A phenomenological connectivity measure for the pore space of rocks
André Rafael Cunha, Celso Peres Fernandes, Luís Orlando Emerich dos Santos, Denise Prado Kronbauer, Iara Frangiotti Mantovani, Anderson Camargo Moreira, Mayka Schmitt
AA phenomenological connectivity measure forthe pore space of rocks ∗André Rafael Cunha † Celso Peres FernandesLuís Orlando Emerich dos SantosDenise Prado Kronbauer Iara Frangiotti MantovaniAnderson Camargo Moreira Mayka SchmittDecember 3, 2020
Abstract
The interconnectivity of the porous space is an important charac-teristic in the study of porous media and their transport properties.Hence we propose a way to quantify it and relate it with the intrinsicpermeability of rocks. We propose a measure of connectivity based ongeometric and topological information of pore-throat network, whichare models built from microtomographic images, and we obtain ananalytical method to compute that property. The method is expandedto handle rocks that present a higher degree of heterogenity in theporous space, which characterization requires images from differentresolutions (multiscale analysis). Trying to expand the methodologybeyond the scope of images, we also propose a new interpretationfor the experiment that generates the mercury intrusion curve andcalculate the permeability. The methodology was applied to imagesof 11 rocks, 3 sandstone and 8 carbonate rock samples, and to theexperimental mercury intrusion curve of 4 tight gas sand rock sam-ples. We observe as result the existence of a correlation between theexperimental and the predicted values. The notions of connectivitydeveloped in this work seek above all to characterize a porous materialbefore a typical macroscopic phenomenology. ∗ Research supported by Brazilian agencies CAPES, CNPq and FAPESC, and Petrobras. † A. R. Cunha. Porous Media and Thermophysical Properties Laboratory, FederalUniversity of Santa Catarina, SC, Brazil. [email protected] . a r X i v : . [ phy s i c s . g e o - ph ] N ov eywords: Porous media. Pore space connectivity. Transportproperties. Intrinsic permeability. Microtomographic images of rocks.Multiscale analysis. Mercury intrusion curve.
Keywords:
The porous space does not have a regular geometry. Nevertheless, it isusual to assign to it someone to allow a mathematical treatment. The mostcommon ones are the capilar tubes and the networks models.The first modeling attempts admitted that the porous medium is formedby capillary tubes (Kozeny, 1927; Carman, 1937). The application of physicallaws to these models is facilitated by the simple geometry. By definition,these models do not contemplate porous space connectivity. Therefore itsapplication is limited to certain classes of materials (Scheidegger, 1963).An alternative idea is to consider the pore space as a network formed by pores , larger spaces that store fluid, and throats , which restrict the flow whileperforming the communication between the pores (Dullien, 1979). Under thisview, two quantities are relevant for the displacement of matter: the radius ofthe pore and the number of throats that leave (or reach) that pore. The firstis of geometric nature, and the second, topological. The number of throats ofa pore is called coordination number of that pore.The representation ways of the porous medium by network are relatedto the development of computation, since a network is formed by manyconstituents, and due to imaging techniques, which can provide informationfrom the material. In the 1950s, some authors used to circumvented theproblem of excessive calculations by means of electromechanical analogies(Bruce, 1943, cited by Scheidegger, 1963; Owen, 1952, cited by Sahimi, 1993;Fatt, 1956a,b,c). At an intermediate stage, The 2D imaging techniquesallowed the introduction of images to the simulation. But a 2D image is notable to adequately represent porous space connectivity (Chatzis and Dullien,1977, cited by Van Marcke et al., 2010). Therefore, criteria were developedto generate new random networks with statistical image information , whichare superimposed to build a 3D volume where the phenomenon is simulated.And even if higher-order statistics (Okabe and Blunt, 2005) or multiscalarschemes (Fernandes et al., 1996) are considered, the generate volume does notproperly express the real pore space. The advent of X-rays microtomographytechnique in the porous media research in the 1980s (Vinegar and Wellington, They are named pixel/voxel based statistics or point-to-point statistics. pore-throat networks or morphological networks, where the phenomenum isdescribed by the conservation laws, i.e., the continuum models . In randomnetworks , in turn, the phenomenon is approached by statistical physicaltheories, based on results of theories of percolation, renormalization, fractalsand cellular automata, for example; they are discrete models (Sahimi, 1993).In this work, we apply the Maximum Ball Algorithm (Dong and Blunt, 2009)to the microtomographic image. The result is a network of spherical poresand cylindrical throats (Fig. 1). In some case, the simulation based onthe discretization of the motion equations is summarized to a linear system(Cunha et al., 2015).
