Article pubs.acs.org/EF
Prediction of Relative Permeability of Unsaturated Porous Media Based on Fractal Theory and Monte Carlo Simulation Boqi Xiao,† Jintu Fan,*,†,‡ and Feng Ding† †
Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Department of Fiber Science and Apparel Design, Cornell University, Ithaca, New York, United States
‡
ABSTRACT: Mass transport through porous media is an important subject to engineers and scientists in various areas including oil engineering, fuel cells, soil science, textile engineering, etc. The relative permeability and capillary pressure are the key parameters that affect liquid transport through porous media. In this paper, the Monte Carlo technique is applied to predict the relative permeability of unsaturated porous media, considering the effect of capillary pressure and tortuosity of capillaries. The relative permeability is expressed as a function of porosity, area fractal dimension of pores, fractal dimension of tortuous capillaries, degree of saturation, and capillary pressure. It is found that the phase fractal dimensions (Df,w and Df,g) strongly depend on porosity. Besides, it is shown that the capillary pressure increases with the decrease of saturation, and at low saturation the capillary pressure increases sharply with the decrease of saturation. There is no empirical constant in the proposed model, and each parameter in the model has a clear physical meaning. The predicted relative permeability obtained by the present Monte Carlo simulation is shown to have a good agreement with the experimental results reported in the literature. The proposed model improved the understanding of the physical mechanisms of liquid transport through porous media. al.7,8 In their experiment7 with altering permeability, an increase of permeability caused an increase in gas production and a decrease in dissociation rate. Recently, Shou and Fan28 applied a difference-fractal approach and unit cell approach29 to model the hydraulic permeability of high porosity porous media. Additionally, Bal and Fan13 analyzed spontaneous flow of liquids through layered heterogeneous porous fibrous media based on fractal theory. The models reported in refs 13, 28, and 29 are, however, limited to saturated porous media. In addition, Hou et al.11,26,27 have experimentally and theoretically investigated water−oil relative permeability. The results11 showed that the calculated water phase relative permeability curve is higher and the calculated oil phase relative permeability curve is lower when compared to the true relative permeability curve if the capillary pressure is neglected. In practice, most porous media are unsaturated in nature. It is still a challenge to predict the relative permeability of unsaturated porous media analytically due to the disordered and extremely complicated microstructures as well as the interrelationship between permeability and capillary pressure and tortuosity of capillaries. In the present paper, we propose an overall mechanistic model to consider the coupled effects of the porosity of porous medium (ϕ), the tortuosity of capillaries (DT), the saturation of the wetting phase (Sw), the pore size (λ), and capillary pressure (Pc) in unsaturated porous media. To elucidate the effect of these parameters on relative permeability, Monte Carlo simulation was performed. For this purpose, the next section is devoted to model description, and then, in section 3, the Monte Carlo method is presented. The results and discussions
1. INTRODUCTION Permeability to fluid flow in porous media is a key parameter to be considered in many fields1−8 including soil science, fuel cells, textile engineering, reservoir engineering, subsurface environmental engineering, and chemical engineering. Past investigations1−13 have shown that the transport processes in porous media are very complex and dependent on the complexity of the microstructure of porous media. Many parameters such as porosity, size of pore, tortuosity of capillaries, and capillary pressure are very important for the permeability of porous media. These parameters, however, are closely related to the geometric architecture of porous media. Besides, the structure of most porous media is highly complex and difficult to describe. It is even more complicated to analyze the transfer behaviors within unsaturated porous systems, especially when different transport mechanisms take part simultaneously. Modeling the permeability of unsaturated porous media therefore presents a great challenge. So far, a number of experimental techniques3−13 have been developed to measure the permeability of porous media. Experiments are, however, generally expensive and time-consuming. Experiments are usually complicated by the interdependence of permeability and capillary pressure, especially in cases where phase change takes place. It is, therefore, always desirable to develop theoretical models. Many of the reported theoretical works14−27 assumed simplified pore geometries, which allowed analytical solutions of the microscopic flow patterns. In early studies, Ayatollahi et al.5 investigated the effects of temperature on the heavy oil relative permeability during the tertiary gas oil gravity drainage mechanism. The results showed that the elevated temperature was the leading factor behind the wettability alteration and heavy oil relative permeability.5 Besides, it was experimentally verified that the permeability of the rock is dependent on the connectivity of pores by Lee et © 2012 American Chemical Society
Received: August 12, 2012 Revised: October 19, 2012 Published: October 20, 2012 6971
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⎛ λmin ⎞ Df ⎜ ⎟ ≅0 ⎝ λmax ⎠
are reported in section 4, and concluding remarks are given in section 5.
