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ORIGINAL RESEARCH

A Genetic Algorithm for Predicting Pore Geometry Based on Air Permeability Measurements

^a Dep. of Agronomy and Soils, 202 Funchess Hall, Auburn Univ., Auburn, AL 36849
^b Dep. of Computer Science and Software Eng., 107 Dunstan Hall, Auburn Univ., Auburn, AL 36849
^* Corresponding author (danejac{at}auburn.edu)

Received 5 August 2004.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Pore size distributions of porous media are of interest to soil scientists, geologists, and engineers with a variety of backgrounds. If known, pore size distributions can be used to determine fluid retention and permeability relationships. In this study, we propose a methodology to predict pore size distributions from air permeability measurements combined with a numerical model representing a porous medium. The model is an extension of the capillary model, which was modified so that the capillaries are composed of sections with different diameters. An optimization scheme that makes use of the measured air permeability values was developed to predict the best possible pore size distribution and pore arrangement. A genetic algorithm, a popular evolutionary computational methodology, was chosen for the optimization process. During our numerical study, we observed that it is not only the pore size distribution that is important, but also how the pores are distributed, in other words, the pore geometry.

Abbreviations: GA, genetic algorithm

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
TRANSPORT OF FLUIDS through porous media, with applications in soils, subsurface hydrology, geology, oil and gas fields, and the medical industry, is of interest to scientists and engineers. Before predicting transport phenomena in a porous medium, its hydraulic properties (i.e., the water retention and hydraulic conductivity relationships) must be determined. This is often time-consuming and expensive if done by currently available methods. For a given porous medium, the values obtained for water retention and hydraulic conductivity at different water content values depend on an intricate system of pores with different size, shape, orientation, and connectivity. This intricate system makes it difficult to express the pore size distribution in a quantitative manner. If known, however, pore size distributions can be used to calculate water retention and hydraulic conductivity relationships. Because of the high degree of complexity of the pore structure, simplifying assumptions need to be made.

Several methods, such as the water-desorption method, visualization method using impregnation, and the Hg injection method, are available for the determination of pore size distributions (Flint and Flint, 2002). Each of these methods, however, has its own disadvantage(s). The water-desorption method is based on calculations of pressures at which water is displaced by air in pores of different radii. Therefore, the obtained water retention curves need to be interpreted to calculate pore sizes using the Young and Laplace equation

[1]

where P_c (Pa) is the capillary pressure, (N m^–1) is the liquid surface tension, is the liquid–material contact angle, and r (m) is the pore radius. This method is elaborate and time-consuming. In fact, it is not uncommon for this test to last 3 to 4 wk.

In the impregnation method, molten paraffin or epoxy is applied to the undisturbed soil sample, which is placed in a sampling tin with both ends open. Once impregnated, the sample is cooled to room temperature, the core is then sawed and sections are photographed. The calculations are based on two-dimensional slices, thus limiting the representativeness. In the Hg injection method, Hg is forced under pressure into a previously oven-dried and air-evacuated sample (Nagpal et al., 1972). The applied pressure is increased in discrete steps, and the volume of pores that is intruded between steps is obtained. The equivalent pore size corresponding to the applied pressure can be calculated with Eq. [1], with being the contact angle between the Hg and soil and being the surface tension of Hg. The advantage of the Hg intrusion method is that it is much faster than the other two techniques previously explained. However, the disadvantage is that pore sizes can change during the injection. It should be mentioned that Hg is a nonwetting fluid in regard to hydrophilic porous media, such as soils and rocks.

