a stress sensitivity model for the permeability of porous media...
TRANSCRIPT
A stress sensitivity model for the permeability of porous
media based on bi-dispersed fractal theory
X.-H. Tan*,‡, C.-Y. Liu*,§, X.-P. Li*,¶, H.-Q. Wang†,|| and H. Deng†,**
*State Key Laboratory of Oil and Gas Reservoir Geology and
Exploitation Southwest Petroleum University
Chengdu 610500, P. R. China†CNPC Southwest Oil and Gas Field Exploration and
Development Research Institute 610041, P. R. China‡[email protected]
§[email protected]¶[email protected]
||[email protected]**[email protected]
Received 2 July 2017Accepted 5 February 2018
Published 12 March 2018
A stress sensitivity model for the permeability of porous media based on bidispersed fractal
theory is established, considering the change of the °ow path, the fractal geometry approach and
the mechanics of porous media. It is noted that the two fractal parameters of the porous media
construction perform di®erently when the stress changes. The tortuosity fractal dimension ofsolid clusterDcT� become bigger with an increase of stress. However, the pore fractal dimension of
solid cluster Dcf� and capillary bundle Dpf� remains the same with an increase of stress. The
de¯nition of normalized permeability is introduced for the analyzation of the impacts of stress
sensitivity on permeability. The normalized permeability is related to solid cluster tortuositydimension, pore fractal dimension, solid cluster maximum diameter, Young's modulus and
Poisson's ratio. Every parameter has clear physical meaning without the use of empirical con-
stants. Predictions of permeability of the model is accordant with the obtained experimentaldata. Thus, the proposedmodel can precisely depict the °ow of °uid in porousmedia under stress.
Keywords: Fractal; porous media; permeability; stress; Darcy's law.
PACS Nos.: 47.53.tn, 47.55.hb, 47.56.tr.
Nomenclature
Dcf : solid cluster fractal dimension
DcT : solid cluster fractal tortuosity dimension
DcT� : solid cluster fractal tortuosity dimension at stress
E : Young's modulus, Pa
‡Corresponding author.
International Journal of Modern Physics C
Vol. 29, No. 1 (2018) 1850019 (12 pages)
#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0129183118500195
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Fc : the shape factor of solid cluster
Fp : the shape factor of capillary bundle
K : the permeability of porous media at the e®ective pressure
K� : the permeability of porous media at the e®ective pressure at stress
K0 : the permeability of porous media at zero stress, m2
Kþ : normalized permeability
k : geometry correction factor
L : representative length
Lc : solid cluster length
Lc� : solid cluster length at stress
Lc0 : original solid cluster length at zero stress
Lp : capillary bundle length
Lp� : capillary bundle length at stress
Lp0 : capillary bundle length at zero stress
L0 : original representative length
Nc : the number of the solid clusters
q : the °ow rate of a capillary in porous media
q� : the °ow rate of a capillary in porous media at stress
Q : the total °ow of porous media
Q� : the total °ow of porous media at stress
" : aspect ratio
� : the porosity of porous media at the e®ective pressure
�c : solid cluster diameter
�cmax : maximum solid cluster diameter
�cmax0 : original maximum solid cluster diameter at zero stress
�cmin0 : original minimum solid cluster diameter at zero stress
�c� : original solid cluster diameter at stress
�c0 : original solid cluster diameter at zero stress
�p� : capillary bundle diameter at stress
� : viscosity
� : Poisson's ratio
� : the stress in cross-sectional area
�eff : e®ective pressure
�p : pressure di®erence
Superscript
þ : normalized
Subscripts
c : solid cluster
e® : e®ective
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p : capillary bundle
T : tortuosity
max : maximum
min : minimum
1. Introduction
The phenomenon of °ow through porous media under stress attracts lots of
researchers' attentions. Such phenomena of °ow universally occurs both in natural
and arti¯cial materials.1–3 The research of the °uid behavior in porous media under
stress crosses the disciplines of science and engineering, such as hydraulics,4,5 phys-
ics,6–9 and chemical engineering,10 etc. Great concerns of the methods, theories and
progresses in this ¯eld have also been arised.
