Fractal Analysis of Permeabilities for Porous Media

Transcription

1 Fractal Analysis o Permeabilities or Porous Media Boming Yu Dept. o Physics and The State Key Laboratory o Plastic Forming and Die & Mold Technology, Huazhong University o Science and Technology, Wuhan, , P. R. China Wei Liu Dept. o Power Engineering, Huazhong University o Science and Technology, Wuhan, , P. R. China A ractal analysis o permeabilities or porous media, both saturated and unsaturated, is presented based on the ractal nature o pores in the media. Both the ractal-phase permeability and the ractal relative permeability are derived and ound to be a unction o the tortuosity ractal dimension, pore-area ractal dimension, phase ractal dimension, saturation, and microstructural parameters. The proposed models or permeabilities both the phase permeability and the relative permeability do not contain any empirical constant. The validity o the present analysis is veriied by a comparison with the existing measurements, and excellent agreement between the model predictions and experimental data is ound. In addition, the present work reveals that the relative permeability depends not only on saturation but also on the two ractal dimensions, pore ractal dimension (at porosity greater than 0.90), and tortuosity ractal dimension, which characterize the ractal characters o capillaries in porous media. The two ractal dimensions may be the two o the important mechanisms aecting the relative permeability in porous media, and this is a supplement to the available conclusion on relative permeability American Institute o Chemical Engineers AIChE J, 50: 46 57, 2004 Keywords: ractal, permeability, porous media, multiphase low Introduction The permeabilities or porous media, both saturated and unsaturated, have received much attention (De Wiest, 1969; Bear, 1972; Bowles, 1984; Jumikis, 1984; Kaviany, 1995; Panilov, 2000) due to practical applications, including chemical engineering, soil science and engineering, oil production, polymer composite molding, and heat pipes. Since the microstructures o porous media are usually disordered and extremely complicated, this makes it very diicult to analytically ind the permeability o the media, especially or unsaturated (or multiphase) porous media. Correspondence concerning this article should be addressed to B. Yu at yu3838@ public.wh.hb.cn American Institute o Chemical Engineers Conventionally, the permeabilities o porous media were ound by experiments (Levec et al., 1986; Sasaki et al., 1987; Wang et al., 1994; Wu et al., 1994; Shih and Lee, 1998; Chen et al., 2000). Besides, much eort was also devoted to numerical simulations o permeabilities or porous media. Simacek and Advani (1996) perormed the numerical solution by reducing a two-dimensional problem to a one-dimensional equation. Adler and Thovert (1998) applied a ourth-order inite dierence scheme or permeabilities o real Fontainebleau sandstone. Although no adjustable parameter was involved, a large discrepancy between the average numerical permeability (plotted against porosity) and the experimental data was observed. Ngo and Tamma (2001) applied the inite-element method to calculate the permeability or the porous iber mat by assuming the Stokes low in the intertow region and the Brinkman s low inside the tow region. Compared with single-phase (or satu- 46 January 2004 Vol. 50, No. 1 AIChE Journal

2 rated) low in porous media, the multiphase (or unsaturated) immiscible lows in porous media are not well understood. The multiphase immiscible lows in porous media are very important in practical applications such as the petroleum industry, chemical engineering, and soil engineering. The lattice Boltzmann method (LBM) (Benzi et al., 1992; Sahimi and Mukhopadhyay, 1996; Martys and Chen, 1996; Chen and Doolen, 1998), based on the Navier-Stokes equation coupled with Darcy s law, has been extensively used to simulate multiphase lows through porous media in order to understand the undamental physics associated with enhanced oil recovery, including relative permeabilities. The LBM is particularly useul or complex geometrical boundary conditions and varying physical parameters. However, the results either rom numerical simulations or rom experiments are usually expressed as correlations with one or more empirical constants, or as curves, and the mechanisms behind the phenomena are thus oten ignored. In order to get a better understanding o the mechanisms or permeability, the analytical solution or permeability o porous media becomes a challenging task. Recently, Yu and Lee (2000) developed a simpliied analytical model or evaluating the permeabilities o porous abrics used in liquid composite molding. This permeability model, which is related to porosity and architectural structures o porous abrics, is based on the one-dimensional (1-D) Stokes low in macropores between iber tows and on the 1-D Brinkman low in micropores inside iber tows. Good agreement between theoretical predictions and experimental results was ound. However, this model may only apply to those media whose macropores can be simpliied as one-dimensional channels. So this and several other models may not be applicable to random/disordered porous media. In addition, this model is only suitable or saturated porous media. Ransoho and Radke (1988) applied the circular and triangle capillary models to numerically simulate the low resistance or laminar low through unsaturated porous media and studied the dependence o low resistance on corner geometry, surace shear viscosity, and contact angle. Katz and Thompson (1985) presented experimental evidence indicating that the pore spaces