Pore sizes of a porous medium have a direct effect on the permeability values; larger pores have a higher permeability. Pore size is of direct importance in understanding transport of volatile organic compounds in the subsurface (Moldrup et al., 1998; Poulsen et al., 1999) or soil–atmosphere gas exchange (Hutchinson and Livingston, 2002), a subject of interest to agriculture and the turf grass industry. Pore size values can also be used as indicators of hydraulic conductivity because pore size affects water run-off (Römkens et al., 2002). The objectives of this study were to (i) determine the air permeability of a consolidated rock sample (experimental study), and (ii) optimize the pore size distribution of the sample by reproducing the air permeability–water content relationship curve that was experimentally determined. The advantage of using air permeability is that it is easy to measure. Because, pore size distributions may have more than one modal pore size, as a result of aggregation (Kutílek and Nielsen, 1994), we used two distribution functions to determine pore sizes: (i) a lognormal distribution and (ii) a bimodal lognormal distribution. If successful for a consolidated rock sample, the method will be extended to unconsolidated soil samples.

Several models have been used to describe porous media. Two types have received great interest, namely, the capillary bundle model and the pore network model (Scheidegger, 1953, 1974; Ball, 1981). In both models, the size of a pore is defined as a radius obtained from the equality of the hydrostatics in the soil and that in the model. Hence, the size of a soil pore is called the "equivalent radius," and no attempt is made to quantify the real radius of a pore. The capillary bundle model is the basis of the so-called "hydraulic radius" theories, which are often used to describe single-phase transport in porous media (Chandler et al., 1982). Such models fail to predict the salient features of capillary displacement such as the known hyteresis between imbibition and drainage. Despite its basic structure and limitations, this simple approach helps to model the porous medium three-dimensionally. D'Hollander (1979) and Wylie and Gardner (1958) developed a capillary bundle model to estimate the pore size distribution of porous media.

The pore network models represent pore spaces as networks of interconnected, simply shaped pores (Lowry and Miller, 1995). Network models are essentially percolation models, in that they involve the percolation of fluid through a network according to specified rules. In a typical network model, the pore structure is assumed to consist of a three-dimensional network of throats (bonds) and pore bodies (nodes) (Dullien, 2000). The throats are subject to the constraint that no bond can have a diameter larger than the smaller of the two nodes that it connects. Evidently, the bond diameters control the drainage process, whereas the volume of nonwetting phase is determined by the nodes. Jerauld and Salter (1990) and Johnson et al. (2003) developed three-dimensional network models to characterize the void structures of soils. Nicholson and Petropoulos (1971)(1973) examined serial and regular network capillary models. Flow in the serial model was dominated by the narrow pores. The main factor causing departure of the network model from this type of behavior was found to be the degree of interconnection between capillaries, or "connectivity." The network models found interest in other areas as well as soils; for example, Bousfield and Karles (2004) developed a network model for ink absorption into coated papers.

Capillary bundle models have relatively more basic structures than network models. However, as mentioned, they fail to predict the known hysteresis between imbibition and drainage. On the other hand, in the network models the throats may have really small diameters compared with the pore bodies. Therefore, they are the ones that determine the flow rather than the pore bodies. This situation may give unrealistic results, especially under conditions where nonwetting fluids are used. Considering all these factors, we chose to use a more sophisticated capillary model, specifically, one in which capillaries are composed of sections, each of different diameter with a different displacement value. The measured air permeability values, as a function of water content, are necessary for the optimization of pore sizes and sequences. A genetic algorithm (GA), a popular evolutionary computational method, was used for the optimization process to determine the pore size distribution of a consolidated rock sample and will be discussed further below.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Experimental
Materials
A Berea sandstone sample, 2.5 cm in diameter and 6 cm long, was used to perform the experimental part of this study. To minimize the effect of organic matter and bacterial growth, the sample was first placed in a muffler oven at 900°C for 24 h. The wetting solution used during the experiment was a 0.005 M CaCl₂ solution with trace amounts of thymol and mercuric chloride to prevent pore structure changes due to bacterial growth.