2. MODEL DESCRIPTION The porous media in nature can be and have been described as fractal objects.1,2,13,18−22,24,28,30−36 This means that the fractal theory may be used to predict the transport property of porous media. Besides, the pore size distribution and tortuosity of capillaries have also been shown to follow the fractal scaling laws;18,20,22,31,34,35 that is, the cumulative number (N) of pores in porous materials whose sizes are greater than or equal to λ can be expressed according to the fractal scaling law18 Df
N (L ≥ λ) = (λmax /λ) with λmin ≤ λ ≤ λmax
(3c)
is satisfied. Equation 3c can be considered as a criterion of whether a porous medium can be characterized by fractal theory and technique. In general,18 most porous media have λmin/λmax = m ≤ 10−2, thus eq 3c approximately holds. Thus, the fractal theory and technique can be used to analyze the characters of porous media. The fractal dimension Df in eqs 1−3 is given by18,30 Df = d − ln ϕ/ln m
(4)
where d is the Euclidean dimension and equals 2 or 3 in the two or three-dimensional spaces, respectively; ϕ is the porosity of the porous medium in Figure 1. In the present work, it is assumed that the real porous media consist of a bundle of tortuous capillaries with different diameters. As shown in Figure 1, the flow rate q(λ) through a single tortuous capillary is given by the well-known Hagen− Poiseulle equation37
(1)
where λmax and λmin are the maximum and minimum diameter of pores in porous media, respectively; λ is the diameter of a pore, and Df is the area fractal dimension of pores. Figure 1 shows a schematic of a porous medium composed of a bundle of tortuous capillaries, whose sizes and tortuosities
q(λ) =
π ΔP λ 4 128 lt(λ) μ
(5)
where μ is the viscosity of the fluid, ΔP is the pressure difference, and λ is the hydraulic diameter of a single capillary tube. Because the tortuosity of capillaries has been proven to follow the fractal scaling laws,18,20,22,31,34,35 the total length of a tortuous capillary can be expressed as18 lt(λ) = λ1 − DTl0DT Figure 1. Schematic of fractal porous media composed of a bundle of tortuous capillaries.
where DT is the tortuosity fractal dimension (1 < DT < 2 in two dimensions), which is given by38
DT = 1 + follow the fractal scaling laws. The applied pressure difference (ΔP) across the straight-line distance (l0) of the porous material have a cross-section area of A, and lt is the actual length for fluid or gas traveling in a fractal porous medium. Differentiating eq 1 with respect to λ yields Df −(Df + 1) −dN = Df λmax λ dλ
Equation 2a gives the pore number between the pore size λ and λ + dλ. The negative sign in eq 2a implies that pore number decreases with the increase of pore size, and −dN > 0. The total number of the pores from the minimum diameter λmin to the maximum diameter λmax can be derived from eq 1 as18,20 Dividing eq 2a by eq 2b results in
(
(3a)
λmax min
⎛λ ⎞ f (λ) dλ = 1 − ⎜ min ⎟ ⎝ λmax ⎠
1 1−ϕ
1−
density function. Patterned after the probability theory, the probability density function f(λ) should satisfy the following normalization relationship or total cumulative probability:18
∫λ
(7)
⎡ ⎢ 1⎢ 1 1−ϕ + τ = ⎢1 + 2 2 ⎢ ⎢⎣
(2b)
Df −(Df + 1) −dN /Nt = Df λmin λ d λ = f (λ ) d λ Df −(Df + 1) where f(λ) = Dfλminλ is the probability
ln τ ln(l0/λav )
