Research ArticleIn Situ Determination of Density Profiles in Complex StrataUsing the Nuclear Density Cone Penetrometer

Rui Jia 12 Huayang Lei 12 Wenjun Zhang12 and Haizuo Zhou 12

1School of Civil Engineering Tianjin University 135 Yaguan Road Jinnan District Tianjin 300350 China2Key Laboratory of Coast Civil Structure Safety of Ministry of Education Tianjin University 135 Yaguan Road Jinnan DistrictTianjin 300350 China

Correspondence should be addressed to Huayang Lei leihuayang74163com

Received 7 February 2019 Accepted 28 May 2019 Published 18 June 2019

Academic Editor Roberto Palma

Copyright copy 2019 Rui Jia et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The nuclear density cone penetrometer (ND-CP) is an in situ testing device that can provide continuous data on soil density butit measures the composite density (120588c) of the soil within a spheroid centered at the midpoint between the gamma ray source andthe detector A theoretical model for predicting 120588c of the ND-CP is proposed and equations for calculating 120588c are derived whenthe ND-CP penetrates into strata with different functions of density distributions The calculated 120588c profiles provide a good fit tothe laboratory-measured 120588c profiles by the ND-CP indicating that the proposed theoretical model can be used to calculate the 120588cwithin the spheroid The comparisons of the density (120588) and 120588c values show that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where density changes suddenly Therefore a method for deducing the actual 120588profiles from themeasured 120588c profiles in complex strata is proposed and this method is used to determine the actual 120588 profiles fromthe ND-CP measured 120588c profiles in the field The research results would be beneficial for the in situ determination of the densityprofile in complex strata using the ND-CP

1 Introduction

Soil density is defined as the mass of the soil per unit ofvolume It depends on the density of the solid constituentsthe porosity of the aggregate and the degree of saturation[1 2] Soil density measurement is important because itdetermines the degree of compactness as a measure of soilstructure and is used for calculating the pore space of soils [3]The density of soil is an indispensable parameter used for theclassification of stratigraphy identification of soft interlayersdetermination of the relative density of sand assessmentof the effect of ground improvement and calculation ofa series of soil physical indexes [4 5] Furthermore thedensity of bed sediments is an important parameter used forpredicting sediment transport determining nautical depthand planning dredging projects [6]

There are mainly two types of methods available forsoil density measurement One is the long-established directmethod which involves the measurement of the sample mass

and volumeThe other is the indirect nondestructivemethodwhich involves measuring the attenuation or scattering ofgamma radiation by the soil and determining the soil densityfrom the intensities of the original and the attenuated gammaradiation The nondestructive method is equal in accuracyto the direct method of density determination and is sim-pler and quicker to use especially where measurements atdifferent depths are required [7] It can be classified into thebackscatter type and direct transmission type In backscattergauges the gamma radiation source and detector are lead-shielded and located on the soil surface or are lowered intoan access hole in the soil The gamma rays emitted from thesource enter into and interact with the soil and attenuate byscattering In this process some of the rays that are scatteredtowards the detector are counted In transmission gaugesthe gamma radiation source emits rays that enter into andinteract with the soil and some of the rays are scatteredaway from the detector The rays passing through the soil aredetected and counted by the detector [8]

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 5804271 14 pageshttpsdoiorg10115520195804271

2 Mathematical Problems in Engineering

The use of gamma radiation to measure density is ageophysical technique that was first applied in oil well logging[9] Subsequently this technique has been used for a numberof years in geotechnical engineering to measure the wetdensity of soils [10ndash14] With recent advances in electronicsand the miniaturization of various electronic sensors anddevices the original bulky nuclear density gauge has beendeveloped and incorporated into the piezocone penetrome-ter which is the most versatile tool currently available for insitu soil investigation [15 16] thus the nuclear density conepenetrometer (ND-CP) was developed [17ndash19] The ND-CPhas been used in soil investigations and it can provide reliablesoil density measurements [20ndash23]

In the ND-CP the density is calculated by the count rateratio (119877120588) of the gamma rays detected by the detector tothe gamma rays emitted from the source Thus the ND-CP measures the composite density (120588c) of the soil withina spheroid centered at the midpoint between the gammaray source and the detector [24] Therefore the measured120588c is not the actual 120588 of the soil at a certain depth in theground There may be a considerable difference between themeasured 120588c profile and the actual 120588 profile at the boundariesof each stratum where the density changes suddenly eg inlayered strata strata with soft interlayers and bed sedimentsTherefore it is necessary to study how to determine the actual120588 profiles in complex strata from the measured 120588c profiles bythe ND-CP

In this study the density measurement principle and thedensity measurement volume of the ND-CP are describedfirst Then a theoretical model for predicting 120588c is proposedwhen the ND-CP penetrates into strata with different densitydistribution functions The theoretical model is verified bycomparing the calculated 120588c profiles by the theoretical modelwith themeasured 120588c profiles by theND-CP in the laboratoryThen a method for deducing the actual 120588 profile from themeasured 120588c profile is proposed based on the differencebetween the 120588 profiles and the 120588c profiles in complex strataFinally the proposed method is used to determine theactual 120588 profiles from ND-CP measured 120588c profiles in thefield

2 Measurement Principle of Nuclear DensityCone Penetrometer

21 Nuclear Density Cone Penetrometer Figure 1 shows theND-CPThe ND-CP mainly consists of a standard piezoconepenetrometer and a nuclear densitometer The piezoconepenetrometer is identical to the standard electronic coneThenuclear densitometer mainly consists of a gamma ray sourceand a gamma ray detector The diameters of the piezoconepenetrometer and the nuclear densitometer are 357 mm and486 mm respectively The distances of the cone resistancesensor the pore pressure sensor the sleeve friction sensorthe inclinometer sensor the gamma ray source the densitycount gamma ray detector and the background count gammaray detector from the apex of the cone are 0 m 004 m011 m 05 m 0721 m 0986 m and 14535 m respectively[24]

22 Density Measurement Principle Gamma rays are a by-product of the natural decay of some radionuclides Thedensity measurement principle of the ND-CP is based on theabsorption of gamma rays (interaction between the rays andthe soil that reduces the intensity of the rays) The photonsof the gamma rays can interact with the electrons of thesoil Three different interaction processes namely the pho-toelectric effect Compton scattering and pair productionare observed The photoelectric effect is the interaction ofa gamma photon with an electron in the inner shell of theatom The gamma photon is completely absorbed and theelectron uses the additional energy to leave its atom as a freeelectron The Compton effect is the scattering of a gammaphoton by an inner shell electron whereby only part of thephoton energy is transferred to the electron The energy anddirection of the gamma photon thus change and the electronleaves its electronic shell Pair production is a process inwhich a gamma ray of sufficient energy is converted intoan electron and a positron Energies of gamma rays rangefrom approximately 10 keV to 10MeVThephotoelectric effectgenerally predominates at lower ray energies the Comptoneffect predominates when the energy of the ray ranges from 1MeV to 5 MeV and pair production occurs at ray energies ofseveral MeVs [25]

The principle of the soil density measurement in the ND-CP utilizes the Compton scattering of gamma rays Whenthe ND-CP penetrates the ground the gamma rays emittedfrom the source pass through the soil particles in the groundinteract with the SiO2 in the soil particles collide with theelectrons in the outer shell and scatter repeatedly with theatomic electrons in the soil (Figure 2) In this process someof the gamma rays are absorbed by the soil while others reachthe detector The absorption depends on the density of thesoil The higher the density is the more the electrons areavailable within a given volume to interact with the gammarays and the more gamma rays are absorbed Hence the soildensity can be determined from the intensities of gamma raysemitted from the source and detected by the detector

The count rate cannot be directly used to calculate densitybecause the radioactive materials decay exponentially withtime The decay rate is characterized by the half-life of aradioisotope which is defined as the time taken for half theoriginal number of radioactive nuclei to decay The gammaray source used in the ND-CP is the radioactive isotopecesium-137 which has a half-life of approximately 30 yearsTherefore the measured count rate must be normalized bythe standard count rate to account for the radioactive decayIn addition to obtain the accurate absorption of gammaradiation in the soil the amount of natural gamma rays inthe ground must be measured and subtracted from the totalcount rate detected by the detector The count rate ratio (119877120588)which is used to calculate the soil density is defined as follows[24]

119877120588 = 119863119862119877 minus 119861119862119877119878119862119877 (1)

whereDCR is the density count rate which ismeasured by thedensity count gamma ray detector BCR is the backgroundcount rate which is measured by the background count

Mathematical Problems in Engineering 3

004

m 011

m0

5 m 0

721

m 0

986

m

145

35 m

Gamma ray detector(Background count)

Gamma ray detector(Density count)

Cone resistance

u Pore pressure

f s

q c

Gamma ray source

Lead shield

Gamma rays

Inclinometer

Gamma rays

Spheroid

Sleeve friction

Figure 1 Nuclear density cone penetrometer

gamma ray detector and SCR is the standard count ratewhich is measured in an inactive material under controlledconditions before each in situ investigation [26] The calibra-tion equation between soil density and 119877120588 is as follows [24]119877120588 = 1198601205882 minus 119861120588 + 119862 (2)

where 120588 is density and A B and C are 06264 40954and 68422 respectively for density values ranging from 10gcm3 to 22 gcm3 Theoretically the ND-CP can be used tomeasure the density of soil up to 22 gcm3 However similarto the piezocone penetrometer the ND-CP is applicable togeneral clay silt and sand especially to loose soil and muckysoil below the groundwater level for which undisturbedsamples are difficult to obtain