Air Permeability Measurements of Consolidated Sandstone
The air permeability method for consolidated samples at different water content values, such as rocks or sandstones, is based on the "stationary liquid method" pioneered by Leas et al. (1950) and Osoba et al. (1951). The permeameter used in this research was developed by A.T. Corey (Jalbert et al., 1999). The instrument measures the upward flux density of air through a porous medium in response to an air pressure gradient equal to the pressure gradient in the static wetting fluid. Equating the air pressure gradient to the wetting fluid pressure gradient results in a uniform capillary pressure throughout the sample. The device, used in conjunction with a soap bubble flowmeter, allows for the determination of air permeability values at different wetting fluid contents. The method assures uniform capillary pressure, and therefore saturation, throughout the sample.

The air permeameter can receive consolidated core samples 2.5 cm in diameter and up to 6 cm long. The central part of the permeameter is a cylinder, with a rubber sleeve mounted to the inside wall. The side port of the permeameter is connected to an air pressure source, so pressure can be applied to the rubber sleeve surrounding the sample to avoid bypass flow during the effective air permeability, k_air, measurements. The rubber sleeve is secured in place at the top and the bottom of the cylinder by two plates connected by two rods and fastened by nuts at the top and bottom. The outflow port can be adjusted to different heights to allow measurements on cores of varying length.

When air pressure is applied to the rubber sleeve, it also presses against the in- and outflow ports to ensure good contact between the ports and the sample. Unrestricted flow is accomplished by the chamfered shape of the ports. Although the inflow port is fixed, it can be removed to install or remove a sample from the permeameter by slightly loosening the two bottom knurled nuts, and turning the locking plate sideways. The in- and outflow ports permit the flow of air through the sample, in the upward direction. The difference in air pressure between the in- and outflow ports is monitored with a water manometer.

Initially, the core sample was fully water saturated with a corresponding effective air permeability value of zero. Subsequent effective air permeability measurements were obtained at reduced water content values by letting a small quantity of water evaporate from the sample at room temperature, after which the sample was kept in a closed glass vial for 24 h to ensure that the water was distributed homogeneously. This process was repeated until the sample was completely dry and the effective air permeability during the initial water drainage curve was formed (Fig. 1) .

View larger version (80K): [in this window] [in a new window]   Fig. 1. Measured air permeability (µm2) as a function of volumetric water content during the initial drainage cycle.

[2]

where µ_air (Pa s) is the air viscosity, A (m²) is the cross-sectional area of the sample, _w (kg m^–3) is the water density, g (m s^–2) is the gravitational acceleration, and V (m³) is the volume of air displaced during time t (s). Equation [2] is equivalent to Eq. [9] of Dane et al. (1998) by assuming the density of the nonwetting fluid (air) to be equal to zero.

Although no direct verification of indirectly determined pore size distributions and configurations of the pores is possible, differences in calculated and measured water retention curves may be the best indicator of how well the pore size distribution was determined. Therefore, the same core was used to determine the initial drainage curve by the hanging water column method (Dane and Hopmans, 2002). Because of the rigidity of the sandstone core, the porous plate of the Buchner funnel was covered with a thin silt layer. Rather than lowering the outflow level in the burette to reduce the soil water pressure, the air pressure in the burette was stepwise changed by a regulated vacuum source. We started with a fully saturated sample and determined the initial drainage curve (Fig. 2) by increasing the vacuum every 24 h. During the first few steps no change in water content was observed. However, when the air entry value was reached and the biggest pores started to drain, the water level in the burette started to rise.

View larger version (90K): [in this window] [in a new window]   Fig. 2. Measured matric head plotted as a function of volumetric water content of the sandstone core sample during the initial drainage cycle.

[3]

where k is the intrinsic permeability and is a constant:

[4]

and is the empirical exponent or pore size distribution index in the Brooks–Corey representation of the water retention curve; that is,

and

[5]

where

[6]

and _r is the residual water content, _s is the saturated water content, h_d (cm) is the displacement pressure head, and h_m (cm) is the matric head. Equation [3] can be fitted through measured values of k_nw at known values of S_e. The resulting value for can then be converted to and the corresponding water retention curve can be compared with the measured one. Consequently, our proposed model can be compared with the Brooks–Corey model.

