Effect of Fracture Dip and Fracture Tortuosity on Petrophysical Evaluation of Naturally Fractured Reservoirs

Abstract A model is developed for petrophysical evaluation of naturally fractured reservoirs where dip of fractures ranges between zero and 90 degrees, and where fracture tortuosity i

PAPER 2008-110

Effect of Fracture Dip and Fracture

Tortuosity on Petrophysical Evaluation

of Naturally Fractured Reservoirs

R AGUILERA University of Calgary This paper is accepted for the Proceedings of the Canadian International Petroleum Conference/SPE Gas Technology Symposium

2008 Joint Conference (the Petroleum Society’s 59th Annual Technical Meeting), Calgary, Alberta, Canada, 17-19 June 2008 This paper will be considered for publication in Petroleum Society journals Publication rights are reserved This is a pre-print and subject to correction

Abstract

A model is developed for petrophysical evaluation of

naturally fractured reservoirs where dip of fractures ranges

between zero and 90 degrees, and where fracture tortuosity is

greater than 1.0 This results in an intrinsic porosity exponent

of the fractures (m f ) that is larger than 1.0

The finding has direct application in the evaluation of

fractured reservoirs and tight gas sands, where fracture dip can

be determined, for example, from image logs In the past, a

fracture-matrix system has been represented by a dual porosity

model which can be simulated as a series-resistance network or

with the use of effective medium theory For many cases both

approaches provide similar results

The model developed in this study leads to the observation

that including fracture dip and tortuosity in the petrophysical

analysis can generate significant changes in the dual porosity

exponent (m) of the composite system of matrix and fractures It

is concluded that not taking fracture dip and tortuosity into

consideration can lead to significant errors in the calculation of

water saturation The use of the model is illustrated with an

example

Introduction

The petrophysical analysis of fractured and vuggy reservoirs has been an area of interest in the oil and gas industry In 1962, Towle1 considered some assumed pore geometries as well as

tortuosity, and noticed a variation in the porosity exponent m in

Archie’s2 equation ranging from 2.67 to 7.3+ for vuggy reservoirs and values much smaller than 2 for fractured reservoirs Matrix porosity in Towle’s models was equal to zero

Aguilera3 (1976) introduced a dual porosity model capable of handling matrix and fracture porosity That research considered

3 different values of Archie’s2 porosity exponent: One for the

matrix (m b ), one for the fractures (m f =1), and one for the

composite system of matrix and fractures (m) It was found that

as the amount of fracturing increased, the value of m became

smaller

Rasmus4 (1983) and Draxler and Edwards5 (1984) presented dual porosity models that included potential changes in fracture

tortuosity and the porosity exponent of the fractures (m f) The models are useful but must be used carefully as they result

incorrectly in values of m > m b as the total porosity increases

PETROLEUM SOCIETY

Serra et al. developed a graph of the porosity exponent m vs

total porosity for both fractured reservoirs and reservoirs with

non-connected vugs The graph is useful but must be employed

carefully as it can lead to significant errors for certain

combinations of matrix and non-connected vug porosities

(Aguilera and Aguilera7) The main problem with the graph is

that Serra’s matrix porosity is attached to the bulk volume of the

“composite system” More appropriate equations should include

matrix porosity (ø b) that is attached to the bulk volume of the

“matrix system” (Aguilera, 1995)

