Open Access Published by De Gruyter Open Access February 26, 2019

Modified Gompertz sigmoidal model removing fine-ending of grain-size distribution

Rui Yuan, Bo Yang, Yingfei Liu and Lingyu Huang
From the journal Open Geosciences

Abstract

Because of the laboratory operating, the fineending of grain-size distribution (GSD) are simply combined as one point, which results in the information loss of the fine and very-fine clastic particles, and affects the geological parameters calculation of GSD. To remove the fine-endings, a modified Gompertz sigmoidal model is proposed in this paper. The first stage is establishing and solving the modified Gompertz sigmoidal model; the second stage is fitting and evaluating the cumulative probability and frequency of GSD; the third stage is calculating the geological parameters. Taking 113 samples for example, coefficients of determination (COD) between measured and fitted individual cumulative probability and frequency are bigger than 0.98980 and 0.97000 respectively, which proves the goodness of fitting results. By moments method using frequency data, the COD between fitted and measured mean is 0.97578, while CODs of sorting, skewness and kurtosis are in low values, which suggest that the fine-endings has little influence on the average grain-sizes of GSD and large influence on its geometry. Besides, modified Gompertz sigmoidal model offers another quick numerical way to calculate median, mean and sorting of GSD by graphical method using cumulative probability data. The proposed method is useful to remove the fine-endings and contribute to calculate the geological parameters of GDS.

1 Introduction

One of the most important and fundamental property of the clastic sediments is the frequency of occurrence of different diameters of particles, called grain-size distribution (GSD) in sedimentology and geology. The abundance of different sized grains reflects a multitude transportation and accumulation factors [1, 2, 3, 4, 5, 6, 7]. In the laboratory measurement of the grain-size, the weights of same very small grains are combined as one data, called fine-ending of the GSD. Oppositely, GSD exists coarse-ending at the coarse portion. However, in the calculating of the parameters of GSD: median, mean, storing, skewness and kurtosis, the fine- or coarse-endings are never considered in previous researches before.

Taking 113 samples from clastic cores of one petroleum well for example, utilizing a modified Gompertz sigmoidal model, the goal of this study is to remove the fine-ending of these GSDs, which comprises three stages. The first stage, based on the related mathematics, involves (1) establishing a modified Gompertz sigmoidal model for the cumulative probability and (2) confirming the least-squares estimate algorithm as solution method for the model parameters. The second stage, known the parameters of Gompertz for an individual GSD, consists in (1) fitting the cumulative probability and then frequency of GSD at the measured

grain-size and fine-endings class interval and (2) evaluating the goodness of the fittings. The third stage, based on fitted frequency or cumulative probability of GSD, embrace (1) calculating the mean, sorting, skewness and kurtosis by moments method using frequency data or (2) calculating the medians, means and sorting numerically by graphical method using cumulative probability data.

2 Material

Collected from clastic cores of one petroleum key well, GSDs of 113 samples are the source material of this paper. The lithology of cores, buried from 1726 to 1140 m depth, is dominate various sands and some conglomeratic sands known as the deposits body of fluvial delta. The samples are consolidated, having to been broken by the core-crusher first to earn the clastic particle. Carbonates and organic matters are removed by HCl and H2O2. Using laser particle size analyzer MS2000, frequencies (Figure 1A) and cumulative probability (Figure 1B) of GDSs are measured in 35 grain-size classes ranging from 0.031 to 8000 μm (15 to -3 Φ). However, the measured gran-sizes class interval is not uniform, 0.25 Φ from -3.125 to 3.875 Φ, 1 Φ from 4.5 to 8.5 Φ and one interval between 8.5 Φ and 15.5 Φ. No diameter of particle is less than -3.125 Φ, all coarse grains are covered. Because of the laboratory operating, the very-fine silts and clays are merged and simplified from 8.5 Φ to 15.5 Φ, called fine-ending of GSD. Their frequencies are increased at the last measured interval point, which does not meet the regulation of log-normal distribution obviously. This operation loses the data detail of silts and clays surely from the GSD, the important part to reveal the depositional setting information of very-fine grains.

