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Journal Article

Closed-Form Model for Hydraulic Properties Based on Angular Pores with Lognormal Size Distribution Available to Purchase

Efstathios Diamantopoulos, Wolfgang Durner
Published: 01 February 2015
Vadose Zone Journal (2015) 14 (2): vzj2014.07.0096.
DOI: https://doi.org/10.2136/vzj2014.07.0096
...Efstathios Diamantopoulos; Wolfgang Durner Abstract We previously presented a mathematical model for hysteretic soil hydraulic properties based on bundles of angular pores. Upscaling to sample-scale hydraulic properties was done analytically, assuming a Gamma density distribution of pore sizes...
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Gamma ray, density, and neutron-logs distribution for the three rock types in the Woodford Shale. Rock type 1 has low density, high gamma ray, and high neutron porosity consistent with high TOC and high laboratory-measured porosity. The TOC can show a high gamma-ray signature at the log due to enhanced radioactivity by the trace uranium generally associated with the organic matter. Rock type 3, on the other hand, has highest density and lowest neutron porosity consistent with high carbonates in laboratory-measured mineralogy.

Gamma ray, density, and neutron-logs distribution for the three rock types ... Available to Purchase

Published: 10 January 2018
Figure 12. Gamma ray, density, and neutron-logs distribution for the three rock types in the Woodford Shale. Rock type 1 has low density, high gamma ray, and high neutron porosity consistent with high TOC and high laboratory-measured porosity. The TOC can show a high gamma-ray signature
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Probability densities of gamma and lognormal distributions under mean 1.0 and variance 0.1 (A), 0.5 (B), 1.0 (C), and 2.0 (D), and probability density of N(1, 1) distribution (truncated at 0) (C). Note that the Gamma(1, 1) distribution (α = 1) is Exp(1) distribution (C).

Probability densities of gamma and lognormal distributions under mean 1.0 a... Open Access

Published: 01 May 2022
Figure 1. Probability densities of gamma and lognormal distributions under mean 1.0 and variance 0.1 (A), 0.5 (B), 1.0 (C), and 2.0 (D), and probability density of N (1, 1) distribution (truncated at 0) (C). Note that the Gamma(1, 1) distribution (α = 1) is Exp(1) distribution (C).
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Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function (Eq. 3). A) Channel-depth probability distribution at T  =  60 hr. B) Channel-depth probability distribution at T  =  120 hr. The χ2 test to quantify goodness of fit indicates that the gamma probability density function is an adequate fit of the data in parts A and B at the p  =  0.05 significance level. Test values: Nbins  =  41, degrees of freedom  =  38,   =  21,   =  0.09,   =  0.04,   =  0.04,   =  55.

Measured values (symbols) of the probability density for channel depth, tog... Available to Purchase

Published: 01 November 2013
Fig. 10.— Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function ( Eq. 3 ). A) Channel-depth probability distribution at T   =  60 hr. B) Channel-depth probability distribution at T   =  120 hr. The χ 2 test
Image
Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function (Eq. 3). A) Channel-depth probability distribution at T  =  60 hr. B) Channel-depth probability distribution at T  =  120 hr. The χ2 test to quantify goodness of fit indicates that the gamma probability density function is an adequate fit of the data in parts A and B at the p  =  0.05 significance level. Test values: Nbins  =  41, degrees of freedom  =  38,   =  21,   =  0.09,   =  0.04,   =  0.04,   =  55.

Measured values (symbols) of the probability density for channel depth, tog... Available to Purchase

Published: 01 November 2013
Fig. 10.— Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function ( Eq. 3 ). A) Channel-depth probability distribution at T   =  60 hr. B) Channel-depth probability distribution at T   =  120 hr. The χ 2 test
Image
Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function (Eq. 3). A) Channel-depth probability distribution at T  =  60 hr. B) Channel-depth probability distribution at T  =  120 hr. The χ2 test to quantify goodness of fit indicates that the gamma probability density function is an adequate fit of the data in parts A and B at the p  =  0.05 significance level. Test values: Nbins  =  41, degrees of freedom  =  38,   =  21,   =  0.09,   =  0.04,   =  0.04,   =  55.

Measured values (symbols) of the probability density for channel depth, tog... Available to Purchase

Published: 01 November 2013
Fig. 10.— Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function ( Eq. 3 ). A) Channel-depth probability distribution at T   =  60 hr. B) Channel-depth probability distribution at T   =  120 hr. The χ 2 test
Image
Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function (Eq. 3). A) Channel-depth probability distribution at T  =  60 hr. B) Channel-depth probability distribution at T  =  120 hr. The χ2 test to quantify goodness of fit indicates that the gamma probability density function is an adequate fit of the data in parts A and B at the p  =  0.05 significance level. Test values: Nbins  =  41, degrees of freedom  =  38,   =  21,   =  0.09,   =  0.04,   =  0.04,   =  55.

Measured values (symbols) of the probability density for channel depth, tog... Available to Purchase

Published: 01 November 2013
Fig. 10.— Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function ( Eq. 3 ). A) Channel-depth probability distribution at T   =  60 hr. B) Channel-depth probability distribution at T   =  120 hr. The χ 2 test
Image
Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function (Eq. 3). A) Channel-depth probability distribution at T  =  60 hr. B) Channel-depth probability distribution at T  =  120 hr. The χ2 test to quantify goodness of fit indicates that the gamma probability density function is an adequate fit of the data in parts A and B at the p  =  0.05 significance level. Test values: Nbins  =  41, degrees of freedom  =  38,   =  21,   =  0.09,   =  0.04,   =  0.04,   =  55.

