracks, and living cells can be properly taken into account, and no generalized permeability estimation method has been established that can be applied to all flow regions including laminar and turbulent flows (Sahimi). , 2011).

Recently, the development of unconventional resources such as tight gas and coal bed methane (CBM), including shale gas, has been greatly affecting the relevant industries as well as related industries. The typical non-traditional storage layer, the shale reservoir layer, is a mixture of dense shale rocks, hydrocracking cracks combined with natural cracks and supports, and thus there is a limit to interpreting it as a general porous flow theory (Cipolla et al., 2010). Therefore, it is possible to consider the geometrical characteristics of various media, including hydraulic fracturing cracks in which props are complex, and derivation of permeability analysis methods that can be applied regardless of the flow area is necessary (Shin et al., 2012a).

The rheological relationships of porous flows are generally represented by the Darcy or Forchheimer equations. However, these relations only suggest a simple proportional relationship between flow rate and permeability, but do not suggest characteristic variables or their correlations that govern permeability. Accordingly, the transmittance is generally measured directly through an experiment, or is generally estimated by contrast with porosity. As a result, this requires the execution of numerous experiments on various types of rock in the target reservoir, which in reality requires considerable cost and time consumption. In particular, in the case of the hydraulic fracturing cracks in which the support is combined, it is very difficult and difficult to identify both the support distribution characteristics, the implementation of the experimental model, and the permeability experiment. In addition, even when experiments are performed on specific porosity, support distribution, and flow conditions, it is a reality to perform another experiment on other geometric or flow conditions. Further, the hydraulic fracturing cracks are gradually closed with cracks or the support is recessed or broken down according to the change of the stress distribution of the strata as the production progresses, accompanied by a change in geometrical specifications such as porosity or gap. This is directly linked to the change in permeability of hydraulic fracturing cracks, the effect of which is known to be quite significant (Shin et al., 2012b). In addition, the hydraulic fracturing cracks are mainly distributed in the vicinity of the production wells and serve to transport a large amount of fluid under high pressure gradient. Accordingly, the flow through the initial crack of production exhibits the characteristics of turbulent flow, often referred to as the Non-Darcy effect, and gradually transitions to laminar flow as the yield decreases. As a result, in order to calculate the permeability of hydraulic fracturing cracks, 1) more precise definitions of permeability characteristics and correlations should be identified, 2) applicable to both turbulent and laminar flow regimes, and 3) geometrical changes due to crack closure. There is an urgent need for derivation of a new permeability relationship to account for the accompanying permeability changes.

The technical problem of the present invention is to provide a method for calculating the permeability of a porous medium using a torsional hydraulic diameter for laminar flow and turbulent flow in consideration of geometric and frictional loss characteristics.

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

Method for calculating the permeability of the porous medium according to the technical idea of the present invention for achieving the above technical problem, providing a porosity of the porous medium; Calculating a hydraulic diameter of the porous medium; Providing a coefficient of friction of the porous medium; Calculating the degree of torsion of the porous medium; Calculating the torsional hydraulic diameter of the porous medium as a function of the degree of torsion using the hydraulic diameter and the degree of torsion; And calculating the permeability of the porous medium using the porosity, the torsional hydraulic diameter, the coefficient of friction, and the degree of torsion.

In some embodiments of the present invention, the hydraulic diameter may have the following relationship with the torsion degree.

Where D _hT is the torsional hydraulic diameter, D _h is the hydraulic diameter without considering the degree of torsion, and T is the degree of torsion

Where D _hT is the torsional hydraulic diameter, D _h is the hydraulic diameter without considering the degree of torsion, L is the straight length of the porous medium, L _e is the length of the internal void flow path, S is the specific surface area, and Φ is the Porosity)

In some embodiments of the present invention, the coefficient of friction may be a function of the degree of torsion.

In some embodiments of the present invention, the friction coefficient may have the following relationship with the torsion degree.

Where f _vT is the coefficient of friction for the flow rate v as a function of torsion, f _uT is the coefficient of friction for the flow rate u as a function of torsion, f _u is the coefficient of friction without considering the degree of torsion, and T is the degree of torsion , D _hT is the torsional hydraulic diameter, ρ is the density, ΔP / L _e is the pressure gradient over the length of the internal void flow path, v is the flow rate of the fluid through the internal void flow path, u is the flow rate of the fluid, ΔP / L is the Pressure gradient over straight length, Φ is the porosity of the porous medium)

Where f _uT is the coefficient of friction for the flow rate u as a function of the degree of torsion, f _u is the coefficient of friction without considering the degree of torsion, L is the straight length of the porous medium, L _e is the length of the internal void flow path, and T is Torsion degree, D _hT is the torsional hydraulic diameter, ρ is the density, ΔP / L is the pressure gradient with respect to the linear length of the porous medium, v is the flow rate of the fluid through the internal void flow path, u is the flow rate of the fluid, ΔP / L is straight Pressure gradient over length, Φ is the porosity of the porous medium)

In some embodiments of the present invention, the transmittance may be configured to include the coefficient of friction as a function of the torsion and the Reynolds number as a function of the torsion.

In some embodiments of the present invention, the transmittance may have the following relationship.

Where k _GEPT is the geometrical permeability considering the degree of torsion, v is the flow velocity of the fluid through the internal pore channel, D _hT is the torsional hydraulic diameter, Φ is the porosity of the porous medium, T is the degree of torsion, and f _vT is the degree of torsion Coefficient of friction for flow rate v as a function of Re _vT is Reynolds number for flow rate v as a function of torsion degree)

Where k _GEPT is the geometric equivalent transmission considering the degree of torsion, u is the flow velocity of the fluid, D _hT is the torsional hydraulic diameter, f _uT is the function of the torsional friction coefficient for the flow rate u, and Re _vT is a function of the torsional Reynolds commission for flow rate u)

In some embodiments of the invention, the transmittance may comprise a Reynolds number that is a function of the torsion.

In some embodiments of the present invention, the transmittance may include a Reynolds number having the following relationship with the torsion.

Where Re _vT is the Reynolds number for flow rate v as a function of torsion, Re _uT is the Reynolds number for flow rate u as a function of torsion, Re _u is the Reynolds number without considering torsion, and T is the degree of torsion , ρ is the density, v is the flow rate of the fluid through the internal void flow path, u is the flow rate of the fluid, D _hT is the torsional hydraulic diameter, μ is the viscosity of the fluid)

In some embodiments of the present invention, the transmittance may have the following relationship with the torsion.

Where Re _uT is the Reynolds number for the flow rate u as a function of the degree of torsion, Re _u is the Reynolds number without considering the degree of torsion, T is the degree of torsion, ρ is the density, u is the flow rate of the fluid, and D _hT is the torsion Hydraulic diameter, μ is the viscosity of the fluid)

In some embodiments of the present invention, the degree of torsion may take into account the length of the internal void flow path of the porous medium.

In some embodiments of the present invention, the degree of twisting may have the following relationship.

Where T is the degree of torsion, L is the straight length of the porous medium, and L _e is the length of the internal void flow path of the porous medium.

In some embodiments of the present invention, the flow rate of the fluid in the porous medium can be expressed as a function of the torsional hydraulic diameter, the coefficient of friction as a function of the degree of torsion, and the Reynolds number as a function of the degree of torsion have.

In some embodiments of the present invention, the flow rate of the fluid in the porous medium may have the following relationship.

Where u is the flow rate of fluid, μ is the viscosity of the fluid, D _hT is the torsional hydraulic diameter, Φ is the porosity of the porous medium, T is the degree of torsion, v is the flow rate of the fluid through the internal pore channel, f _vT is the torsional Coefficient of friction for flow rate v as a function of degrees, Re _vT is the Reynolds number for flow rate v as a function of torsion degree, ΔP / L is the pressure gradient for the straight length)

Where u is the fluid velocity, μ is the viscosity of the fluid, D _hT is the torsional hydraulic diameter, Φ is the porosity of the porous medium, T is the degree of torsion, u is the flow rate of the fluid, and f _uT is the function of the degree of torsion the coefficient of friction for u, Re _uT is the Reynolds number for the flow rate u as a function of the degree of torsion, ΔP / L is the pressure gradient for the straight length)

Where k _GEP is the geometric equivalent transmittance, D _hT is the torsion hydraulic diameter, f _uT is the coefficient of friction as a function of torsion, Re _uT is the Reynolds number as a function of torsion, and subscript "1" is the first model Subscript "2" is the second model)

Where k _GEP is the geometrical permeability, D _hT is the torsional hydraulic diameter, f is the coefficient of friction, Re _uT is the Reynolds number as a function of torsion, T is the degree of torsion, Φ is the porosity of the porous medium, and the subscript "1" is the first model, subscript "2" is the second model)

( _Wherein k _GEP is the geometric equivalent transmittance, D _hT is the torsion hydraulic diameter, subscript "1" is the first model and subscript "2" is the second model)

In some embodiments of the present invention, the porosity may have a value ranging from 40% to 60%.