Figure 1:
Result of the application of the Maximum Ball Algorithm.
Although the idea of porous space connectivity is an intuitive truth, thereis no single definition, and the attempts to quantify it vary according tothe branch of research. In mathematics, connectivity is synonymous with topology (Flegg, 2001). Thus, the first attempt to describe it goes throughtopological definitions. Therefore, in general, from a theoretical point of view,the coordination number has been the basis for quantifying connectivity and3enerally the only parameter explored. According to Sahimi (1993), Bettinumbers are the most accurate way to characterize connectivity. As thereference, we restrict ourselves to the first two numbers to ilustrate. Thefirst Betti number B is the number of separate components that make up astructure. A value B > B is the number of holes ina structure. B is equivalent to the genus of a surface, which is the maximumnumber of closed curves that do not intercept and can be built on a surfacewithout dividing it into distinct regions (Vasconcelos, 1997). In this paper,we propose a phenomenological definition to connectivity as a conceptuallyless sophisticated alternative. And before continue, we explore some examplesfrom literature.Vasconcelos (1997, 1998), for example, adopting a 3D network model ofcylindrical tubes, associates the genus per unit volume G V to the specificsurface S and the volume V , both experimentaly determined. Then G V isintroduced as a multiplicative factor directly into the permeability expression.When dealing with random networks, the coordination number is aninherent and constant information of spatial organization. Therefore, themore complex the network, the more it tends to be useful for describing realsituations (Efros, 1986). In morphological networks, in turn, the coordinationnumber is not fixed, and we can know a distribution. Mason (1982), forexample, interpreting the media as a random network, estimates the coordi-nation number from adsorption isotherms. For the author, connectivity iscoordination number by definition, and also comments on the limitation ofconsidering a constant connectivity to what would probably be a distribution.From the pixel based statistical view, the connectivity of a random networkcan be defined from higher order moments of the phase function. The intentionis the 3D volume reconstruction.Other examples of interpretation of connectivity in porous media are cited.Glover (2009) uses electrical parameters to propose a measure of connectivity,more precisely to the inverse of the resistivity of the rock formation. Montaron(2009), in the same domain as the previous work, associates the connectivity ofa random network to the conductivity equations obtained from percolation andmedium field theories. Trinchero et al. (2008), in a groundwater perspective,consider the lack of a univocal concept for connectivity in this domain andadopt, as a measure of connectivity to an aquifer, the hydraulic responsetime between two points after the injection of markers in one. Bernabé et al.(2010, 2011, 2016) define connectivity as the mean coordination number frompore-throat networks. 4 new approach Considering a pore-throat network model, we propose a measure of con-nectivity in which the coordination number is not the only input data, theother is the radius of the pore. In other words, topology is not the onlyrelevant information for connectivity, so geometry is. It is a phenomenolog-ical perspective, based on the weighting of the contribution of each objectof the network to a flow. What is more important: a large pore with lowcoordination number, or a small pore with high coordination number? Wepropose that the interaction between the two informations can be the answer.And from the quantification of the connectivity, we propose a quantitativecorrelation with the intrinsic permeability of the porous medium.We start from the observation that the pore-throat networks exhibitinteresting patterns: the pore size distribution can be fitted by the gammadistribuition, and the mean coordination number of a pore increases linearlywith its radius.
It is very difficult to define an ideal volume size at which physical propertiestend to stabilize, and even if a certain physical property does, there are noguarantees that others will do so (Dvorkin and Nur, 2009), and as the largerscales are considered, it contributes with the increase in the physical-chemicalheterogeneity of the porous formation. In practice, the elementary volume (which may or may not be representative) is determined by the limitationsof the equipment used. It is part of this work to assume that the patternspresented by the pore-throat network serve as criteria for determining anelementary volume that is representative before the phenomena in question,a monophasic flow.