o a set o porous sandstone samples (in nature) are ractals and are sel-similar over 3 to 4 orders o magnitude in length extending rom 10 Å to 100 m. They argued that the pore volume is a ractal with the same ractal dimension as the pore rock interace. This conclusion was supported by correctly predicting the porosity rom the ractal dimension, which was measured by a log-log plot o a number o pores vs. the pore size, and a ractal correlation, C(l 1 /l 2 ) 3D, is presented to correlate the measurements on a variety o porous sandstone samples (pores). In the correlation, is the porosity o porous sandstone, D ( 2 3 in three dimensions) is the ractal dimension o pores, C is a constant o order one, and l 1 and l 2 are the lower and upper limits, respectively, o the sel-similar regions. Krohn and Thompson (1986) also carried out the experiments on sandstone, and the ractal dimensions o the ive sandstone pores were ound to be in the range o in three dimensions. Figure 1 (Krohn and Thompson, 1986) displays one o the ractal scaling laws o ive sandstone pores. In the igure, the negative slope o the solid line gives the ractal dimension D 2.75 obtained by itting the measurements rom the automatic technique. This igure shows that the sandstone pores are ractal objects in Figure 1. Fractal scaling law rom measurement o Coconino sandstone pores by the automatic technique (Krohn and Thompson, 1986). nature. Readers may also consult the paper by Krohn and Thompson (1986) or more evidence that the porous media are ractals in nature. Smidt and Monro (1998) perormed experimental investigations on the images o laboratory-made synthetic sandstone and on modeled sandstone. Their results showed that the pore space o both the synthetic and the modeled sandstone was ound to be ractals and the ractal scaling laws were obtained by the box-counting method (see Figure 2). Figure 2 presents the ractal scaling law or the laboratory synthetic stone pores, and the slope o the log(n d ) log(1/d) plot shows ractal dimension 1.89 (in two dimensions) obtained by the box-counting method (count the number, N d, o boxes o side length d or covering the pore space). This suggests that the laboratory synthetic stone pores are also ractal objects. According to the ractal character o real porous media, Yu and Cheng (2002) developed a ractal permeability model or bidispersed saturated porous media (see Figure 3), and this ractal model is also applicable to porous abrics (Yu et al., 2002) (see Figure 4; Yu et al., 2001). It is seen that the random porous abric is also a ractal medium. For more ractal microstructures o abrics, readers may consult the paper by Yu et al. (2001). Although this model does not contain any empirical constant, and good agreement is ound between the model predictions and experimental data, it does not apply to unsaturated porous media. The saturated porous medium is, in act, only the special case o the unsaturated porous medium. It is thereore more meaningul or practical applications to develop an analytical solution or the permeability o unsaturated (or multiphase) porous media. Once the credible permeability is obtained, it also can be used to analyze the heat and mass transer in unsaturated porous media such as soil (Liu et al., 1995, 1998). In this article, we ocus our attention on the derivation o an analytical ractal model or both the phase and relative permeability o two-phase porous media based on the available evidence that porous media in nature are ractal objects (Katz and Thompson, 1985; Krohn and Thompson, 1986; Young and Craword, 1991; Perect and Kay, 1991; Smidt and Monro, 1998; Yu and Li, 2001; Yu et al., 2001, 2002; Yu and Cheng, 2002). This article is organized as ollows: the ollowing section describes the ractal characteristics o microstructures o AIChE Journal January 2004 Vol. 50, No. 1 47

3 over the considered medium, and the signiicant inluence o microstructures on low is thus ignored. Fortunately, the ractal nature o pores and pore luids may provide us with a better understanding o the mechanisms o low and transport properties, such as the permeability in porous media. It is known that the cumulative size-distribution o islands on the earth s surace ollows the power law N( A a) a D/2 (Mandelbrot, 1982), where N is the total number o islands o area ( A) greater than a, and D is the ractal dimension o the surace. Mandelbrot (1982) also pointed out that the sponge gap s (i.e., pores) size distribution is analogous to islands and clusters and satisies the cumulative distribution unction, N(L ) D. Majumdar and Bhushan (1990) used this Figure 2. (a) Image o the laboratory-made synthetic sandstone (the white are stones and the black are pores), and (b) the ractal scaling law obtained by the box-counting method applied to the image (Smidt and Monro, 1998). porous media, which are the theoretical bases or the present ractal analysis o permeability or porous media. The complete ractal permeability models or both saturated and unsaturated porous media are given in the third section. The results and discussions are arranged in the ourth section, and then come the concluding remarks. The Fractal Description o Microstructures o Porous Media Porous media such as soil, sandstones in an oil reservoir, packed beds in chemical engineering, abrics used in liquid composite molding, and wicks in heat pipes consist o numerous irregular pores o dierent sizes spanning several orders o magnitude in length scales. The pore in porous media plays a remarkable role in luid low and heat transer in porous media. The conventional method or description o characteristics o porous media is based on the volume average (Kaviany, 1995) Figure 3. (a) Image photo (Yu and Cheng, 2002) o a bidispersed medium at porosity 0.52, where the black and the white regions are pores and clusters ormed by agglomeration o copper particles; since the micropores inside clusters are very small and the copper particles are sot, it is diicult to see the micropores inside clusters ater the sample has been polished, and (b) the ractal scaling law obtained by the box-counting method applied to (a) or box number N(L ) vs. box size covering the space. 48 January 2004 Vol. 50, No. 1 AIChE Journal