23 DensityMeasurement Volume The absorption of gammarays and the measured DCR depend on the density of thesoil within a spheroid between the gamma ray source and

the detector as shown in Figure 1 The measurement volumeis an oblate spheroid centered at the midpoint between thegamma ray source and detector (Figure 3) The length ofsemiaxis b which is the distance from the center to the polealong the symmetry axis is half of the gamma ray source-detector separation distance (2b) of the ND-CP In this studythe value of 2b is 265 cm The length of semiaxis a whichis the equatorial radius of the spheroid decreases with anincreasing density of the surrounding soil [10]The lengths ofa are approximately 24 cm 20 cm and 17 cm in water clayand sand respectively [24] Notably the measured DCR ismore sensitive to the density of the soil closer to the centerof the spheroid

The density measured by the ND-CP is the compositedensity (120588c) of the soil within a spheroid as shown in Figure 3Thus themeasured density (120588c) is not the actual density (120588) ata certain depth If the density changes suddenly with deptheg as in layered soils themeasured120588c profilewill be differentfrom the actual 120588 profile

4 Mathematical Problems in Engineering

O

Si

Electron

Soil particle

Gamma rays passthrough soil particles in the ground

Gamma rays

Gamma rays interact with SiO in soil 2

particles

Gamma rays collide with the electrons in the outer shell

Figure 2 Schematic diagram of Compton scattering of gamma rays in the soil (modified from httpwwwsoilandrockcojp)

Gamma ray source

Gamma ray detector

aba

b

Figure 3 Density measurement volume of the ND-CP

3 Theoretical Model for CalculatingComposite Density

31 Theoretical Model for Predicting Count Rate Ratio Amathematical model is proposed for the ND-CP to predictthe count rate ratio (119877120588) when the ND-CP spans two ormore strata Then the composite density (120588c) within themeasurement spheroid can be calculated by (2) Figure 4shows the spheroid coordinates for the theoretical modelused to calculate 119877120588 [24] The adopted assumptions are asfollows

(1) It is assumed that the scattering and attenuation ofgamma rays occur over a spheroid volume thus the volumeaffecting the measured 119877120588 is a spheroid [27] The center ofthe spheroid is at the midpoint between the source and thedetectorThe vertical section of the spheroid is an ellipse andthe horizontal section of the spheroid is a circle(2) The spheroid volume is constant The length ofsemiaxis b is equal to half of the gamma ray source-detectorseparation distance (265 cm) of the ND-CP The length ofsemiaxis a decreases with increasing density of the surround-ing soil [10] For simplification it is assumed to be a constant

Mathematical Problems in Engineering 3

004

m 011

m0

5 m 0

721

m 0

986

m

145

35 m

Gamma ray detector(Background count)

Gamma ray detector(Density count)

Cone resistance

u Pore pressure

f s

q c

Gamma ray source

Lead shield

Gamma rays

Inclinometer

Gamma rays

Spheroid

Sleeve friction

Figure 1 Nuclear density cone penetrometer

gamma ray detector and SCR is the standard count ratewhich is measured in an inactive material under controlledconditions before each in situ investigation [26] The calibra-tion equation between soil density and 119877120588 is as follows [24]119877120588 = 1198601205882 minus 119861120588 + 119862 (2)

where 120588 is density and A B and C are 06264 40954and 68422 respectively for density values ranging from 10gcm3 to 22 gcm3 Theoretically the ND-CP can be used tomeasure the density of soil up to 22 gcm3 However similarto the piezocone penetrometer the ND-CP is applicable togeneral clay silt and sand especially to loose soil and muckysoil below the groundwater level for which undisturbedsamples are difficult to obtain

23 DensityMeasurement Volume The absorption of gammarays and the measured DCR depend on the density of thesoil within a spheroid between the gamma ray source and

the detector as shown in Figure 1 The measurement volumeis an oblate spheroid centered at the midpoint between thegamma ray source and detector (Figure 3) The length ofsemiaxis b which is the distance from the center to the polealong the symmetry axis is half of the gamma ray source-detector separation distance (2b) of the ND-CP In this studythe value of 2b is 265 cm The length of semiaxis a whichis the equatorial radius of the spheroid decreases with anincreasing density of the surrounding soil [10]The lengths ofa are approximately 24 cm 20 cm and 17 cm in water clayand sand respectively [24] Notably the measured DCR ismore sensitive to the density of the soil closer to the centerof the spheroid

The density measured by the ND-CP is the compositedensity (120588c) of the soil within a spheroid as shown in Figure 3Thus themeasured density (120588c) is not the actual density (120588) ata certain depth If the density changes suddenly with deptheg as in layered soils themeasured120588c profilewill be differentfrom the actual 120588 profile

4 Mathematical Problems in Engineering

O

Si

Electron

Soil particle

Gamma rays passthrough soil particles in the ground

Gamma rays

Gamma rays interact with SiO in soil 2

particles

Gamma rays collide with the electrons in the outer shell

Figure 2 Schematic diagram of Compton scattering of gamma rays in the soil (modified from httpwwwsoilandrockcojp)

Gamma ray source

Gamma ray detector

aba

b

Figure 3 Density measurement volume of the ND-CP

3 Theoretical Model for CalculatingComposite Density

31 Theoretical Model for Predicting Count Rate Ratio Amathematical model is proposed for the ND-CP to predictthe count rate ratio (119877120588) when the ND-CP spans two ormore strata Then the composite density (120588c) within themeasurement spheroid can be calculated by (2) Figure 4shows the spheroid coordinates for the theoretical modelused to calculate 119877120588 [24] The adopted assumptions are asfollows

(1) It is assumed that the scattering and attenuation ofgamma rays occur over a spheroid volume thus the volumeaffecting the measured 119877120588 is a spheroid [27] The center ofthe spheroid is at the midpoint between the source and thedetectorThe vertical section of the spheroid is an ellipse andthe horizontal section of the spheroid is a circle(2) The spheroid volume is constant The length ofsemiaxis b is equal to half of the gamma ray source-detectorseparation distance (265 cm) of the ND-CP The length ofsemiaxis a decreases with increasing density of the surround-ing soil [10] For simplification it is assumed to be a constant

4 Mathematical Problems in Engineering

O

Si

Electron

Soil particle

Gamma rays passthrough soil particles in the ground

Gamma rays

Gamma rays interact with SiO in soil 2

particles

Gamma rays collide with the electrons in the outer shell

Figure 2 Schematic diagram of Compton scattering of gamma rays in the soil (modified from httpwwwsoilandrockcojp)

Gamma ray source

Gamma ray detector

aba

b

Figure 3 Density measurement volume of the ND-CP

3 Theoretical Model for CalculatingComposite Density

31 Theoretical Model for Predicting Count Rate Ratio Amathematical model is proposed for the ND-CP to predictthe count rate ratio (119877120588) when the ND-CP spans two ormore strata Then the composite density (120588c) within themeasurement spheroid can be calculated by (2) Figure 4shows the spheroid coordinates for the theoretical modelused to calculate 119877120588 [24] The adopted assumptions are asfollows

(1) It is assumed that the scattering and attenuation ofgamma rays occur over a spheroid volume thus the volumeaffecting the measured 119877120588 is a spheroid [27] The center ofthe spheroid is at the midpoint between the source and thedetectorThe vertical section of the spheroid is an ellipse andthe horizontal section of the spheroid is a circle(2) The spheroid volume is constant The length ofsemiaxis b is equal to half of the gamma ray source-detectorseparation distance (265 cm) of the ND-CP The length ofsemiaxis a decreases with increasing density of the surround-ing soil [10] For simplification it is assumed to be a constant

Mathematical Problems in Engineering 5

x

Gamma ray source

Gamma ray detector

b

b

aa

z = 0

z

z = z i

z = z i-b

z = z i+b

2b =

26

5 cm

Figure 4 Spheroid coordinates for the theoretical model to calculate the count rate ratio

(3) All parts of the spheroid contribute equally to thescattering and attenuation of radiation and the contributionof the count rate ratio from a small volume (Δ119877120588) in thespheroid to the total count rate ratio (119877120588) is proportional toits volume This assumption is not strictly true but does notlead to a large error [28](4) The calibration equation between 119877120588 and density isdefined by (2) for the soils with densities ranging from 10gcm3 to 22 gcm3

The volume of the spheroid is as follows119881 = 431205871198862119887 (3)

The vertical cross-section of the spheroid in the x-z plane isan ellipse and its equation is11990921198862 + (119911 minus 119911119894)1198872 = 1 (4)

where a is the length of the major semiaxis b is the lengthof the minor semiaxis z is the depth and 119911i is the depthcorresponding to the midpoint of the ellipse The horizontalcross-section of the spheroid at any point is a circle of radiusx where 0 lt 119909 lt 119886 A small volume of the spheroid 120597119881 is120597119881 = 1205871199092120575119911 (5)

where 120575119911 is the differential of zThe x calculated by (4) can besubstituted into (5) to give

120597119881 = 12058711988621198872 [1198872 minus (119911 minus 119911119894)2] 120575119911 (6)

The assumption is that the contribution of the small volume120597119877120588 to 119877120588 is proportional to its volume120597119877120588119877120588 = 120597119881119881 (7)

Substituting (2) (3) and (6) into (7) gives

120597119877120588 = 3 (1198601205882 minus 119861120588 + 119862)41198873 [1198872 minus (119911 minus 119911119894)2] 120575119911 (8)