It is the numerical methods that have exerted vital function in analyzing the °ow
through porous media.11,12 Kupkov�a and Kupka13 derived an expression for veloci-
ties of elastic waves in a medium with su±ciently high number of pores with power-
law distribution of their sizes. Schreyer-Bennethum14 described a phenomenon of
°owing through a swelling porous material on the assumption that this media swelled
when whetted and shrunk when dried. A novel comprehensively coupled °ow de-
formation method was presented by Khoshghalb and Khalili15 and the validity of the
meshfree arithmetic of analyzing °ow-deformation feature in porous media full of
°uid was con¯rmed. Sabetamal et al.16 outlined a numerical procedure in order to
analyze dynamic coupled problems in geomechanics. However, the application of
numerical methods to delineate this phenomenon is constrained for these methods
require tedious calculations and necessitates precise model.17
The extensive investigation of °uid °owing through porous material, using fractal
theory, has lasted for more than two decades, such as the seepage characters of
°ow,18–20 thermal conductivity,21,22 non-Darcy °ow,23,24 capillary driven °ow25–27
and multiphase °ow,28–30 as well as electrical conduction.31 Xiaoet al.32 developed a
model on the °uid penetration behavior of ¯brous GDL in PEMFCs. Validating the
experimental data very well, this con¯rmed model indicated that the water and gas
relative permeabilities were merely in terms of water saturation as the porosity is
irrelevant. Utilizing both the fractal theory and Monte Carlo technique Xu et al.33
constructed a probabilistic model to represent the °uid behavior of radial direction
through fracture system of porous material. In comparison with usual numeric
model, this one excelled and can possibly be applied to forecast other motion features
of fracture system.
Consequently, it is noted that the not well-developed researches on the perme-
ability of porous media under stress should be attributed to the di±culty of calcu-
lating the permeability of porous media under stress, because of the disorder and
exceeding complication of microstructures of porous media.34,35 Hence, fractal the-
ory, to characterize the microstructures of porous media, is introduced. Sukop et al.36
BDFT stress sensitivity model for permeability
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thought that highly complex and rich fractal models can be produced by structured
geometries. Based on the fractal model, water retention is researched considering the
connections between pores and atmosphere. The fractal model can readily determine
the relations between fractal parameters utilized to describe the porous media and
the water retention behavior of the media. According to the assumption of fractal
pore-space geometry and the Kozeny–Carman approach, a model to represent how
the permeability of porous media was related to the porosity of that, was detected by
Costa.37 It was simple for the equation to depict the permeability of porous media.
A novel model, to describe the penetration behavior of porous media, was proposed
by Tan et al.38 in consideration of stress sensitivity. But the experimental data
showed that the samples in porous media reservoirs may obey Hook law stress–strain
diagram. Because the assumption of the extended length of a solid cluster under
stress may not accord with physical fact,3 an improved model for the permeability
under stress needs to be researched, considering the constant length and the
changing tortuosity fractal dimension of a solid cluster under stress.
2. Theoretical Model
The assumption of the porous media considered in present work is that a set of
crooked solid clusters comprises the porous media and the interspaces of solid clusters
are capillary bundles. What is featured in Fig. 1 is the porous media that consists of a
large number of solid clusters. There is a transversal shrinkage of solid clusters when
exerting the stress on a set of solid clusters. The diameters of capillary bundles and
solid clusters in porous media diminish, but the shapes of those are the same, as is
shown in Fig. 2. The fractal model for permeability of porous media will be derived
on the basis of the mechanics of solid clusters.
The cumulative distributions of solid clusters in porous media with a greater
diameter or an equal one as diameter �c obey the fractal scaling law:39
Ncðl � �cÞ ¼�cmax
�c
� �Dcf
; ð1Þ
Fig. 1. (Color online) A porous medium formed by all kinds of solid clusters with di®erent diameters.