4 porous medium samples with signiicantly dierent microstructures (Yu et al., 2001; Yu and Li, 2001; Yu and Cheng, 2002; Yu et al., 2002). For more about the nature o ractals o porous media, readers may also consult the reerences by Katz and Thompson (1985), Krohn and Thompson (1986), Young and Craword (1991), Perect and Kay (1991), Smidt and Monro (1998). Dierentiating Eq. 2 with respect to results in the number o pores whose sizes are within the ininitesimal range to d D dn D max D1 d (3) where d 0. The negative sign in Eq. 3 implies that the island or pore number decreases with the increase in island or pore size, and dn 0. The number o pores rom Eq. 2 becomes ininite as 3 0, which is one o the properties o ractal objects (Mandelbrot, 1982). Equation 2 describes the scaling relationship o the cumulative pore population. The total number o pores or islands or spots, rom the smallest diameter min to the largest diameter max, can be obtained rom Eq. 2 as (Yu and Li, 2001; Yu and Cheng, 2002) Dividing Eq. 3 by Eq. 4 gives N t L min max mind (4) Figure 4. (a) Image photo o the U750 random porous abric ater magniication o 100, and (b) the ractal scaling law obtained by the box-counting method applied to (a) or box (pore) number N(L ) vs. box (pore) size covering the space (Yu et al., 2001). power law to describe the contact spots on engineering suraces, and the power-law relation is N A a a D max a /2 2 where a max g max, a g 2, with being the diameter o a spot and g being a geometry actor. Compared with the islands on earth or spots on engineering suraces, the pores in porous media are also analogous to the islands on earth and to the spots on engineering suraces. The cumulative size-distributions o pores whose sizes are greater than or equal to have been proven to ollow the ractal scaling law (1) NL max D (2) where D is the pore-area ractal dimension, 1 D 2, and max is the maximum pore size. The ractal power-law behavior (Eq. 2) has been proven to be correct by correlating the data rom the box-counting method applied to a number o real dn D D N min D1 d d (5) t D where () D min (D1) is the probability density unction, which satisies the ollowing condition 0 (6) Patterned ater probability theory, the probability density unction, (), should satisy the ollowing relationship d 0 min max d 1 min maxd 1 (7) It is clear that Eq. 7 holds i and only i (Yu and Li, 2001) min maxd 0 (8) is satisied. Equation 8 implies that min max must be satisied or ractal analysis o a porous medium; otherwise, the porous medium is a nonractal medium. For example, i min max, both Eqs. 7 and 8 do not hold. Equation 8 can be considered to be a criterion o whether a porous medium can be characterized by ractal theory and technique (Yu and Li, 2001). This means that i Eq. 8 does not hold, the porous medium is a nonractal medium, and the ractal theory and technique are not applicable to the medium. However, in general, min / max 10 2 or 10 2 in porous media, and Eq. 8 AIChE Journal January 2004 Vol. 50, No. 1 49

5 holds approximately. Thus, the ractal theory and technique can be used to analyze the characteristics o porous media. For saturated porous media, the pore area ractal dimension D is given by (Yu and Li, 2001) D d ln ln min max, (9) where is the eective porosity o porous media, d 2 in two dimensions and d 3 in three dimensions. Equation 9 exactly holds or exactly sel-similar ractal geometries, such as Sierpinski carpet and Sierpinski gasket ( max and min are the upper and lower limits o sel-similarity, respectively). However, Eq. 9 approximately holds or random or disordered porous media. For porous media, max and min are the maximum and minimum pore diameters, respectively, in a unit cell or in a sample, implying that the statistical sel-similarity exists in the range o max min in a porous medium. Equation 9 shows that the pore area ractal dimension is a unction o porosity and microstructures, min and max. A porous medium with various pore sizes can be considered as a bundle o tortuous capillary tubes with variable crosssectional areas. Let the diameter o a capillary in the medium be and its tortuous length along the low direction be L t (). Due to the tortuous nature o the capillary, L t () L 0, with L 0 being the representative length. For a straight capillary, L t () L 0. Wheatcrat and Tyler (1988) developed a ractal scaling/tortuosity relationship or low through heterogeneous media, and the scaling relationship is given by L t () 1D T D L T 0 where is the length scale o measurement. We argue that the diameters o capillaries are analogous to the length scales,, which means that the smaller the diameter o a capillary, the longer the capillary. Thereore, the relationship between the diameter and length o capillaries also exhibits a similar ractal scaling law L t 1DT L 0 D T (10) where D T is the tortuosity ractal dimension, with 1 D T 2 in two dimensions, representing the extent o the convolutedness o capillary pathways or luid low through a medium. Note that D T 1 represents a straight capillary path, and a higher value o D T corresponds to a