where τ is the average tortuosity of tortuous capillaries and λav is the average diameter of capillaries. The purpose of introducing the tortuosity of tortuous capillaries is to include the effect of the complexity of the geometrical shape on fluid permeability. For flow paths in porous media, an approximate relation between the average tortuosity and porosity can be obtained as38
(2a)
Nt(L ≥ λmin) = (λmax /λmin)Df
(6)
2
)
−1
1−ϕ
+
1 4
1−ϕ
⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦
(8)
The average diameter of capillaries can be found with the aid of eq 3
Df
≡1 (3b)
λav =
The integration result of eq 3b shows that eq 3b holds if and only if
∫λ
λmax min
⎡ ⎛ λmin ⎞ Df − 1⎤ Df ⎢ ⎥ λ f (λ ) d λ = λmin 1 − ⎜ ⎟ ⎢⎣ ⎥⎦ Df − 1 ⎝ λmax ⎠ (9)
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Because λmin/λmax = m, eq 9 can be further modified as λav =
mDf λmax (1 − m Df − 1) Df − 1
Pc = (10)
A=
λmax
π (λ /2)2 ( −dN )
min
ϕ
=
πDf (1 − m2 − Df ) 2 λmax 4(2 − Df )ϕ
(11)
By inserting eq 4 into eq 11, eq 11 can be further modified as A=
πDf (1 − ϕ) 2 λmax 4(2 − Df )ϕ
(12)
The cross-section area can be also expressed as18
A = l02
(13)
Comparing eqs 12 and 13, the straight-line distance (l0) can be written as l0 =
λmax 2
πDf (1 − ϕ) (2 − Df )ϕ
(14)
By inserting eq 10 into eq 14, a relation between the straightline distance and the average diameter of capillaries can be further modified as l0 Df − 1 = λav 2mDf (1 − m Df − 1)
πDf (1 − ϕ) (2 − Df )ϕ
(15)
Inserting eqs 8 and 15 into eq 7, The tortuosity fractal dimension for tortuous capillaries in porous media can be further modified as ⎧ ⎡ ⎛ ⎪ ⎢ ⎜ ⎪ ⎢1⎜ 1 DT = 1 + ⎨ln⎢ ⎜1 + 1−ϕ + 2 ⎪ ⎢2⎜ ⎪ ⎢ ⎜ ⎩ ⎣ ⎝ ⎫ ⎪ ⎧ ⎪ ⎪ ⎡ Df − 1 ⎬ /⎨ln⎢ ⎪ ⎢ 2 mD (1 − m Df − 1) ⎪ ⎩ ⎣ f ⎪ ⎭
1−ϕ
(
1 1−ϕ
1−
2
)
−1
1−ϕ
+
1 4
⎞⎤ ⎟⎥ ⎟⎥ ⎟⎥ ⎟⎟⎥ ⎠⎦⎥
qw (λ w ) =
4 4 π ΔPw λ w π (Pm + Pc,w ) λ w = 128 lt(λ w ) μw 128 lt(λ w ) μw
(19)
where λw is the equivalent diameter for the wetting phase, and the capillary pressure for wetting phase in unsaturated porous media is obtained by modifying eq 18 as
⎪ πDf (1 − ϕ) ⎤⎫ ⎥⎬ ⎪ (2 − Df )ϕ ⎥⎦⎭
Pc,w =
(16)
where Df is obtained from eq 4. Equation 16 indicates that the tortuosity fractal dimension for tortuous capillary in porous media is a function of porosity (ϕ) and the area fractal dimensions of pore (Df). In general, the pressure difference may include mechanical pressure or injection pressure (Pm), gravitational (Pg), and capillary pressures (Pc) due to surface tension. Thus, the total pressure difference may be expressed as4 ΔP = Pm + Pg + Pc
(18)
In eq 18, T is surface tension of fluids, θ is contact angle between liquid and solid, and F is shape factor depending on geometry of a medium and on flow direction (F = 4 when the capillary is cylindrical, see Ahn et al.4). Equation 18 can be used to predict the capillary pressure for different pore sizes.4 Several studies13,35,36 reported the transport properties of porous media based on eq 18, and good agreement between the model (eq 18) predictions and experimental data were obtained by using eq 18. So, eq 18 is also used in our work. It is known that the contact angle is related to wettability. Wettability is defined as the tendency of one fluid to spread on or adhere to a solid surface in the presence of other immiscible fluids. Wettability affects the relative permeability because it is a factor in control of location, flow, and distribution of fluid in porous media. The results showed that39 when the relative permeability for water is increased, the relative permeability for oil