Assuming the density 120588 is a function of depth 120588 = 120588(119911)integrating (8) over the spheroid volume gives 119877120588119877120588 = 341198873 int(1198601205882 (119911) minus 119861120588 (119911) + 119862) [1198872 minus (119911 minus 119911119894)2] 120575119911 (9)

32 Equations for Calculating the Count Rate Ratio If thedensity function 120588(119911) of the strata is known (9) can beused to calculate the count rate ratio (119877120588) when the ND-CPpenetrates into the strata The probable density profiles inthe ground are described by constant linear parabolic andsquare root functions (Figure 5) the equations of which areas follows

120588 (119911) = 1205880 (10)

120588 (119911) = 1205881 + 1205731 (119911 minus 1199111) (11)

120588 (119911) = 1205721 (119911 minus 1199111)2 + 1205881 (12)

120588 (119911) = 1205722 (119911 minus 1199111)12 + 1205881 (13)

If the density and depth at the top of the stratum (1205881 z1)and at the bottomof the stratum (1205882 z2) are known the values

6 Mathematical Problems in Engineering

Dep

thz

(m)

Density (gcm3)

(z)= 0

(a)D

epth

z(m

)

Density (gcm3)

(1z1)

(z)= 1 + 1(z minus z1)

(2z2)

(b)

Dep

thz

(m)

Density (gcm3)

(1z1)

(2z2)

(z)= 1(z minus z1)2+ 1

(c)

Dep

thz

(m)

Density (gcm3)

(1z1)

(2z2)

(z)= 2(z minus z1)12 + 1

(d)

Figure 5 Assumed density profiles in the ground (a) constant (b) linear (c) parabolic (d) square root

of 1205731 1205721 and 1205722 in (11) (12) and (13) can be determined asfollows

1205731 = (1205882 minus 1205881)(1199112 minus 1199111) (14)

1205721 = (1205882 minus 1205881)(1199112 minus 1199111)2 (15)

1205722 = (1205882 minus 1205881)(1199112 minus 1199111)05 (16)

Eqs (17) (18) (19) and (20) can be used to calculate 119877120588when the density functions are constant linear parabolic andsquare root respectively

119877120588 = 3 (11986012058820 minus 1198611205880 + 119862)41198873 int [1198872 minus (119911 minus 119911119894)2] 120575119911 (17)

119877120588 = 341198873 int119860 [1205881 + 120573 (119911 minus 1199111)]2minus 119861 [1205881 + 120573 (119911 minus 1199111)] + 119862 [1198872 minus (119911 minus 119911119894)2] 120575119911 (18)

119877120588 = 341198873 int119860 [1205721 (119911 minus 1199111)2 + 1205881]2minus 119861 [1205721 (119911 minus 1199111)2 + 1205881] + 119862 [1198872 minus (119911 minus 119911119894)2] 120575119911 (19)

119877120588 = 341198873 int119860 [1205722 (119911 minus 1199111)12 + 1205881]2minus 119861 [1205722 (119911 minus 1199111)12 + 1205881] + 119862 [1198872 minus (119911 minus 119911119894)2] 120575119911 (20)

The solutions for (17) to (20) are derived in (21) to (24)respectively

119877120588 = 341198873 (11986012058820 minus 1198611205880 + 119862) minus11991133 + 1199111198941199112 + (1198872 minus 1199112119894 ) 1199111003816100381610038161003816100381610038161003816100381610038161199112

1199111

(21)

119877120588 = 341198873

minus119860120573211991155 + (minus21198601205881120573 + 21198601199111198941205732 + 211986012057321199111 + 119861120573) 11991144+( 11988721198601205732minus11986012058821+41198601199111198941205881120573minus1198601199112119894 1205732+211986012058811205731199111minus119860120573211991121minus411986011991111989412057321199111+1198611205881minus1198611205731199111minus2119861119911119894120573minus119862

) 11991133+( 2119887

21198601205731205881minus2119887211986012057321199111minus21198601205881120573119911

2

119894+2119860119911119894120588

2

1minus411986011991111989412058811205731199111+2119860119911119894120573

211991121

+2119860119911211989412057321199111minus119887

2119861120573+1198611199112119894120573minus21198611199111198941205881+21198611199111198941205731199111+21198621199111198942 )1199112

+(119887211986012058821 minus 2119887211986012058811205731199111 + 1198872119860120573211991121 minus 1198601199112119894 12058821 + 2119860120588112057311991111199112119894 minus 1198601199112119894 120573211991121minus11988721198611205881 + 11988721198611205731199111 + 1198611199112119894 1205881 minus 1198611199112119894 1205731199111 + 1198621198872 minus 1198621199112119894 )119911

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(22)

Mathematical Problems in Engineering 7

119877120588 = 341198873

(minus11986012057221) 11991177 + (4119860120572211199111 + 211986011991111989412057221) 11991166+(119887211986012057221 minus 61198601205722111991121 minus 211986012057211205881 minus 8119860119911119894120572211199111 minus 119860120572211199112119894 + 1198611205721) 11991155+( minus41198872119860120572211199111+41198601205722111991131+4119860120572112058811199111+121198601199111198941205722111991121+411986011991111989412057211205881+41198601199112119894 120572211199111minus211986111991111205721minus21198611199111198941205721) 11991144

+( 611988721198601205722111991121+2119887211986012057211205881minus1198601205722111991141minus21198601205721120588111991121minus11986012058812minus81198601199111198941205722111991131minus8119860119911119894120572112058811199111minus611986011991121198941205722111991121minus21198601199112

11989412057211205881minus119887

21198611205721+1198611199112

11205721+1198611205881+411986111991111989411991111205721+119861119911

2

1198941205721minus119862

) 11991133+( minus41198872119860120572112058811199111minus411988721198601205722111991131+21198601199111198941205722111991141+211986011991111989412058821+41198601199111198941205721120588111991121+41198601199112119894 1205722111991131+41198601199112

119894120572112058811199111+2119887

211986111991111205721minus21198611199111198941199112

11205721minus21198611199111198941205881minus2119861119911

2

11989411991111205721+2119862119911119894

) 11991122+(211988721198601205721120588111991121 + 119887211986012058821 + 11988721198601205722111991141 minus 1198601199112119894 1205722111991141 minus 21198601199112119894 1205721120588111991121 minus 1198601199112119894 12058821 minus 1198872119861119911211205721minus11988721198611205881 + 1198611199112119894 119911211205721 + 1198611199112119894 1205881 + 1198621198872 minus 1198621199112119894 )119911

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(23)

119877120588 = 341198873

minus1411986012057222 (119911 minus 1199111)4 + 27 (1198611205722 minus 211986012058811205722) (119911 minus 1199111)72+13 [minus211986012057222 (1199111 minus 119911119894)] (119911 minus 1199111)3 minus 13 (11986012058821 minus 1198611205881) (119911 minus 119911119894)3 minus 131198621199113+25 [21198611205722 (1199111 minus 119911119894) minus 411986012058811205722 (1199111 minus 119911119894)] (119911 minus 1199111)52+12 [119887211986012057222 minus 11986012057222 (1199111 minus 119911119894)2] (119911 minus 1199111)2 + 1198621199111198941199112+23 [2119887211986012058811205722 minus 211986012058811205722 (1199111 minus 119911119894)2 minus 11988721198611205722 + 1198611205722 (1199111 minus 119911119894)2] (119911 minus 1199111)32+ [119887211986012058821 minus 11988721198611205881 + 119862 (1198872 minus 1199112119894 )] 119911

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(24)

If the gamma ray source and detector are in strata thatcan be described by one function of density 119877120588 is calculatedas an integral of that function If the gamma ray source anddetector are in strata described by two or more functions119877120588 is calculated as a piecewise integration of those functionsThen the composite density (120588c) which is referenced to themidpoint of the spheroid can be calculated by (25) Finallythe 120588c profile can be obtained as the midpoint of the spheroidmoves with depth along with the penetration of the ND-CP

120588119888 = 40954 minus radic25056119877120588 minus 0371512528 (25)

33 Verification of the Theoretical Model The ND-CP wasused to measure the density profiles of two kinds of layeredsoils by Karthikeyan and Tan in the laboratory [29] Theexperimental setup consists of a loading frame and a cubicsteel tank with dimensions of 15 mtimes15 mtimes15 mThe loadingframe is placed on top of the steel tank to push the ND-CPusing a hydraulic piston The ND-CP measurement and thedepth meter readings were recorded through a data loggerFigure 6 shows schematic configurations for layered soilsin the steel tank One configuration is with a top layer ofwater and bottom layer of soil (Figure 6(a)) and the otherconfiguration is with a top layer of kaolin clay slurry andbottom layer of soil (Figure 6(b)) The measured 120588c profilesdo not show a sharp transition at the interface of the waterand soil layers (Figure 6(a)) or at the interface of the slurryand soil layers (Figure 6(b)) The change in density from the

waterslurry to the soil is extended over a distance equal tothe source-detector separation (2b) This transition occurs asthe gamma ray source enters the soil layer and continues untilthe gamma ray detector enters the soil layer The actual 120588profile shows a step change in density while the measured120588c profile shows a continuous change in density The changepattern in themeasured 120588c profile indicates that theND-CP ispenetrating the interface of two layers with different densityvalues The 120588c profiles predicted by the theoretical model arealso shown in the figure and these predicted results showa close agreement with the measured 120588c profiles suggestingthat the theoretical model is able to describe the signature ofthe 120588c profilesmeasured by theND-CPThe theoreticalmodelwill be used to investigate the differences between the actual120588 profiles in the ground and the measured 120588c profiles by theND-CP

