WO2014178505A1

WO2014178505A1 - Method for determining permeability and flow velocity of porous medium by using dispersion number of pores

Info

Publication number: WO2014178505A1
Application number: PCT/KR2013/010511
Authority: WO
Inventors: Chang Hoon Shin
Original assignee: Korea Gas Corporation
Priority date: 2013-04-30
Filing date: 2013-11-19
Publication date: 2014-11-06
Also published as: WO2014178504A1

Abstract

The present invention relates to a method for more accurately analyzing flow characteristics of porous media with various pores including fractures. The present invention suggests engineering concept of the permeability more systematically, and thereby it is expected to be linked to a new method, which can classify and compound various flows of shale gas reservoir more properly, and identify characteristics.

Description

METHOD FOR DETERMINING PERMEABILITY AND FLOW VELOCITY OF POROUS MEDIUM BY USING DISPERSION NUMBER OF PORES

The present invention relates to a method for determining permeability of porous media with various pores including fractures, and a method for measuring flow velocity of fluid in the porous media by using the same.

Recently, unconventional gas resources such as shale gas, tight gas and coalbed methane (CBM) are receiving much attention as a new natural gas energy source. In particular, the shale gas is receiving more global attention thanks to huge original gas in place (OGIP) and rapid commercial development centering in North America. For this development of the shale gas, it is absolutely needed to form a fracture zone through extended reach horizontal drilling and multi-stage hydraulic fracturing to a shale formation. Namely, the shale gas field has a characteristic that natural gas production centering at artificial fractures formed through horizontal drilling and hydraulic fracturing is possible. At this time, unlike the previous natural gas reservoir, gas flow in shale gas reservoir exhibits unique flow characteristics, compounded with flow characteristics of fractures and propant banking area, and various porous regions such as sand stones. In other words, the flow in a shale gas reservoir exhibits a very complex flow behavior, compounded with pipe flows in vertical and horizontal wells, reaching generally 4 to 7 km in distance, diffusion flows in shale rock matrices, porous flows in a geological formation such as sand stones, and fracture network flows through both natural fractures and artificial hydraulic fractures in a shale formation (see Non-patent documents 1 and 2).

In order to analyze the flow in the shale gas reservoir having various pore distribution and shapes in these various media and flow paths, it is needed to draw and apply new approach, which can appropriately consider geometric characteristics of each flow path (see Non-patent document 3). General previous porous media such as sand stone layers also exhibit various pore size, shape, and distribution depending on geological characteristics of each geological formation, and separate considerations and estimation methods are required to properly classify and analyze the flow characteristics.

On the other hand, permeability determination at the previous fields was calculated based on well testing, special core test and the like. To these tests, there were problems that high cost and long term tests were needed, and it was not easy to secure reliability to the result due to many limitations regarding to measuring condition at real fields. In addition, systematic and technological approaches and analysis, which can select pores representing real formations having various layer distribution and lithological characteristics, and estimate sectional area, were impossible.

Hence, the present invention conducted studies for the purpose of investigating proper consideration methods for effects of various pore size, shape and flow paths, and newly establishing concept of permeability closer to flow characteristics of actual porous media. Namely, in addition to the concept of the equivalent pore sectional area derived from another application of the present applicant, Korean Patent Application No. 10-2013-0060903, the present invention aims to draw a method, which can properly describe the flow of porous media showing distribution and shape characteristics of various pores including fractures such as shale gas reservoir, by defining and suggesting a new concept of equivalent permeability. This is ultimately, suggests engineering concept of the permeability more systematically, and thereby it is expected to be linked to a new method, which can classify and compound various flows of shale gas reservoir more properly, and identify characteristics.

[Prior Art]

[non-patent document]

(Non-patent document 1) Shin C. H., Lee S. M., Kwon S. I., Park D. J., and Lee Y. S., 2012, "A Classification and a Survey on the Core Technology for Shale Gas Development", Trans. of KSGE, Vol. 49, No. 3 pp. 395-410.