Aguilera and Aguilera7 published rigorous equations for dual

porosity systems that were shown to be valid for all

combinations of matrix and fractures or matrix and

non-connected vugs The non-non-connected vugs and matrix equations

were validated using core data published by Lucia.8 The

fractures and matrix equations were validated originally with

data from the Altamont trend in Utah and the Big Horn Basis in

Wyoming (Aguilera3) Subsequently, Aguilera9 illustrated the

use of these equations with core data from Abu Dhabi

limestones and dolomites (Borai,10 Aguilera11), and carbonates

from various locations in the USA and the Middle East

(Ragland12) The models can also be shown to be valid with

published data from vuggy carbonates from the Lower Congo

Basin of Angola13, vuggy dolomites and limestones from the

Simonette area, Swann Hills formation of Alberta14

Aguilera and Aguilera15 researched instances where the

reservoir is composed mainly by matrix, fractures and

non-connected vugs, which are sometimes observed in cores, or

deduced from micro-resistivity and/or sonic images In these

cases a triple porosity model is more suitable for petrophysical

evaluation of the reservoir

In the above cases, it has been assumed that the flow of current

is parallel to the fractures More recently Aguilera and

Aguilera16 investigated the effect on m of current flow that is

not parallel to the fractures This type of anisotropy, which can

be correlated with fracture dip, is important to avoid potential

errors in the calculation of water saturation This model

assumed a fracture tortuosity is equal to 1.0 A comparison of

results with those obtained by Berg 17 using effective medium

theory yields an excellent agreement for fracture angles of zero

and ninety degrees The comparison for other angles is

reasonable but there are some differences that will be evaluated

based on results from core laboratory work The present paper

extends the Aguilera and Aguilera16 model to cases where

tortuosity is larger than 1.0

THEORETICAL MODEL

Figure 1 shows schematics of the fracture dip model considered

in this study Schematics 1-A through 1-D assume that fracture

porosity is equal to 1% and that current flow direction is

horizontal in all cases thus the angle corresponds to fracture dip

Schematic 1-A displays a horizontal fracture with tortuosity

equal to 1.0 In this case the porosity exponent of the fractures

(m f) is also equal to 1.0 and fracture dip is equal to zero

Schematic 1-B presents a horizontal fracture with a fracture

tortuosity greater than 1.0 In this case the tortuosity leads to

porosity exponent of the fractures (m f) equal to 1.3 It is

important to note that although fracture dip is equal to zero, as

in the case of schematic 3-A, the porosity exponent (m f) is larger than 1.0 due to tortuosity

Schematic 1-C is for a fracture with a dip equal to 50° Tortuosity is equal to 1.0, and as a result the porosity exponent

of the fractures (m f) is equal to 1.0 However, the 50° angle

leads to a pseudo fracture porosity exponent (m fp) equal to 1.19

Schematic 1-D shows a non-horizontal fracture (dip = 50°) with a certain amount of tortuosity that leads to m f = 1.3 The 50° angle leads to a pseudo fracture porosity exponent (m fp) equal to 1.49

Aguilera and Aguilera16 have presented results associated with schematics 1-A and 1-C This paper presents research results for schematics 1-B and 1-D when tortuosity greater than 1.0 is taken into account

Permeability of idealized fracture rock, including fluid flow through anisotropic media, has been discussed in detail by Parsons18 and need not be repeated here Although Parson’s model is strictly for fluid flow, we have used it for current flow with reasonable results.16 Parsons fluid flow anisotropy concepts can be combined with Equations A-4 and A-5 in Appendix A and the formation factor for calculating the

porosity exponent m of the composite system at any angle of

interest

Sihvola19 considers the flow of fluids through a host medium, and how the addition of an inclusion would affect the flow

Figure 2 shows a mixture with aligned ellipsoidal inclusions

The host environment has a permittivity ε e and the ellipsoidal

inclusion has a permittivity ε i The mixture effective

permittivity ε eff is anisotropic as on the different principal directions the mixture possesses different permittivity components For these conditions the dual porosity exponent,

m, is given by:16

( ) ( ) ( ) ( )

φ

θ

log

F / sin F

/ cos log

2 0 2

=

where,

b m m

b m m

f

f b

2

2

1

'

φ

φ φ φ

−

−

=

m f is the porosity exponent of the fractures and,

2

ln

ln ) 1 (

φ

φ

−

−

=m f m f f

………… ……… (5)

Equation 5 is valid for ø 2 >0; f has been found to range exponentially between 1.0 at ø = ø2, and m f at ø = 1.0, using numerical experimentation.20

Development of the above equations is presented in Appendix

A The total porosity of the system is represented by ø The

angle between the fracture and the current flow direction is

equal to θ If the flow of current is horizontal the angle

corresponds to fracture dip The formation factor F θ=0 applies to

a systems in parallel (zero angle) The formation factor F θ=90

applies to systems in series (90-degree angle) This study also

presents cases for various intermediate values of θ between 0

and 90 degrees The equation for total porosity is:7, 15

( 2) 2

b

2

φ

where ø m is matrix porosity attached to the bulk volume of the

composite system, and ø b is matrix porosity attached to bulk

volume of only the matrix block

RESULTS

Figure 3 shows a crossplot of the porosity dual porosity

exponent m vs total porosity calculated from equations 1 to 6

for angles θ equal to 0 and 90 degrees The graph is constructed

for a constant value of m f equal to 1.3, a porosity exponent of

the matrix m b equal to 2.0 and fracture porosity (PHI2 or ø 2)