is decreased as the system becomes more oil-wet. Besides, for a two-fluid system, the model40 predicted that an increase in the contact angle (measured through water) or organic-wet fraction of a medium would be accompanied by an increase in the relative permeability (krw) for the wetting phase (such as water), and a decrease in the relative permeability (krg) for nonwetting phase (such as organic). This result occurred owing to the change in the roles (wetting versus nonwetting) of water and organic as θ increased.40 These trends are consistent with the experimental results.39 The difference between saturated and unsaturated porous media is that, in saturated porous media, there is only a single fluid such as water filled in pores or capillaries and, in unsaturated porous media, at least two different fluids coexist. For example, pores are partially filled with water and gas. In unsaturated fractal porous media, for the wetting phase, eq 5 can be modified as
Equation 10 depicts a fractal expression of the average diameter of capillaries in porous media. In Figure 1, the pores in the cross-section can be considered as circles with different diameters λ; consequently, the crosssection area can be obtained with the aid of eq 2a:
∫λ
FT cos θ 1 − ϕ λ ϕ
FT cos θ 1 − ϕw λw ϕw
(20)
In a porous medium, due to a distribution of different pore sizes, the concept of an average capillary pressure is more useful.13,35 The average capillary pressure for the wetting phase can be obtained with the help of eq 3a: Pc,w =
(17)
=
In practical applications, the gravitational pressure, which is relatively small compared with the applied mechanical pressure and capillary pressure, can be neglected. Thus, here, we assume ΔP = Pm + Pc. The capillary pressure is given by4
∫λ
λmax ,w
Pc,wf (λ w ) dλ w
min ,w
Df,w Df,w
FT cos θ 1 − ϕw + 1 λmin ,w ϕw
⎡ ⎛ λmin ,w ⎞ Df,w + 1⎤ ⎢ ⎥ ⎟⎟ × 1 − ⎜⎜ ⎢ ⎥ λ ⎝ ⎠ max ,w ⎣ ⎦ 6973
(21)
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where Df,w is the fractal dimension for the wetting phase. Since λmin/λmax ≤ 10−2, 0 < Df,w < 2, so (λmin,w/λmax,w)Df,w + 1 ≪ 1, eq 21 can be further reduced to Pc,w =
Df,w 1 + Df,w
∑ λi2 + D ⎟⎟ T
(22)
ln
λmin ,w
=d−
Pc =
λmax ,w
λmin λmax
(23)
Similarly, for the nonwetting phase, we have Df,g = d −
ln ϕg ln
λmin ,g
=d−
(24) 19
λmin λmax
(25)
∑
i=1
i=1
4 π (Pm + Pc,w ) λ w, i 128 lt(λ w ) μw
J
Qg =
(27)
K wA(Pm + Pc,w ) K wAΔP = μw l0 μw l0
Kg =
(28)
1 − Swϕ Swϕ
i=1
⎠⎥⎦
i=1
1−D π l0 T 128 A
(33)
J
∑ λg,3i+ D
T
(34)
i=1
J ⎡ 1−ϕ /⎢Pm ∑ λi3 + DT + FT cos θ ⎢⎣ i = 1 ϕ
T
⎠
∑ qg(λg,i) = ∑
4 π ΔPm λg, i 128 lt(λg ) μg
J ⎡⎛ Df FT cos θ 1 − ϕ ⎞ ⎢ k rg = ⎜Pm + ⎟ ∑ λg,3i+ DT ⎤⎦ ⎢⎣⎝ λmin ϕ ⎠ i=1 1 + Df
⎞
J
∑ λw,2+i D ⎟⎟ i=1
T
According to the definition for relative permeability krg = Kg/K, dividing eq 30 by eq 34, we can obtain the relative permeability for nonwetting phase in unsaturated porous media as follows
J ⎛ l01 − DT π ⎜⎜Pm ∑ λ w,3+i DT 128 A(Pm + Pc,w ) ⎝ i = 1
+ FT cos θ
∑ λi2 + D ⎟⎟⎥
where λg is the equivalent diameter for nonwetting phase in unsaturated porous media. The permeability for the nonwetting phase is
where Kw is the permeability for wetting phase. Comparing eq 27 with eq 28, we obtain the permeability for wetting phase in unsaturated porous media as Kw =
J
i=1
For creeping flow through a porous medium, Darcy’s law14 also applies, thus the total flow rate for wetting phase can also be expressed by Qw =
⎞⎤
J