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where Dcf , Nc, �cmax and �c are the fractal dimension, the number, the maximum
diameter and the diameter of solid cluster, respectively.
The di®erential form of Eq. (1) with regard to �c obeys the quantity of solid
clusters with the sizes varying in the extremely small domain of �c to �c þ d�c on
intersection surface area.
�dNc ¼ Dcf�Dcfcmax�
�ðDcfþ1Þc d�c: ð2Þ
In Eq. (2), �dNc > 0, which suggests a decrease of number of solid clusters occuring
as the solid cluster diameter increases.
The fractal scaling law for the solid clusters in porous media can be
expressed as:40,41
Lc ¼ � 1�DcTc LDcT ; ð3Þ
where Lc, L, DcT is the length, straight distance and tortuosity fractal dimension of
solid cluster, respectively. It is noteworthy that Eq. (3), based on the e®ort of
Wheatcraft and Tyler,42 is a totally empirical relationship.
According to experimental data, before the solid cluster of the crude porous
material becomes invalid, it yields little or even not at all. Moreover, one of the
concrete, of which the value of tensile strength is low, is put in a category of
fragile material. So Hook43 described the stress–strain illustration for solid cluster in
porous media. The diameter and length of a solid cluster in porous media can be
expressed as:3
�c� ¼ �c0 1� ��
E
� �; ð4Þ
and
Lc� ¼ Lc0 1þ 1
�
��
E
� �; ð5Þ
where �� is the stress, E is Young's modulus, � is the ratio Poisson, �c� is the solid
cluster diameter under stress, �c� ¼ �c0 as the value of stress is 0, Lco is solid cluster
length under stress, Lco is solid cluster length when the value of stress in transversal
surface is 0.
Fig. 2. (Color online) Comparisons of solid cluster diameters under di®erent stress.
BDFT stress sensitivity model for permeability
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Rewriting Eq. (3) yields
Lco ¼ �1�DcT0
c0 LDcT0 ; ð6Þ
and
Lc� ¼ �1�DcT�c� LDcT� ; ð7Þ
where DcT� represents the fractal tortuosity dimension of solid cluster of porous
media under stress.
When a radial stress is exerted on a solid cluster, the latter wrinkles because of
the axial stress contracts laterally (Fig. 3). It is notable that the length Lc� and the
fractal tortuosity dimension DcT0 of solid cluster will have an increase while the solid
cluster straight distance L gets no change, as the solid cluster is compressed.
Inserting Eqs. (4), (5) and (6) into Eq. (7), the solid cluster fractal tortuosity
dimension of porous media under stress can be presented.
DcT� ¼ log��; ð8Þ
where � ¼ L�c0 1þ��
Eð Þ and � ¼ �E����Eþ���
L�c0
� �DcT0
.
In Eq. (8), DcT� is a function of the stress ��, the mechanics parameters � and E.
The DcT� increase with an increase of the stress ��. When the �� equals 0, the DcT�
equals theDcT0. As the values of the Poisson's ratio � and the Young's modulusE get
higher, the DcT� is smaller. It is the responsible phenomenon because the resistance
which wrinkles solid clusters is bigger when the Poisson's ratio � is bigger. In addi-
tion, the contract of solid clusters is larger with the higher Young's modulus E.
This validates the physical situation.
Approximating the length of the solid clusters, comprised of capillary bundle,
the Capillary bundle's length Lp� is
Lp� ¼ �1�DcT�
c0 LDcT� 1� ��
E
� �1�DcT�
: ð9Þ
The capillary bundle diameter of porous media can be expressed as38
�p� ¼ �c�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
p � 2Fc
4Fp
s¼ �c0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
p � 2Fc
4Fp
s1� ��
E
� �: ð10Þ
Fig. 3. (Color online) Comparisons of solid cluster lengths under di®erent stress.