highly tortuous capillary. In the limiting case o D T 2, we have a highly tortuous line that ills a plane (Wheatcrat and Tyler, 1988). Equation 10 diverges as 3 0, which is one o the properties o ractal streamlines (Wheatcrat and Tyler, 1988). Equations 2 3 and 8 10, which provide a complete description o the ractal characteristics o porous media, orm the basis o the present ractal analysis o permeabilities, and this will be derived in the ollowing sections. rates through all the individual capillaries. The low rate through a single tortuous capillary is given by modiying the well-known Hagen-Poiseulle equation (Denn, 1980) to give q G P 4 L t (11) where G /128 is the geometry actor or low through a circular capillary, is the hydraulic diameter o a single capillary tube, is the viscosity o the luid, P is the pressure gradient, and L t is the length o the tortuous capillary tube. The total low rate Q can be obtained by integrating the individual low rate, q(), over the entire range o pore sizes rom the minimum pore min to the maximum pore max in a unit cell. According to Eqs. 3, 10, and 11, we have 1D T max Q qdn G P A L 0 L 0 A min 1 D min max T D 3D T 3 D T D max 3D T2D min max (12) where D is the pore-area ractal dimension, and 1 D 2 in two dimensions. Since 1 D T 2 and 1 D 2, the exponent 3 D T 2D 0 and 0 ( min / max ) 3D T2D 1. Also, according to the Yu and Li s criterion (Yu and Li, 2001), ( min / max ) D 0 (because ( min / max ) 10 2 ). It ollows that Eq. 12 can be reduced to 1D T max Q qdn G P A L 0 L 0 A min D 3D T 3 D T D max (13) Using Darcy s law, we obtain the expression or the permeability o a porous medium as ollows K L 0Q PA G L 1D T 0 D 3D T A 3 D T D max (14) which indicates that the permeability is a unction o the pore-area ractal dimension D, tortuosity ractal dimension D T and structural parameters, A, L 0 and max. Equations 13 and 14 indicate that the total low rate and total permeability are very sensitive to the macropore max, and the total low rate and total permeability are mainly determined by the maximum/macro pore max. This is consistent with the practical situation. For straight capillaries, D T 1, Eqs. 13 and 14 can be reduced to Fractal Permeabilities or Porous Media Fractal permeability or saturated porous media Consider a unit cell consisting o a bundle o tortuous capillary tubes with variable cross-sectional area. The total volumetric low rate Q through the unit cell is a sum o the low Q G P K G 1 A A 1 L 0 A D 4 3 D max D 4 4 D max (15) (16) 50 January 2004 Vol. 50, No. 1 AIChE Journal

6 respectively. Equations present the single-phase low rates and single-phase permeabilities (also called absolute permeabilities). Equations indicate that the low rate and permeability are very sensitive to the maximum pore size max. It is also shown that the higher the ractal dimension D, the larger the low rate and the permeability value. From Eqs , it can be seen that the low rate and the permeability will reach the maximum possible values as the pore-area ractal dimension approaches its maximum possible value o 2. This is consistent with ractal theory. Equations are valid not only or isotropic porous media but also or anisotropic porous media. For anisotropic porous media, we only need to calculate the principal permeabilities (Yu et al., 2002) in the principal directions by separately using Eqs. 14 or 16, then inding the other permeability components. Fractal permeability or unsaturated porous media We now extend the preceding analysis to the permeability or unsaturated porous media. The unique dierence between the saturated and unsaturated porous media is that or saturated porous media there is only a single luid such as water illed with pores or capillary pathways. In an unsaturated porous media there are at least two dierent luids, such as water and gas. This means that pores are partially illed with water and gas. Figures 5a and 5b display those typical pores, and Figure 5c shows a simpliied model or the cross section o a capillary tube partially illed with water and gas. From Figure 5c, we can obtain the pore volume, V p, and the volume, V w, occupied by water or wetting phase as V p 2 /4 (17) and V w V p V g 2 /4 g 2 /4 (18) respectively, where and g are the diameter o a capillary pathway and the diameter o the nonwetting (such as gas) phase pathway, and V g is the volume occupied by a nonwetting luid (such as gas). According to the deinition or saturation, S w,we have and S w V w 1 V p 2 g S g V g V p 2 g (19) (20) Obviously, Eqs. 19 and 20 satisy S w S g 1, which is expected. From Eqs. 19 and 20 the diameter or nonwetting luid (such as gas) can be expressed as Figure 5. (a) Spontaneous spreading o oil blobs in a capillary (Chatzis et al., 1988), (b) possible luid saturation state in sandstone (Bear, 1972), and (c) a simpliied model or the cross section o a capillary tube partially illed with water and gas. Equation 21 denotes that g 0asS w 1 and g as S w 0, and vice versa. This is expected and is consistent with the physical situation. The volume, V w, occupied by the wetting luid (such as water) can be written as g S g 1 S w (21) V w 2 /4 g 2 /4 w 2 /4 (22) AIChE Journal January 2004 Vol. 50, No. 1 51