(Non-patent document 2) Shin C. H., Lee Y. S., Lee J. H., Jang H. C. and Baek Y. S., 2012, "Global Distribution of Shale Gas and Its Industrial Trend", Trans. of KSGE, Vol. 49, No. 4 pp. 571-589.

(Non-patent document 3) Cipolla, C. L., Williams M. J., Weng X., Mack M. and Maxwell S., 2010, "Hydraulic Fracture Monitoring to Reservoir Simulation : Maximizing Value", SPE ATCE 2010, SPE 133877.

(Non-patent document 4) Nield D.A, and Adrian B., 2006, "Convection in Porous Media" 3rd ed., Springer, New York, pp. 1-16.

(Non-patent document 5) Burmeister L. C., 1993, "Convective Heat Transfer" 2^nded., John Wiley & Sons, Inc., New York, pp. 44-51.

(Non-patent document 6) White, F. M., 2001, "Fluid Dynamics", 4th ed., McGraw-Hill, New York, pp. 325-404.

(Non-patent document 7) Ahmed S., Islam M. Q. and Jonayat A. S. M., 2011, "Determination of Loss Coefficient for Flow through Flexible Pipes and Bends", Proceeding of ICME 2011, ICME 11-FL-045.

In order to solve the above-described problems associated with prior art, the present invention provides a method for determining a method for calculating permeability, which is more reliable method for properly reflect size, distribution and shape of pores, and a method for measuring flow velocity of fluid in the porous media by using the same.

In order to accomplish one object of the present invention, the present invention provides a method for calculating an equivalent permeability (K ^*) of porous media comprising the steps of:

(i) selecting a porous medium containing a plurality of micropores having a certain porosity( ) and a total pore sectional sectional area(A);

(ii) calculating a permeability for an ideal single cylindrical pore (K) by the following formula (1):

K= ØA _p /8π (1)

wherein, A _pis a pore sectional area determined by A _p = ØA;

(iii) calculating an equivalent permeability coefficient (C _K) by the following formula (2):

(2)

wherein, C _S is a pore shape coefficient, D is a total pore diameter, D _P ^* is an equivalent pore diameter, f is a friction coefficient, f ^* is an equivalent friction coefficient, l is an equivalent areal circumference and l ^*isanequivalentperimetriccircumference; and

(iv) calculating the equivalent permeability(K ^*) by the following formula (3):

K ^* =C _K K (3).

According to one embodiment of the present invention, the pore shape coefficient Cs may be 1 when the pore shape is circular cylinder.

According to another embodiemtn of the present invention, the pore shape coefficient Cs may be 4/π when the pore shape is regular rectangle.

According to another embodiemtn of the present invention, the pore shape coefficient Cs may be 5.83/ when the pore shape is isosceles triangle .

While the invention has been described with respect to the above specific embodiments, it should be recognized that various modifications and changes may be made for any geometrical figures whose perimeter and area can be arithmetically calculated.

Further, the present invention provides a method for calculating a flow velocity (u) of a fluid in a porous medium by using the equivalent permeability (K ^*) according to claim 1 by the following formula (4):

(4)

wherein, μ is fluid viscosity, and P is pressure.

Furthermore, the present invention provides a method for calculating an equivalent permeability coefficient (C _K) of porous media by the following formula (2):

(2)

wherein, C _S is a pore shape coefficient, D is a total pore diameter, D _P ^* is an equivalent pore diameter, f is a friction coefficient, f ^* is an equivalent friction coefficient, l is an equivalent areal circumference and l ^*is an equivalent perimetric circumference.

According to the present invention, more reliable flow analysis is possible even when flow characteristics of various porous media including fractures like gas flow of shale gas reservoir are mixed, by investigating relationship with all geometric factors such as size, shape and distribution of pores and flow paths in various porous media including fractures, and suggesting concept of equivalent permeability and dispersion number for various pore shapes, which can properly reflect thereof.