In some embodiments of the present disclosure, within the range of the porosity, f _uT Re _uT may have a numerical value ranging from 156 to 171.

In some embodiments of the invention, the porosity, the hydraulic diameter, and the coefficient of friction are equivalent variables from the coefficient of friction derived for a straight cylindrical flow path to exhibit the geometric and frictional loss characteristics of the porous medium. Can be converted to

In some embodiments of the present invention, the step of calculating the hydraulic diameter may be performed by using a cylindrical capillary tube that generates an equivalent frictional loss with respect to the frictional loss of the pores of the porous medium.

In some embodiments of the present invention, the friction coefficient may be a Darcy friction coefficient.

In the method of calculating the permeability of a porous medium according to the technical idea of the present invention, the permeability may be calculated by applying variables representing the geometric and friction loss characteristics of the porous medium, and the permeability may be calculated for not only laminar flow but also turbulent flow. Can be. The method of calculating permeability according to the technical concept of the present invention enables more reliable permeability analysis, and in particular, it is possible to properly consider geometric characteristics of each rock such as friction coefficient, hydraulic diameter, and torsion degree, thereby allowing porosity according to each rock phase. Permeability correlations can be presented as appropriate. In particular, it is possible to successfully present the geometrical permeability relationship to define the correlation between the characteristic variables of porous flows by newly defining the friction coefficient and torsional hydraulic diameter of porous media. Based on this, it can be extended to the generalized Darcy friction flow relation that can be applied to porous flow. In addition, it is possible to confirm the validity of the relational expressions derived from the present invention by performing computational fluid analysis on the hydraulic fracturing cracks and to present examples of application thereof.

In addition, the method of calculating the permeability of the porous medium according to the technical concept of the present invention can be applied more effectively in the analysis of the strata where the geometrical characteristics are distinguished from the general sandstone, such as dense sandstone, natural cracks, fracture cracks, and the like. In addition, the method for calculating the permeability of the porous medium according to the technical idea of the present invention can be applied to various porous media, for example, the porous medium is sandstone, silt, carbonate rock, cracked rock, porous biological tissue, porous mechanical components, or porous It may include an electronic component.

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

1 shows three analytical models assuming a simple crack filled with spherical particles in a shale structure.

FIG. 2 shows the pressure contours and streamlines of the three simple crack models at an average velocity of 0.0822 m / s from CFD simulation.

3 is a graph comparing permeability of CFD simulations, Kozeni-Carmen analysis, and GEP analysis.

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings. Embodiments of the present invention are provided to more fully explain the technical idea of the present invention to those skilled in the art, and the following embodiments may be modified in many different forms, and The scope of the technical idea is not limited to the following examples. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the inventive concept to those skilled in the art.

In this study, the main theories of previous researchers were first investigated and reviewed to properly identify and supplement the characteristic variables affecting the permeability. Next, in order to derive permeability relations that can consider the geometric characteristics of various media and can be applied regardless of flow conditions, the geometric equivalent permeability (GEP) based on the newly defined friction coefficient and hydraulic diameter of porous media Geometry Equivalent Permeability relationship was derived. Then, to verify the definitions of the derived characteristic variables and the validity of the relations and to present examples of application, CFD analysis was performed on simple hydraulic fracture cracks with support, and the results were compared and examined.

Investigation and review of permeability relations and characteristic variables

The theoretical approach of the previous researchers on the permeability is to infer the permeability based on other relatively easy-to-measure rock properties such as porosity (Nelson, 1994). Historically, this first approach has been an empirical relationship to the permeability measurements for saturated sandstone by Hazen (1892). Later, Kozeny (1927) and Carmen (Carman, 1937, 1938, 1956), based on a model resembling a rock cavity as a tube, widely known as the Kozeny-Carman relation, A relationship has been presented. In addition, Paterson (1983), Walsh (Balsh), and Brace (1984) derive an equivalent channel model that assumes that the flow through the porous flows through a related bundle with different diameters. have. In addition, Archdou and Azellaneda (1992) proposed a method of estimating the 'dc-permeability' to further solidify the Cogeni-Carmen model. The effects of pore roughness and pore size distribution on the static permeability were analyzed. In addition to the various attempts to correlate permeability with the various characteristic variables of rock and voids as mentioned above, rock particles and mineralogy, specific surface area and saturation, and logarithmic log model (Nelson, 1994) Many other classes of approaches have been tried several times. In other respects, the physical meaning of permeability and the mathematical basic relations and approaches to their expansion have also been proposed by several previous researchers (Bear, 1975; Burmeister, 1993; Kaviany, 1995). Rubinstein and Torquato (1989) used an ensemble-average formulation to derive a rigid energy representation of the permeability by deriving the Darcy law in the micro realm. Whitaker (1996) also found a volume averaged Navier-Stokes that can determine the permeability tensor of the Darcy's law and the correction tensor of the forkheimer. The relationship was derived. Mei and Auriault (1991) have considered rigid porous media as incompressible Newtonian fluids with constant density, suggesting that the inertial forces are local, although weak, rather than globally.

Derivation of the most basic definition of permeability begins with the equivalence modeling of pores in a porous medium into horizontal, straight cylindrical capillaries, and the development of the momentum equation for porous flows (Bear, 1975; Burmeister, 1993; Kaviany, 1995). The equation of the x-axis momentum of the fully developed fluid flow through the void introduced in the previous study can be expressed as Equation 1. This equation is obtained by integrating this expression with respect to the cross-sectional area of the void, "A _p = Φ A", and then multiplying by "dA = 2πrdr". Substituting equations for wall shear stress based on laminar flow into Equation 2 and dividing it by A _p yields the x-axis direction conservation momentum equation of the porous flow as shown in Equation 3 below.

Where u is

Is the area-averaged velocity, u _p is

Average pore velocity, defined by. In this case, τ _w is the shear shear stress based on laminar flow as shown in Equation 2 (Burmeister, 1993). Equation 3 is composed of six terms from the left side, the abnormal term on the left side, the pressure term on the convective and right side, the gravity term, the friction loss term, and the diffusion term. The last two terms on the right side are known as Darcy and Brinkmann, respectively. Here, the unsteady term on the left side and the gravity term on the right side can be eliminated for steady-state horizontal flow, and in general porous flow analysis, the convection term on the left side and the Brinkman diffusion term on the right side have very small effects and can be ignored. have. As a result, as shown in Equation 4, Equation 3 can be arranged in a form very similar to the Darcy relation, and through comparison with the Darcy relation, the basic mathematical definition of the transmittance is presented as Equation 5 (Bear, 1975; Burmeister, 1993; Kaviany, 1995).

In the development process above, we should note that the derivation of the Darcy term results from the friction loss term due to the wall shear stress. This means that from the physical point of view, the permeability represents the equivalent cross-sectional area of the pore channel through which the wall shear stress of the porous flow is transmitted, and as a result, the permeability becomes a function of the wall shear stress. The development of Equation 5 has been presented in other previous studies and documents, including Brumeister (1993), and has been widely used for the theoretical review and expansion of permeability based on laminar flow (Bear, 1975; Kaviany, 1993). 1995; Jurgawczynski, 2007). Nevertheless, the reason why Equation 5 is not widely used as a relation for calculating the actual permeability can be presumed to be largely due to two reasons. First, Equation 5 introduces an incompressible laminar flow relation, known as Hagen-Poiseuille, for the wall shear stress analysis of Equation 2, and thus has a limitation in that it cannot be applied to turbulent flow. Next, in Equation 5, the transmittance is simply ΦA _p , which is in the form of the product of the porosity and the cross-sectional area of the medium. This satisfies only a very simple quantitative relationship with respect to the porosity of the porous medium and cannot reflect the various geometric and rheological properties of the actual pores. In other words, Equation 5 can be applied only to a very simple form of laminar flow which can be completely approximated by a horizontal straight cylinder introduced during its development.

Since the particles that make up the porous medium have various mineral composition, size distribution and heterogeneity, porous pores composed of them are not only curvature or tortuosity, but also divergence and conjunction. Same very complex structural and geometrical characteristics. The geometrical properties of these porous media, in addition to the flow resistance due to the turbulence effect of the transition of flow conditions, are a key factor in directly affecting the frictional losses of the fluid. As a result, the basic definition of permeability of Equation 5 is only meaningful in terms of presenting a physical concept of permeability based on an ideal flow model, and can be regarded as an ideal maximum permeability of the target porous medium. Kozeny (1927) quantified the hydraulic diameter of a porous medium defined as a function of porosity and specific surface area in order to quantitatively consider the flow loss due to the geometrical characteristics of the porous medium. 6 is presented. Here, Kozeny (1927) presented a specific surface area as a function of particle shape coefficient and average diameter, assuming a particle of a porous medium as a sphere, as shown in Equation 7 below. Kozeny (1927) suggested '6 as the effective particle shape factor (C _s ) for spherical particles of the same size in the process of deriving this equation, but this is a recent experimental study on actual sandstones. The approach suggested that '4.27' is more appropriate for general porous rocks such as sandstone (Engler, 2010). As a result, the relation of the final hydraulic diameter expressed by introducing the relation of Equation 7 into Equation 6 and the transmittance relation of Cogeny applied to Equation 5 can be summarized as in Equation 8.