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The following is the expression of the °ow rate of a capillary, whose aperture shapes
are various, in porous media:44
q� ¼ k��4
p��p
128�Lp�
; ð11Þ
where q�, �p, � and k is the °ow rate of a capillary in porous media, the pressure
di®erence, the viscosity and a geometry factor, respectively. When the intersection
surface of a capillary bundle is circular, square and regular triangular, the value of
k is separately 1, 1.43 and 1.98.45
Substituting Eqs. (9) and (10) into Eq. (11), the °ow rate of a capillary in porous
media is acquired.
q� ¼�k�p� 3þDcT�
c0
ffiffi3
p �2Fc
4Fp
� �21� ��
E
� �3þDcT�
128�LDcT�: ð12Þ
Integrating Eq. (12) from the minimum value to maximum one of solid cluster
diameter, the total °ow Q� can be acquired, when the stress is considered.
Q� ¼ �kDcf�p�3þDcT�
cmax0
128�ð3þDcT� �DcfÞLDcT�
ffiffiffi3
p � 2Fc
4Fp
� �2
1� ��
E
� �3þDcT�
: ð13Þ
By making an integration of the transversal surface area of the regular triangular
unit cell. Au� from the minimum value to the maximum one of solid cluster diameter,
the transversal surface area of the porous media A� can be calculated.
A� ¼ �Z �cmax�
�cmin�
Au�dNc ¼ffiffiffi3
pDcf�
2cmax0
4ð2�DcfÞ1� ��
E
� �2ð1� 2�Dcf Þ; ð14Þ
where ¼ �cmin0
�cmax0.
The permeability of porous media can be expressed by Darcy.
K� ¼ Q��L
A��p: ð15Þ
Substituting Eqs. (13) and (14) into Eq. (15), the permeability of porous media is
acquired.
K� ¼ ð2�DcfÞ�kL1�DcT��1þDcT�
cmax0
32ffiffiffi3
p�ð3þDcT� �DcfÞ
ffiffiffi3
p � 2Fc
4Fp
� �2
1� ��
E
� �1þDcT�
: ð16Þ
The permeability of porous media K� tends to 0 with the solid cluster fractal di-
mension Dcf approximating 2 in Eq. (16), which indicates the porous media is ¯lled
up with solid clusters. However, as Dcf approaches 0, K� approximates maximum,
which suggests the porous media is ¯lled up with pores. This validates the physical
situation.
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In Eq. (16), when the value of stress is 0, the original permeability of porous media
K0 can be expressed as
K0 ¼ð2�DcfÞ�kL1�DcT0�
1þDcT0
cmax0
32ffiffiffi3
p�ð3þDcT0 �DcfÞ
ffiffiffi3
p � 2Fc
4Fp
� �2
: ð17Þ
The de¯nition of the normalized permeability is the permeability ratio of K to K0,
which is obtained.
Kþ ¼ K�
K0
¼ ð3þDcT0 �DcfÞð3þDcT� �DcfÞ
�cmax0
L
� �DcT��DcT0
1� ��
E
� �1þDcT�
; ð18Þ
where DcT� can be calculated by Eq. (8).
3. Results and Discussion
Compared to experimental data, the validity of the model which is used to charac-
terize the permeability under stress of porous media is discussed. The mechanics
parameters of the porous media are E ¼ 2:79� 4:76� 109 Pa, � ¼ 0:08� 0:22,46 so
E ¼ 2:79� 109 Pa, � ¼ 0:08 and E ¼ 4:76� 109 Pa, � ¼ 0:22 are decided based on
the properties of porous media, and � ¼ 0:1, k ¼ 1, Fp ¼ �=4, Fc ¼ �=4; �eff ¼ �� are
used in this study. As is shown in Fig. 4, the predictions of the presented fractal
model (Eq. (18)) with extremum values of Young's modulus and Poisson's ratio are
compared with the available experimental data.47–49 The dots, of which the e®ective
stress domain varies from 0 to 120 �106 Pa, can represent the experimental data of
normalized permeabilities. The samples 1–6 are from Chierici et al.'s data,48 which is
marked as solid points. The samples 7–12 are chosen from Yale's paper,49 marked as
void points. The samples 13–15 are selected because of Hsu's contribution47 with half
solid points. The two solid lines represent the predicted normalized permeabilities of
Fig. 4. (Color online) Comparisons of the permeability of porous media between the prediction value and
experimental data.47–49
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the proposed model with extremum values of Young's modulus and Poisson's ratio.