The images are: 3 sandstone rocks, named A1, A2 and A3, observedwith the respective resolutions of 2.40 µ m, 3.40 µ m e 3.90 µ m, from which acubic volume of edges 300 voxels were cropped. 8 carbonate rocks, namedsequentially from C1 to C8, observed with the respective resolutions of 5.90and 20.0 µ m to C1, 0.500, 1.20, 1.50, 1.69, 4.57 and 19.0 µ m to C2, 1.40, 2.96and 20.0 µ m to C3, 1.50, 5.90 and 20.0 µ m to C4, 1.00, 5.00 and 13.0 µ m toC5, 1.20, 3.48 and 13.0 µ m to C6, 1.93, 5.10 and 13.0 µ m to C7, 4.00 and53.0 µ m to C8, from which a cubic volume of edges 500 voxels were cropped.The experimental permeability values are (in milliDarcy): to A1, 2.45; A2,5.00; A3, 4.00; C1, 105; C2, 0.117; C3, 11.6; C4, 0.987; C5, 0.232; C6, 0.209;C7, 0.173; C8, 4.65.In a second moment, we work with experimental data from mercuryporosimetry. They are 4 tight gas sand samples, named sequentially fromT1 to T4. The length L of the samples follow respectivelly: 0.0340, 0.0325,0.0315 and 0.0331 m; and the diameter D : 0.0370, 0.0380, 0.0375 and 0.0380m; and the permeability ones: 4.00 × − , 6.60 × − , 4.60 × − e 1.01 × − mD. It is observed that the Spherical Pore Size Distribution (S-DTP) R ( r ) forthe sandstones can be approximated by a gamma distribution R ( r ) = 1Γ( α ) β α r α − e − βr , r ≥ . (1)where α and β the parameters of the distribution, and Γ( x ) is the gammafunction. raio (voxel) D en s i dade de p r obab ili dade ( ba s eada e m nú m e r o ) A1 A2 A3 ajuste observado Figure 2:
Spherical Pore Size Distribution (S-PSD) R ( r ) of sandstone samplesfitted by the gamma distribution. It is also observed the existence of a linear correlation between the meancoordianation number n ∗ of a pore and its radius r (Fig. 3), i.e., n ∗ ∼ r ,n ∗ = ar + b . r = 0 does not exist, and implies no connectedthroat, i.e., b = 0. Then, n ∗ = ar . (2) raio (voxel) n ∗ ℜ ² = 0.984 A1 ℜ ² = 0.978 A2 ℜ ² = 0.926 A3 ajuste observado Figure 3:
Correlation between mean coordination number n ∗ of a pore and itsradius r for sandstone samples. The observed correlation implies that one can express N ∗ ( n ∗ ) in terms of R ( r ) (Kay, 2005): N ∗ ( n ∗ ) = 1 a R (cid:18) n ∗ a (cid:19) . Faced with a flow in the pore-throat network, two quantities are relevantfor the mass displacement: the pore radius and its coordination number.As said before, the first has a geometric nature and the second topological.Highlighting the interaction between these entities of spatial configuration,we define the connectivity function ξ as ξ ( n, r ) ∼ N ( n ) R ( r ) , (3)where r is the pore radius with distribution R , and n is the coordinationnumber with distribuition N . The expression can be rewritten for the meancoordination number n ∗ : ξ ( n ∗ , r ) ∼ N ∗ ( n ∗ ) R ( r ) . (4)The observed linear correlation, n ∗ ∼ r , allows to rewrite the connectivity ξ as a function only of radius r , ξ ( r ) ∼ R ( r ) . (5)7hat is, the connectivity function ξ , which covers geometric and topologicalinformations of the network, is completely characterized by only one of thevariables, n ∗ or r . In the case of eq. (5), the radius r was chosen because itcan be measured by different techniques. Normalizing the equation , ξ ( r ) = 2 α − β Γ( α ) Γ(2 α − R ( r ) . (6)Fig. 4 shows an example of R ( r ) and its respective connectivity density ξ ( r ) for α = 2 and β = 2. The R ( r ) curve shows that the smaller pores aremore abundant than the larger ones. The discussed correlation establishes,in turn, that the larger pores have more connections. Finally, the ξ ( r ) curvemediates these contributions and reveals the pores that most contribute tothe network connectivity. raio D en s i dade de p r obab ili dade R ( r ) ξ ( r ) Figure 4: R ( r ) and ξ ( r ) for α = 2 and β = 2. k The permeability k is given by (Scheidegger, 1963): k = − ηLQA ( p out − p in ) , (7)where η is the viscosity of the fluid, L is the length of the material, A is thearea of the section, p in and p out are the pressures applied at the inlet andoutlet ends, respectively, and Q is the flow through A . The flow Q is given by Q = −