Despite the introduction of Cozeni's hydraulic diameter, the permeability of most actual porous media will be much lower than the permeability value given by Equation 8. This is because the actual porous flow depends not only on the influence of the pore cross-sectional area such as the hydraulic diameter, but also on the geometrical influence of the flow path such as the bending and torsion of the air gap. Kozeny (1927) proposed the relational equation (9), which first defined the torsion degree (T) and introduced it to the pressure gradient term, in order to take into account the effects of such flow changes (Carman, 1937). . Later, Carmen (1937) argued that Cogeni's torsional degree should be considered as well as the pressure gradient term as well as the velocity term and suggested modified equation (10). In other words, Carmen finally presented the so-called Kozeny-Carman relation, which is well known as the Kozeny-Carman relation, considering the influence of the degree of torsion on both the pressure gradient and the flow velocity. Here, the definition of the degree of torsion is mixed with several definitions according to previous researchers using the relevant theory (Bear, 1975), and in order to avoid confusion, the definition of Equation 10 first presented by Cogeny in this study is used. Was used. As a result, the definition of the transmittance proposed and modified by Carmen becomes a form of multiplying the transmittance definition of Cogeny of Equation 8 by the torsion degree, and can be summed up as shown in Equation 11 by introducing the hydraulic diameter definition again. In this case, the Darcy relational expression may be modified and presented as in Equation 12 by introducing a transmission definition of Equation 11.

Combining the derivation process of Equation 11, it can be seen that the cozeni-carmen relation is defined by combining the concept of hydraulic diameter and torsion of Kozeni with the basic definition of permeability of Equation 5 derived from the laminar flow theory. have. The Cozeni-Carmen relation was used as the theoretical basis of many previous studies, and is more widely known in the form of Equation 13 (Carman, 1937; Bear, 1975; Carrier, 2003). Equation 13 assumes spherical particles having the same mean diameter d _m in Kozeny's (1927) 's specific surface area equation as shown in Equation 7, and the shape coefficient relationship is set to C _S = 6 and the torsion degree is Hitchcock ( Based on Hitchcock's home

Is substituted into Equation 11 (Carman, 1937). However, many recent studies have found that the degree of torsion is more appropriately expressed as a function of porosity, so a method of estimating torsion based on porosity may be more useful (Bear, 1975). In addition, the measurement of specific surface area required for derivation of the porous hydraulic diameter also includes the statistical method proposed by Chalkett et al. (1949) and the like, the gas adsorption test method, the PETA (Petrographic Image Analysis) method, and the NMR. Various methods such as (Nuclear Magnetic Resonance) measurement have been proposed. Therefore, it is more preferable to use the prototype of Equation 11 than the form having the constant coefficient of Equation 13 in terms of reliability of the result or applicability to various media.

Definition of geometric equivalent transmission based on coefficient of friction and hydraulic diameter of porous media

The cozeni-carmen relationship is one of the most widely used permeability relations and is the theoretical basis of many subsequent studies. Nevertheless, the Cogeny-Carmen relation still has limitations in reflecting the various geometrical characteristics of the actual porous media. The Cozeni-Carmen equation expresses the characteristics of porous flows only as a function of the receiving diameter, torsion and porosity, and only these variables affect the flow loss such as roughness, arrangement, size distribution, anisotropy and heterogeneity of the actual porous particles. This is because it is impossible to consider all geometric properties. For example, suppose that two media with irregular, varying shape, porous media with various particle size distributions, and relatively simple and homogeneous particle distributions have the same specific surface area and porosity. The Cozeni-Carmen relation will exclude the particle surface roughness or structural features of the two media and will calculate the same hydraulic diameter and torsion based on the same specific surface area and porosity, resulting in the same permeability values. Of course, this is not appropriate, and ultimately, the permeability should be expressed in a form that allows for proper consideration of porosity, torsion, hydraulic diameter, and various geometrical characteristics that affect the flow loss of each medium.

In terms of the flow model, the Cogeny-Carmen relation assumes that the porous flow is a laminar flow model through a capillary bundle with an equivalent hydraulic diameter and torsion. In this case, the porosity, the average particle diameter, and the shape coefficient were introduced into the hydraulic diameter definition to reflect the geometric characteristics of the voids. It is possible to estimate the cross-sectional area of equivalent pore channel, which is a key variable of permeability, by definition of hydraulic diameter and flow model, but it is still insufficient to consider the flow resistance characteristics that depend on the structure, arrangement and roughness of particles. Moreover, it is very difficult to measure or quantitatively measure all the flow loss factors individually, and the definition and introduction of characteristic variables that can be taken into account are urgently needed. Therefore, the permeability relation needs to be presented on the basis of new characteristic variables that can be measured in an easier and more universal way and can consider various factors related to flow loss. Ultimately, it needs to be derived in a generalized form including all permeability characteristic variables.

To this end, in the present study, "transmittance is equivalent to the equivalent cross-sectional area of voids that physically generates the wall shear stress, and can be treated as a function of the wall shear stress". The results of the review were noted. In particular, the laminar pipe flow relation, known as the Hagen-Poishley relation, was introduced for the treatment of the wall shear stress term, and it was recalled that the resulting relation was valid only for laminar flow. On the contrary, this problem is overcome by introducing the Darcy friction coefficient equation of Equation 14, which can be applied to the turbulent region instead of the Hagen-Poisley relation that is effective only in the laminar flow region. It means you can. In other words, in this study, the conventional permeability relations, such as the Cogeny-Carmen relations, were derived based on the theory of common piping flows, such as the Hagen-Poisley relations. We tried to find a way to introduce the Darcy friction coefficient relationship to the porous flow analysis. As a result, this study introduced the Darcy's friction factor (f) definition, which is presented in Equation 14, to properly consider the geometric characteristics of the porous flow and the flow loss due to the change of the flow region. At this time, in addition to the Darcy friction coefficient, the theoretical approach describing the friction loss of internal flow is Fanning's friction coefficient (Bear, 1975; Muskat, 1946). However, the two theories differ only in that each relation is defined for the hydraulic diameter and the hydraulic radius, and the Darcy friction coefficient can be treated as simply four times the panning friction coefficient. Proceeded.

Table 1 shows equivalent flow models for each material used in the previous studies and the present invention.

At this point, the equivalent flow model proposed for the theoretical analysis of porous flows in previous studies can be summarized as shown in Table 1 below. Representative flows expressed through the definition of hydraulic diameter and coefficient of friction include concentric pipe flows shown in Table 1 (A) as well as pipe flows in the form of flat, square, triangular channels and ellipses. As an example, model (A) in Table 1 defines the hydraulic diameter and friction coefficient-Reynolds number relationship (f Re) of a pipe flow path for concentric pipes originally in the form of (AI), such as (A-II). It can be expressed as cylindrical pipe flow. In other words, the analysis of pipe flows with various geometric cross-sectional shapes can be achieved by representing equivalent flow models through the definition of the hydraulic diameter and friction coefficient of each flow path, which is very common and widely used in the field of pipe flow analysis. It is utilized. In fact, this concept has already been introduced in the Representative Elementary Volume (REV) model, which derives the basic definition of permeability of Equation 5 (Burmeister, 1993). That is, as shown in B of Table 1, a porous medium composed of solid particles having various sizes, shapes and structures is equivalently represented as a straight cylindrical capillary tube having an equivalent pore diameter, D _p , most simply set to have the same pore volume. Can be. As a result, the basic definition of permeability of Equation 5 is derived based on this simplest and ideal flow model, and as a result, the maximum value of permeability for the corresponding porosity is calculated.

In order to overcome the limitations of the oversimplification of the flow model (B), Cogeni and Carmen introduced the concept of hydraulic diameter (D _h ) for porous media and added the definition of the torsion degree (T). C) A flow model was proposed. Here, the actual velocity of the fluid passing through the void flow path is u _e , which is significantly increased by the flow path length increased with the introduction of a much smaller hydraulic diameter and torsional degree compared to the equivalent void diameter D _p of the model (B). Be careful to have For convenience of understanding, the model (C) shown in Table 1 is expressed based on the length (L _e ) of the internal void flow path, not the straight line length (L) of the medium, as shown in Table 1 (D). As a result, Cogeni and Carmen presented the (D) flow model of Table 1, but there is still a limit in expressing all the geometric characteristics of each medium flow path. In this study, we propose a flow model, model (E), which assumes a cylindrical capillary tube with multiple corrugations presented at the end of Table 1. It basically has the same concept as the Cozeni-Carmen relational flow model based on the concept of hydraulic diameter and torsion, and (D) model, but a number of corrugations are considered to account for the additional flow loss due to the pore geometry such as roughness. In addition, the model is presented.