4 Method for Deducing Actual 120588 Profile

41 Comparison of 120588 and 120588119888 Profiles The density profiles ofcomplex strata which are common in engineering practiceare assumed in Figure 7 Figure 7(a) shows the density profilesof layered strata The density of the first layer is 16 gcm3and the densities of the second layer are 15 gcm3 17 gcm3or 18 gcm3 Figure 7(b) shows the density profiles of stratawith soft interlayers The density of the strata is 17 gcm3Three different soft interlayers are assumed For the first casethe soft interlayer has a density of 13 gcm3 and a thicknessof 015 m for the second case the density is 15 gcm3 and

8 Mathematical Problems in Engineering

8 12 16 20

Soil

Measured by ND-CPPredicted by model

Water

Actual density profile

09

07

05

03

01

Dep

th (m

)

Wet density (kNm3)

(a)

12 14 16 18 20

Soil

Measured by ND-CPPredicted by model

Kaolin clay slurry

Actual density profile

09

07

05

03

0110

Dep

th (m

)

Wet density (kNm3)

(b)

Figure 6 Verification of the theoretical model with laboratory test results

the thickness is 015 m for the third case the density is13 gcm3 and the thickness is 03 m Figure 7(c) shows thedensity profiles of bed sediments The density of the wateris 10 gcm3 and the density of the bed mud increases withdepth in a linear fashion Three different layers of fluid mudare assumed For the first case the fluid mud has a maximumdensity of 12 gcm3 and thickness of 02 m for the secondcase the maximum density is 13 gcm3 and the thickness is02 m for the third case the maximum density is 115 gcm3and the thickness is 025 m

The 120588c profiles are calculated by the theoretical modelwhen the ND-CP penetrates into the assumed complex strata(Figure 7) and the results are compared with the 120588 profilesThe 120588c profiles are clearly different from the 120588 profiles nearthe interface of the layered strata near the soft interlayer andnear the upper and lower boundaries of the fluid mud Thedifferences between 120588c and 120588 near the interface of the layeredstrata in Figure 7(a) are explained as follows (1)When boththe gamma ray source and detector are in the top layer 120588cis equal to the density of the top layer (2) Once the gammaray source enters the bottom layer 120588c will be the weightedaverage density of the two layers while 120588 is still the densityof the first layer (3) When the midpoint of the gamma raysource and detector enters the bottom layer 120588 will change tothe density of the bottom layer while 120588c is still the weightedaverage density of the two layers because the gamma raydetector is still in the first layer (4) When the gamma raydetector also enters the bottom layer 120588c is equal to 120588 of thebottom layer Similar to the profiles for the case in which theND-CP passes through the interface of layered strata the 120588cprofiles are different from the 120588 profiles when the ND-CP

passes through a soft interlayer and the fluid mudThereforethe 120588c profile measured by the ND-CP must be interpretedand the actual 120588 profile must be deduced from the measured120588c profile42 Method for Deducing Actual 120588 Profile from Measured 120588119888Profile A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed as follows(1)Determining the depths of and densities at the bound-aries of each stratum based on the features of the measured120588c profile (2) Assuming the probable density function ofeach stratum and the probable 120588 profile of the strata (3)Back-calculating the 120588c profile by using the theoretical modelfor the case of the ND-CP passing through layered strata(4) Comparing the calculated 120588c profile with the measured120588c profile and adjusting the assumed 120588 profile until thecalculated 120588c profile matches the measured 120588c profile well (5)Determining the actual 120588 profile to be the assumed 120588 profilewhen the calculated 120588c profile fits the measured 120588c profile

For the layered strata as shown in Figure 7(a) themeasured 120588c profile can be divided into three sectionsnamely the first constant section the second curved sectionand the third constant section The measured 120588c values aredifferent from the actual 120588 values in the range of 2b whichis the distance between the gamma ray source and detectorFrom the 120588c profile the boundary between layered strata canbe determined ie themidpoint of the second curved sectionof the 120588c profileThe actual 120588 value of the top layer is assumedto be the measured 120588c value of the first constant section andthe actual 120588 value of the bottom layer is assumed to be themeasured 120588c value of the third constant sectionThe 120588c profile

Mathematical Problems in Engineering 9

13 14 15 16 17 18 19 20

16

14

12

10

08

06

(4)

(3)

(2)

Bottom layer

Dep

th (m

)Top layer

(1)

Assumed and calculated ＝ (gcm3)

(16 to 15)Assumed

(16 to 15)Calculated ＝(16 to 17)Assumed

(16 to 17)Calculated ＝(16 to 18)Assumed

(16 to 18)Calculated ＝

(a) Layered strata

12 13 14 15 16 17 18

18

16

14

12

10

08

06

Bottom layer

Top layer

(13 015) (13 015)

(15 015) (15 015)

(13 03) (13 03)

Dep

th (m

)

Soft interlayer

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(b) Strata with soft interlayers

09 10 11 12 13 14

16

14

12

10

08

06

Bed mud

Fluid mud

(12 02) (12 02)

(13 02) (13 02)

(115 025) (115 025)

Dep

th (m

)

Water

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(c) Bed sediments

Figure 7 Comparison of assumed 120588 and calculated 120588c profiles

is back-calculated assuming that the ND-CP passes throughthe layered strata If the calculated120588c profilematcheswell withthe measured 120588c profile the actual 120588 profile of the layeredstrata was determined

For the strata with a soft interlayer as shown in Fig-ure 7(b) the measured 120588c profile can be divided into threesections namely the first constant section the second curvedsection and the third constant section The measured 120588cvalues are different from the actual 120588 values near the softinterlayer (2b plus the thickness of the soft interlayer) From

the 120588c profile the upper and lower boundaries of the softinterlayer can be determined that is the depth of the startpoint of the second curved section plus b and the depth of theend point of the second curved section minus b The actual120588 of the top layer is assumed to be the measured 120588c of thefirst constant section and the actual 120588 of the bottom layer isassumed to be the measured 120588c of the third constant sectionThe actual 120588 of the soft interlayer can be assumed based onthe minimum 120588c in the second curved section The 120588c profileis back-calculated assuming that the ND-CP passes through

10 Mathematical Problems in Engineering

Tokyo

JAPANA1

A2 Osaka

Kyushu

Isahaya Bay

Ariake Sea Sapporo

5 km

N

Figure 8 Investigation locations

0

1

2

3

4

5

Dep

th fr

om w

ater

surfa

cez

(m)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

Middlesection

Finalsection

(a)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

section

Finalsection

Dep

th fr

om w

ater

surfa

ce (m

)

8

9

10

11

12

(b)

Figure 9 Investigation results (a) at location A1 (b) at location A2

the strata with a soft interlayer The actual 120588 profile of stratawith a soft interlayer is determined when the calculated 120588cprofile matches the measured 120588c profile well

For bed sediments as shown in Figure 7(c) the measured120588c profile can be divided into three sections namely the firstconstant section the second curved section and the thirdlinear section The measured 120588c values are different fromthe actual 120588 values near the upper and lower boundariesof the fluid mud From the 120588c profile the upper and lowerboundaries of the fluid mud can be determined that is thedepth of the point from which density increases plus b andthe depth of the point from which density slightly increaseslinearly minus b The actual 120588 of the water layer is assumed tobe the measured 120588c of the first constant section The slope ofthe actual 120588 profile of the bed mud layer is assumed to be theslope of themeasured120588c profile of the third linear sectionTheactual 120588 distribution of the fluidmud layer can be assumed to

be a parabolic linear or square root function The 120588c profileis back-calculated assuming that the ND-CP passes throughbed sediments The actual 120588 profile of bed sediments can bedetermined when the calculated 120588c profile fits the measured120588c profile5 An Example of Practical Application

51 ND-CP Measured 120588119888 Profile The density profiles of bedsediments at Isahaya Bay Japan were investigated using theND-CP The details on the equipment and methods used inthis investigation can be found in a prior publication [23]TheND-CP measured density profiles at two locations (Figure 8)are used to illustrate how to use the proposed method todetermine the actual 120588 profile from the measured 120588c profile

The measured BCR and DCR and the calculated 120588cvalues at locations A1 and A2 are shown in Figure 9 BCR

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

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6 Mathematical Problems in Engineering

Dep

thz

(m)

Density (gcm3)

(z)= 0

(a)D

epth

z(m

)

Density (gcm3)

(1z1)

(z)= 1 + 1(z minus z1)

(2z2)

(b)

Dep

thz

(m)

Density (gcm3)

(1z1)

(2z2)

(z)= 1(z minus z1)2+ 1

(c)

Dep

thz

(m)

Density (gcm3)

(1z1)

(2z2)

(z)= 2(z minus z1)12 + 1

(d)

Figure 5 Assumed density profiles in the ground (a) constant (b) linear (c) parabolic (d) square root

of 1205731 1205721 and 1205722 in (11) (12) and (13) can be determined asfollows

1205731 = (1205882 minus 1205881)(1199112 minus 1199111) (14)

1205721 = (1205882 minus 1205881)(1199112 minus 1199111)2 (15)

1205722 = (1205882 minus 1205881)(1199112 minus 1199111)05 (16)

Eqs (17) (18) (19) and (20) can be used to calculate 119877120588when the density functions are constant linear parabolic andsquare root respectively

119877120588 = 3 (11986012058820 minus 1198611205880 + 119862)41198873 int [1198872 minus (119911 minus 119911119894)2] 120575119911 (17)

119877120588 = 341198873 int119860 [1205881 + 120573 (119911 minus 1199111)]2minus 119861 [1205881 + 120573 (119911 minus 1199111)] + 119862 [1198872 minus (119911 minus 119911119894)2] 120575119911 (18)