values of 0.001, 0.01, and 0.1 The same type of graph is

presented in Figure 4 for a constant fracture porosity ø 2 equal

to 0.01 and values of m f equal to 1.0, 1.3 and 1.5 The values of

the dual porosity exponent m increase significantly for a given

total porosity as the values of m f become larger Not taking this

into account can lead to significant errors in the calculation of

water saturation

Figure 5 shows values of the dual porosity exponent, m, vs

total porosity calculated from equations 1 to 6 for different

angles, for constant porosity exponent of the matrix m b equal to

2.0, and for a constant porosity exponent of the fractures m f

equal to 1.3 Note that if the current flow is horizontal, the angle

corresponds to the fracture dip The larger the angle, the bigger

is the value of m for a given total porosity All curves eventually

converge at a porosity exponent m b of the matrix equal to 2.0

EXAMPLE 1

Given an angle θ of 50 degrees between the direction of current

flow and the fracture, what is the value of m for a dual-porosity

system, if total porosity equals 0.05, fracture porosity is 0.01,

the porosity exponent (m b) of only the matrix is 2.0, and the

porosity exponent of the fractures (m f) affected by tortuosity is

1.3?

The first step is calculating matrix porosity, ø m, which is equal

to total porosity minus fracture porosity (ø m = 0.05 – 0.01 =

0.04); matrix porosity, ø b, which is equal to 0.040404 from

equation 6 (ø b = 0.04/(1 – 0.01) = 0.040404); f that is equal to

1.104845 from equation 5, and ø’ b that is equal to 0.0441017

from equation 4 The inverse of the formation factor, 1/F θ=0, is

equal to 0.004452 from equation 2 The inverse of the formation

factor, 1/F θ=90, is equal to 0.00195 from equation3 Finally, the

value of m for the composite system is calculated to be 1.941

from equation 1

EXAMPLE 2

What is the error in m and water saturation if θ is assumed to be

equal to zero and m f is assumed to be equal to 1.0 in the

previous example? What is the value of the pseudo porosity

exponent of the fractures (m fp) resulting from the 50-degree

angle?

If anisotropy and tortuosity are ignored leading to θ = 0 and m f

= 1.0, the value of m is calculated to be 1.487 following the

procedure explained in Example 1 This corresponds to an error

of 23.4% The error in the calculated water saturation is determined from:7

] ) (

1 [

100 (m m) 1/n

If the water saturation exponent, n, is 2.0 the error in the calculated water saturation is 100[1-(0.05 1.721-1.487 ) 1/2 ] = 49.3% Finally the pseudo porosity exponent of the fractures (m fp) resulting from the 50-degree angle between the fracture orientation and direction of current flow, and the tortuous value

of m f (1.3) is m fp = 1.49 This is calculated repeating the same steps shown above but assuming matrix porosity equal to zero (in reality use a very small of fracture porosity for the equations

to work For example, I have used ø b = 1E-12) In this case the

inverse of the formation factor, 1/F θ=0, is equal to 0.002512

from equation 2 The inverse of the formation factor, 1/F θ=90, is essentially equal to 0.0 (in reality 1.69E-24) from equation 3

Finally, the value of m fp for the fractures is calculated to be 1.49 from equation 1

Conclusions

1) The effect of current flow that is not parallel to fractures has been investigated for cases where the porosity exponent of the

fractures, m f, is greater than 1.0 due to fracture tortuosity It has been found that the larger the amount of fracture tortuosity, the

greater is the dual porosity exponent, m, of the composite

system of matrix and fractures

2) Not taking into account variations in fracture dip and fracture tortuosity can lead in some cases to significant errors in the

calculations of the dual porosity exponent, m, of matrix and

fractures; and water saturation For the examples presented in this paper the water saturation error is 49.3%

Acknowledgements

Parts of this work were funded by the Natural Sciences and Engineering Research Council of Canada (NSERC agreement 347825-06), ConocoPhillips (agreement 4204638) and the Alberta Energy Research Institute (AERI agreement 1711) Their contributions are gratefully acknowledged