7 where w is the eective diameter o wetting luid occupying the cross section o a capillary pathway. Again, according to the deinition o saturation This results in S w V w /V p w 2 / 2 (23) w S w (24) Equation 24 indicates that w 0, as S w 0, and w, as S w 1, and vice versa. This is again expected and is consistent with the physical situation. Usually, 0 S w 1, such as saturation in soil, in which pores are partially illed with luid such as water, that is, the two phases (such as water and gas) coexist in soil. From Eqs. 21 and 24, we can directly write the eective maximum and the smallest diameters or wetting and nonwetting luids in the largest and smallest pores (or capillary tubes) as max,w max S w (25) min,w min S w (26) max,g max 1 S w (27) min,g min 1 S w (28) The volume ractions, w and g, or the wetting phase and the nonwetting phase luids in a unit cell are given by (Bear, 1972) w S w (29) g 1 S w (30) respectively, and clearly w g. The permeations o both wetting and nonwetting luids play important roles in unsaturated (or multiphase) porous media. Muskat and Meres (1936) recommended that the phase permeabilities K w and K g be treated as isotropic and given by or K w Kk rw (31) K g Kk rg (32) k rw K w /K (33) k rg K g /K (34) where K is the absolute permeability (given by Eq. 14) used in the single-phase lows, and k rw and k rg are the relative permeabilities o the w and g phases, respectively. In this work, the single-phase ractal permeability Eq. 14 is extended to the phase ractal permeabilities, K w and K g (k rw Figure 6. (a) One o possible tortuous streamlines passing through the bidispersed porous medium, and (b) tortuosity ractal dimension, D T 1.12, measured or (a) by the box-counting method at porosity o 0.52 (Yu and Cheng, 2002). and k rg ), under the ollowing assumptions similar to those by Kaviany (1995): (1) The Darcy (Stokes) low is applicable with a negligible interacial drag in two-phase porous media. (2) The body orce is neglected. (3) The liquid low is not coupled with gas low. (4) The viscosities o the liquid and gas phases are independent o each other. The tortuosity ractal dimension, D T, is usually determined by the box-counting method or Monte Carlo method, and an analytical expression or D T has not yet been developed. In this work, we also assume that the wetting and the nonwetting phases low through tortuous paths have approximately the same tortuosity as the single-phase low, that is, D T D T,w D T,g 1.10 (Yu and Cheng, 2002) measured using the box-counting method (see Figure 6). Figure 6a depicts one o the possible tortuous streamlines passing through the bidispersed porous medium, and Figure 6b is the ractal scaling law measured by the box-counting method applied to the tortuous streamline. Using the Monte Carlo simulation, Wheatcrat and Tyler (1988) obtained the averaged tortuous streamline ractal dimension D T or low through heterogeneous media (see Figure 7). Figure 7a demonstrates the simulation results or tortuous streamlines by a ractal random-walk model, and Figure 7b is a plot o ractal travel distance, L F, vs. the scale o observation, L s, or the ractal random-walk model. Their Monte Carlo simulation result, D T 1.081, is very close to the averaged result, D T 1.10, obtained by the box-counting method (Yu and Cheng, 2002). This work uses D T 1.10, in calculating permeability. Applying the analogy between the Darcean single-phase and 52 January 2004 Vol. 50, No. 1 AIChE Journal

8 K g G L 1D T 0 D,g 3D T A 3 D T D max,g,g G L 1D T 0 D,g A 3 D T D max 1 S w 3DT (36),g respectively. Equations 35 and 36 reveal that the phase permeability or wetting and nonwetting luids in porous media is a unction o saturation (S w ), ractal dimensions (D T, D,w,or D,g ), and microstructure parameters ( max, A, and L 0 ). It can be seen that the present ractal phase permeabilities do not contain any empirical constant. In Eqs. 35 and 36, the ractal dimensions D,w and D,g can be obtained by extending Eq. 9 and inserting Eqs as D,w d D,g d ln w ln min,w max,w ln g ln min,g max,g d lns w ln min max d ln1 S w ln min max (37) (38) Compared with Eq. 37, it is seen that Eq. 9 is only a speciic case o S w 1, and Eq. 37 is a more general orm or area ractal dimension in porous media, including the saturated and unsaturated porous media. We can now turn our attention again on the relative permeabilities or the wetting and nonwetting phases. Noting Eqs. 14, 25 28, 33 36, we arrive at k rw K w K 3 D T D D,w 3D S T/2 3 D T D,w D w (39) Figure 7. (a) Fractal random-walk model to simulate the low through heterogeneous medium, and (b) ractal travel distance, L F, vs. scale o observation L s or the ractal random-walk model with the averaged tortuosity ractal dimension, D T (Wheatcrat and Tyler, 1988). two-phase lows, and modiying the single-phase ractal permeability model, Eq. 14, by replacing max in Eq. 14 with Eqs , and ractal dimension D (Eq. 9) with D,w and D,g,we can obtain the phase ractal permeabilities or the wetting and nonwetting luids K w G L 1D T 0 D,w 3D T A 3 D T D max,w,w G L 1D T 0 D,w A 3 D T D max S w 3DT (35),w k rg K g K 3 D T D 3 D T D,g D,g D 1 S w 3DT/2 (40) It is evident that the relative permeability is a unction o saturation S w and ractal dimensions D, D T, and D,w (or D,g ), and there is no empirical constant in this ractal relative permeability model. Results and Discussion Phase ractal dimensions The ractal theory requires that the values o ractal dimensions D,w and D,g be in the range o 1 and 2 in two dimensions based on the deinition given by Eq. 2. To be valid, we irst check/calculate the phase ractal dimensions, D,w and D,g. For this purpose, we take the bidispersed porous media (Yu and Cheng, 2002) as samples or study, because they have been proven to be ractal media. The bidispersed porous structure, as shown in Figure 3a, is composed o clusters (at the macro level), which are agglomerated by small particles (at the micro level). Figure 3b demonstrates the cumulative distribution or pore sizes. The pore-area ractal dimension D can be determined by the value o the slope o a linear it through data on AIChE Journal January 2004 Vol. 50, No. 1 53