Accordingly, the present invention defines and suggests a new concept of equivalent permeability and dispersion number for various pore shapes, which is closer to flow characteristics of actual porous media by investigating proper consideration method to effect of size, shape and flow paths of various pores; and more systematically suggests engineering concept of permeability in order to properly describe the flow of porous media showing distribution and shape characteristics of various pores including fractures such as shale gas reservoir. Thereby it is expected to be linked to a new method, which can classify and compound various flows of shale gas reservoir more properly, and identify characteristics.

The above and other objects and features of the present invention will become apparent from the following description of the invention taken in conjunction with the following accompanying drawings, which respectively show:

FIG. 1 schematically illustrates various pore structures of porous media various pore structures of porous media.

For estimation of reserves and development of shale gas reservoirs, it is required to deduce a method, which can properly distinguish flow characteristics depending on various size, shape and distribution of pores and flow paths in various porous media including hydraulic fractures, and complexly consider thereof. Hence, in the present invention, investigation regarding to relation with all geometric factors such as size, shape and distribution of pores and flow paths in various porous media including fractures; a concept of equivalent permeability and dispersion numbers for various pore shapes, which can properly reflect thereof, is newly defined; and a concept of pore equivalent sectional area is combined. Thereby, pore equivalent permeability relation in the porous media is established and suggested.

The present invention can properly describe the flow of porous media exhibiting distribution and shape characteristics of various pores including fractures like shale gas reservoir by defining and suggesting a concept of equivalent permeability and dispersion number of various pore shapes, which can consider geometric influences directly related to shape characteristics of each pore and flow path characteristics in the porous media, or influence of additional flow loss such as turbulent effect or viscosity increase according to size effect and the like.

Hereinafter, the present invention will be described in detail. Symbols used herein refer as follows:

A : sectional area

C _K : equivalent permeability coefficient

C _S : pore shape coefficient

D : diameter

h _f : friction loss head

f : friction coefficient (friction factor)

g : gravity

K : permeability

L : length

l : perimeter/circumference

n : dispersion number

P : pressure

Q : volume flow rate

R, r : radius

V : velocity(average flow velocity)

u : space averaged flow velocity

x : position

z : height

μ : viscosity

ρ : density

τ_w: shear stress at wall

Ø : porosity

* : equivalent

e : effective

p : pore

From the previous study (Non-patent document 5) done by Burmeister, we can recognize that the equation of motion for porous flow is expressed as a function of wall shear stress as follows.

The above equation can be rearranged as a function of the permeability for an ideal single cylindrical pore after integrating the wall shear stress term by Burmeister as follows.

From the above two equations, wall shear stress and permeability can be directly related as below.

This means if we can find a more appropriate expression of wall shear stress for various and complicated pore shapes, and turbulent effects, then we can substitute the Darcy term with same to get a more reliable and exact permeability definition, which would be very useful to consider unusual shaped pore flows like fracture flows. That is to say, a much more reasonable permeability definition which is applicable to turbulent flow can be derived without Forchheimer term for various actual pores having different geometric factors such as pore shapes, sizes and distributions.

Before proceeding with a solution to the equations of motion, a control-volume analysis of the flow between inlet and outlet sections is explained with an incompressible flow analysis of micro cylindrical pipe, driven by pressure or gravity or both, with the steady state energy equation.

The continuity relation reduces to the equation below since the pipe is of constant area.

or

Then, the steady-flow energy equation reduces to the following equation since there are no shaft-work or heat-transfer effects.

Now it is assumed that the flow is fully developed and, then, the kinetic-energy correction factor α ₁ = α ₂. Since V ₁ = V ₂, the above equation can be reduced to a simple expression for the friction-head loss h _f as follows.

The pipe-head loss equals the change in the sum of pressure and gravity head, i.e., the change in height of the hydraulic grade line (HGL). Since the velocity head is constant through the pipe, h _f also equals the height change of the energy grade line (EGL). The EGL decreases downstream in a flow with losses unless it passes through an energy source, e.g., as a pump or heat exchanger.