Applying the above analysis to the gas flow in the nonwetting phase of unsaturated porous media, we can obtain the total gas flow rate and the permeability for nonwetting phase. Note that, unlike that in the wetting phase, the capillary pressure Pc is in nonwetting phase is zero. The total flow rate Qg for nonwetting phase in a unit cell is
(26)
The total flow rate Qw for wetting phase in a unit cell can be obtained by summing up the flow rates through all capillaries, that is J
(31)
(32)
λmax ,g = λmax Sg = λmax 1 − Sw
∑ qw (λi) =
min
Df FT cos θ 1 − ϕ 1 + Df λmin ϕ
J ⎛ ⎜Pm ∑ λi3 + DT + FT cos θ 1 − ϕ ⎜ ϕ ⎝ i=1
λmin ,g = λmin Sg = λmin 1 − Sw ,
J
Pcf (λ)dλ =
J J ⎛ ⎤ ⎜Pm ∑ λ w,3+i DT + FT cos θ 1 − Swϕ ∑ λ w,2+i DT)⎥ ⎜ ⎥⎦ Swϕ i = 1 ⎝ i=1 ⎡⎛ Df,w FT cos θ 1 − Swϕ ⎞ ⎟⎟ /⎢⎜⎜Pm + ⎢⎣⎝ Swϕ ⎠ 1 + Df,w λmin ,w
where Df,g is the fractal dimension for nonwetting phase in unsaturated porous media and λmax,g and λmin,g are the maximum and minimum equivalent diameters for nonwetting phase in unsaturated porous media, which are related by19
Qw =
λmax
⎡⎛ Df FT cos θ 1 − ϕ ⎞ k rw = ⎢⎜Pm + ⎟ ⎢⎣⎝ λmin ϕ ⎠ 1 + Df
ln(Sgϕ) ln
λmax ,g
∫λ
According to the definition of relative permeability, krw = Kw/ K, dividing eq 30 by eq 29, we can obtain the relative permeability for the wetting phase in unsaturated porous media
where λmin,w and λmin,w are the maximum and minimum equivalent diameters for the wetting phase in unsaturated porous media, which are related by19 λmin ,w = λmin Sw , λmax ,w = λmax Sw
(30)
where the average capillary pressure in saturated porous media can be obtained with the aid of eq 3a:
ln(Swϕ) ln
⎠
i=1
where Sw is the saturation of the wetting phase and is related to porosity by ϕw = Swϕ (see ref 14), Df,w can be obtained by extending the fractal dimension for saturated fluid as19 Df,w = d −
⎞
J
FT cos θ 1 − Swϕ λmin ,w Swϕ
ln ϕw
J l01 − DT ⎛ 1−ϕ π ⎜⎜Pm ∑ λi3 + DT + FT cos θ ϕ 128 A(Pm + Pc) ⎝ i = 1
K=
(29)
where Pc,w is the average capillary pressure for wetting phase which is given by eq 22. If we apply the above analysis to saturated porous media, we can obtain the permeability in saturated porous media as
⎤
J
∑ λ i2 + D ⎥ T
i=1
⎥⎦
(35)
Equation 32 and 35 indicate that the relative permeability is a function of parameters Pm, F, T, θ, ϕ, and the microstructural parameters (Df, DT, and λmin) of a medium. 6974
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3. MONTE CARLO SIMULATION The present Monte Carlo method is used to simulate the fractal distribution of the sizes and tortuosity of capillaries for calculation of relative permeability. The cumulative probability (R) in the range λmin∼λ can be found by
∫λ
R (λ ) =
∫λ
=
σ=
f (λ ) d λ Df −(1 + Df ) Df λmin λ
⟨k r2 ⟩ =
dλ
min
Df
λ = λmin /(1 − R )1/ Df = m
λmax (1 − R )1/ Df
λi = λmin /(1 − R i)1/ Df = m[λmax /(1 − R i)1/ Df ]
(37)
(38)
where Ri is the cumulative probability in the range of λmin∼λi. By randomly assigning a value of Ri within [0, 1], we can simulate the capillary size distribution, provided that we know the maximum and minimum capillary sizes. For the bidispersed porous media, the maximum capillary size and λmin/λmax = m can be respectively estimated by18,20 ⎡ ⎞ R̅ c ⎢ ⎛ 1 − ϕc = − 1⎟ + 2⎜ 2 ⎢⎣ ⎝ 1 − ϕ ⎠
⎤ 2π 1 − ϕc − 2⎥ ⎥⎦ 3 1−ϕ
d
1−ϕ 1 − ϕc
1 N
N
∑ kr2n n=1
4. RESULTS AND DISCUSSION Figure 2 gives the capillary sizes randomly chosen by the probability model described by eq 38 in the present simulations
(39)
2