Numerical
Model
The main objective of the model is to indirectly predict the pore size distribution from air permeability measurements. Since there is no direct verification of the indirectly measured pore size distribution, the model included the prediction of the water retention curve of the sample. The pores were modeled as capillaries; each capillary was composed of a number of sections (i.e., 5, 10, etc.; Fig. 3) .

View larger version (14K): [in this window] [in a new window]   Fig. 3. Structure of capillaries (pore section sequences).

Once air flow occurs through a sequence of sections, we only know the pressure difference between the top and the bottom. As during the experimental measurements, we will assume a unit head gradient. To calculate the flow through a given sequence, we need to know how the pressure changes from section to section, as smaller sections offer greater resistance to flow. The volumetric flow rate, Q, however, is the same through each section and is given by Poiseulle's Law:

[7]

where r_i is the pore radius of section i, i = 1,2,..,5 (m), P_i–1 is the pressure at the pore section entrance (Pa), P_i is the pressure at the pore section exit (Pa), is the air viscosity, and L is the pore section length (m). If L is assumed to be the same for all sections, then

[8]

where C is a constant. Hence, the flow equation is written for all sections as

[9]

In Eq. [9], Q and P₁ through P₄ are the unknowns, whereas P₀ (pressure at the bottom of the pore sequence), P₅ (pressure at the top of the sequence), and the radii are known variables. The unknown variables can be calculated as we can obtain five equations with five unknowns from Eq. [9]. Introducing a variable tortuosity factor, , to better represent a naturally occurring porous medium, the total flow rate, Q_total (m³ s^–1), is

[10]

where is the number of pore conducting sequences, Q_j is the flow rate for the jth conducting pore sequence, and _j is a variable tortuosity factor depending on the water content. Dividing Q_total by the cross-sectional area, A (m²), of the porous medium (summation of solids and pores), we obtain the flux density, q, analogous to the Darcy flux:

[11]

According to Darcy's Law, the air flux density is proportional to the air pressure gradient by means of the air conductivity (K_a, m² Pa^–1 s^–1) of the porous medium; that is,

[12]

Equation [12] allows K_a to be calculated for different water content values because both q and the overall air pressure gradient, P/z, are known. The relationship between air conductivity and effective air permeability, k_a (m²), is given by

[13]

where is the air viscosity (Pa s).

To produce the water retention curve, we tracked the amount of water drained from the sections and the corresponding capillary matric head values. Because the porosity of the sample was determined, the total volume of sample pores was known. The water content, , of the sample was calculated from

[14]

where V_T (m³) is the total soil volume, and V_wp (m³) is the volume of water-filled pore sections.

Using the capillary approach and the Darcy equation, the Kozeny–Carman equation for the intrinsic permeability, k (m²), was derived as (Wylie and Gardner, 1958)

[15]

where is the porosity or internal volume per unit bulk volume, ß (L₀/L)² is the Kozeny–Carman constant, ß is a shape factor, L₀ (m) is the apparent flow path length in the direction of macroscopic flow, L (m) is the actual length of the porous medium through which the fluid flows, and S (m^–1) is the internal surface area per unit bulk volume. The tortuosity is described as (L₀/L)². Kozeny suggested using ß = 2.5 and = 2; that is, a constant of 5 for ß (L₀/L)². As was pointed out by Corey (1994)(p. 93), this same value cannot be applied to porous media other than sand, nor to less than fully saturated media. Burdine (1953) noted that the factor that has caused the greatest amount of comment, and which is probably least understood, is the tortuosity of the fluid path in the porous sample. Because it is saturation dependent, Corey (Brooks and Corey, 1964; Corey, 1994) suggested a function for the tortuosity rather than a constant value for all saturation stages. The tortuosity of a wetting phase ("w") is inversely related to the effective saturation squared:

[16]

in which _1.0 is the tortuosity of the wetting phase when S_e = 1.0. For the nonwetting phase ("nw"), they suggested

[17]

As the wetting-phase saturation of a porous medium decreases, fluid particles of the wetting phase must take an increasingly longer path in moving between two points. This is because the particles cannot take a direct route across pore spaces because the central portion of the spaces are occupied by the nonwetting phase. After sufficient desaturation takes place, the nonwetting phase (air) has a finite tortuosity, and this tortuosity decreases with decreasing liquid saturation.