Finally, the momentum relation is applied to the control volume, accounting for applied forces due to pressure, gravity, and shear stress at wall.

This equation relates to the wall shear stress.

Wherein,

for a horizontal pipe.

So far, we have not assumed either laminar or turbulent flow. τ_w can be correlated with flow conditions to solve the problem of head loss in pipe flow. Functionally, we can assume that

where

is the wall-roughness height. Then dimensional analysis tells us that:

.

The dimensionless parameter f is called the Darcy friction factor, after Henry Darcy(1803-1858), a French engineer whose pipe-flow experiments in 1857 first established the effect of roughness on pipe resistance.

Combining above equations, the desired expression for finding pipe-head loss can be obtained.

This is the Darcy-Weisbach equation, valid for duct flows of any cross section and for laminar and turbulent flow. It was proposed by Julius Weisbach, a German professor who in 1850 published the first modern textbook on hydrodynamics. Here, it is noted that we can find the form of the function f in the definition of Darcy friction factor and plot it in the Moody chart.

As a result, we can confirm that the head loss in pipe flow has major relations among Darcy friction factor or wall shear stress, geometric factors and other flow properties as arranged below.

Consequently, we have another proper wall shear stress definition of the Darcy friction factor which is applicable to both laminar and turbulent. Here, it is a function of the wall roughness in particular and flow velocity. Then we can characterize a porous flow through a very small porous media by regarding it as a similar pipe flow blocked several walls. For example, for a flow through a square duct, a small pipe can be assumed to be surrounded by four very close walls having a certain Darcy friction value. Of course, the flow aspect through a pipe wall is not same to that though a porous media in general but the concept of the Darcy friction factor must also be valid for the cases that the pipe is very small enough or the walls are very close like pore throats. Hence, even though we need to obtain the Darcy friction factors for various porous medium through experimental or theoretical approaches later, we can determine the wall shear stress through them and apply them for permeability estimation ultimately.

The wall shear stress in the following equation can be changed by using the Darcy friction factor to a very small cylindrical, capillary tube.

Firstly, the wall shear stress in the Darcy term is substituted with the definition of Darcy friction factor.

At this point of time, we need to think what values of the pore section area, perimeter and Darcy friction factor will be the proper for actual porous flow permeability. Of course, the actual values cannot be setermiedn yet, then we just suppose the equivalent values for those variables here by adding superscript '*' like

as follows.

Therefore, the actual permeability K _actual can be calculated through the above relation if we determine those equivalent variables needed for it even though we cant yet in reality. Here, the equivalent permeability, K ^* is newly introduced and also assumed to have equivalent values to the actual permeability values. Additionally, the velocity in the relation is changed to the space-averaged velocity, u from the average flow velocity V.

Then, the above equation is rearranged to a similar form of Darcy term to derive the exact definition of the equivalent permeability under the assumption of horizontal flow path as follows:

Here, we adopted the basic definition of the Darcy friction factor for laminar cylindrical duct flow like

and substituted velocity V in the definition through that of the Hagen-Poiseulle equation like

.

Now, we are ready to substitute the wall shear stress term in the fundamental equation of motion for porous flow through the derived relations here, neglecting unsteady, convection, gravitation and diffusion terms. That is to say, we will only consider only the wall shear stress term and pressure gradient term. The wall shear stress term is subsitutued to devise better definition of the equivalent permeability by being arranged to a very similar form of Darcys equation as follows.

The only variable we cant determine yet for a calculation of the equivalent permeability, ultimately to solve the modified Darcys equation is dispersion number,

. It represents the number of dispersed pores having the same shape and averaged size corresponding to both cross sectional area and perimeter of the horizontal micro cylindrical pipe used as a basic model through the derivation of the equation of motion for porous flow.

Accordingly, a new method for estimating the dispersion number, n is to be outlined which can include the effects due to pore distribution and size for more reliable permeability estimations.