1Ω ( p out − p in ) , The normalization condition requires that: α > / e β > k = LηA Ω . (8)In the right side, L , A and Ω are all macroscopic ( η is a fluid property). Butthe hydraulic resistance Ω is affected by the microscopic characteristics of theporous space. We then consider Ω as a mean from microscopic informations.Strictly speaking, the hydraulic resistance of a cell in the pore-throatnetwork has two parts: Ω = Ω p + Ω g , where Ω p is the contribution due to the spherical pores, and Ω g is due tocylindrical throats.The connectivity ξ ( r ) explicitly considers only the radius r of the poresand the average number of throats that depart from that pore, but not thegeometry of those links. However the Maximum Ball Algorithm establishes arelation between the geometries of the radius and its connected throat (Dongand Blunt, 2009; Cunha et al., 2015). Then we can rewrite,Ω = Ω p (1 + τ ) , Ω ∝ Ω p . And since the objective of this work is to demonstrate the existence of acorrelation, we write, without loss of generality,Ω = Ω p , therefore (Cunha et al., 2015) Ω = 81 ηπr . (9)At this point the connectivity function ξ is used to weight an expectedvalue of r in eq. (9) (Cunha, 2011) h r i ξ = Z ∞ ξ ( r ) r dr , h r i ξ = 14 α (cid:16) α − (cid:17) β . Replacing eq. (9) in eq. (8), k = πL A α (4 α − β . (10)9he described methodology has two main limitations. The first is relatedto the anisotropy of the medium; it is known that the permeability is atensor entitity (Scheidegger, 1954; Liakopoulos, 1965; Szabo, 1968; Durlofsky,1991), i.e., it depends on the flow direction; in this work, however, we departsfrom a PSD characteristic of the volume, and that’s why cubic volumes areconsidered. The second limitation is related to the normalization procedureof eq. (6), which can not be reached for α ≤ / . For carbonates, which show a high degree of heterogeneity in their porousstructure, we propose to interpret the porous medium as a succession ofthe involved scales in such a way that the equivalent hydraulic resistance isunderstood as a serial association of the resistances of each scale. We meansΩ eq = X i Ω i = 81 ηπ X i h r i i , (11)and L = X i L i . where i is the number of scales.Observation at different scales results in distributions that overlap in someregion (Fig. 5). This means that some pores have been counted more thanonce, and their contributions to the flow are overestimated. Therefore, in thecalculation of the permeability, we will avoid to consider very close resolutionsand consider the connectivity function ξ to give the proper weight of theradius measured by each scale. A pore size distribution resulted from a mercury porosimetry is constructedfrom capillary tubes model through the Young-Laplace equation, PSD
Y L , p = 2 σ cos θλ . (12)This model does not consider connectivity by definition. Therefore we proposeto calculate R ( r ) from PSD Y L before estimates the permeability.A porous medium is considered to be completely saturated by a fluid. Andfor each pressure p applied, we measure the cumulative volume V that leavesthe structure. In the n -th measurement, the medium reached the irreducible10 uperposição de DTP diferentes raio D en s i dade de p r obab ili dade DTP
Conectividades diferentes raio D en s i dade de p r obab ili dade ξ ( r ) Figure 5:
Example of two spherical pore size distribution overlapping and theirrespectives connectivity functions. saturation, i.e., the curve has n points. And through Young-Laplace equation,we calculate the n values of radius Λ n = { λ , . . . , λ n } .Let ( p , V ) a point of the experimental curve. The Young-Laplace equationassociates p to λ , meaning that all capillaries with radii greater than orequal to λ are accessible at this pressure. Thus, the pressure p can expelan amount of fluid from all connected spherical pores whose radii are greaterthan or equal to λ . Mathematically it is expressed: V = 4 π Z ∞ λ ξ ( r ) r dr = 4 π β Γ (cid:16) α + 2 , λβ (cid:17) α − . Analogously, high pressures can reach the capillaries with smaller radii.Theoretically, when pressure p ∗ → ∞ , the radius λ ∗ →
0. In this case all thecapillaries are accessible to the pressure p ∗ , as long as they are connected.Hence the total accumulated volume is V ∗ = 4 π Z ∞ ξ ( r ) r dr = 4 π αβ (4 α − . The cumulative volume density v is given by the division of V by V ∗ , v ( r ) = Γ (cid:16) α + 2 , λ β (cid:17) α − α (4 α − . (13)The previous equation is one of the n equations, and is a transcendentalequation of α and β , and its numerical solution is presented in the Ap. A.With the values of the parameters, we can go back to eq. (10).11 Results and discussion
For the sandstone samples, the hypotheses have already been observed inFigs. 2 and 3.For the carbonates ones, follow Figs. 6 and 7. It is noted a weakeningof the linear correlation for certain resolutions, which are deprecated for thecalculation of the permeability, unless for C1, since they are the only onesavailable. Here there is an implicit consideration: the hypotheses that anelementary volume is representative when it presents the explored patterns,i.e., a gamma distribuition to S-PSD and the linear correlation between thepore radius and its mean coordination number.Tab. 1 and Tab 2 show the parameters obtained from the pore-throatnetworks and mercury intrusion curves, respectively.12 aio (voxel) D en s i dade de p r obab ili dade ( ba s eada e m nú m e r o ) C1:5.90
C1:20.0
C2:0.500
C2:1.20
C2:1.50
C2:1.69
C2:4.57
C2:19.0
C3:1.40
C3:2.96
C3:20.0
C4:1.50
C4:5.90
C4:20.0
C5:1.00
C5:5.00
C5:13.0
C6:1.20
C6:3.48
C6:13.0
C7:1.93
C7:5.10
C7:13.0
C8:4.00
C8:13.0
Figure 6:
Spherical Pore Size Distribution (S-PSD) R ( r ) of carbonate samplesfitted by the gamma distribution. aio (voxel) nú m e r o de c oo r dena ç ão m éd i o n ∗ ℜ ² = 0.665 C1:5.90 ℜ ² = 0.866 C1:20.0 ℜ ² = 0.919 C2:0.500 ℜ ² = 0.968 C2:1.20 . . . . . . . ℜ ² = 0.685 C2:1.50 ℜ ² = 0.961 C2:1.69 ℜ ² = 0.768 C2:4.57 ℜ ² = 0.696 C2:19.0 ℜ ² = 0.888 C3:1.40 ℜ ² = 0.928 C3:2.96 ℜ ² = 0.808 C3:20.0 ℜ ² = 0.826 C4:1.50 ℜ ² = 0.986 C4:5.90 ℜ ² = 0.765 C4:20.0 ℜ ² = 0.840 C5:1.00 ℜ ² = 0.644 C5:5.00 ℜ ² = 0.889 C5:13.0 ℜ ² = 0.937 C6:1.20 ℜ ² = 0.696 C6:3.48 ℜ ² = 0.739 C6:13.0 ℜ ² = 0.621 C7:1.93 ℜ ² = 0.952 C7:5.10 ℜ ² = 0.863 C7:13.0 ℜ ² = 0.952 C8:4.00 ℜ ² = 0.916 C8:13.0
Figure 7:
Correlation between mean coordination number n ∗ of a pore and itsradius r for carbonate samples. able 1: Pore-throat network parameters.
Sample Resolution ( µ m) α θ < A1 2,40 3,43 0,223 0,984A2 3,40 3,05 0,119 0,978A3 3,90 3,36 0,171 0,926C1 5,90 2,90 8,25 e +04 0,66520,0 3,89 6,65 e +04 0,866C2 0,500 2,92 1,90 e +06 0,9191,20 4,42 2,12 e +06 0,9681,50 4,56 1,96 e +06 0,6851,69 3,00 9,47 e +05 0,9614,57 6,46 1,06 e +06 0,76819,0 2,24 4,34 e +04 0,696C3 1,40 2,04 1,96 e +05 0,8882,96 2,07 1,72 e +05 0,92820,0 1,94 2,83 e +04 0,808C4 1,50 2,40 5,18 e +05 0,8265,90 3,03 1,90 e +05 0,98620,0 3,19 5,45 e +04 0,765C5 1,00 4,07 1,17 e +06 0,8405,00 2,52 2,18 e +05 0,64413,0 3,48 1,24 e +05 0,889C6 1,20 4,02 1,19 e +06 0,9373,48 5,20 9,40 e +05 0,69613,0 3,10 1,00 e +05 0,739C7 1,93 4,97 1,38 e +06 0,6215,10 4,19 4,33 e +05 0,95213,0 3,09 1,02 e +05 0,863C8 4,00 1,95 1,88 e +05 0,95213,0 4,06 1,28 e +05 0,91615 able 2: Calculated parameters from the mercury intru-sion curves.
Sample α β
T1 2,10 5,92 e − e − e − e − Table 3:
Calculated permeability values.