A number of distribution functions have been used to represent pore size distributions of porous media, including distribution, Gaussian distribution, and lognormal distribution. The lognormal distribution gained more popularity than the others for two reasons. First, a phenomenon is lognormally distributed whenever its magnitude depends on the product of a very large number of independent factors, which may, in turn, each follow their own distribution (Brutsaert, 1966). Second, the particle sizes of many soils and rocks have lognormal distributions (Brutsaert, 1966; Shirazi and Boersma, 1984; Kosugi et al., 2002). In addition, previous studies showed that the water retention models are more realistic if bimodality is introduced because pore size distributions may have more than one modal pore size as a result of aggregation (Kutílek and Nielsen, 1994). Therefore, we developed two pore size distribution models using two different distribution functions: (i) lognormal distribution and (ii) bimodal lognormal distribution. The radius value of each capillary section was randomly assigned based on the corresponding distribution function in each model.

Genetic Algorithms
Genetic algorithms use computational models of evolutionary processes as key elements in the design and implementation of computer based problem solving systems (Spears et al., 1993). This approach is becoming the method of choice for complex problem solving, especially when more traditional methods cannot be efficiently applied or produce unsatisfactory solutions. Recently, GAs have been frequently applied to solve both science and engineering problems (Wang, 1991; Gwo, 2001; Unsal et al., 2002; Karpouzos et al., 2001).

Genetic algorithms differ from more traditional search algorithms in that they work with a number of candidate solutions rather than just one or a partial solution. Each candidate solution of a problem is represented by a data structure known as an individual, which has two parts: a chromosome and a fitness (Fig. 4) . The chromosome is composed of genes, and the values that can be assigned to a gene of a chromosome are the alleles of that gene. If the genes have only two alleles (0s and 1s), the chromosome is called a binary coded chromosome. If the genes are assigned real values, then the chromosome is called a real coded chromosome.

View larger version (8K): [in this window] [in a new window]   Fig. 4. Structure of an individual.

After initialization, parents are selected according to a probabilistic function based on relative fitness and allowed to create offspring (Spears et al., 1993). In other words, those individuals with higher relative fitness are more likely to be selected as parents. The tournament selection is the most popular selection method; one parent is selected by randomly comparing b individuals in the current population (Dozier et al., 2001). Among these b individuals, the individual with the best fitness is selected as a parent. A second parent may be selected by repeating the same procedure. The most common type of tournament selection is the binary tournament selection where b is equal to 2.

After selection, reproductive operators such as crossover and mutation are applied to the parents. In crossover, parents contribute copies of their genes to create a chromosome for an offspring. The created offspring is a mixture of its parents. Mutation requires only one parent. An offspring created by mutation resembles its parent with the exception of a few altered genes.

After children are created (new individuals), the candidate solutions that they represent are evaluated and each child (individual) receives its own fitness. Before they can be added to the population, some individuals in the current population must die and be removed to make room for the new children. That way, the population size can remain constant. Usually, individuals are removed based on their fitness, with below average individuals having an above average chance of being selected to die. This process is called the natural selection; individuals that are better fit are allowed to live longer and procreate more often.

An interesting aspect of GAs is that the initial population of individuals need not be very good. In fact, each individual of an initial population usually represents a randomly generated candidate solution. By repeatedly applying selection and reproduction, GAs produce satisfactory solutions quickly and efficiently.

Genetic Algorithm Attributes
In this section, we present some of the basic attributes of our GA. Genetic algorithms can be characterized in terms of six basic attributes: (i) genetic representation of candidate solutions (individuals), (ii) population size, (iii) evaluation function, (iv) genetic operators, (v) selection algorithm, and (vi) type of reproduction used (Eshelman and Schaffer, 1993).