119877120588 = 341198873 int119860 [1205721 (119911 minus 1199111)2 + 1205881]2minus 119861 [1205721 (119911 minus 1199111)2 + 1205881] + 119862 [1198872 minus (119911 minus 119911119894)2] 120575119911 (19)

119877120588 = 341198873 int119860 [1205722 (119911 minus 1199111)12 + 1205881]2minus 119861 [1205722 (119911 minus 1199111)12 + 1205881] + 119862 [1198872 minus (119911 minus 119911119894)2] 120575119911 (20)

The solutions for (17) to (20) are derived in (21) to (24)respectively

119877120588 = 341198873 (11986012058820 minus 1198611205880 + 119862) minus11991133 + 1199111198941199112 + (1198872 minus 1199112119894 ) 1199111003816100381610038161003816100381610038161003816100381610038161199112

1199111

(21)

119877120588 = 341198873

minus119860120573211991155 + (minus21198601205881120573 + 21198601199111198941205732 + 211986012057321199111 + 119861120573) 11991144+( 11988721198601205732minus11986012058821+41198601199111198941205881120573minus1198601199112119894 1205732+211986012058811205731199111minus119860120573211991121minus411986011991111989412057321199111+1198611205881minus1198611205731199111minus2119861119911119894120573minus119862

) 11991133+( 2119887

21198601205731205881minus2119887211986012057321199111minus21198601205881120573119911

2

119894+2119860119911119894120588

2

1minus411986011991111989412058811205731199111+2119860119911119894120573

211991121

+2119860119911211989412057321199111minus119887

2119861120573+1198611199112119894120573minus21198611199111198941205881+21198611199111198941205731199111+21198621199111198942 )1199112

+(119887211986012058821 minus 2119887211986012058811205731199111 + 1198872119860120573211991121 minus 1198601199112119894 12058821 + 2119860120588112057311991111199112119894 minus 1198601199112119894 120573211991121minus11988721198611205881 + 11988721198611205731199111 + 1198611199112119894 1205881 minus 1198611199112119894 1205731199111 + 1198621198872 minus 1198621199112119894 )119911

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(22)

Mathematical Problems in Engineering 7

119877120588 = 341198873

(minus11986012057221) 11991177 + (4119860120572211199111 + 211986011991111989412057221) 11991166+(119887211986012057221 minus 61198601205722111991121 minus 211986012057211205881 minus 8119860119911119894120572211199111 minus 119860120572211199112119894 + 1198611205721) 11991155+( minus41198872119860120572211199111+41198601205722111991131+4119860120572112058811199111+121198601199111198941205722111991121+411986011991111989412057211205881+41198601199112119894 120572211199111minus211986111991111205721minus21198611199111198941205721) 11991144

+( 611988721198601205722111991121+2119887211986012057211205881minus1198601205722111991141minus21198601205721120588111991121minus11986012058812minus81198601199111198941205722111991131minus8119860119911119894120572112058811199111minus611986011991121198941205722111991121minus21198601199112

11989412057211205881minus119887

21198611205721+1198611199112

11205721+1198611205881+411986111991111989411991111205721+119861119911

2

1198941205721minus119862

) 11991133+( minus41198872119860120572112058811199111minus411988721198601205722111991131+21198601199111198941205722111991141+211986011991111989412058821+41198601199111198941205721120588111991121+41198601199112119894 1205722111991131+41198601199112

119894120572112058811199111+2119887

211986111991111205721minus21198611199111198941199112

11205721minus21198611199111198941205881minus2119861119911

2

11989411991111205721+2119862119911119894

) 11991122+(211988721198601205721120588111991121 + 119887211986012058821 + 11988721198601205722111991141 minus 1198601199112119894 1205722111991141 minus 21198601199112119894 1205721120588111991121 minus 1198601199112119894 12058821 minus 1198872119861119911211205721minus11988721198611205881 + 1198611199112119894 119911211205721 + 1198611199112119894 1205881 + 1198621198872 minus 1198621199112119894 )119911

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(23)

119877120588 = 341198873

minus1411986012057222 (119911 minus 1199111)4 + 27 (1198611205722 minus 211986012058811205722) (119911 minus 1199111)72+13 [minus211986012057222 (1199111 minus 119911119894)] (119911 minus 1199111)3 minus 13 (11986012058821 minus 1198611205881) (119911 minus 119911119894)3 minus 131198621199113+25 [21198611205722 (1199111 minus 119911119894) minus 411986012058811205722 (1199111 minus 119911119894)] (119911 minus 1199111)52+12 [119887211986012057222 minus 11986012057222 (1199111 minus 119911119894)2] (119911 minus 1199111)2 + 1198621199111198941199112+23 [2119887211986012058811205722 minus 211986012058811205722 (1199111 minus 119911119894)2 minus 11988721198611205722 + 1198611205722 (1199111 minus 119911119894)2] (119911 minus 1199111)32+ [119887211986012058821 minus 11988721198611205881 + 119862 (1198872 minus 1199112119894 )] 119911

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(24)

If the gamma ray source and detector are in strata thatcan be described by one function of density 119877120588 is calculatedas an integral of that function If the gamma ray source anddetector are in strata described by two or more functions119877120588 is calculated as a piecewise integration of those functionsThen the composite density (120588c) which is referenced to themidpoint of the spheroid can be calculated by (25) Finallythe 120588c profile can be obtained as the midpoint of the spheroidmoves with depth along with the penetration of the ND-CP

120588119888 = 40954 minus radic25056119877120588 minus 0371512528 (25)

33 Verification of the Theoretical Model The ND-CP wasused to measure the density profiles of two kinds of layeredsoils by Karthikeyan and Tan in the laboratory [29] Theexperimental setup consists of a loading frame and a cubicsteel tank with dimensions of 15 mtimes15 mtimes15 mThe loadingframe is placed on top of the steel tank to push the ND-CPusing a hydraulic piston The ND-CP measurement and thedepth meter readings were recorded through a data loggerFigure 6 shows schematic configurations for layered soilsin the steel tank One configuration is with a top layer ofwater and bottom layer of soil (Figure 6(a)) and the otherconfiguration is with a top layer of kaolin clay slurry andbottom layer of soil (Figure 6(b)) The measured 120588c profilesdo not show a sharp transition at the interface of the waterand soil layers (Figure 6(a)) or at the interface of the slurryand soil layers (Figure 6(b)) The change in density from the

waterslurry to the soil is extended over a distance equal tothe source-detector separation (2b) This transition occurs asthe gamma ray source enters the soil layer and continues untilthe gamma ray detector enters the soil layer The actual 120588profile shows a step change in density while the measured120588c profile shows a continuous change in density The changepattern in themeasured 120588c profile indicates that theND-CP ispenetrating the interface of two layers with different densityvalues The 120588c profiles predicted by the theoretical model arealso shown in the figure and these predicted results showa close agreement with the measured 120588c profiles suggestingthat the theoretical model is able to describe the signature ofthe 120588c profilesmeasured by theND-CPThe theoreticalmodelwill be used to investigate the differences between the actual120588 profiles in the ground and the measured 120588c profiles by theND-CP

4 Method for Deducing Actual 120588 Profile

41 Comparison of 120588 and 120588119888 Profiles The density profiles ofcomplex strata which are common in engineering practiceare assumed in Figure 7 Figure 7(a) shows the density profilesof layered strata The density of the first layer is 16 gcm3and the densities of the second layer are 15 gcm3 17 gcm3or 18 gcm3 Figure 7(b) shows the density profiles of stratawith soft interlayers The density of the strata is 17 gcm3Three different soft interlayers are assumed For the first casethe soft interlayer has a density of 13 gcm3 and a thicknessof 015 m for the second case the density is 15 gcm3 and

8 Mathematical Problems in Engineering

8 12 16 20

Soil

Measured by ND-CPPredicted by model

Water

Actual density profile

09

07

05

03

01

Dep

th (m

)

Wet density (kNm3)

(a)

12 14 16 18 20

Soil

Measured by ND-CPPredicted by model

Kaolin clay slurry

Actual density profile

09

07

05

03

0110

Dep

th (m

)

Wet density (kNm3)

(b)

Figure 6 Verification of the theoretical model with laboratory test results

the thickness is 015 m for the third case the density is13 gcm3 and the thickness is 03 m Figure 7(c) shows thedensity profiles of bed sediments The density of the wateris 10 gcm3 and the density of the bed mud increases withdepth in a linear fashion Three different layers of fluid mudare assumed For the first case the fluid mud has a maximumdensity of 12 gcm3 and thickness of 02 m for the secondcase the maximum density is 13 gcm3 and the thickness is02 m for the third case the maximum density is 115 gcm3and the thickness is 025 m