Measured values (symbols) of the probability density for channel depth, tog... Available to Purchase

Published: 01 November 2013
Fig. 10.— Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function ( Eq. 3 ). A) Channel-depth probability distribution at T   =  60 hr. B) Channel-depth probability distribution at T   =  120 hr. The χ 2 test
Image
Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function (Eq. 3). A) Channel-depth probability distribution at T  =  60 hr. B) Channel-depth probability distribution at T  =  120 hr. The χ2 test to quantify goodness of fit indicates that the gamma probability density function is an adequate fit of the data in parts A and B at the p  =  0.05 significance level. Test values: Nbins  =  41, degrees of freedom  =  38,   =  21,   =  0.09,   =  0.04,   =  0.04,   =  55.

Measured values (symbols) of the probability density for channel depth, tog... Available to Purchase

Published: 01 November 2013
Fig. 10.— Measured values (symbols) of the probability density for channel depth, together with a fitted gamma probability density function ( Eq. 3 ). A) Channel-depth probability distribution at T   =  60 hr. B) Channel-depth probability distribution at T   =  120 hr. The χ 2 test
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Inter-event time distributions: observed density distributions and gamma distribution fits (equation 1). Continuous or dash-dotted lines, normalized probability distribution function of inter-event times; thick dotted line, gamma distribution fit. On abscissa the inter-event times are multiplied by the seismicity rate 〈R〉; on ordinates the density is divided by the seismicity rate 〈R〉 (Molchan, 2005; Corral and Christensen, 2006; Hainzl et al., 2006). (a)–(e) The chi-squared goodness of fit test allows us to accept the null hypothesis that empirical data follow a gamma law with 95% confidence level. Estimation of the background event rate is thus computed as 1/a in equation (1). (f) The null hypothesis is rejected with 99% confidence level. 〈R〉 is the observed average daily seismicity rate; 〈R*〉 is the average daily seismicity rate normalized by , ΔM* 1.8 ,and Vseimogenic of each volcano (see Table 1).

Inter-event time distributions: observed density distributions and gamma di... Available to Purchase

Published: 01 August 2010
Figure 5. Inter-event time distributions: observed density distributions and gamma distribution fits (equation 1 ). Continuous or dash-dotted lines, normalized probability distribution function of inter-event times; thick dotted line, gamma distribution fit. On abscissa the inter-event times
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Inter-event time distributions: observed density distributions and gamma distribution fits (equation 1). Continuous or dash-dotted lines, normalized probability distribution function of inter-event times; thick dotted line, gamma distribution fit. On abscissa the inter-event times are multiplied by the seismicity rate 〈R〉; on ordinates the density is divided by the seismicity rate 〈R〉 (Molchan, 2005; Corral and Christensen, 2006; Hainzl et al., 2006). (a)–(e) The chi-squared goodness of fit test allows us to accept the null hypothesis that empirical data follow a gamma law with 95% confidence level. Estimation of the background event rate is thus computed as 1/a in equation (1). (f) The null hypothesis is rejected with 99% confidence level. 〈R〉 is the observed average daily seismicity rate; 〈R*〉 is the average daily seismicity rate normalized by , ΔM* 1.8 ,and Vseimogenic of each volcano (see Table 1).

Inter-event time distributions: observed density distributions and gamma di... Available to Purchase

Published: 01 August 2010
Figure 5. Inter-event time distributions: observed density distributions and gamma distribution fits (equation 1 ). Continuous or dash-dotted lines, normalized probability distribution function of inter-event times; thick dotted line, gamma distribution fit. On abscissa the inter-event times
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Density distribution of tsunami interevent times from 1960 to 2012 and sizes greater than 0.1 m (filled circles). (Heavy solid line; the best‐fit exponential distribution; light solid line, best‐fit gamma distribution; dashed line, best‐fit generalized gamma distribution.) The color version of this figure is available only in the electronic edition.

Density distribution of tsunami interevent times from 1960 to 2012 and size... Available to Purchase

Published: 01 July 2014
Figure 2. Density distribution of tsunami interevent times from 1960 to 2012 and sizes greater than 0.1 m (filled circles). (Heavy solid line; the best‐fit exponential distribution; light solid line, best‐fit gamma distribution; dashed line, best‐fit generalized gamma distribution.) The color
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Schematic diagram of the clustered sample selection: (a) 2D feature plane of natural gamma and density, (b) data distribution for cluster 1, and (c) data distribution for cluster 2.

Schematic diagram of the clustered sample selection: (a) 2D feature plane o... Available to Purchase

Published: 15 February 2024
Figure 1. Schematic diagram of the clustered sample selection: (a) 2D feature plane of natural gamma and density, (b) data distribution for cluster 1, and (c) data distribution for cluster 2.
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The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape
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The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape
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The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape
Image

The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape
Image

The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape
Image

The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape
Image

The Statistical Distributions Considered in This Study Distributi... Available to Purchase

Published: 12 July 2016
Table 1 The Statistical Distributions Considered in This Study Distribution f ( x ) 1− F ( x ) Exponential h exp(− hx ) exp(− hx ) Gamma Lognormal Truncated exponential Weibull exp[−( x / c ) h ] The scale and shape