Figure 1 GSDs of 113 samples from cores of one well in phi-size scale, measured by laser particle size analyzer MS2000. In the fine-ending, from 8.5 Φ to 15.5 Φ, the frequency is merged. (A) Frequency of GSDs. The frequencies are increased at the fine-ending. (B) Cumulative probability of GSDs, the curves are sigmoidal monotone increasing from zero to one hundred percent.

Figure 1

GSDs of 113 samples from cores of one well in phi-size scale, measured by laser particle size analyzer MS2000. In the fine-ending, from 8.5 Φ to 15.5 Φ, the frequency is merged. (A) Frequency of GSDs. The frequencies are increased at the fine-ending. (B) Cumulative probability of GSDs, the curves are sigmoidal monotone increasing from zero to one hundred percent.

The frequency curves of GSD are most unimodal or bimodal, dominate diameter between 4 Φ and 7 Φ (Figure 1A) The cumulative probability distributions of GSD are typical sigmoidal curve, monotone increasing from zero to one hundred percent (Figure 1B) The parameters of every GDSs, median, mean, storing, skewness and kurtosis, are offered by the laboratory assistants using moments and graphical respectively, called measured values here. All the corresponding grain-size data are given in Supplemental Material 1. These will be contrasted by that of fitted GDS in scatter cross-plots as results.

3 Methods

3.1 Modified Gompertz sigmoidal model

The Gompertz model [8] was a long history to be a growth curve for biological [9, 10, 11, 12], economic [13] and industrial [14] studies and so on. It is the aim of researches to indicate to some extent its usefulness and limitations as a growth model (Figure 2), which is similar to the cumulative probability of GSD. For actual purposes, the asymmetric Gompertz growth curve is more generally convenient to write in [8]

Figure 2 The Gompertz growth model. The curve is sigmoidal monotone increasing from zero to the upper limit.

Figure 2

The Gompertz growth model. The curve is sigmoidal monotone increasing from zero to the upper limit.

(1) y = f x = a e x p ( b e x p ( k ( x x c ) ) )

in which a, k and b are essentially positive quantities: a is the saturation value, which is the upper limit of the data set; k and b are growth velocity factor, which controls the slope of the curve. The xc implies the point of inflection of the curve. It is clear that Gompertz is monotonic increasing function, as x becomes negatively infinite y will approach zero, and as x becomes positively infinite y will approach a.

For the GSD of clastic rocks, the lower and upper limit of cumulative probability are just right zero and hundred percent respectively. Therefore, we modify the Gompertz model as

(2) C ^ = f m = 100 e x p ( b e x p ( k ( m x c ) ) )

which is called modified grain-size distributional Gompertz model (GSD-GM) here. The independent variable, m, is the middle-point of each class interval in phi-size scale; dependent variable, ĉC, is the estimated cumulative probability of GSD.

3.2 Fitting GSD

In order to understand the great significance of GSD both in modern systems and ancient deposits, many mathematic statistical methods and models have been used to fit the frequency of GSD. Log-normal distributional model, based on the normal distribution, the earliest and widespread method, analyzed the GSDin Log base 2 in phisize scale [16]. Therefore, multiple log-normal distribution functions are first used to fit the GSD which is considered to be a sum of several log-normal distribution [17, 18, 19]. Benefit from the advancements of mathematic statistical models, many other methods, such as Rosin [20, 21], loghyperbolic [22, 23], Weibull [24, 25, 26], log-skew-Laplace [27], gamma [28], tangential hyperbolic (tanh) [29] and skewnormal distribution [30, 31] are attempted as well. However, all these methods just inspect and fit the frequency of GSD at measured grain-size class interval, not extending to the coarse-or fine-endings of GSD. As a matter of fact, the cumulative probability is another expression of GSD. It is typical sigmoidal distribution in the phi-size scale.