In our GA, each candidate solution (individual) was represented by real-value coded chromosomes. Each gene represented a pore with a number of sections of different diameters (r_x,y,z) (Fig. 5) . The index x represents the individual, the index y represents the gene, and the index z represents the section within a gene or sequence (Fig. 5 shows 10 sections per sequence). The radii were assigned randomly between a previously determined upper and lower limit. A total of 500 genes (pore sequences) made up a chromosome, and each chromosome was assigned a fitness value based on the evaluation function. Our population size was 25.

View larger version (21K): [in this window] [in a new window]   Fig. 5. Structure of the genetic algorithm.

[18]

where n is the number of air permeability measurements. In our case, n = 10. It should be mentioned that all air permeability measurements were limited to the drying cycle of the sample.

Our crossover procedure was that of blend crossover. Assume that the real-coded allele values of two parents are X and Y, where X < Y (Parent 1: X and Parent 2: Y). We define = (Y – X), where is a number between 0 and 1.0. Then the offspring will be created randomly between the range of (X – , Y + ). Our mutation rate was 0.01. Tournament selection was used as the selection algorithm. The GA used in this study is referred to as a steady-state GA; that is, the worst individuals (lowest fitness values) were replaced by new children.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
On the basis of Eq. [3], we determined = 4.59 and subsequently = 0.56 from Eq. [4]. Using the calculated value and assuming the same air entry value as determined for the experimental data, the Brooks–Corey curve and our experimental water retention curve are shown in Fig. 6 . By most measures, the two curves do not agree very well.

View larger version (102K): [in this window] [in a new window]   Fig. 6. Comparison of the experimental water retention curve and the predicted water retention curve based on the Brooks-Corey model.

View larger version (89K): [in this window] [in a new window]   Fig. 7. Model results for effective air permeability (µm2, logarithmic) during initial water drainage.

View larger version (83K): [in this window] [in a new window]   Fig. 8. Comparison of experimental water retention curve and the and water retention curves predicted according to Models I and II.

The genetic algorithm ran 10000 generations to find the best solution for both models by repeatedly applying selection and reproduction to a randomly created initial candidate population. Model I predicted the best pore size distribution for a mean radius value of 2.89 µm and a standard deviation () of 0.53. Model II resulted in two mean values with corresponding standard deviations; the best solution was obtained at ( = 0.95 µm, _l = 0.45) and ( = 2.7 µm, ₂ = 0.2).

	SUMMARY AND CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 
Two numerical models for determining pore size distribution and pore geometry of a consolidated rock sample were presented. Model I used a lognormal distribution. Even though the air permeability results were rather satisfactory, the generated water retention curve did not match the measured curve very closely. Previous studies have shown that water retention models are often more realistic if bimodality is introduced because pore size distributions may have more than one modal pore size as a result of aggregation (Kutílek and Nielsen, 1994). Therefore, we developed Model II, which used a bimodal lognormal pore size distribution. This second model was more successful in predicting the correct water retention curve. Therefore, we concluded that Model II was more successful than Model I to determine pore size distributions and their configurations. We also observed from the model results that it is not only the pore section sizes that are important, but also the sequences in which they occur. For a pore section to drain (or fill), it is not only its radius that matters but also the state of the neighbors to which it is connected.

The optimization process was performed using a GA because GAs give satisfactory results very quickly and efficiently. Genetic algorithms find more and more applications in different engineering and science areas, and to the best of our knowledge, they have not been used to determine pore size distributions. Once the pore size distributions are known, hydraulic conductivity functions can be calculated as well.

In ongoing research we are developing a similar experimental and numerical approach for undisturbed, unconsolidated soil samples.

	ACKNOWLEDGMENTS

 
We would like to thank Dr A.T. Corey (Professor Emeritus at Colorado State University, Fort Collins, CO), for making the consolidated core available.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS REFERENCES

 

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