It is noticed that almost all the experimental data locate between the two lines,
the vast majority of the experimental data distributed in the intermediate
region which is between the maximum values and minimum ones of the predictive
data. That means the model predictions validate the available experimental
data very well.
Figures 5 and 6 show the normalized permeabilityKþ of porous media a®ected by
mechanics parameters, the Young's modulus E and Poisson's ratio �, respectively.
Fig. 5. (Color online) The permeability of the porous media versus e®ective stress with di®erent Young's
modulus, E:
Table 1. The details of samples.
Sample no.
Minimum
pressure (MPa)
Maximum
pressure (MPa)
Permeabilitywith minimum
pressure (mD)
Permeabilitywith maximum
pressure (mD) Reference
1 0.1 45 67.7 58.97 Ref. 48
2 0.1 45 72.3 62.613 0.1 45 400 369.2
4 0.1 45 23 20.59
5 0.1 45 31.9 29.226 0.1 45 352 329.47
7 1 49 30.3 27.61 Ref. 49
8 1 49 129.6 116.12
9 1 48 49.0 45.04
10 1 49 901.9 842.2811 1 49 152.2 131.27
12 1 49 494.6 442.22
13 4.6 120 0.58 0.48 Ref. 47
14 4.7 120 0.34 0.24
15 4.9 120 0.64 0.48
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When the e®ective stress �eff increases, the normalized permeability Kþ reduces
gently. Meanwhile, as the e®ective stress �eff increases, the pore space decreases and
the °owing path increases as well as the solid cluster fractal tortuosity dimension
DcT�, which indicates the decrease of the cross-sectional area of capillary bundles.
Then, the °ow resistance increases with the increase of the e®ective stress �eff , re-
ducing the permeability under stress, so the normalized permeability Kþ of porous
media is always less than 1.0. Figure 5 reveals that the permeability of porous media
K� is smaller as the Young's modulus E gets higher. Two factors contribute to this
phenomenon. One is the higher resistance that contracts solid clusters with the
higher Young's modulus E and the other is the pore space of porous media which gets
harder to compress with the increase of Young's modulus E. From Fig. 6, we can also
see that the permeability of porous media K� increases as the Poisson's ratio �
increases. This can be explained by the fact that the solid cluster fractal tortuosity
dimension DcT� of porous media becomes less easy to increase when the Poisson's
ratio � increases. So, the permeability of porous mediaK� increases with the increase
of the Young's modulus E and the Poisson's ratio �.
4. Conclusion
The previous research for permeability of porous media is reviewed ¯rst, and then
based on the stress e®ect, the predicted permeability method of porous media is
proposed considering the mechanics of porous media and using the fractal statistics
method. The predictions of the present model, as tested by experimental data, val-
idate the experimental data very well. Thus, the validity of the model for the pre-
diction of permeability in porous media is veri¯ed. Consequently, the fractal
characters of porous media have an e®ect on the permeability of porous media.
Considering the fractal properties of microstructure of porous media and based on
Fig. 6. (Color online) The permeability of porous media versus e®ective stress with di®erent
Poisson's ratio.
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the stress e®ect and basic °uid dynamics, an entire analytical expression of perme-
ability of porous media can be constructed.
Acknowledgments
This work was jointly supported by Open Fund (PLN201721) of State Key Labo-
ratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum
University), National Natural Science Foundation of China (51704246), the National
Science and Technology Major Project (2016ZX05052 -002-04, 2016ZX05024-005-
008), and PetroChina Innovation Foundation (2016D-5007-0209).
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