A selection of three sample porous media sections are provided in Table 1. Porous media section ② consists of a number of identical circular micro-pores. Porous media section ① has an equivalent total sectional area to media section ②, where section ① is equal to the sum of n sectional areas of micro-pores in section ②. Porous media section ③ has an equivalent total perimeter as the sum of pore perimeters of section ②, where the perimeter length of circular section ③ is equal to the sum of n perimeter lengths of micro-pores in section ②. In addition, the principal geometrical relations among these three sections are presented in Table 1 below the figures to aid in the comparison.

Table 1

Geometric value relations among equivalent sections

Section ①(Equivalent areal section)	Section ②(Simple circular pore section)	Section ③(Equivalent perimetric section)

Hence, n, the dispersion number of the identical circular micro-pores, can be defined as the ratio of

, the micro-pore sectional area of section ②, and A _p, the equivalent sectional area of section ①. Then, the dispersion number can then be substituted to the ratio of the equivalent sectional area of section ①, A _p and the equivalent perimeter length of section ③, A ₃ from the relations shown in Table 1. Consequently, n, the dispersion number of the identical circular micro-pores can be expressed as the following equation.

Here, the total perimeter length of the equivalent perimetric section,

can be directly measured through recent optical experiments and geometrical analysis with numerical methods. Accordingly, we can determine the last unknown value, the dispersion number of pores through the above equation for the equivalent permeability estimation. Therefore, the final definition of equivalent permeability with the dispersion number of the identical circular micro-pores can be arranged as follows.

At last, we succeeded to derive the modified Darcys equation by defining the equivalent permeability with the dispersion number as shown above.

But one thing we should note in applications of the dispersion number is the shapes of the actual micro pores because we assumed only a circular shaped cylindrical pore for derivation of the dispersion number. So, we need to consider additional determination methods for the dispersion number of other shapes like rectangular, triangular pores and all other shapes. Of course, it is impossible to consider an infinite number of all actual pore shapes then we have no other ways of simplifying them into just a few representative shapes. As an example, we will continue to examine for the dispersion number of two representative shapes of regular tetragonal and isosceles triangular pores, and use almost same procedures as that for circular pore previously.

Table 2

Regular tetragon

Section (Equivalent areal section)	Section (Simple square pore section)	Section (Equivalent perimetric section)

Therefore, the dispersion numberof the identical squire micro-poresis changed as follows;

Table 3

Isosceles triangle

Section (Equivalent areal section)	Section (Simple triangular pore section)	Section (Equivalent perimetric section)

And the dispersion numberof the identical isosceles triangular micro-poresis changed as follows;

After all, the modified Darcys equation andthe final definition of equivalent permeability with the dispersion numbergeneralized for the pore shape coefficient is finally rearranged as follows;

Table 4

*Pore Shape*	*Circular Cylinder*	*Regular Rectangle*	*Isosceles Triangle*
C _k

Finally, we can present the modified Darcys equation and the definition of equivalent permeability, K ^* which is used for the modification and more accurate and representative by considering actual pore geometry and turbulent effects through the introduction of Darcy-Weisbach relation into the wall shear stress term in a porous flow equation.

or

Herein, C _K is defined as the equivalent permeability coefficient.

The fundamental variables in C _K like D,F can be easily calculated from the definitions for laminar flow in a circular cylindrical duct individually. The equivalent diameter of a pore,

can be estimated through hydraulic diameter concepts widely used for internal viscous flow analyses or the relation with perimeters, which can be measured through optical experiments and geometrical analysis with numerical methods recently developed.

The actual or equivalent Darcy friction factor f ^*, which can reflect the influences due to wall roughness and turbulent flow must be determined through experimental or analytic approaches similar to those for the conventional Moody chart, Colebrooks equation or others.