9 a log-log plot o the cumulative number o box numbers (pores), N(L ), vs. the box (pore) size,. Since the clusters and particles within the clusters are randomly distributed, this leads to macropores and micropores o various sizes in a bidispersed porous medium. The bidispersed porous media are widely used as wicks in the evaporators o heat pipes. For saturated (or single-phase) bidispersed porous media, the pore-area ractal dimension, D, also can be given by Eq. 9 with (Yu et al., 2001; Yu and Cheng, 2002) min 2 max d 1 (41) 1 i where i is the porosity inside the clusters, d is the ratio o average cluster size to the minimum particle size, and d 24 (Yu and Cheng, 2002). I we set i 0, the saturated bidispersed porous media become the saturated monodispersed porous media (such as packed beds), and Eq. 41 is reduced to min 2 max d 1 (42) Substituting Eq. 42 into Eq. 9 yields D or monodispersed porous media, and inserting Eq. 42 into Eqs. 37 and 38 yields the phase ractal dimensions D,w and D,g or unsaturated monodispersed porous media. It should be noted that the monodispersed porous media reer to the materials that consist o particles and pores with dierent sizes, but these particles do not agglomerate to orm clusters. On the other hand, or bidispersed porous media, these particles agglomerate to orm clusters with dierent sizes. The pores in the monodispersed porous media are nonuniorm and still ollow the ractal scaling law. The algorithm or determination o the phase ractal dimensions or monodispersed porous media is summarized as ollows: (1) Select a porosity, ; (2) Find min / max rom Eq. 42; (3) Select a saturation, S w, ind D,w and D,g (let d 2) rom Eqs. 37 and 38. Procedure 3 is repeated to ind the phase ractal dimensions, D,w and D,g, or a given porosity. Figure 8 gives the phase ractal dimensions, D,w and D,g, vs. the saturation, S w, at dierent porosities. It is seen rom Figure 8 that the phase ractal dimension D,w increases monotonously with saturation, and as saturation tends to 1, the ractal dimension D,w reaches its maximum possible value o about 1.80 at porosity 0.54, approximately the same value as that (1.81) or the bidispersed medium (Yu and Cheng, 2002) at porosity At present, the explanation or this is that or the monodispersed porous media, Eqs. 37 and 42 are applied to calculate the phase ractal dimension, while or the bidispersed porous media, Eqs. 9 and 41 were used to ind the pore area ractal dimension. A similar phenomenon can be observed or the nonwetting luid. The phase ractal dimension, D,g, reaches its maximum possible value, 1.80, as saturation is zero at porosity This means that as saturation tends to zero, the medium is ully illed with a nonwetting luid (or single-phase luid), so it is expected that the ractal dimension is exactly the Figure 8. Phase ractal dimensions vs. saturation. same as that or the saturated porous medium. Figure 8 also shows that the phase ractal dimensions depend on porosity. The higher the porosity, the higher the ractal dimension. One interpretation or this might be that the higher porosity implies larger pore area, and the larger pore area leads to the higher ractal dimension. In the limiting case, as porosity tends to 1, a unit cell o the medium becomes a smooth plane, whose ractal dimension is 2. Thereore, the present results are reasonable. An important phenomenon can be also ound rom Figure 8. That is when saturation S w 0.1, the phase ractal dimension D,w 1. This reveals that when saturation S w 0.1, the wetting phase distribution in porous media is nonractal (in two dimensions), according to ractal theory. Similarly, when saturation S w 0.9, D,g 1. This means that the nonwetting phase distribution is also nonractal (in two dimensions) when S w 0.9. This suggests that only when S w 0.1 and S w 0.9, the wetting and nonwetting phases are ractal objects, respectively. On the other hand, according to an article by Kaviany (1995), at very low saturations the wetting phase becomes disconnected (or immobile). At very high saturations, the nonwetting phase becomes disconnected. This means that at low saturation or the wetting phase and at high saturation or the nonwetting phase, the pore luid is embedded in one dimension and the ractal dimension is less than one, as the luid is disconnected. Usually, the experimentally relative permeability data (De Wiest, 1969; Bear, 1972; Kaviany, 1995) were also reported in the range o about S w 0.1 or the wetting phase. Thus, the present ractal analysis o permeability is restricted in the ranges o S w 0.1 or the wetting phase and S w 0.9 or the nonwetting phase by the requirements rom both ractal theory and experimental observations. Fractal relative permeabilities According to the preceding analysis, the present ractal relative permeabilities are given in the ranges o S w 0.1 or the wetting phase and S w 0.9 or the nonwetting phase. The algorithm or determining the relative permeabilities or monodispersed porous media is summarized as ollows: (1) Select a porosity, ; (2) Find D rom Eq. 9 with min / max rom Eq. 42; (3) Select a saturation, S w, ind D,w and D,g rom Eqs. 37 and 38; 54 January 2004 Vol. 50, No. 1 AIChE Journal