For example, suppose two kinds of porous media show the same flow loss (permeability) under certain specific flow conditions. One is a medium composed of the same spherical particle size of an ideal spherical shape, which can be assumed as Table 1 (D), and the other is composed of particles of heterogeneous, large size and anisotropy, which can be represented by Table 1 (E). Medium. If, by chance, the same mean velocity and pressure loss occur under certain porosity and flow conditions, the permeability of the two media is estimated to be the same under these conditions. However, if the two media are present at different identical porosity or flow conditions, it is very unlikely that under these new conditions the permeability of the two media will be the same. This is because the characteristics of the flow loss (friction resistance) are different depending on the geometric characteristics of each medium flow path. In order to reflect the characteristics of the flow loss, a separate characteristic variable called friction coefficient is added to the concept of hydraulic diameter. It is necessary to introduce. As a result, this study proposes a cylindrical capillary tube with a large number of corrugations as a flow model of porous media, and attempts to introduce the concept of equivalent hydraulic diameter and torsion, as well as coefficient of friction for proper interpretation and expression.

In order to introduce the friction coefficient, the Darcy friction coefficient definition of Equation 14 is substituted into Equation 2, which is a momentum equation of porous flow, and both sides are divided by A _p , and Equation 2 can be summarized as Equation 15 below. At this time, in the same manner as the derivation process of Equation 4, when the steady state porous flow is removed and the abnormal and gravity terms are removed, the convection term and the Brinkmann diffusion term are ignored. It can be reduced to the form of Equation 16 and summarized by the coefficient of friction for the porous flow. Here, Equation 17 can be seen to have the same form as the Darcy-Weissbacher equation representing the loss relation of the general pipe flow, and accordingly Equation 17 can be thought of as a representation of the porous flow of the Darcy-Weissbacher equation have. Next, the pore channel diameter (D _p ), flow velocity (v), and pressure gradient (dP / dx) of Equation 17 are as shown on the right side of Equation 17 for the models (D) and (E) of Table 1 below. Similarly, Cogeni's hydraulic diameter and torsion may be replaced by modified pressure gradients and flow rates of Cogeni and Carmen, depending on consideration. By substituting these variables into the friction coefficient relational expression of Equation 17, it is possible to obtain the relational expression (18) which defines the Friction Factor of Porous Flow (f) of the porous flow. Here, comparing Equation 18 with the Darcy-Weissbacher equation of general pipe flow, the hydraulic diameter (D _h ), the average flow rate (u) and the linear length pressure gradient (P / L) of the porous medium as a basic term, Porosity Φ and torsion degree T are added. In summary, Equation 18 is a relation that can express the loss characteristics of various media based on the friction coefficient calculated based on the physical quantity of the porous flow which can be measured by the general method. This is a way to overcome the limitations of the application or the limitation of reliability, as the existing theories are expressed on the basis of constant coefficients, difficult to measure, or expressed on the basis of low physical reliability.

In this study, the friction coefficient definition of the porous flow based on the (E-III) model of Table 1 was derived as shown in Equation 18. At this time, the coefficient of friction derived must have the same rheological correlation as the original porous medium, that is, (CI) of Table 1. Therefore, if the friction coefficient for the Darcy relation and the (E-III) model, which are general relations representing the (CI) flow in Table 1, are combined as in Equation 19 for the same pressure gradient condition, Equation 20 is derived. can do. If this is again summarized as a function of the friction coefficient f and the Reynolds number Re, it can be summarized in the form of Equation 21 below. Here, Equation 21 (b) is a relational expression expressed on the basis of the relationship between the friction coefficient f and the Reynolds number (Re _ue ) in terms of equivalent void flow paths presented in the model (E-III) of Table 1. Needs to be. As a result, Equation 21 is a permeability relationship newly derived through the introduction of the Darcy friction coefficient so that it can be applied irrespective of various geometrical characteristics and flow ranges of the porous medium. It is named as GEP (Geometry Equivalent Permeability) relation.

In this study, the Darcy friction coefficient for porous flow was introduced to the hydraulic diameter and torsion degree of the porous medium proposed in the previous study, and the friction coefficient of the porous flow of Equation 18 and the geometric equivalent permeability relationship of Equation 21 Presented. Since the derived relations are derived through strict mathematical development based on the well-known relations, their validity may be sufficient. However, to confirm this again, and to examine the relationship between the characteristic variables constituting the relation, the comparison with the cozeni-carmen relation was performed. To this end, Equation 21 (b) expressed by the relationship between the friction coefficient f and the Reynolds number Re _v in terms of the equivalent void flow path based on the Cozeni-Carmen flow model Model D is used as the basic relational expression. Introduced. Equation (22) can be obtained by incorporating Hizecock's assumption of the hydraulic diameter definition and Carmen's introduction of the torsion degree in the same manner as the cogene-Carmen relation derivation process of Equation (13). In Equation 22, if Cogeni applies the particle shape coefficient relationship C _S = 6 and f Re _v = 64, which is a combination of the friction coefficient and Reynolds number of the laminar flow for the cylindrical flow path (capillary), Can be obtained. Here, Equation 23 can be confirmed that the Cogeny-Carmen relation is the same as Equation 13, which is an expression for the case that is applied to the laminar flow of the porous medium consisting of spherical particles of the same size.

As a result, it is further verified through the above comparison that the geometric equivalent transmission definition of Equation 21 derived from this study is appropriate. On the contrary, the Cogeny-Carmen relation is a result that is applicable only to a specific flow which can define the friction coefficient and Reynolds number relation (f Re _v = 64) based on the cylindrical laminar flow. Actual pipe flows vary considerably depending on the shape of the flow path, roughness of the flow path, and structural changes in the flow path, such as shrinkage or expansion of the flow path (White, 2001). Rheologically, the most representative way to express the loss characteristics of internal flow is based on the relationship between hydraulic diameter and friction coefficient. That is, if the hydraulic diameter is quantitatively expressing the surface area of the flow path subjected to flow resistance, the friction coefficient and the Reynolds number relationship are quantitatively expressing the strength of the flow resistance. Rheological analysis of flat plates, triangular channels, concentric tubes, etc., which are generally related with relatively simple geometrical characteristics, requires quantitative representation of each friction loss as well as hydraulic diameter. Furthermore, it is very difficult for the rheological representation of a porous medium with much more complicated geometrical characteristics to be expressed only by hydraulic diameter, and it is necessary to express the friction coefficient as well as the torsion of the porous flow path. In conclusion, this study proposes a new method of calculating the permeability independent of the flow region by reflecting the geometric characteristics of porous media by deriving the geometric equivalent permeability relationship based on the relationship between friction coefficient and Reynolds number.

However, here, despite the derivation of the geometric equivalent transmittance relation of Equation 21, there is one more matter that should be considered important for the related characteristic variables. The fundamental reason for introducing friction coefficient of porous flow in this study was to characterize the rheological properties of each porous medium in relation to the hydraulic diameter of the medium. For this purpose, the relationship between the coefficient of friction and the Reynolds number represented by f _u Re _u in Equation 21 must be able to be specified as a unique constant value for each porous medium. For example, the relation between Darcy's coefficient of friction and Reynolds number of cylindrical pipes for laminar flow area is generally '64', and the rectangular channel has its own constant value such as '57' and triangular channel '51'. . In addition, each of the flow paths having the same shape is maintained at the same constant value without change even in the laminar flow region as well as when the flow path size changes, so that the flow rate changes only in accordance with the difference in hydraulic diameter. Employing this concept in porous flows, the f _u Re _u relation of any particular porous flow should have the same constant value regardless of the size change of the porous flow path (hydraulic diameter) in the laminar flow region, and a partial change in geometric characteristics such as porosity. The same kind of media should be maintained at similar values for. However, looking at the friction coefficient definitions presented in Equations (17) and (18), it can be concluded that the friction coefficient relationship based on Cogeni's hydraulic diameter definition is difficult to satisfy. This is because the definition of the hydraulic diameter constituting Equation 18 does not strictly correlate with the porosity and the degree of torsion and thus cannot satisfy this relationship.

Looking back at the derivation process of Equation 18, Equation 18 introduces Cogeni's hydraulic diameter definition, pressure gradient relationship, and Carmen's torsion degree to the friction coefficient relationship of the porous flow derived from Equation 16. It is the result derived from applying the modified flow rate relationship. In the related derivation history, Cogeni first defined the hydraulic diameter of the porous medium and newly proposed the concept of the torsion degree to modify the pressure gradient term. Later, Carmen applied Cozeni's torsional concept to pressure gradient terms as well as flow rate terms to suggest the Cozeni-Carmen relation. As a result, Cogeni and Carmen successfully introduced the concept of torsion for porous media into the velocity and pressure gradient terms, but the definition of hydraulic diameter must be closely correlated with the torsionality as well as the porosity. A new definition of hydraulic diameter was not presented. Therefore, in this study, a new definition is proposed in which the hydraulic diameter of a porous medium is correlated with the torsional degree, and through this, an attempt was made to improve the completeness of the friction coefficient and equivalent permeability relationship of the porous medium represented by Equations 18 and 21. .