NOMENCLATURE

f - volume fraction which the inclusions occupy

F - formation factor of the matrix system

F t - formation factor of the composite system

F θ=0 - formation factor of composite system at θ = 0°

F θ=90 - formation factor of composite system at θ = 90°

m – dual porosity exponent (cementation factor) of composite

system of matrix and fractures

m b - porosity exponent (cementation factor) of the matrix block

m c – correct dual porosity exponent (cementation factor) of

composite system

m i – incorrect dual porosity exponent (cementation factor) of

composite system

m θ=0 – dual porosity exponent (cementation factor) of the

composite system at θ = 0°

m θ=90 – dual porosity exponent (cementation factor) of the

composite system at θ = 90°

m f - porosity exponent (cementation factor) of the fracture

system

m fp - pseudo porosity exponent of the fractures (cementation

factor) resulting from θ

n - water saturation exponent

N x - depolarization factor in x direction

R o - matrix resistivity when it is 100% saturated with water

(ohm-m)

Roθ=0 - resistivity of the composite system (matrix plus

fractures) at θ = 0 when it is 100% saturated with water

(ohm-m)

Roθ=90 - resistivity of the composite system (matrix plus

fractures) at θ = 90 when it is 100% saturated with water

(ohm-m)

R w - water resistivity at formation temperature (ohm-m)

S w – water saturation, fraction

ε e - host environment permittivity

ε i - inclusion permittivity

ε eff - effective permittivity

ε effx - effective permittivity in x direction

ø - total porosity

ø b - matrix block porosity attached to bulk volume of the matrix

system

ø m - matrix block porosity attached to bulk volume of the

composite system

ø 2 - porosity of natural fractures

θ - angle between fracture and current flow direction

REFERENCES

1 Towle, G., An analysis of the formation resistivity

factor-porosity relationship of some assumed pore

geometries; Paper C presented at Third Annual

Meeting of SPWLA, Houston, 1962

2 Archie, G E., The electrical resistivity log as an aid in

determining some reservoir characteristics;” Trans

AIME, vol 146, p 54-67, 1942

3 Aguilera, R., Analysis of naturally fractured

reservoirs from conventional well logs: Journal of

Petroleum technology; v XXVIII, no.7, p 764-772,

1976

4 Rasmus, J C., A variable cementation exponent, m,

for fractured carbonates; The Log Analyst, vol 24, no

6, p 13-23, 1983

5 Draxler, J K and Edwards, D P., Evaluation

procedures in the Carboniferous of Northern Europe;

Ninth International Formation Evaluation

Transactions, Paris, 1984

6 Serra, O et al, Formation Micro Scanner image

interpretation; Schlumberger Educational Service,

Houston, SMP-7028, 117 p, 1989

7 Aguilera, R and Aguilera, M.S., Improved models for

petrophysical analysis of dual porosity reservoirs;

Petrophysics, Vol 44, No 1, p 21-35,

January-February, 2003

8 Lucia, F J., Petrophysical parameters estimated from

visual descriptions of carbonate rocks: A field

classification of carbonate pore space; Journal of

Petroleum Technology, v 35, p 629-637, 1983

9 Aguilera, R., 2003, Discussion of trends in

cementation exponents (m) for carbonate pore

systems; Petrophysics, Vol 44, No 1, p 301-305,

September-October, 2003.

10 Borai, A M., A new correlation for cementation factor in low-porosity carbonates; SPE Formation Evaluation, vol 4, no 4, p 495-499, 1985

11 Aguilera, R., Determination of matrix flow units in naturally fractured reservoirs; Journal of Canadian Petroleum Technology, vol 12, pp 9-12, December

2003

12 Ragland, D A., Trends in cementation exponents (m) for carbonate pore systems; Petrophysics, vol 43, no

5, p 434-446, 2002

13 Guillard, P and Boigelot, J., Cementation factor

analysis – a case study from Albo-Cenomanian dolomitic reservoir of the lower Congo basin in

Angola; SPWLA, circa 1990

14 Bateman, P W., Low resistivity pay in carbonate rocks and variable “m”; The CWLS Journal, vol 21,

p 13-22, 1988

15 Aguilera, R F and Aguilera, R, A Triple Porosity Model for Petrophysical Analysis of Naturally

Fractured Reservoirs; Petrophysics, vol 45, No 2, pp 157-166, March-April 2004

16 Aguilera, C G and Aguilera, R.: “Effect of Fracture

Dip on Petrophysical Evaluation of Naturally

Fractured Reservoirs,” paper CICP 2006-132 presented at the Petroleum Society’s 7 th Canadian International Petroleum Conference (57 th Annual Technical Meeting), Calgary, Alberta, Canada, June