10 Figure 10. Comparison o the relative permeabilities between the present ractal model predictions and experimental data (see Table 1 or experimental descriptions). Figure 9. Relative permeabilities predicted by the present ractal model at D T 1.10: (a) the wetting phase, and (b) the nonwetting phase. (4) Find the relative permeabilities rom Eqs. 39 and 40. Procedures 3 and 4 are repeated to ind the relative permeabilities or a given porosity. In the preceding calculations, we have assumed the tortuous ractal dimension D T D T,w D T,g [ 1.10 rom the box-counting method (Yu and Cheng, 2002)]. We have ound that this computation o relative permeabilities takes less than one second in a microcomputer, and no grid generation and no boundary conditions are needed. While applying any numerical method such as the inite dierence method, inite-element method, lattice-boltzmann method, and Monte Carlo simulation, grid generation and/or boundary conditions are needed, and thus much more computer time is oten required. Thereore, the advantage o the present ractal analysis o permeabilities or porous media over numerical methods is evident. Figure 9 presents the relative permeabilities, k rw and k rg, vs. saturation at dierent porosities/pore-area ractal dimensions and at given tortuosity ractal dimension D T 1.10, calculated rom Eqs. 39 and 40, respectively. From Figure 9 it can be ound that k rw k rg 1, which agrees with the general observations, and the shapes o the relative permeability curves are also in agreement with reports in the literature (De Wiest, 1969; Bear, 1972; Kaviany, 1995). Figure 9 indicates that, although the phase ractal dimensions (D,w and D,g ) depend on porosity (see Figure 8), the relative permeabilities predicted by Eqs. 39 and 40 are independent o the porosity/pore-area ractal dimension, D (through Eq. 9), in the porosity range o , the usual porosity range, and they depend only on saturation, although the phase permeabilities are dependent upon phase ractal dimensions D,w and D,g (see Eqs. 35 and 36). This is consistent with the available conclusions: the relative permeability depends only on the saturation and available experimental evidence indicates that this ormal extension and concept o relative permeability that depends only on saturation is a good approximation or all practical purpose (Bear, 1972). The present ractal relative permeability results are also consistent with the existing correlations (Kaviany, 1995), which are expressed as a unction o saturation only, with one or more empirical constants. Figure 10 compares the predictions rom the present ractal permeability model or monodispersed porous media (corresponding to packed beds) to the experimental data (Charpentier and Favier, 1975; Specchia and Baldi, 1977; Saez and Carbonnel, 1985; Levec et al., 1986) or packed beds. See Table 1 or experimental descriptions. It is ound that excellent agreement between the predictions rom the ractal relative permeability model and the experimental data is obtained at D 1.80 (that is, 0.54; note that Eq. 9 relates porosity to pore-area ractal dimension) and D T This veriies the validity o the present ractal analysis o permeabilities or these porous media. However, it should be noted that we choose D 1.80 (that is, 0.54) as a reerence or comparison because Figure 9 has shown that the relative permeability is indepen- Table 1. Experimental Descriptions Exp. Reerences Systems Packing 1 Levec et al. (1986) Air Water Spheres 2 Charpentier and Favier (1975) Air Water Spheres 3 Favier (1975) Air Cyclohexane Cylinders 4 Specchia and Baldi (1977) Air Water Cylinders 5 Saez and Carbonnell (1985) Air Water Raschig AIChE Journal January 2004 Vol. 50, No. 1 55

11 dent o porosity/ractal dimension D in the usual porosity range o Figure 10 urther indicates that when ractal dimension D is less than 1.97 (i.e., porosity 0.90 calculated by Eq. 9), the relative permeability is independent o porosity/ractal dimension D at a given D T This is consistent with the available conclusion, because the available conclusion states that relative permeability depends only on saturation (Bear, 1972). However, at higher porosity, D 1.97 (that is, 0.90), it is ound that the relative permeability is a weak unction o porosity at a given tortuosity D T In the limit o D 2 (that is, 1) and D T 1.10, we have the maximum possible relative permeability, 9% higher than that in the usual porosity range. An explanation or this is that when porosity increases up to 1 (i.e., D 2), a unit cell consists o only one capillary tube with no any impermeable substance inside the unit cell (Yu, 2001; Yu et al., 2002; Yu and Cheng, 2002). So, in this situation, the low resistance reaches the minimum value and the relative permeability arrives at the maximum value. The present result reveals that at higher porosity, 0.90, the relative permeability weakly depends on porosity/ractal dimension D, and increases with porosity/ractal dimension D. This dependence o relative permeability on porosity/ractal dimension D at higher porosity, however, was not reported by experiments and numerical simulations (such as the lattice Boltzmann method), and this article reports such dependence or the irst time. Figure 10 also compares the present model predictions to the experimental data in the limit cases o tortuosity ractal dimensions D T 2 and D T 3. It is known that the streamlines/capillary tubes/ low ields in porous media exhibit multidimensionality and are similar to turbulence and are oten approximately described by turbulence theory (Bear, 1972; Kaviany, 1995). While turbulence can be characterized by ractal theory (Falconer, 1985; Feder, 1988; Turcotte, 1988; Screenivasan, 1991). Thus, the tortuosity ractal dimension, D T, deined by Eq. 10 may be in the range o 1 2 in a two-dimensional section or in the range o 2 3 in a three-dimensional space. In the limit case o D T 2, the streamlines/capillary tubes are so tortuous that they ill a plane, while in the limit case o D T 3, the streamlines/ capillary tubes are so tortuous that they ill a