Looking at the flow models presented in Table 1 (C), (D) or (E), the increase in torsion means the increase in the length of the equivalent pore channel, and the porosity of the target medium cannot be changed. Must result in a change of Therefore, the degree of torsion of the porous medium should be correlated not only with the flow velocity and pressure gradient, but also with the hydraulic diameter as suggested by Cogeny and Carmen. As a result, the hydraulic diameter D _hT in consideration of the degree of torsion of the porous medium may be represented by Equation 24 from the basic correlation between the porosity and the degree of torsion and the hydraulic diameter. In other words, the porosity (Φ = A _h L / V _b ) in a straight cylindrical tube, which is the equivalent flow model of porous flow (model B and symbol Φ1 in Equation 24), is a model that considers torsional variation (models C and D). , E, and even when applied to the symbol Φ 2 of Equation 24, the same porosity (Φ = A _hT L _e / V _b ) must be maintained, so the hydraulic diameter D _hT of Equation 25 is derived through this relationship. . In this study, the equation (25) is a newly defined in consideration of the twist of the porous medium is also the hydraulic diameter, was named the "twist hydraulic diameter (D _hT, Tortuous Hydraulic Diameter), in order to distinguish it from general hydraulic diameter.

For example, in the definition of the hydraulic diameter (D _h ) of the porous medium included in the torsional hydraulic diameter (D _hT ) definition, the co-Zeni hydraulic diameter, such as Equation 6, Equation 7, and Equation 8, modified In case of calculating the torsional hydraulic diameter by applying one type and other previously disclosed hydraulic diameter, it is as follows.

Accordingly, Equation 17 regarding the friction coefficient definition should be modified again based on the relationship of Equation 25. The final definition of the Friction Factor of Porous Flow (f), which is the definition of the friction coefficient from the rheological point of view of the porous medium, can be derived as shown in Equation 26. In this case, the apparent friction coefficient may be defined based on the average flow velocity (u) of the medium and the pressure drop per linear length (P / L) of the medium in the macroscopic view of the porous medium (the apparent flow rate through the medium). In this study, it separately, of the porous medium friction coefficient (f _uT) (Friction Factor of Porous Media) "defined by the friction coefficient of the flow based on the average flow velocity (v) of the porous void passage in (f _vT) and separated And as a new characteristic variable for a more practical approach. In addition to the coefficient of friction, an essential factor in characterizing porous flow is the Reynolds number. Thus, re-introducing the torsional hydraulic diameter definition of Eq. 25, the Reynolds number for the porous flow can be defined as Eq.

Next, the relationship between the friction coefficient and the Reynolds number of the porous medium based on the torsional hydraulic diameter can be summarized as shown in Equation 28. Equation 28, which is the Weisbach friction flow relation (Darcy-Weisbach relation) can be derived. Finally, in order to derive the final geometric equivalent transmission relation, Equation 27 representing the final geometric equivalent transmission relation can be obtained by correlating Equation 27 with the Darcy-Weisbach relation in the same manner as in the derivation of Equation 20.

As a result, it can be seen that Equation 28 (b) has the same form as the Darcy frictional flow relation in general pipe flow. This means that the Darcy friction coefficient-Reynolds number relationship presented in the form of f _uT Re _uT has a unique constant value for pipes or media having the same geometrical properties. In other words, Equation 28 derived from this study is a generalized frictional flow equation that can be extended to porous flows, and the flow loss characteristics of any porous medium or pipe can be specified by the unique f _uT Re _uT value and D _hT . have. This is meaningful in that it additionally presents the characteristic variables that govern the porous flow characteristics such as permeability and strictly suggests each definition and correlation. For example, when porosity and specific surface area are known, variables such as hydraulic diameter, f _uT Re _uT value and permeability, which are variously required in the porous flow analysis, can be estimated approximately. Furthermore, it is significant to provide a theoretical basis to approximate changes in permeability even in the case of changing geometric characteristics such as porosity or flow conditions in the same (similar) media. This is possible because Equation 28 and Equation 29 are based on the 'f _uT Re _uT ' relationship, which can express the flow loss characteristics of the target medium regardless of the flow size as well as the size of the flow path. In the next chapter, we propose the process of calculating permeability through geometric equivalent transmission.

Examination of the validity and applicability of geometrically equivalent transmittance relations through CFD simulation

Figure 1 shows three analytical models assuming a simple crack filled with spherical particles in a shale structure. (a) is a 50% void model, (b) is a 40% void model, and (c) is a 60% void model.

FIG. 2 shows the pressure contours and streamlines of the three simple crack models at an average velocity of 0.0822 m / s from CFD simulation. (a) is a 50% void model, (b) is a 40% void model, and (c) is a 60% void model.

In this chapter, we propose a method to calculate and correlate the characteristic variables of porous flows based on the friction coefficient, torsional hydraulic diameter, and geometric equivalent permeability relationship of the porous media derived from this study. . To this end, we simulate a simple plate-type hydraulic fracturing crack in which a support is combined, and perform Computational Fluid Dynamics (CFD) analysis on the change of porosity according to the support density and the change of flow characteristics according to the pressure gradient. After that, the results of the analysis of geometric equivalent permeability for the same object were compared. The hydraulic fracturing crack analysis model was configured to have three different porosities by filling smooth spherical beads between two parallel plates as shown in FIG. 1. At this time, the particle arrangement, distribution, and size were set to be kept as similar as possible in order to examine only the effect of the porosity change and the flow zone transition in the same type of media (rock phase, fracture crack). As shown in FIG. 1, a porous medium having a porosity of 50% filled with spherical particles having the same diameter between two parallel plates is used as a reference crack model (a), and a crack model having a porosity of 40% filled with three large and small spherical particles ( b) and spherical particles of the same size as model (a) were coarsely filled into three models of crack model (c) having a porosity of 60%. The dimensions of the two parallel plates are equal in width and length in the length of 1 mm (L) x 1 mm (W), and the upper and lower plates are vertically 0.1 mm (H) in all three models. However, the size and quantity of spherical particles filled in the plate were set differently to satisfy the porosity condition of each model. The model (a) is a reference model assuming that there are 100 0.098 mm single diameter beads and 50% porosity.In the case of model (b), 0.098 mm diameter is 100, 0.048 mm is 162 and 0.014 mm 324 pieces, a 40% porosity model using a total of three beads, model (c) was set to a model with a porosity of 60% using 81 0.098 mm single diameter beads, the same as the reference model.

The target fluid was assumed to be gaseous pure methane, considering that this study focuses on the shale gas reservoir, the density was set to 0.6679 kg / s and the viscosity to 0.00001087 kg / ms. CFD modeling and analysis is based on steady-state analysis based on Ansys-Fluent commercial CFD simulation software, and the DNS (Direct Numerical Simulation) method is introduced because the size of the analysis grid is very small at micrometer level. . At this time, since the theoretical formulation of the surface roughness consideration at the micro-scale is incomplete until now, an appropriate model cannot be used, this analysis assumes that the surfaces of the solid particles and the flat plate are completely smooth and isothermal. The 2nd Order Upwind Spatial Discretization Scheme was applied, and the SIMPLE method was used as the Pressure-Velocity Coupling Scheme. CFD analyzes were performed for five Reynolds number ranges from laminar to turbulent, adding the same five flow conditions to three types of hydraulic fracturing crack models with different porosities. FIG. 2 is a CFD analysis result of each model in the case of a flow condition having an average flow rate of 0.0822 m / s, and shows pressures, streamlines, and flow rate distributions to help understand the flow patterns of the models.

Table 2 shows the CFD simulation and GEP analysis results of the reference model (a) having a porosity of 50%.

In order to consider the analysis results for the hydraulic fracturing crack model and to examine the application of geometric equivalent permeability, the analysis results of the model (a) with a porosity of 50%, which is the reference model of the present study, will be reviewed first. The major CFD analysis results for model (a) are presented in the left part of Table 2, and the geometric equivalent permeability analysis results are presented in the right part of the table. Through CFD analysis, the pressure drop of each of the five mass flows in the medium was derived first. Based on this, the permeability of each analysis case was calculated using the differential equation. Next, in order to analyze the geometric equivalent permeability relations derived from this study, the torsional hydraulic diameter-based Darcy friction coefficient, Reynolds number relation, and f _uT Re _uT must be estimated. For this purpose, this study utilized the CFD analysis results for the model (a) with a porosity of 50%. In the future, for real field analysis, it is essential that f _uT Re _uT for at least one reference medium (stratum) be calculated through experimental or analytical methods. If this reference value cannot be calculated, an approximate estimate can be made using existing experimental and analytical results for a medium similar to that medium or for strata in nearby areas.