The 120588c profiles are calculated by the theoretical modelwhen the ND-CP penetrates into the assumed complex strata(Figure 7) and the results are compared with the 120588 profilesThe 120588c profiles are clearly different from the 120588 profiles nearthe interface of the layered strata near the soft interlayer andnear the upper and lower boundaries of the fluid mud Thedifferences between 120588c and 120588 near the interface of the layeredstrata in Figure 7(a) are explained as follows (1)When boththe gamma ray source and detector are in the top layer 120588cis equal to the density of the top layer (2) Once the gammaray source enters the bottom layer 120588c will be the weightedaverage density of the two layers while 120588 is still the densityof the first layer (3) When the midpoint of the gamma raysource and detector enters the bottom layer 120588 will change tothe density of the bottom layer while 120588c is still the weightedaverage density of the two layers because the gamma raydetector is still in the first layer (4) When the gamma raydetector also enters the bottom layer 120588c is equal to 120588 of thebottom layer Similar to the profiles for the case in which theND-CP passes through the interface of layered strata the 120588cprofiles are different from the 120588 profiles when the ND-CP

passes through a soft interlayer and the fluid mudThereforethe 120588c profile measured by the ND-CP must be interpretedand the actual 120588 profile must be deduced from the measured120588c profile42 Method for Deducing Actual 120588 Profile from Measured 120588119888Profile A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed as follows(1)Determining the depths of and densities at the bound-aries of each stratum based on the features of the measured120588c profile (2) Assuming the probable density function ofeach stratum and the probable 120588 profile of the strata (3)Back-calculating the 120588c profile by using the theoretical modelfor the case of the ND-CP passing through layered strata(4) Comparing the calculated 120588c profile with the measured120588c profile and adjusting the assumed 120588 profile until thecalculated 120588c profile matches the measured 120588c profile well (5)Determining the actual 120588 profile to be the assumed 120588 profilewhen the calculated 120588c profile fits the measured 120588c profile

For the layered strata as shown in Figure 7(a) themeasured 120588c profile can be divided into three sectionsnamely the first constant section the second curved sectionand the third constant section The measured 120588c values aredifferent from the actual 120588 values in the range of 2b whichis the distance between the gamma ray source and detectorFrom the 120588c profile the boundary between layered strata canbe determined ie themidpoint of the second curved sectionof the 120588c profileThe actual 120588 value of the top layer is assumedto be the measured 120588c value of the first constant section andthe actual 120588 value of the bottom layer is assumed to be themeasured 120588c value of the third constant sectionThe 120588c profile

Mathematical Problems in Engineering 9

13 14 15 16 17 18 19 20

16

14

12

10

08

06

(4)

(3)

(2)

Bottom layer

Dep

th (m

)Top layer

(1)

Assumed and calculated ＝ (gcm3)

(16 to 15)Assumed

(16 to 15)Calculated ＝(16 to 17)Assumed

(16 to 17)Calculated ＝(16 to 18)Assumed

(16 to 18)Calculated ＝

(a) Layered strata

12 13 14 15 16 17 18

18

16

14

12

10

08

06

Bottom layer

Top layer

(13 015) (13 015)

(15 015) (15 015)

(13 03) (13 03)

Dep

th (m

)

Soft interlayer

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(b) Strata with soft interlayers

09 10 11 12 13 14

16

14

12

10

08

06

Bed mud

Fluid mud

(12 02) (12 02)

(13 02) (13 02)

(115 025) (115 025)

Dep

th (m

)

Water

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(c) Bed sediments

Figure 7 Comparison of assumed 120588 and calculated 120588c profiles

is back-calculated assuming that the ND-CP passes throughthe layered strata If the calculated120588c profilematcheswell withthe measured 120588c profile the actual 120588 profile of the layeredstrata was determined

For the strata with a soft interlayer as shown in Fig-ure 7(b) the measured 120588c profile can be divided into threesections namely the first constant section the second curvedsection and the third constant section The measured 120588cvalues are different from the actual 120588 values near the softinterlayer (2b plus the thickness of the soft interlayer) From

the 120588c profile the upper and lower boundaries of the softinterlayer can be determined that is the depth of the startpoint of the second curved section plus b and the depth of theend point of the second curved section minus b The actual120588 of the top layer is assumed to be the measured 120588c of thefirst constant section and the actual 120588 of the bottom layer isassumed to be the measured 120588c of the third constant sectionThe actual 120588 of the soft interlayer can be assumed based onthe minimum 120588c in the second curved section The 120588c profileis back-calculated assuming that the ND-CP passes through

10 Mathematical Problems in Engineering

Tokyo

JAPANA1

A2 Osaka

Kyushu

Isahaya Bay

Ariake Sea Sapporo

5 km

N

Figure 8 Investigation locations

0

1

2

3

4

5

Dep

th fr

om w

ater

surfa

cez

(m)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

Middlesection

Finalsection

(a)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

section

Finalsection

Dep

th fr

om w

ater

surfa

ce (m

)

8

9

10

11

12

(b)

Figure 9 Investigation results (a) at location A1 (b) at location A2

the strata with a soft interlayer The actual 120588 profile of stratawith a soft interlayer is determined when the calculated 120588cprofile matches the measured 120588c profile well

For bed sediments as shown in Figure 7(c) the measured120588c profile can be divided into three sections namely the firstconstant section the second curved section and the thirdlinear section The measured 120588c values are different fromthe actual 120588 values near the upper and lower boundariesof the fluid mud From the 120588c profile the upper and lowerboundaries of the fluid mud can be determined that is thedepth of the point from which density increases plus b andthe depth of the point from which density slightly increaseslinearly minus b The actual 120588 of the water layer is assumed tobe the measured 120588c of the first constant section The slope ofthe actual 120588 profile of the bed mud layer is assumed to be theslope of themeasured120588c profile of the third linear sectionTheactual 120588 distribution of the fluidmud layer can be assumed to

be a parabolic linear or square root function The 120588c profileis back-calculated assuming that the ND-CP passes throughbed sediments The actual 120588 profile of bed sediments can bedetermined when the calculated 120588c profile fits the measured120588c profile5 An Example of Practical Application

51 ND-CP Measured 120588119888 Profile The density profiles of bedsediments at Isahaya Bay Japan were investigated using theND-CP The details on the equipment and methods used inthis investigation can be found in a prior publication [23]TheND-CP measured density profiles at two locations (Figure 8)are used to illustrate how to use the proposed method todetermine the actual 120588 profile from the measured 120588c profile

The measured BCR and DCR and the calculated 120588cvalues at locations A1 and A2 are shown in Figure 9 BCR

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

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Mathematical Problems in Engineering 7

119877120588 = 341198873

(minus11986012057221) 11991177 + (4119860120572211199111 + 211986011991111989412057221) 11991166+(119887211986012057221 minus 61198601205722111991121 minus 211986012057211205881 minus 8119860119911119894120572211199111 minus 119860120572211199112119894 + 1198611205721) 11991155+( minus41198872119860120572211199111+41198601205722111991131+4119860120572112058811199111+121198601199111198941205722111991121+411986011991111989412057211205881+41198601199112119894 120572211199111minus211986111991111205721minus21198611199111198941205721) 11991144

+( 611988721198601205722111991121+2119887211986012057211205881minus1198601205722111991141minus21198601205721120588111991121minus11986012058812minus81198601199111198941205722111991131minus8119860119911119894120572112058811199111minus611986011991121198941205722111991121minus21198601199112

11989412057211205881minus119887

21198611205721+1198611199112

11205721+1198611205881+411986111991111989411991111205721+119861119911

2

1198941205721minus119862

) 11991133+( minus41198872119860120572112058811199111minus411988721198601205722111991131+21198601199111198941205722111991141+211986011991111989412058821+41198601199111198941205721120588111991121+41198601199112119894 1205722111991131+41198601199112

119894120572112058811199111+2119887

211986111991111205721minus21198611199111198941199112

11205721minus21198611199111198941205881minus2119861119911

2

11989411991111205721+2119862119911119894

) 11991122+(211988721198601205721120588111991121 + 119887211986012058821 + 11988721198601205722111991141 minus 1198601199112119894 1205722111991141 minus 21198601199112119894 1205721120588111991121 minus 1198601199112119894 12058821 minus 1198872119861119911211205721minus11988721198611205881 + 1198611199112119894 119911211205721 + 1198611199112119894 1205881 + 1198621198872 minus 1198621199112119894 )119911

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(23)

119877120588 = 341198873

minus1411986012057222 (119911 minus 1199111)4 + 27 (1198611205722 minus 211986012058811205722) (119911 minus 1199111)72+13 [minus211986012057222 (1199111 minus 119911119894)] (119911 minus 1199111)3 minus 13 (11986012058821 minus 1198611205881) (119911 minus 119911119894)3 minus 131198621199113+25 [21198611205722 (1199111 minus 119911119894) minus 411986012058811205722 (1199111 minus 119911119894)] (119911 minus 1199111)52+12 [119887211986012057222 minus 11986012057222 (1199111 minus 119911119894)2] (119911 minus 1199111)2 + 1198621199111198941199112+23 [2119887211986012058811205722 minus 211986012058811205722 (1199111 minus 119911119894)2 minus 11988721198611205722 + 1198611205722 (1199111 minus 119911119894)2] (119911 minus 1199111)32+ [119887211986012058821 minus 11988721198611205881 + 119862 (1198872 minus 1199112119894 )] 119911

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199112

1199111

(24)

If the gamma ray source and detector are in strata thatcan be described by one function of density 119877120588 is calculatedas an integral of that function If the gamma ray source anddetector are in strata described by two or more functions119877120588 is calculated as a piecewise integration of those functionsThen the composite density (120588c) which is referenced to themidpoint of the spheroid can be calculated by (25) Finallythe 120588c profile can be obtained as the midpoint of the spheroidmoves with depth along with the penetration of the ND-CP

120588119888 = 40954 minus radic25056119877120588 minus 0371512528 (25)