(45)
The algorithm for determination of the relative permeability in fractal porous media is summarized as follows: 1. Df and DT are obtained from eqs 4 and 16, respectively. Measure/determine the pressure ΔP, surface tension T, and contact angle θ (in this simulation, we used the parameters reported in Amico et al.,9 i.e. T = 0.044 N·m, θ = 57°, ΔP = 20000.0 Pa). 2. Produce a random number Ri of 0−1 by the Monte Carlo method. 3. Calculate λi by eqs 38−40. 4. If λi > λmax, return to step 2; otherwise, continue to the next step. 5. Calculate the relative permeability (krw and krg) from eqs 32 and 35, respectively. Steps 3−5 are repeated for calculation of relative permeability until a converged value is obtained at a given porosity. Step 4 means that the randomly produced pore size λi in the Monte Carlo simulation is not allowed to exceed the maximum pore size λmax in order to comply with the physical situation.
(36)
where λmin ≤ λ ≤ λmax. For the ith capillary, eq 37 can be written as
+
(44)
where
Equation 36 indicates that R = 0 as λ → λmin and R ≈ 1 as λ → λmax. From eq 36, we can obtain
m=
n=1
⟨k r2⟩ − ⟨k r⟩2
min
λ
N
∑ krn
where N is the total number of runs for a given porosity. The variance is defined by
λ
= 1 − (λmin /λ)
λmax
1 N
⟨k r⟩ =
(40)
where R̅ c = 0.30 mm is the cluster mean radius,18 d+ = 24 is the ratio of the cluster mean size to the minimum particle size,18 and ϕc is the microporosity in the cluster and given by41 ϕc = 0.342ϕ
(41)
The convergence criterion is as follows: AJ = A p /ϕ > A
(42)
where Ap is the total pore area in a unit cell, that is, J
Ap =
∑ πλi2 /4 i=1
Figure 2. Capillary sizes simulated by the Monte Carlo simulations. (43)
for bidispersed porous media at the porosity of 0.25. It is found from Figure 2 that the minimum pore size is about 7.2 μm. Estimation from eqs 24, 39, and 40 gives λmin,w = 7.3 μm at the porosity of 0.25. The error is only 1.4%, which means that Monte Carlo simulations are accurate. From Figure 2, it is also seen that the number of larger capillaries is much less than that of smaller ones; this is consistent with the fractal theory. Figure 3 shows a comparison of the average capillary pressure predicted by the present model (by eq 22) and that from
The simulation was stopped, and the final/convergent relative permeability and the total simulated number (J) in one run for a given porosity were recorded. In eq 42, AJ is the total area calculated after the Jth computation in one run. If a converged relative permeability is obtained in one run, the relative permeability is set as krn (n = 1, 2, 3, ..., N). Then, the averaged relative permeability for a given porosity can be calculated by 6975
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saturation decreases to 0.34 at ϕ = 0.4. So, the effect of the capillary pressure on the relative permeability cannot be ignored at low saturation. Figure 6 shows that the phase fractal
Figure 3. Influence of porosity on the average capillary pressure.
available experiments with noncrimp fabrics.23 It can be seen that there is good agreement between the model predictions and the experimental data.23 Figure 3 also shows that the average capillary pressure increases with the decrease of porosity; this is consistent with the physical situation. Figure 4 shows a comparison on the capillary pressure versus
Figure 6. Effects of porosity on the phase fractal dimensions (Df,w and Df,g).