As shown in Equation 28 or Equation 29, another essential factor required for the calculation of permeability based on the geometrical permeability relationship is the torsional hydraulic diameter D _hT of the target medium. The torsional hydraulic diameter is a combination of the torsion degree and the hydraulic diameter definition of the basic porous media defined by Cogeni, and can be derived by using the relational formula for calculating the general hydraulic diameter. The definition of the hydraulic diameter of the basic porous medium is shown in Equation 6, and the definition of this as a spherical particle has been presented in Equation 8. In this analysis, to introduce these definitions, (a), (b) and (c) calculate the void volume (V _p ) and the particle surface area (E _S ) for each model, and the specific surface area and torsional hydraulic diameter of each model. Was calculated and summarized in Table 3. In this case, the surface area (E _b ) of each model was calculated to include the upper and lower plates as well as the particle surface area in the medium, and the particle shape coefficient for calculating the average particle diameter, C _S, was determined based on the spherical shape. It is set to '6' in the same way as relational expression.)

Table 3 shows the geometric parameters including the torsional hydraulic diameter for each model.

To calculate the final torsional hydraulic diameter, the torsion degree relationship must be added to the basic hydraulic diameter definition derived as shown in Eq. Carmen based on Hitchcock's assumption that the torsion for the target medium at the time,

The relationship of was applied. In recent studies, however, it has been found that the degree of torsion is more appropriately expressed as a function of porosity, and various methods have been proposed by many leading researchers until recently. Among them, the most reliable method is to calculate the correlation between Formation Factor and Resistance Index defined by Archie, and the proposed torsional relationship is shown in Equation 30. (Bear, 1975). In this analysis, the definition of Equation 30 was also introduced in this analysis, and considering the cementation factor (m) value, the support particles considered in this study were similar in size to the sandstone particles and set to an unsolidified state. , m = 2. As a result, the degree of torsion of this analysis results in the relationship T = Φ, which is the case with Hitchcock's assumptions about 40% porosity media introduced by Carmen (Carman, 1937).

In relation, the same results were judged to be appropriate.

Finally, based on the torsional hydraulic diameter for each porosity model derived from Table 3, the Reynolds number, Re _uT , for each analysis case of each model can be calculated. In addition, in connection with flow variables such as pressure gradient and flow velocity calculated through CFD analysis, the coefficient of friction, f _uT , of model (a) with a porosity of 50%, which is the reference model of this analysis, can be derived from Equation 26. have. The f _uT Re _uT and f _u Re _u analysis results for each analysis condition of the model (a) are presented in the right part of Table 2, and the transmittances are based on the calculated f _uT Re _uT and f _u Re _u , respectively. Equation 29 and Equation 21 can be recalculated. It is natural here that the recalculated permeability has the same result as the CFD analysis, and is presented only to verify the calculations performed and to confirm the correlation between each characteristic variable. Rather, it should be noted here that the calculated f _uT Re _uT and f _u Re _u values remain at their respective constants substantially constant in the laminar flow region (Re _uT ≦ 1) of the porous flow. Conversely, this _{assumes that} the critical Reynolds number of the crack model (a) is approximately in the range of Re _{uT ˜1} , which can be seen to be in the range similar to the critical Reynolds number of porous flows presented in several previous studies (Rose). , 1945; Bear, 1975; Nield & Bejan, 1992).

Through the review process of the reference model with the porosity of 50%, the torsional hydraulic diameter, which is a key characteristic variable of porous flow, and the correlation between the friction coefficient and Reynolds number based on the equations 25 and 26 are properly It was confirmed that the permeability can be properly calculated through Equation 28 or Equation 29. Next, the application method for the case where the porosity of the medium to be analyzed is changed, that is, having similar particle shape and distribution characteristics but different porosity is examined. This simulates the case where the hydraulic fracturing cracks are gradually closed due to the change of the strata stress over the course of the production period, or the support is crushed due to the decrease of the hardness of the surrounding rock. As shown in Fig. 1, models (b) and (c) are models that simulate the case where the shape and arrangement of the particles (a) and particles are almost similar, but the porosity is different. Model (b) adds two small spherical particles inside to reduce porosity, and model (c) has the same size as model (a) but reduces the number of beads to achieve an increase in porosity. The model you placed. The actual distribution of the support of the hydraulic fracturing cracks or the particle distribution of the general porous medium will be much more structurally complex than the models. However, due to this complexity, the same kind of actual media (stratum, fracture cracks) will have a higher similarity in pore structure such as particle size, shape, arrangement, etc. It is not extreme that this approach is considered to be valid enough for the actual medium.

Table 4 shows the results of CFD simulation and GEP analysis of the reference model (b) with a porosity of 40%.

Table 5 shows CFD simulation and GEP analysis results of the reference model (c) with a porosity of 60%.

Table 4 summarizes the CFD analysis and the geometric equivalent transmittance analysis results for the model (b) with 40% porosity, and Table 5 summarizes the analysis results for the model (c) with 60% porosity. The left part of Table 4 and Table 5 summarizes the CFD analysis results for each flow condition of the two models, and the right part is the torsion hydraulic diameter and coefficient of friction for each analysis case through the same process as the model (a). -The Reynolds number relationship is calculated and presented. In addition, the results of calculating the permeability for the flow conditions of each model to which the geometric equivalent permeability relation equation presented by Equation 29 is applied are shown on the right side of each table. Here, the f _uT Re _uT relationship representing the flow loss of each analytical model for the laminar flow region is 165 in model (a), model (b) is 169, model (c) is 157, and 10 to 20% between each model. It can be seen that, despite the significant porosity change of the level, it remains almost constant. At this time, the transmission rate calculated based on the GEP relational expression is a little subtle differences in 2-3% levels based on the geometry of the difference is the indicated or, derived f _uT Re _uT relationship among the similar model, according to the porosity difference similar model Sufficient results were confirmed to confirm that they remained similar. On the other hand, it should be noted that f _u Re _{u, which} is the relationship between the friction coefficient of the porous flow and the Reynolds number, varies greatly by about 30% from 203 to 268 in the laminar flow region. This is due to the fact that the initial definition of the geometric equivalent transmittance presented by Equation 21 is based on the Cozeni hydraulic diameter, which does not strictly satisfy the relation between the torsion and the hydraulic diameter for the porosity. As a result, this study successfully derived geometric permeability relations and related characteristic variable definitions and relations based on the f _uT Re _uT relationship in order to extend and apply the f Re-based flow analysis technique to the general flow analysis. In addition, it was confirmed through this CFD review that the newly defined f _uT Re _uT is properly specified as a unique constant for similar media.

Based on the above discussion, it can be assumed that f _uT Re _uT of models (b) and (c) having geometrical characteristics similar to the reference medium of model (a) is maintained at a value similar to that of model (a). Therefore, the geometric equivalent transmittances of models (b) and (c) apply approximately 165, which is the value of f _uT Re _uT of model (a), and calculate only the torsional hydraulic diameter change according to the change in porosity of each model by using Equation 25 separately. Combined, the geometric equivalent transmission estimates presented in Tables 4 and 5 can be derived. At this time, in order to calculate the geometric equivalent transmittance considering the case where the porosity is changed for the same or similar medium as in the analysis of the models (b) and (c), it is preferable to use the equation (29) by modifying the equation (29). It is convenient. That is, assuming that the same medium is geometrically or rheologically different from Condition 1 and Condition 2, Equation 31 can be obtained by concatenating the geometric equivalent transmittance relations for the two conditions based on Equation 29. . For example, Equation 31 assumes that the model (a), which is the reference model of the present study, is the case of condition 1, and the model (b), which is the analysis target, is the case of condition 2, and Equation 31 is based on the model (a). It is a relational equation of geometric equivalent transmittance which can calculate the transmittance of the model (b). Here, Equation 31 can be finally summarized as Equation 32 since the f _uT Re _uT relationship must remain the same (similar) for the same (similar) medium.

The geometric equivalent transmission analysis results for models (b) and (c) based on Equation 32 are finally shown in the right part of Table 4 and Table 5. In this case, k _GEP and k _GEP _{_old} are calculated based on the values of f _uT Re _uT and f _u Re _u, respectively, and the numbers in parentheses indicate an error from the CFD analysis result. As a result, it can be confirmed that the k _GEP results based on the f _uT Re _uT based on the newly defined torsional hydraulic diameter in this study are much closer than those of k _GEP _{_old} .