13 – 15, 2006

17 Berg, C R., Dual and Triple Porosity Models from Effective Medium Theory, SPE 101698-PP presented

at the Annual Technical Conference and Exhibition held in San Antonio, Texas, Sept 24-27, 2006

18 Parsons, R W., Permeability of Idealized Fractured Rock; Society of Petroleum Engineers Journal, p 126-136, June 1966

19 Sihvola, A., Electromagnetic Mixing formula and

Applications; The Institution of Electrical Engineers,

London, United Kingdom, 1999

20 Aguilera, R.: “Role of Natural Fractures and Slot

Porosity on Tight Gas Sands,” SPE paper 114174 presented at at the 2008 SPE Unconventional Reservoirs Conference held in Keystone, Colorado,

U.S.A., 10–12 February 2008

APPENDIX A

The development presented here assumes that fluid flow equations though porous media have application in the flow of current through porous media Equations published originally

by Parsons18 for fluid flow through anisotropic porous media are used as a base for developing the model presented in this paper that permits evaluating the effect of fracture dip and fracture tortuosity on the petrophysical evaluation of dual porosity naturally fractured reservoirs

Figure 2 shows a mixture with aligned ellipsoidal inclusions

The host environment has a permittivity e and the ellipsoidal inclusion has a permittivity i The mixture effective permittivity eff is anisotropic as on the different principal directions the mixture possesses different permittivity components In this case, the Maxwell Garnett formula for the x-component is given by:19

( i e)

x e

e i e

e

x

,

ε ε ε

ε

ε

−

− +

− +

=

where f is the volume fraction which the inclusions occupy and

N x is the depolarization factor in the x direction In the case of

naturally fractured reservoirs, f is the equivalent of fracture

porosity (ø 2) The balance (1-f) is equivalent to the summation

of matrix porosity and solid rock

Making the depolarization factor (N x) in equation (A-1) equal to

zero results in:

i

….… (A-2)

Making the depolarization factor (N x) equal to one leads to:

i e

e i

ε ε ε

) 1 (

min

For the case at hand, the permittivity concept is associated with

the dielectric constant for mixtures of particles (rock crystals

and grains) and water Permittivity19 has also been called

dielectric permeability Permittivity equals the conductivity of

the composite system of matrix and fractures

Since resistivity is the inverse of conductivity, equations A-2

and A-3 can be re-written in more standard oil and gas notation

as:

⎠

⎞

⎜⎜

⎝

⎛

− +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

1 1

R

1

R

1

0

φ φ

( − )⎜⎜⎛ ⎟⎟⎞

+

⎟⎟

⎞

⎜⎜

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

=

w

2 o

2

o w o

R

1 1

R

1

R

1 R 1 R

1

θ

Equations A-4 and A-5 are for a system consisting of

matrix-fractures at zero and ninety degrees, respectively The situation

is presented schematically in Figure 6

0

o

Rθ=

represents the resistivity of the composite system at zero

degrees when it is 100% saturated with water of resistivity R w

90

o

Rθ=

is the resistivity of the composite system at ninety

degrees when it is 100% saturated with water of resistivity R w

ø 2 represents the porosity of fractures; this porosity is attached

to the bulk volume of the composite system, i.e., it is equal to

fracture void space divided by the bulk volume of the composite

system R w is water resistivity at reservoir temperature, and R o is

the resistivity of the matrix (when Sw=100%)

The formation factor F =0 of a system in parallel is given by:

F

0 0

=

= =

= −

The formation factor F =90 of a system in series is given by:

=

= =

= −

The formation factor F of only the matrix is given by:

( )b m R o / R w

F= φ − b = ……… (A-8)

Combining equations (A-4), (A-6) and (A-8) leads to:

( ) ( )

b 2 2

Fθ= = φ + −φ φ ……… (A-9) Combining equations (A-5), (A-7) and (A-8) leads to:

b

Fθ=90 = φ2 + 1 − φ2 φ − .…… (A-10) Equations A-9 and A-10 assume that the fracture porosity

exponent, m f, is equal to 1.0 The equations can be extended to

the case where m f is greater than 1.0 as follows:

b m m

Fθ=0 = 1 / φ2 + 1 − φ2 φ ' ……… (A-11)

b m m

Fθ=90 = φ2 + 1 − φ2 φ ' − ……… (A-12)

where a modification is entered from ø b to ø’b for taking into

account the possibility of an m f >1.0 The modification is:

f

f b

2

2

1

'

φ

φ φ φ

−

−

=

2

ln

ln ) 1 (

φ

φ

−

−

=m f m f f

……… (1-14)

The equation is valid for ø 2 >0; f has been found to range exponentially between 1.0 at ø = ø2, and m f at ø = 1.0, using numerical experimentation

Equations (A-11) and (A-12) can be combined as follows for

calculating the porosity exponent m for current flowing at any

angle with respect to the fractures:

θ θ

θ θ

2 90

2 0 t

sin F

1 cos

F

1 F

1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

⎟⎟

⎠

⎞

⎜⎜

⎛

=

Knowing that F t = ø -m leads to:

θ θ

2 90

2 0

F

1 cos

F

1 1

⎟⎟

⎞

⎜⎜

⎛ +

⎟⎟

⎞

⎜⎜

⎛

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

=

−

… (A-16) Solving for m of the composite system at any angle, we obtain:

( ) ( ) ( ) ( )

φ

θ

log

F / sin F

/ cos log

=

…… (A-17) which is the same as equation (1) in the main body of the text

Θ = 50°

mf= 1.0

mfp= 1.19

Θ = 50°

mf= 1.3

mfp= 1.49

Θ = 0°

mf= 1.0

mfp= 1.0

Θ = 0°

mf=1.3

mfp= 1.3

CURRENT DIRECTION IN ALL CASES

DUAL POROSITY

Ø 2 = 0.01

FIGURE 1 Schematics assume that current direction is horizontal in all cases, thus the angle θ in the schematic corresponds to fracture dip Fracture porosity (Ø2) = 0.01 (A) horizontal fracture with unity tortuosity (m f = 1.0), (B) horizontal fracture with tortuosity larger than 1.0 that leads to a porosity exponent of the fractures (m f ) equal to 1.3, (C) non-horizontal fracture (θ = 50°) with unity tortuosity (m f

= 1.0); the 50° angle leads to a pseudo fracture porosity exponent (m fp ) equal to 1.19, (D) non-horizontal fracture (θ = 50°) with tortuosity (m f = 1.3) The 50° angle leads to a pseudo fracture porosity exponent (m fp) equal to 1.49 If the flow of current is vertical, the angle corresponds to 90 minus fracture dip This paper discusses research associated with cases (B) and (D) Research associated with cases (A) and (C) were discussed previously.16

FIGURE 2 Schematic of mixture and aligned ellipsoidal inclusions The host environment has a permittivity ε e and the ellipsoidal

inclusion has a permittivity ε i The mixture effective permittivity ε eff is anisotropic as on the different principal directions the mixture possesses different permittivity components (Source: Sihvola19)

0.010

0.100

1.000

PHI2 = 0.001 PHI2=0.01 PHI2=0.1

FIGURE 3 Total porosity versus dual porosity exponent (m) for different values of fracture porosity (PHI2) The matrix porosity exponent (m b = 2.0) and the fracture porosity exponent (m f = 1.3) are constant

0.01

0.10

1.00

mf = 1.0

mf = 1.3

mf = 1.5

FIGURE 4 Total porosity versus dual porosity exponent (m) for different values of the fracture porosity exponent (m f ) Fracture porosity (Ø2 = 0.01) and the matrix porosity exponent (m b = 2.0) are constant

0.10

1.00

0 degrees

50 degrees

70 degrees

80 degrees

90 degrees

FIGURE 5 Total porosity versus dual porosity exponent (m) for different fracture angles ( θ ) Fracture porosity (Ø2 = 0.01), matrix

porosity exponent (m b = 2.0) and fracture porosity exponent (m f = 1.3) are constant

FIGURE 6 Systems where host and inclusion run (A) parallel and (B) perpendicular to flow (Source: Sihvola19) In these cases fracture

tortuosity is equal to 1.0 and the fracture porosity exponent m f = 1.0 In cases C and D, object of this study, the values of m f are larger than 1.0 due to tortuous paths of the fractures