Sample k (mD) Used resolution ( µ m)A1 194 2,40A2 634 3,40A3 249 3,90C1 1,70 e +03 5,90 ; 20,0C2 1,90 0,500; 1,20C3 188 1,40 ; 2,96C4 16,0 1,50 ; 5,90C5 3,76 1,00 ; 13,0C6 3,39 1,20 ; 13,0C7 2,80 1,93 ; 5,10 ; 13,0C8 75,3 4,00 ; 13,0T1 2,25 e − Hg ¶ T2 8,34 e − Hg T3 1,97 e − Hg T4 2,35 e − Hg ¶ Value obtained from the mercury intrusion curves. xperiemental C a l c u l ado - - - - . . . A1 A2A3C1C2 C3C4C5 C6C7 C8T1 T2T3 T4
ANOVAMultiple R-squared: 0.9833, Adjusted R-squared: 0.982F-statistic: 766.7 on 1 and 13 DF, p-value: 6.064e-13 arenito carb. (multiescala) t.g. sand (intrusão Hg) (a)
Residuals vs. Fitted
Valores ajustados R e s í duo s -0.50.00.5 10^-6 10^-4 10^-2 10^0 10^2 A1A2A3C1C2 C3C4C5C6C7 C8T1T2T3 T4 (b)
Normal Q-Q Plot
Quantis teóricos R e s í duo s pad r on i z ado s -2-101 -2 -1 0 1 2 A1A2A3C1 C2C3C4 C5C6 C7C8 T1T2T3 T4 (c)
Figure 8: a) Comparison between calculated and experimental permeability values. b) Apparently random layout of the residuals. c) Comparison of the standardizedresiduals with the theoretical quantiles. Summary and conclusions
In this paper we discuss some ideia related to the connectivity of porousmedia, which is an important intrinsic feature in the study of the transportproperties of those materials. The main objetive was to quatify it and relatedit to the intrinsic permeability of rock samples.We start from the observation that the pore-throat networks exhibitinteresting patterns: the pore size distribution can be fitted by the gammadistribuition, and the mean coordination number of a pore increases linearlywith its radius.Our thesis was to suppose that both geometry and topology of the networkare important for the mass displacement before a monophasic flow. Then, wepropose a phenomenological connectivity function ξ ( n, r ) ∼ N ( n ) R ( r ) , (3)that assumed the form ξ ( r ) = 2 α − β Γ( α ) Γ(2 α − R ( r ) . (6)That equation can quantify how a pore is connected only by its radius, whichcan be know by different experimental techniques.We used it to calculate the permeability from a single network and for sev-eral networks coming from differente resolution images (multiscalar analysis).We still extrapolate its use beyound the scope of images, proposing a newinterpretation of the mercury intrusion curves. Those expressions gave us aanalytical formula for the intrinsic permeability, which results are consistentlycorrelated with the experimental values.Those results make us affirm that the defined connectivity function is arelevant entity before a monophasic flow.During the study we have established two important characteristics tothe pore space. The first is that, after observe the above explored patterns insome microtomographic images, we operate reciprocally and imposed themas quantitative criteria to evaluate if a volume is representative from itsoriginal material. The second is the new interpretation to the experimentthat generates the mercury intrusion curves and how to build the ideal PSDfrom the PSD Y L . Perspectives
In a first moment, it is expected that the observed patterns can beexplored in other rocks, even in other porous materials, and in other scales ofobservations. 18econdly, we imagine that the connectivity function ξ can be applied tothe other phenomena. If no, we still expect that one can start from a generic ξ ( n, r ) ∼ f ( N ( n ) , R ( r ) ) (14)to propose alternative expressions most suitable. We thank Prof. Carlos Appoloni for the valuable comments.
A Resolution of the transcendental equation
We search the values of α and β that satisfy the eq. (13). Therefore anadditional equation is required. It is related to the expected value of gammadistribuition (Kay, 2005): αβ = h r i R = Z ∞ R ( r ) r dr . (15)None of the three terms in the preceding equation is still known. Therefore,the experimental data DTP-YL is used to approximate the right side ofeq. (15). To emphasize the central tendency of values, we choose to use themedian of the set Λ n , denoted by Λ M . We write αβ = Λ M . (16)An implicit consideration of the previous equation is that the DTP-YLexperimental curve must also be close to a gamma distribution. We can nowreplace β = Λ M α , (17)in the eq. (13), and have a transcendental equation only for α , whose solutioncan be searched numerically.Ideally the parameters should be unique for the existing n equations;But in practice only m < n equations have a solution. And the value of theparameter will be the mean of the set of m elements.Since α is known, we return to eq. (17) to determine β . And now we areable to know R ( r ) and ξ ( r ). 19 eferences A.S. Al-Kharusi and M. J. Blunt. Network extraction from sandstone andcarbonate pore space images.
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