FIG. 3 is a graph of CFD analysis results (k), geometric equivalent transmittance analysis (GEP), and Coseny-Carmen relations of Equation 13 for all analysis models and flow conditions performed in this review. It is presented. In FIG. 3, a solid line indicates a CFD analysis, a thick dotted line indicates a geometric equivalent transmittance analysis result, and a thin dotted line indicates a Cozeni-Carmen relation analysis result. Model (a) with 50% porosity is a reference model, so the geometric equivalent transmittance (k _GEP ) analysis for model (b) with CFD and GEP analysis showing the same permeability values and the porosity reduced to 40%. Compared with CFD analysis results, the error level was about 2% in the laminar flow region and 14% in the whole region including the turbulent region. In the case of model (c), which has a porosity of 60% and an increase of about 10% compared to the reference model, the error was about 5% in the laminar flow region and up to 22% in the turbulent flow region. As a result, the geometric equivalent transmittance relation yields a very good analysis result in the laminar flow region, but the error is somewhat increased in the turbulent region. However, the pressure gradient in the flow zone that produces the maximum error is the result of analysis at unrealistically high pressure conditions of about 650 bar and 1300 bar per unit length (1m), respectively. Considering this, the error in the pressure range of the actual strata is much smaller, and the geometric equivalent permeability relation in the realizable turbulent region yields relatively good results of around 10%. In this chapter, the computational fluid analysis of hydraulic fracturing cracks was conducted to confirm the usefulness of the definitions and relational expressions derived from this study.

conclusion

With regard to recent shale reservoirs, the permeability of hydraulic fracturing cracks, which is a major research topic, is very important to derive new permeability relations that can be considered in the geometrical characteristics of porous media and applicable to both laminar and turbulent flow zones. Required. Therefore, this study attempts to derive a method to extend Darcy friction flow equation based on the Darcy friction coefficient and Reynolds number relationship (f Re) widely used for the analysis of general pipe flows to porous flow analysis. It was. To this end, based on Kozeni's and Carmen's previous studies, the friction coefficient (f _uT ) and the torsional hydraulic diameter (D _hT ) of porous media were newly defined, and the geometric equivalent transmittance (k _GEP ) relation was derived based on them. . Finally, they are presented as a generalized Darcy frictional flow equation that can be extended and applied to porous flow as aimed at in this study. This suggests that the present study successfully derived the permeability relationship based on the coefficient of friction and Reynolds number, that is, f _uT Re _uT , by rigorously presenting the characteristic variable definition and correlation of porous flow.

Next, the computational fluid dynamics (CFD) analysis of the hydraulic fracture crack models with the support was performed to verify the validity of the relations derived from this study and to present an example of geometric permeability application. Through this, it was confirmed that f _uT Re _{uT, which} is a relation between the torsion hydraulic diameter-based Darcy friction coefficient and Reynolds number, was maintained at similar values for similar media. On the contrary, f _u Re _{u, which} is related to the friction coefficient of porous flow based on the existing hydraulic diameter definition of Cogeny, also confirmed that it did not satisfy this relationship. In addition, the CFD analysis for each model with varying porosity showed good agreement with the analysis results of the geometric equivalent permeability relationship that defined the correlation with the torsion hydraulic diameter based on the f _uT Re _uT relationship. The new torsional hydraulic diameter and the friction coefficient definition of the porous media can be used to properly express the porous flow characteristics, and the geometric equivalent permeability relations based on this method can be used in the case of changing geometric characteristics such as porosity or flow conditions. To show that it can.

In conclusion, in this study, we have successfully defined the geometrical permeability relationship that can correlate the characteristic variables of porous flows by newly defining the friction coefficient and torsional hydraulic diameter of porous media. Based on this, the Darcy frictional flow equation, which is widely used in general pipe flow analysis, is extended to the generalized Darcy frictional flow equation that can be applied to porous flows. Finally, the computational fluid analysis of hydraulic fracturing cracks was conducted to confirm the validity of the definitions and relations derived from this study. For reference, the relations derived from this study are expected to be widely used in the analysis of various kinds of porous flows with various geometrical characteristics such as biological tissues, porous filters, and reactors as well as other types of rocks and layers.

summary

The permeability of the hydraulic fracturing cracks in the shale reservoir is very important in terms of the productivity analysis of the photosphere. Hydraulic fracturing cracks have a large geometric specification, and because of the high pressure gradient, the flow rate through the fracturing cracks is also relatively high. In addition, the permeability changes continuously as the geometrical conditions of the crack change due to the change of the strata stress during the production period. Therefore, it is very urgent to derive a new method of calculating the permeability to consider the non-dachy effect of the geometrical change of the hydraulic fracturing crack and the high pressure gradient around the production well. In this study, a new definition of the friction coefficient and hydraulic diameter of porous media was derived based on the previous studies of Cogeni and Carmen. These definitions are presented in a Darcy friction flow relationship that is transformed into a geometric equivalent permeability (GEP) relation and that can be extended to porous flows in conjunction with the Darcy equation. As a result, it is expected that the derived relations can take into account various geometrical characteristics and characteristic changes, and can effectively calculate the permeability that can be applied regardless of the flow region. In order to validate the derived relations and examine their application, a series of computational fluid dynamics (CFD) models were developed for a simple hydraulic fracturing crack model filled with beads of some size and set to 40%, 50% and 60% porosity. ) The simulation was performed. As a result, the difference between CFD and GEP analysis showed about 2% error in the 40% porosity model and about 5% error in the 60% porosity model and about 10% error in the realizable turbulence region. It is suggested that. In addition, by conducting a comparison and review through the Cogeni-Carmen relations, it was confirmed that the relations derived in this study are valid and yield much closer results. In conclusion, this study newly defined the friction coefficient and torsional hydraulic diameter of porous media, and successfully presented a new permeability calculation method based on the geometric permeability relation and generalized Darcy friction flow relation that can be extended to porous media. It was.

The porosity, the hydraulic diameter, and the coefficient of friction can be converted into equivalent variables from the coefficient of friction derived for a straight cylindrical flow path to exhibit the geometrical and frictional loss characteristics of the porous medium.

The calculating of the hydraulic diameter may be performed by using a cylindrical capillary tube generating friction loss equivalent to the friction loss of the pores of the porous medium.

The friction coefficient may be a Darcy friction coefficient.

The porous medium may include materials having various porosities, and may include, for example, sandstone, silt, carbonate rock, cracked rock, porous biological tissue, porous mechanical components, or porous electronic components.

Based on the above research, the method of calculating the permeability of the porous medium according to the technical idea of the present invention can be implemented as follows.

Method for calculating the permeability of the porous medium according to an embodiment of the present invention, providing a porosity of the pores of the porous medium; Calculating the hydraulic diameter of the pores of the porous medium; Providing a coefficient of friction of the porous medium; Calculating the degree of torsion of the porous medium; Calculating a torsion hydraulic diameter which is a function of the torsion degree using the hydraulic diameter and the torsion degree; And calculating the permeability of the porous medium using the porosity, the torsional hydraulic diameter, the coefficient of friction, and the degree of torsion.

The above-described technical spirit of the present invention can also be embodied as computer-readable code on a computer-readable storage medium. Computer-readable storage media includes all types of storage devices on which data readable by a computer system is stored. Examples of computer-readable storage media include ROM, RAM, CD-ROM, DVD, magnetic tape, floppy disks, optical data storage, flash memory, and the like, and also in the form of carrier waves (for example, transmission over the Internet). It also includes implementations. The computer readable storage medium can also be distributed over network coupled computer systems so that the computer readable code is stored and executed in a distributed fashion. Here, the program or code stored in the storage medium means that a computer or the like is expressed as a series of instruction commands used directly or indirectly in an apparatus having an information processing capability to obtain a specific result. Thus, the term computer is used to mean all devices having an information processing capability for performing a specific function by a program including a memory, an input / output device, and an arithmetic device, despite the name actually used.

The storage medium comprises the steps of providing a porosity of the pores of the porous medium; Calculating the hydraulic diameter of the pores of the porous medium; Providing a coefficient of friction of the porous medium; Calculating the degree of torsion of the porous medium; Calculating a torsion hydraulic diameter which is a function of the torsion degree using the hydraulic diameter and the torsion degree; And calculating the permeability of the porous medium using the porosity, the torsional hydraulic diameter, the friction coefficient, and the degree of torsion, when performing the method of calculating the permeability of the porous medium in a computer. You can store programmed instructions to execute. Further, optionally, the storage medium may further store a programmed instruction to perform the step of correcting the design pattern to prevent collapse of the actual patterns.

The technical spirit of the present invention described above is not limited to the above-described embodiment and the accompanying drawings, and various substitutions, modifications, and changes can be made without departing from the technical spirit of the present invention. It will be apparent to those of ordinary skill in the art.