As can be seen from the above, in accordance with the present invention, a method for calculating an equivalent permeability (K ^*) of porous media is provided. The method comprises the steps of:

(i) selecting a porous medium containing a plurality of micropores having a certain porosity(Ø) and a total pore sectional sectional area(A);

K = ØA _p /8π (1)

wherein, A _pis a pore sectional area determined by A _p = ØA;

(2)

wherein, C _S is a pore shape coefficient, D is a total pore diameter, D _P ^* is an equivalent pore diameter, f is a friction coefficient, f ^* is an equivalent friction coefficient, l is an equivalent areal circumference and l ^*is an equivalent perimetric circumference; and

K ^* = C _K K (3).

According to another embodiemtn of the present invention, the pore shape coefficient Cs may be 5.83/π when the pore shape is isosceles triangle.

Further, the present invention provides a method for calculating a flow velocity (u) of a fluid in a porous medium by using the equivalent permeability (K ^*) by the following formula (4):

(4)

wherein, μ is fluid viscosity, and P is pressure.

(2)

wherein, C _S is a pore shape coefficient, D is a total pore diameter, D _P ^* is an equivalent pore diameter, f is a friction coefficient, f ^* is an equivalent friction coefficient, l is an equivalent areal circumference and l ^*isanequivalentperimetriccircumference.

As a conclusion, we successfully devised the notion of the equivalent permeability to include geometric and turbulent effects for porous flow analysis and properly defined it with reasonable basic theories and exact mathematical approaches. The main idea in deriving it is based on adopting the concept of Darcy friction factor for general viscous pipe flow and extending to very small pore flows, especially for including turbulent effects with a wall roughness concept. Therefore, the equivalent permeability is successfully defined by adopting the concept of the Darcy friction factor.

While the invention has been described with respect to the above specific embodiments, it should be recognized that various modifications and changes may be made and also fall within the scope of the invention as defined by the claims that follow.

The present invention suggests engineering concept of the permeability more systematically, and thereby it is expected to be linked to a new method, which can classify and compound various flows of shale gas reservoir more properly, and identify characteristics.

Claims

A method for calculating an equivalent permeability (K ^*) of porous media comprising the steps of:

(i) selecting a porous medium containing a plurality of micropores having a certain prosity(Ø) and a total pore sectional sectional area(A);

(ii) calculating a permeability for an ideal single cylindrical pore (K) by the following formula (1):

K = ØA _p /8π (1)

wherein, A _pis a pore sectional area determined by A _p = ØA;

(iii) calculating an equivalent permeability coefficient (C _K) by the following formula (2):

(2)

wherein, C _S is a pore shape coefficient, D is a total pore diameter, D _P ^* is an equivalent pore diameter, f is a friction coefficient, f ^* is an equivalent friction coefficient, l is an equivalent areal circumference and l ^*is an equivalent perimetric circumference; and

(iv) calculating the equivalent permeability(K ^*) by the following formula (3):

K ^* = C _K K (3).
The method for calculating an equivalent permeability (K ^*) of porous media accoding to claim 1, wherein the pore shape coefficient Cs is 1 when the pore shape is circular cylinder.
The method for calculating an equivalent permeability (K ^*) of porous media accoding to claim 1, wherein the pore shape coefficient Cs is 4/π when the pore shape is regular rectangle.
The method for calculating an equivalent permeability (K ^*) of porous media accoding to claim 1, wherein the pore shape coefficient Cs is 5.83/π when the pore shape is isosceles triangle.
A method for calculating a flow velocity (u) of a fluid in a porous medium by using the equivalent permeability (K ^*) according to claim 1 by the following formula (4):

(4)

wherein, μ is fluid viscosity, ρ is fluid density, and P is pressure.
A method for calculating an equivalent permeability coefficient (C _K) of porous media by the following formula (2):

(2)

wherein, C _S is a pore shape coefficient, D is a total pore diameter, D _P ^* is an equivalent pore diameter, f is a friction coefficient, f ^* is an equivalent friction coefficient, l is an equivalent areal circumference and l ^*is an equivalent perimetric circumference.