Parameters b, k and xc of Equation (2) could be estimated from measured cumulative probability of GSD. There are several known methods for parameter estimation, among which least-squares estimate algorithm is the common fitting method [32, 33, 34]. The least-squares is integrated in many commercial software, such as Origin, developed by OriginLab Corporation (https://www.originlab.com/), which is used in this study directly. In order to distribute the GSD into 1 Φ interval between 8 Φ and 15 Φ, the independent variable of Equation (2) have to be changed to be different form the measured grain-size classes. After fitting the cumulative probability of GDS, the fitted frequency distribution could be computed further more easily.

In additional, to evaluate the goodness of fitting result of the GSD-GM, residual (R) and coefficient of determination (COD) are involved. The residual at induvial grain-size classes is defined as the difference of measured and fitted frequency or cumulative probability. The COD, between 0 and 1, is defined by the variance of measured values and the residual sum of square divided by the number of grainsize classes.

{ r e s i d u a l ( R ) = m e a s u r e d v a l u e f i t t e d v a l u e c o e f f i c i e n t o f d e t e r m i n a t i o n ( C O D ) = 1 s u a d r a t i c s u m o f r e s i d u a l n u m b e r o f g r a i n s i z e c l a s s e s v a r i a n c e o f m e a s u r e d v a l u e s

In general, the larger the COD, the better the fitting. For the convenient of writing here, the residual and COD of cumulative probability are abbreviated to Rc and CODc, and residual and COD of frequency are abbreviated to Rf and CODf here.

3.3 Calculating parameters of GSD

There are main two principles to calculate the geological parameters of GDS, median (Md), mean (Mz), storing (standard deviation, ĉ), skewness (SK) and kurtosis (KG): (1) moments method, using the frequency distribution of GSD [35]; and (2) graphical measures, using cumulative probability of GSD [36]. The Folk-Ward Formula of graphical measures just inspect a subset of cumulative probability between 5% and 95%, ignoring cumulative probability of grain-size class that less than 5% and larger than 95% [36]. While the moments method takes all frequency into calculation. After removing the fine-ending of the frequency distribution, we deservedly choose moments method to estimate involved parameters ofGSD. The moments functions are given by [35]

(3) M z = P i m i 100 σ = P i m i M z 2 100 S k = P i m i M z 3 100 σ 3 K G = P i m i M z 4 100 σ 4

where themi is the ith middle-point of class interval in phisize scale and the Pi is the corresponding fitted frequency.

4 Results

4.1 Fitting GSD

113 samples’ GSD have been fitted to remove the fineendings utilizing the GSD-GM. For each individual GSD, there are four steps to finish the fitting. Taking 1451.78 m sample for example, its GSD is used to introduce the fitting process and results detailly. The measured and fitted frequency and cumulative probability of 1451.78 m sample are shown in the Figure 3. The first step, utilizing least-squares method to estimate the parameters of GSDGM using the measured cumulative probability, b=1.10729, k=0.96534 and xc=1.78878. Equation (2) of the GSD-GM are obvious and clear. The second step, utilizing Equation (2) to estimate the fitted value of the cumulative probability (Figure 3A) The third step, utilizing the estimated cumulative probability to calculate the fitted value of the frequency (Figure 3B) The fourth step, utilizing residual and COD to evaluate goodness of fit. The absolute residual at induvial grain-size classes of cumulative probability and frequency are less than 0.7% and 0.4% respectively, very closed to zero; and CODc=0.99996, CODf =0.94013 very closed to 1 (Figure 3A, B) which suggest the excellent goodness of fit. The fitting of the 1451.78 m is the best result among 113 samples.

Figure 3 Fitting result of 1451.78 m sample. (A) Fitting result of cumulative probability and its residual. The cumulative probability between 8 Φ and 15 Φ has been estimated. The COD between the measured and fitted data is 0.99996 and the max absolute residual is 0.7%, which explains the well goodness of fitting. (B) Fitting result of frequency and its residual. The frequency of last grain sizes class interval is distributed into 8 Φ to 15 Φ. The COD between the measured and fitted data is 0.94013 and the max absolute residual is 0.4%, well goodness of fitting.