dimension (Df,w and Df,g) depends on porosity according to eqs 23 and 25. The figure denotes that the higher the porosity, the higher the fractal dimension. This can be explained by the fact that higher porosity means larger pore area, leading to a larger phase area/volume and higher fractal dimension. It is further seen that the fractal dimension for nonwetting phase decreases with the increase of saturation. This may be explained that an increase in saturation results in the area of nonwetting phase decreasing and the decrease of the fractal dimension for nonwetting phase. Figure 7a compares the predictions from the simulations (by eq 32) for monodispersed porous media (similar to packed beds) and the experimental data3,6 (the experimental data are from Figure 5 in ref 3), and good
Figure 4. Average capillary pressure of wetting phase in unsaturated porous media versus saturation for three different porosities.
saturation at three different porosities. Figure 4 shows that the capillary pressure increases with the decrease of saturation and, at low saturation, the capillary pressure increases sharply with the decrease of saturation. Figure 5 compares the capillary pressure from the present model predictions by eq 22 and those from experiments.12 It is seen that there is a fair agreement between the model predictions and the experimental data.12 The capillary pressure can reach 5.34 × 104 Pa when
Figure 5. Comparison between the average capillary pressure for wetting phase in unsaturated porous media by the present model and experimental data.12
Figure 7. Comparison between the present Monte Carlo simulations and experimental data3,6,10 for wetting phase (krw) and nonwetting phase (krg), respectively. 6976
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agreement between the model predictions from the simulations and the experimental data is found. Li and Hone10 discussed the calculated results and the comparison in nitrogen−water systems. Figure 7b shows a comparison of the relative permeability predicted by the present Monte Carlo simulations (by eq 35) and the experimental data,10 and good agreement between them is again observed. Figure 7 shows that the relative permeability of wetting phase (krw) increases as the saturation of the wetting phase increases. On the contrary, from Figure 7 it can be seen that the relative permeability of nonwetting phase (krg) decreases as the saturation of the wetting phase increases. This is understandable as higher saturation of the wetting phase (Sw) means more fluid and less gas. These are all expected and consistent with the practical physical phenomena.
5. CONCLUSIONS The relative permeability of unsaturated porous media is modeled and predicted by using the Monte Carlo simulation technique in this paper. The proposed model is found to be a function of porosity, the area fractal dimension for pores, the fractal dimension of tortuous capillaries, saturation, capillary pressure and microstructural parameters of a medium. It is found that the capillary pressure increases with the decrease of saturation, and at low saturation, the capillary pressure increases sharply with the decrease of saturation. In addition, it is found that the phase fractal dimensions (Df,w and Df,g) depend on porosity. The present model contains no additional or empirical constant, which is normally required in conventional models. The model predictions are compared with the available experimental data, and good agreement between the model predictions and experimental data is found. The validity of the present model is thus verified. Therefore, the proposed probability model can reveal the physical mechanisms of relative permeability in porous media. Besides, the present technique may also have the general interests/potentials in analysis of other transports such as gas diffusivity and thermal conductivity of porous media. So, the proposed technique may provide us with a new approach in addition to the analytical and other numerical methods.
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Kw = permeability for wetting phase in unsaturated porous media Kg = permeability for the nonwetting phase in unsaturated porous media krw = relative permeability for the wetting phase in unsaturated porous media krg = relative permeability for nonwetting phase in unsaturated porous media l0 = straight-line distance lt = actual length ΔP = pressure difference Pc = capillary pressures q(λ) = flow rate through a single tortuous capillary T = surface tension of fluid λ = pore/capillary diameter λav = average pore/capillary diameter λmax = maximum diameter of pore λmin = minimum diameter of pore Sw = saturation of the wetting phase τ = average tortuosity of tortuous capillaries μ = viscosity of the fluid ϕ = porosity θ = contact angle
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors would like to acknowledge the funding support of Research Grant Council of Hong Kong SAR through a GRF project (PolyU 5158/10E) and the National Natural Science Foundation of China (Grant No. 11102100).
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NOMENCLATURE A = cross-section area D = Euclidean dimension Df = area fractal dimension of pore Df,w = fractal dimension for the wetting phase Df,g = fractal dimension for nonwetting phase DT = tortuosity fractal dimension of tortuous capillaries K = absolute permeability in saturated porous media 6977
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