NOMENCLATURE

A: Cross Sectional Area, A _p = ΦA,

C: Coefficient or Factor,

C _S : Shape Factor,

C ₁ : Carman's Sectional Shape Factor,

D, d: Diameter,

D _h : Hydraulic Diameter,

E: Surface Area,

E _S : E on Solid,

F: Formation Factor,

f: Friction Factor,

g: Gravity,

g _x : Gravity in x-direction,

h: Loss Head,

h _f : Friction Loss Head,

k: Permeability,

L: Length,

L _e : Real Flow Path Length,

P: Pressure,

q: Flow Rate,

R, r: Radius,

R _h : Hydraulic Radius,

Re: Reynolds Number,

S: Specific Surface Area,

S _S : S on solid,

s: Surface Area of Effective Pores,

T: Tortuosity,

t: Time,

u: Average Porous Flow Velocity,

u _e : Fluid Velocity through a Pore Path,

u _f : Flow Velocity through a Flow Path,

V: Volume,

V _S : Volume on Solid,

v: Average Flow Velocity through a Pore,

x, y, z: Position in x, y, z-direction,

α: Conversion Factor between Hydraulic and Matrix values,

μ: Viscosity,

ρ: Density,

τ: Shear Stress,

τ _w : Shear Stress at Wall,

Φ: Porosity,

Super-Script

m: Cementation Factor,

Sub-script

b: bulk,

e: Real, Pore Equivalent,

f: Fluid flow, Friction,

GEP: Geometry Equivalent Permeability,

H, h: Hydraulic,

h _T : Tortuous Hydraulic,

m: Matrix,

Name, Number: Defined State or Case Number or Name,

old: Previous Definition,

GEP _old : Previous Definition of GEP,

p: Pore, Porous,

S: Solid, Shape,

T: Tortuosity, Tortuous,

u: Average Porous Flow Velocity,

u _T : u based on D _hT

v: Average Flow Velocity of Pore Paths in a Porous Medium

v _T : v based on D _hT

References

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Claims

Providing a porosity of the porous medium;

Calculating a hydraulic diameter of the porous medium;

Providing a coefficient of friction of the porous medium;

Calculating the degree of torsion of the porous medium;

Calculating the torsional hydraulic diameter of the porous medium as a function of the degree of torsion using the hydraulic diameter and the degree of torsion; And

Calculating the permeability of the porous medium using the porosity, the torsional hydraulic diameter, the coefficient of friction, and the torsion degree;

Comprising a method for calculating the permeability of the porous medium.
The method of claim 1,

The hydraulic diameter has the following relationship with the torsion, the method of calculating the permeability of the porous medium.

Where D hT is the torsional hydraulic diameter, D h is the hydraulic diameter without considering the degree of torsion, and T is the degree of torsion
The method of claim 1,

The hydraulic diameter has the following relationship with the torsion, the method of calculating the permeability of the porous medium.

Where D hT is the torsional hydraulic diameter, D h is the hydraulic diameter without considering the degree of torsion, L is the straight length of the porous medium, L e is the length of the internal void flow path, S is the specific surface area, and Φ is the Porosity)
The method of claim 1,

Wherein the coefficient of friction is a function of the torsion degree.
The method of claim 1,

The coefficient of friction has a relationship between the torsion and the following, the method of calculating the permeability of the porous medium.

Where f vT is the coefficient of friction for the flow rate v as a function of torsion, f uT is the coefficient of friction for the flow rate u as a function of torsion, f u is the coefficient of friction without considering the degree of torsion, and T is the degree of torsion , D hT is the torsional hydraulic diameter, ρ is the density, ΔP / L e is the pressure gradient over the length of the internal void flow path, v is the flow rate of the fluid through the internal void flow path, u is the flow rate of the fluid, ΔP / L is the Pressure gradient over straight length, Φ is the porosity of the porous medium)
The method of claim 1,

The coefficient of friction has a relationship between the torsion and the following, the method of calculating the permeability of the porous medium.

Where f uT is the coefficient of friction for the flow rate u as a function of the degree of torsion, f u is the coefficient of friction without considering the degree of torsion, L is the straight length of the porous medium, L e is the length of the internal void flow path, and T is Torsion degree, D hT is the torsional hydraulic diameter, ρ is the density, ΔP / L is the pressure gradient with respect to the linear length of the porous medium, v is the flow rate of the fluid through the internal void flow path, u is the flow rate of the fluid, ΔP / L is straight Pressure gradient over length, Φ is the porosity of the porous medium)
The method of claim 1,

Wherein the permeability comprises the friction coefficient as a function of the torsion and the Reynolds number as a function of the torsion.
The method of claim 1,

The permeability has a relationship as follows, a method for calculating the permeability of the porous medium.

Where k GEPT is the geometrical permeability considering the degree of torsion, v is the flow velocity of the fluid through the internal pore channel, D hT is the torsional hydraulic diameter, Φ is the porosity of the porous medium, T is the degree of torsion, and f vT is the degree of torsion Coefficient of friction for flow rate v as a function of Re vT is Reynolds number for flow rate v as a function of torsion degree)
The method of claim 1,

The permeability has a relationship as follows, a method for calculating the permeability of the porous medium.

Where k GEPT is the geometric equivalent transmission considering the degree of torsion, u is the flow velocity of the fluid, D hT is the torsional hydraulic diameter, f uT is the function of the torsional friction coefficient for the flow rate u, and Re vT is a function of the torsional Reynolds commission for flow rate u)
The method of claim 1,

Wherein the permeability comprises a Reynolds number as a function of the torsion.
The method of claim 1,

Wherein the permeability comprises a Reynolds number having the following relationship with the torsion.

Where Re vT is the Reynolds number for flow rate v as a function of torsion, Re uT is the Reynolds number for flow rate u as a function of torsion, Re u is the Reynolds number without considering torsion, and T is the degree of torsion , ρ is the density, v is the flow rate of the fluid through the internal void flow path, u is the flow rate of the fluid, D hT is the torsional hydraulic diameter, μ is the viscosity of the fluid)
The method of claim 1,

Wherein the permeability comprises a Reynolds number having the following relationship with the torsion.

Where Re uT is the Reynolds number for the flow rate u as a function of the degree of torsion, Re u is the Reynolds number without considering the degree of torsion, T is the degree of torsion, ρ is the density, u is the flow rate of the fluid, and D hT is the torsion Hydraulic diameter, μ is the viscosity of the fluid)
The method of claim 1,

The torsion degree is a method of calculating the permeability of the porous medium in consideration of the length of the internal void flow path of the porous medium.
The method of claim 1,

The torsion degree has the following relationship, the method of calculating the permeability of the porous medium.

Where T is the degree of torsion, L is the straight length of the porous medium, and L e is the length of the internal void flow path of the porous medium.
The method of claim 1,

And a flow rate of the fluid in the porous medium is expressed as a function of the torsional hydraulic diameter, the coefficient of friction as a function of the degree of torsion, and the Reynolds number as a function of the degree of torsion.
The method of claim 1,

The flow rate of the fluid in the porous medium has the following relationship, the permeability calculation method of the porous medium.

Where u is the flow rate of fluid, μ is the viscosity of the fluid, D hT is the torsional hydraulic diameter, Φ is the porosity of the porous medium, T is the degree of torsion, v is the flow rate of the fluid through the internal pore channel, f vT is the torsional Coefficient of friction for flow rate v as a function of degrees, Re vT is the Reynolds number for flow rate v as a function of torsion degree, ΔP / L is the pressure gradient for the straight length)
The method of claim 1,

The flow rate of the fluid in the porous medium has the following relationship, the permeability calculation method of the porous medium.

Where u is the fluid velocity, μ is the viscosity of the fluid, D hT is the torsional hydraulic diameter, Φ is the porosity of the porous medium, T is the degree of torsion, u is the flow rate of the fluid, and f uT is the function of the degree of torsion the coefficient of friction for u, Re uT is the Reynolds number for the flow rate u as a function of the degree of torsion, ΔP / L is the pressure gradient for the straight length)
The method of claim 1,

The permeability has a relationship as follows, a method for calculating the permeability of the porous medium.

Where k GEP is the geometric equivalent transmittance, D hT is the torsion hydraulic diameter, f uT is the coefficient of friction as a function of torsion, Re uT is the Reynolds number as a function of torsion, and subscript "1" is the first model Subscript "2" is the second model)
The method of claim 1,

The permeability has a relationship as follows, a method for calculating the permeability of the porous medium.

Where k GEP is the geometrical permeability, D hT is the torsional hydraulic diameter, f is the coefficient of friction, Re uT is the Reynolds number as a function of torsion, T is the degree of torsion, Φ is the porosity of the porous medium, and the subscript "1" is the first model, subscript "2" is the second model)
The method of claim 1,

The permeability has a relationship as follows, a method for calculating the permeability of the porous medium.

( Wherein k GEP is the geometric equivalent transmittance, D hT is the torsion hydraulic diameter, subscript "1" is the first model and subscript "2" is the second model)
The method of claim 1,

The porosity is a method of calculating the permeability of the porous medium having a value in the range of 40% to 60%.
The method of claim 21,

The f uT Re uT within the range of the porosity has a value in the range of 156 to 171, the method of calculating the permeability of the porous medium.
The method of claim 1,

Wherein the porosity, the hydraulic diameter, and the coefficient of friction are converted into equivalent variables from the coefficient of friction derived for a straight cylindrical flow path to represent the geometric and frictional loss characteristics of the porous medium. .
The method of claim 1,

The calculating of the hydraulic diameter is performed by using a cylindrical capillary tube which generates friction loss equivalent to the frictional loss of the pores of the porous medium.
The method of claim 1,

The friction coefficient is a Darcy friction coefficient applied method of calculating the permeability of the porous medium.