Figure 3

Fitting result of 1451.78 m sample. (A) Fitting result of cumulative probability and its residual. The cumulative probability between 8 Φ and 15 Φ has been estimated. The COD between the measured and fitted data is 0.99996 and the max absolute residual is 0.7%, which explains the well goodness of fitting. (B) Fitting result of frequency and its residual. The frequency of last grain sizes class interval is distributed into 8 Φ to 15 Φ. The COD between the measured and fitted data is 0.94013 and the max absolute residual is 0.4%, well goodness of fitting.

All parameters of 113 GSD-GMs, b, k and xc, have been calculated. Based on these parameters, the cumulative probability and frequency of 113 GSDs are fitted, and the residual and COD are evaluated (Figure 4). The fine-ending of the GSDs are distributed to seven grain-size class intervals, without increasing at the last measured interval point. The max absolute Rc and Rf are no more than 5%, the CODc are between 0.98980 and 0.99996, and the CODf are between 0.97000 to 0.99900. The integral goodness of fitting results of GDS are proved well, and their fineendings have been removed, which would be used to calculate the geological parameters of GDS. All the calculated parameters of GSD-GMs, fitted cumulative probability, frequency, residual and COD are given in Supplemental Material 2.

Figure 4 Fitting result of all 113 samples. (A) Fitting result of cumulative probabilities and their residuals. The fine-endings of cumulative probabilities are supplemented. (B) Fitting result of frequencies and their residuals. There are no increases at the fine-ending.

Figure 4

Fitting result of all 113 samples. (A) Fitting result of cumulative probabilities and their residuals. The fine-endings of cumulative probabilities are supplemented. (B) Fitting result of frequencies and their residuals. There are no increases at the fine-ending.

4.2 Calculating parameters of GSD

After removing the fine-ending of GSD, the mean, storing, skewness and kurtosis would be estimated by the moments method. Taking the fitting result of 1451.78 m for example again, the fitted Mz=2.501 Φ, σ=1.344 Φ, SK=1.135, KG=5.288 using the Equation (3) and the measured M z=2.527 Φ, σ=1.409 Φ, SK=1.500, K G=7.483. In the same way, four parameters of all 113 samples’ GDS are estimated (Figure 5). All the calculated results are given in Supplemental Material 3. The COD of means is 0.97578, which implies that the fitted and measured means coincident (Figure 5A) Because of the increased fine grain-size element into the GSD, the fitted sorting is not well coincident with measured values (Figure 5B)COD of storing in 0.27389. The measured skewnesses are between 0.5 and 2.5, which suggests the fine- and very-fine-skewed of the GSD. However, the fitted skewnesses are 0.5 to 1.5, mainly fine-skewed (Figure 5C) The measured kurtosises are between 2.5 and 13, which implies the mesokurtic, leptokurtic and very-leptokurtic. Whereas, the fitted kurtosises are 1.0 to 6.0, leptokurtic dominantly (Figure 5D) There are large differences of skewness and kurtosis between measured and fitted values. Mathematically, when the measured skewness is larger than 1.2, the fitted skewness reduces to about 1.0 (Figure 5C); and if the measured kurtosis is larger than 5.3, the fitted kurtosis reduces to about 4.8 (Figure 5D) Geologically, the added fine-ending has large influence on the geometry of the GSD. Many veryfine-skewed samples are fine-skewed and very-leptokurtic samples are leptokurtic in fact.

5 Discussions: A new numerical method to calculate median, mean and sorting

On the other hand, once the b, k and xc are solved, the GSD-GM is a clear analytic expression, which offers another numerical approach, based on graphical method, to calculate the median, mean, sorting of the GSD. The inverse function of Equation (2) is given by

(4) m = f 1 C ^ = x c 1 k ln ln 100 C ^ ln b

Taking the ĉC is 5, 16, 50, 84 and 95 respectively, the corresponding grain-size of f−1(5), f−1(16), f−1(50), f−1(84) and f−1(95) can be calculated using Equation (4) Then, based on the Folk-Ward grain-size formula, median, mean and sorting are given by [36]

(5) M d = f 1 50 M z = f 1 16 + f 1 50 + f 1 84 3 σ = f 1 84 f 1 16 4 + f 1 95 f 1 5 6.6

For example, for the b, k and xc of 1451.78 m, the f−1(5)=0.75777 Φ, f−1(16)=1.26688 Φ, f−1(50)=2.27403 Φ, f−1(84)=3.70374 Φ and f−1(95)=4.97119 Φ, then Md=2.27403 Φ, Mz=2.41488 Φ and σ= 1.24761 Φ.