33 Verification of the Theoretical Model The ND-CP wasused to measure the density profiles of two kinds of layeredsoils by Karthikeyan and Tan in the laboratory [29] Theexperimental setup consists of a loading frame and a cubicsteel tank with dimensions of 15 mtimes15 mtimes15 mThe loadingframe is placed on top of the steel tank to push the ND-CPusing a hydraulic piston The ND-CP measurement and thedepth meter readings were recorded through a data loggerFigure 6 shows schematic configurations for layered soilsin the steel tank One configuration is with a top layer ofwater and bottom layer of soil (Figure 6(a)) and the otherconfiguration is with a top layer of kaolin clay slurry andbottom layer of soil (Figure 6(b)) The measured 120588c profilesdo not show a sharp transition at the interface of the waterand soil layers (Figure 6(a)) or at the interface of the slurryand soil layers (Figure 6(b)) The change in density from the

waterslurry to the soil is extended over a distance equal tothe source-detector separation (2b) This transition occurs asthe gamma ray source enters the soil layer and continues untilthe gamma ray detector enters the soil layer The actual 120588profile shows a step change in density while the measured120588c profile shows a continuous change in density The changepattern in themeasured 120588c profile indicates that theND-CP ispenetrating the interface of two layers with different densityvalues The 120588c profiles predicted by the theoretical model arealso shown in the figure and these predicted results showa close agreement with the measured 120588c profiles suggestingthat the theoretical model is able to describe the signature ofthe 120588c profilesmeasured by theND-CPThe theoreticalmodelwill be used to investigate the differences between the actual120588 profiles in the ground and the measured 120588c profiles by theND-CP

4 Method for Deducing Actual 120588 Profile

41 Comparison of 120588 and 120588119888 Profiles The density profiles ofcomplex strata which are common in engineering practiceare assumed in Figure 7 Figure 7(a) shows the density profilesof layered strata The density of the first layer is 16 gcm3and the densities of the second layer are 15 gcm3 17 gcm3or 18 gcm3 Figure 7(b) shows the density profiles of stratawith soft interlayers The density of the strata is 17 gcm3Three different soft interlayers are assumed For the first casethe soft interlayer has a density of 13 gcm3 and a thicknessof 015 m for the second case the density is 15 gcm3 and

8 Mathematical Problems in Engineering

8 12 16 20

Soil

Measured by ND-CPPredicted by model

Water

Actual density profile

09

07

05

03

01

Dep

th (m

)

Wet density (kNm3)

(a)

12 14 16 18 20

Soil

Measured by ND-CPPredicted by model

Kaolin clay slurry

Actual density profile

09

07

05

03

0110

Dep

th (m

)

Wet density (kNm3)

(b)

Figure 6 Verification of the theoretical model with laboratory test results

the thickness is 015 m for the third case the density is13 gcm3 and the thickness is 03 m Figure 7(c) shows thedensity profiles of bed sediments The density of the wateris 10 gcm3 and the density of the bed mud increases withdepth in a linear fashion Three different layers of fluid mudare assumed For the first case the fluid mud has a maximumdensity of 12 gcm3 and thickness of 02 m for the secondcase the maximum density is 13 gcm3 and the thickness is02 m for the third case the maximum density is 115 gcm3and the thickness is 025 m

The 120588c profiles are calculated by the theoretical modelwhen the ND-CP penetrates into the assumed complex strata(Figure 7) and the results are compared with the 120588 profilesThe 120588c profiles are clearly different from the 120588 profiles nearthe interface of the layered strata near the soft interlayer andnear the upper and lower boundaries of the fluid mud Thedifferences between 120588c and 120588 near the interface of the layeredstrata in Figure 7(a) are explained as follows (1)When boththe gamma ray source and detector are in the top layer 120588cis equal to the density of the top layer (2) Once the gammaray source enters the bottom layer 120588c will be the weightedaverage density of the two layers while 120588 is still the densityof the first layer (3) When the midpoint of the gamma raysource and detector enters the bottom layer 120588 will change tothe density of the bottom layer while 120588c is still the weightedaverage density of the two layers because the gamma raydetector is still in the first layer (4) When the gamma raydetector also enters the bottom layer 120588c is equal to 120588 of thebottom layer Similar to the profiles for the case in which theND-CP passes through the interface of layered strata the 120588cprofiles are different from the 120588 profiles when the ND-CP

passes through a soft interlayer and the fluid mudThereforethe 120588c profile measured by the ND-CP must be interpretedand the actual 120588 profile must be deduced from the measured120588c profile42 Method for Deducing Actual 120588 Profile from Measured 120588119888Profile A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed as follows(1)Determining the depths of and densities at the bound-aries of each stratum based on the features of the measured120588c profile (2) Assuming the probable density function ofeach stratum and the probable 120588 profile of the strata (3)Back-calculating the 120588c profile by using the theoretical modelfor the case of the ND-CP passing through layered strata(4) Comparing the calculated 120588c profile with the measured120588c profile and adjusting the assumed 120588 profile until thecalculated 120588c profile matches the measured 120588c profile well (5)Determining the actual 120588 profile to be the assumed 120588 profilewhen the calculated 120588c profile fits the measured 120588c profile

For the layered strata as shown in Figure 7(a) themeasured 120588c profile can be divided into three sectionsnamely the first constant section the second curved sectionand the third constant section The measured 120588c values aredifferent from the actual 120588 values in the range of 2b whichis the distance between the gamma ray source and detectorFrom the 120588c profile the boundary between layered strata canbe determined ie themidpoint of the second curved sectionof the 120588c profileThe actual 120588 value of the top layer is assumedto be the measured 120588c value of the first constant section andthe actual 120588 value of the bottom layer is assumed to be themeasured 120588c value of the third constant sectionThe 120588c profile

Mathematical Problems in Engineering 9

13 14 15 16 17 18 19 20

16

14

12

10

08

06

(4)

(3)

(2)

Bottom layer

Dep

th (m

)Top layer

(1)

Assumed and calculated ＝ (gcm3)

(16 to 15)Assumed

(16 to 15)Calculated ＝(16 to 17)Assumed

(16 to 17)Calculated ＝(16 to 18)Assumed

(16 to 18)Calculated ＝

(a) Layered strata

12 13 14 15 16 17 18

18

16

14

12

10

08

06

Bottom layer

Top layer

(13 015) (13 015)

(15 015) (15 015)

(13 03) (13 03)

Dep

th (m

)

Soft interlayer

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(b) Strata with soft interlayers

09 10 11 12 13 14

16

14

12

10

08

06

Bed mud

Fluid mud

(12 02) (12 02)

(13 02) (13 02)

(115 025) (115 025)

Dep

th (m

)

Water

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(c) Bed sediments

Figure 7 Comparison of assumed 120588 and calculated 120588c profiles

is back-calculated assuming that the ND-CP passes throughthe layered strata If the calculated120588c profilematcheswell withthe measured 120588c profile the actual 120588 profile of the layeredstrata was determined

For the strata with a soft interlayer as shown in Fig-ure 7(b) the measured 120588c profile can be divided into threesections namely the first constant section the second curvedsection and the third constant section The measured 120588cvalues are different from the actual 120588 values near the softinterlayer (2b plus the thickness of the soft interlayer) From

the 120588c profile the upper and lower boundaries of the softinterlayer can be determined that is the depth of the startpoint of the second curved section plus b and the depth of theend point of the second curved section minus b The actual120588 of the top layer is assumed to be the measured 120588c of thefirst constant section and the actual 120588 of the bottom layer isassumed to be the measured 120588c of the third constant sectionThe actual 120588 of the soft interlayer can be assumed based onthe minimum 120588c in the second curved section The 120588c profileis back-calculated assuming that the ND-CP passes through

10 Mathematical Problems in Engineering

Tokyo

JAPANA1

A2 Osaka

Kyushu

Isahaya Bay

Ariake Sea Sapporo

5 km

N

Figure 8 Investigation locations

0

1

2

3

4

5

Dep

th fr

om w

ater

surfa

cez

(m)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

Middlesection

Finalsection

(a)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

section

Finalsection

Dep

th fr

om w

ater

surfa

ce (m

)

8

9

10

11

12

(b)

Figure 9 Investigation results (a) at location A1 (b) at location A2

the strata with a soft interlayer The actual 120588 profile of stratawith a soft interlayer is determined when the calculated 120588cprofile matches the measured 120588c profile well

For bed sediments as shown in Figure 7(c) the measured120588c profile can be divided into three sections namely the firstconstant section the second curved section and the thirdlinear section The measured 120588c values are different fromthe actual 120588 values near the upper and lower boundariesof the fluid mud From the 120588c profile the upper and lowerboundaries of the fluid mud can be determined that is thedepth of the point from which density increases plus b andthe depth of the point from which density slightly increaseslinearly minus b The actual 120588 of the water layer is assumed tobe the measured 120588c of the first constant section The slope ofthe actual 120588 profile of the bed mud layer is assumed to be theslope of themeasured120588c profile of the third linear sectionTheactual 120588 distribution of the fluidmud layer can be assumed to

be a parabolic linear or square root function The 120588c profileis back-calculated assuming that the ND-CP passes throughbed sediments The actual 120588 profile of bed sediments can bedetermined when the calculated 120588c profile fits the measured120588c profile5 An Example of Practical Application

51 ND-CP Measured 120588119888 Profile The density profiles of bedsediments at Isahaya Bay Japan were investigated using theND-CP The details on the equipment and methods used inthis investigation can be found in a prior publication [23]TheND-CP measured density profiles at two locations (Figure 8)are used to illustrate how to use the proposed method todetermine the actual 120588 profile from the measured 120588c profile

The measured BCR and DCR and the calculated 120588cvalues at locations A1 and A2 are shown in Figure 9 BCR

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

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Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

8 12 16 20

Soil

Measured by ND-CPPredicted by model

Water

Actual density profile

09

07

05

03

01

Dep

th (m

)

Wet density (kNm3)

(a)