three-dimensional space. These limit cases lead to the highest resistance or low, and thus to the lowest permeabilities. The results show (see Figure 10), that the relative permeabilities are lower (19% and 37%) than the experimental data and the model predictions at D 1.80 and D T 1.10 when the tortuosity ractal dimension D T increases up to 2 and 3, respectively. This shows that the higher the tortuosity ractal dimension, D T, the lower the relative permeability. This can be interpreted as the higher tortuosity ractal dimension implies the higher tortuosity o streamlines/capillary tubes, causing the higher low resistance, and thus resulting in the lower relative permeability. This trend also can be seen rom Eq. 39, because relative permeability is proportional to S w (3D T )/ 2, and note that saturation 0 S w 1. The present work thus also quantitatively explains why 3-dimensional permeability is much lower than 2-dimensional permeability (Adler and Thovert, 1993). It is seen in Figures 9 and 10 that although the present relative permeability model is expressed as a unction o the tortuosity ractal dimension, D T, pore-area ractal dimension, D, phase ractal dimensions, D,w and D,g, and saturation, S w, it turns out that the model predictions depend only on saturation S w and on tortuosity ractal dimension D T in the porosity range o 0.90 (i.e., D 1.97). At higher porosity 0.90 (that is, D 1.97), the relative permeability increases with porosity/pore area ractal dimension D. It is also known that the existing relative permeability correlations contain one or more empirical constants with no physical meaning, and the mechanisms behind these constants were ignored. The present work demonstrates that the tortuosity ractal dimension D T o streamlines/capillaries and pore-area ractal dimension D or characterization o the ractal natures o porous media may be two o the important mechanisms aecting the relative permeability in porous media. Thereore, this work is an addition to the conclusion by Bear (1972) and existing correlations (Kaviany, 1995), which indicate that relative permeability depends only on saturation. The relative permeability, o course, may be aected by many other actors, such as viscosity ratio, surace tension, wettability, density ratio, capillary pressure, hysteresis, and contact angle. This article presents a simple model. A more sophisticated model, which is expected to include one or more other actors, might be studied in the uture. Concluding Remarks A complete ractal analysis o permeabilities or porous media, both saturated and unsaturated, is presented in this article. The phase ractal permeability models, given by Eqs. 35 and 36, are in terms o the tortuosity ractal dimension D T, pore-area ractal dimension, D, phase ractal dimensions, D,w and D,g, saturation, S w, and the structural parameters, A, max, L 0. The ractal relative permeability models given by Eqs. 39 and 40 are expressed as a unction o tortuosity ractal dimension D T, pore-area ractal dimension D, phase ractal dimensions D,w and D,g, and saturation. There is no empirical constant in the present relative permeability models. The ractal permeability model Eq. 14 can be considered as a special case o the unsaturated porous medium by setting S w 1in Eqs. 35 and 37 or by setting S w 0 in Eqs. 36 and 38. The results (at a given D T ) rom the present relative permeability models are ound to be independent o porosity/pore-area ractal dimension, D, and to depend only on saturation in the porosity range o 0.90 (that is, D 1.97), which is consistent with the available conclusion and existing experimental data. At a higher porosity o 0.90 (that is, D 1.97), the relative permeability is a weak unction o the porosity/pore ractal dimension, D, and increases with porosity/pore ractal dimension D. The present results also show that the higher the tortuosity ractal dimension, D T, the lower the permeability. The present work reveals that ractal dimension D and tortuosity ractal dimension D T may be two o the important mechanisms aecting the relative permeability in porous media. Thereore, this work is a supplement to the available conclusion by Bear (1972) and existing correlations (Kaviany, 1995), which indicate that relative permeability depends only on saturation. I the tortuosity ractal dimension D T 1.10 is used or real monodispersed porous media (similar to packed beds), the predictions o the relative permeabilities based on the proposed ractal model in the porosity range o 0.90 (that is, D 1.97) are ound to be in excellent agreement with experimental data. This veriies the 56 January 2004 Vol. 50, No. 1 AIChE Journal

12 validity o the present ractal analysis o the permeability or porous media. However, it should be noted that while the ractal model is able to capture the data shown in Figure 10, it might not be the only model. It should also be noted that not all porous media are ractals. The present model is only applicable to the porous media whose pore-size distributions ollow ractal scaling laws. Thereore, one must ascertain experimentally that the porous media used in the experiments (reported in Figure 10) are indeed ractal in nature to conclusively demonstrate that the assumption o ractal geometry is indeed correct. Since many other actors, such as viscosity ratio, surace tension, wettability, density ratio, capillary pressure, hysteresis, and contact angle, may aect the relative permeability, a more complicated model than the present one may need to be developed. This article presents a preliminary work, and urther study in this area is in progress. 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