All medians, means and sortings of 113 GDSs are numerically computed by the GSD-GM, given in Supplemental Material 3. The results are contrasted with that of the laboratory graphical measurement (Figure 6). The COD of medians, means and storings are 0.99222 (Figure 6A)

0.988034 (Figure 6B) and 0.78191 (Figure 6C) respectively, which suggests the equality of the both. Therefore, com- bining GSD-GM and Equation (5), the three parameters of massive GSDs would be fast achieved, without the tedious plotting and reading of the graphical method. However, the GSD-GM cannot used to estimate the skewness and kurtosis regrettably, for the internal constant value of both for Gompertz sigmoidal model.

Figure 5 The mean, storing, skewness and kurtosis scatter cross-plot of measured and fitted GSD. The measured parameters are come from moments method using measured data, while he fitted parameters are used fitted data. (A) Scatter cross-plot of mean. (B) Scatter crossplot of storing. The fitted sorting is most less than measured values. (C) Scatter cross-plot of skewness. (D) Scatter cross-plot of kurtosis.

Figure 5

The mean, storing, skewness and kurtosis scatter cross-plot of measured and fitted GSD. The measured parameters are come from moments method using measured data, while he fitted parameters are used fitted data. (A) Scatter cross-plot of mean. (B) Scatter crossplot of storing. The fitted sorting is most less than measured values. (C) Scatter cross-plot of skewness. (D) Scatter cross-plot of kurtosis.

Figure 6 The median, mean and storing scatter cross-plot of measured and fitted GSD. The measured parameters are come from graphical method using measured data, while the fitted parameters are come from Equation (5). (A) Scatter cross-plot of median. (B) Scatter crossplot of mean. (C) Scatter cross-plot of storing.

Figure 6

The median, mean and storing scatter cross-plot of measured and fitted GSD. The measured parameters are come from graphical method using measured data, while the fitted parameters are come from Equation (5). (A) Scatter cross-plot of median. (B) Scatter crossplot of mean. (C) Scatter cross-plot of storing.

6 Conclusions

A modified Gompertz sigmoidal model is proposed to remove fine-endings of 113 clastic GSDs using cumulative probability data based on least-squares estimate algorithm. The COD of individual cumulative probability and frequency is bigger than 0.98980 and 0.97000 respectively, which proves the well goodness of fitting results. By moments method using frequency data, the COD of fitted and measured means is 0.97578, while CODs of sorting, skewness and kurtosis is in low values, which suggest that the fine-endings has little influence on the average grain-sizes of GSD and large influence on its geometry.

Based on graphical method using cumulative probability, GSD-GMoffers another quick and automatic numerical way to calculate median, mean and sorting of GSD, without plotting data and reading parameters. The COD between fitted and measured medians, means and sortings are 0.99222, 0.988034 and 0.78191 respectively, suggests the availability and efficiency of the method. The proposed method can be used to remove the fine-ending and calculate the geological parameters of GDS.

Acknowledgement

We are grateful to anonymous reviewers for their constructive reviews on the manuscript, and the editors for carefully revising the manuscript. This research is financially supported by Scientific Research Project of Hubei Provincial Department of Education (No. Q20181310), Open Fund of Key Laboratory of Exploration Technologies for Oil and Gas Resources (Yangtze University), Ministry of Education (No. K2018-21), Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology) (No. PLC20180605), Scientific Research Project of Educational Commission of Sichuan Province of China (No. 18ZB0076). The supports are gratefully acknowledged.

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Received: 2018-09-20
Accepted: 2018-12-05
Published Online: 2019-02-26

© 2019 Rui Yuan et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.