12 14 16 18 20

Soil

Measured by ND-CPPredicted by model

Kaolin clay slurry

Actual density profile

09

07

05

03

0110

Dep

th (m

)

Wet density (kNm3)

(b)

Figure 6 Verification of the theoretical model with laboratory test results

the thickness is 015 m for the third case the density is13 gcm3 and the thickness is 03 m Figure 7(c) shows thedensity profiles of bed sediments The density of the wateris 10 gcm3 and the density of the bed mud increases withdepth in a linear fashion Three different layers of fluid mudare assumed For the first case the fluid mud has a maximumdensity of 12 gcm3 and thickness of 02 m for the secondcase the maximum density is 13 gcm3 and the thickness is02 m for the third case the maximum density is 115 gcm3and the thickness is 025 m

The 120588c profiles are calculated by the theoretical modelwhen the ND-CP penetrates into the assumed complex strata(Figure 7) and the results are compared with the 120588 profilesThe 120588c profiles are clearly different from the 120588 profiles nearthe interface of the layered strata near the soft interlayer andnear the upper and lower boundaries of the fluid mud Thedifferences between 120588c and 120588 near the interface of the layeredstrata in Figure 7(a) are explained as follows (1)When boththe gamma ray source and detector are in the top layer 120588cis equal to the density of the top layer (2) Once the gammaray source enters the bottom layer 120588c will be the weightedaverage density of the two layers while 120588 is still the densityof the first layer (3) When the midpoint of the gamma raysource and detector enters the bottom layer 120588 will change tothe density of the bottom layer while 120588c is still the weightedaverage density of the two layers because the gamma raydetector is still in the first layer (4) When the gamma raydetector also enters the bottom layer 120588c is equal to 120588 of thebottom layer Similar to the profiles for the case in which theND-CP passes through the interface of layered strata the 120588cprofiles are different from the 120588 profiles when the ND-CP

passes through a soft interlayer and the fluid mudThereforethe 120588c profile measured by the ND-CP must be interpretedand the actual 120588 profile must be deduced from the measured120588c profile42 Method for Deducing Actual 120588 Profile from Measured 120588119888Profile A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed as follows(1)Determining the depths of and densities at the bound-aries of each stratum based on the features of the measured120588c profile (2) Assuming the probable density function ofeach stratum and the probable 120588 profile of the strata (3)Back-calculating the 120588c profile by using the theoretical modelfor the case of the ND-CP passing through layered strata(4) Comparing the calculated 120588c profile with the measured120588c profile and adjusting the assumed 120588 profile until thecalculated 120588c profile matches the measured 120588c profile well (5)Determining the actual 120588 profile to be the assumed 120588 profilewhen the calculated 120588c profile fits the measured 120588c profile

For the layered strata as shown in Figure 7(a) themeasured 120588c profile can be divided into three sectionsnamely the first constant section the second curved sectionand the third constant section The measured 120588c values aredifferent from the actual 120588 values in the range of 2b whichis the distance between the gamma ray source and detectorFrom the 120588c profile the boundary between layered strata canbe determined ie themidpoint of the second curved sectionof the 120588c profileThe actual 120588 value of the top layer is assumedto be the measured 120588c value of the first constant section andthe actual 120588 value of the bottom layer is assumed to be themeasured 120588c value of the third constant sectionThe 120588c profile

Mathematical Problems in Engineering 9

13 14 15 16 17 18 19 20

16

14

12

10

08

06

(4)

(3)

(2)

Bottom layer

Dep

th (m

)Top layer

(1)

Assumed and calculated ＝ (gcm3)

(16 to 15)Assumed

(16 to 15)Calculated ＝(16 to 17)Assumed

(16 to 17)Calculated ＝(16 to 18)Assumed

(16 to 18)Calculated ＝

(a) Layered strata

12 13 14 15 16 17 18

18

16

14

12

10

08

06

Bottom layer

Top layer

(13 015) (13 015)

(15 015) (15 015)

(13 03) (13 03)

Dep

th (m

)

Soft interlayer

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(b) Strata with soft interlayers

09 10 11 12 13 14

16

14

12

10

08

06

Bed mud

Fluid mud

(12 02) (12 02)

(13 02) (13 02)

(115 025) (115 025)

Dep

th (m

)

Water

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(c) Bed sediments

Figure 7 Comparison of assumed 120588 and calculated 120588c profiles

is back-calculated assuming that the ND-CP passes throughthe layered strata If the calculated120588c profilematcheswell withthe measured 120588c profile the actual 120588 profile of the layeredstrata was determined

For the strata with a soft interlayer as shown in Fig-ure 7(b) the measured 120588c profile can be divided into threesections namely the first constant section the second curvedsection and the third constant section The measured 120588cvalues are different from the actual 120588 values near the softinterlayer (2b plus the thickness of the soft interlayer) From

the 120588c profile the upper and lower boundaries of the softinterlayer can be determined that is the depth of the startpoint of the second curved section plus b and the depth of theend point of the second curved section minus b The actual120588 of the top layer is assumed to be the measured 120588c of thefirst constant section and the actual 120588 of the bottom layer isassumed to be the measured 120588c of the third constant sectionThe actual 120588 of the soft interlayer can be assumed based onthe minimum 120588c in the second curved section The 120588c profileis back-calculated assuming that the ND-CP passes through

10 Mathematical Problems in Engineering

Tokyo

JAPANA1

A2 Osaka

Kyushu

Isahaya Bay

Ariake Sea Sapporo

5 km

N

Figure 8 Investigation locations

0

1

2

3

4

5

Dep

th fr

om w

ater

surfa

cez

(m)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

Middlesection

Finalsection

(a)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

section

Finalsection

Dep

th fr

om w

ater

surfa

ce (m

)

8

9

10

11

12

(b)

Figure 9 Investigation results (a) at location A1 (b) at location A2

the strata with a soft interlayer The actual 120588 profile of stratawith a soft interlayer is determined when the calculated 120588cprofile matches the measured 120588c profile well

For bed sediments as shown in Figure 7(c) the measured120588c profile can be divided into three sections namely the firstconstant section the second curved section and the thirdlinear section The measured 120588c values are different fromthe actual 120588 values near the upper and lower boundariesof the fluid mud From the 120588c profile the upper and lowerboundaries of the fluid mud can be determined that is thedepth of the point from which density increases plus b andthe depth of the point from which density slightly increaseslinearly minus b The actual 120588 of the water layer is assumed tobe the measured 120588c of the first constant section The slope ofthe actual 120588 profile of the bed mud layer is assumed to be theslope of themeasured120588c profile of the third linear sectionTheactual 120588 distribution of the fluidmud layer can be assumed to

be a parabolic linear or square root function The 120588c profileis back-calculated assuming that the ND-CP passes throughbed sediments The actual 120588 profile of bed sediments can bedetermined when the calculated 120588c profile fits the measured120588c profile5 An Example of Practical Application

51 ND-CP Measured 120588119888 Profile The density profiles of bedsediments at Isahaya Bay Japan were investigated using theND-CP The details on the equipment and methods used inthis investigation can be found in a prior publication [23]TheND-CP measured density profiles at two locations (Figure 8)are used to illustrate how to use the proposed method todetermine the actual 120588 profile from the measured 120588c profile

The measured BCR and DCR and the calculated 120588cvalues at locations A1 and A2 are shown in Figure 9 BCR

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

13 14 15 16 17 18 19 20

16

14

12

10

08

06

(4)

(3)

(2)

Bottom layer

Dep

th (m

)Top layer

(1)

Assumed and calculated ＝ (gcm3)

(16 to 15)Assumed

(16 to 15)Calculated ＝(16 to 17)Assumed

(16 to 17)Calculated ＝(16 to 18)Assumed

(16 to 18)Calculated ＝

(a) Layered strata

12 13 14 15 16 17 18

18

16

14

12

10

08

06

Bottom layer

Top layer

(13 015) (13 015)

(15 015) (15 015)

(13 03) (13 03)

Dep

th (m

)

Soft interlayer

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(b) Strata with soft interlayers

09 10 11 12 13 14

16

14

12

10

08

06

Bed mud

Fluid mud

(12 02) (12 02)

(13 02) (13 02)

(115 025) (115 025)

Dep

th (m

)

Water

Assumed and calculated ＝ (gcm3)

Assumed

Calculated ＝Assumed

Calculated ＝Assumed

Calculated ＝

(c) Bed sediments

Figure 7 Comparison of assumed 120588 and calculated 120588c profiles

is back-calculated assuming that the ND-CP passes throughthe layered strata If the calculated120588c profilematcheswell withthe measured 120588c profile the actual 120588 profile of the layeredstrata was determined

For the strata with a soft interlayer as shown in Fig-ure 7(b) the measured 120588c profile can be divided into threesections namely the first constant section the second curvedsection and the third constant section The measured 120588cvalues are different from the actual 120588 values near the softinterlayer (2b plus the thickness of the soft interlayer) From

the 120588c profile the upper and lower boundaries of the softinterlayer can be determined that is the depth of the startpoint of the second curved section plus b and the depth of theend point of the second curved section minus b The actual120588 of the top layer is assumed to be the measured 120588c of thefirst constant section and the actual 120588 of the bottom layer isassumed to be the measured 120588c of the third constant sectionThe actual 120588 of the soft interlayer can be assumed based onthe minimum 120588c in the second curved section The 120588c profileis back-calculated assuming that the ND-CP passes through

10 Mathematical Problems in Engineering

Tokyo

JAPANA1

A2 Osaka

Kyushu

Isahaya Bay

Ariake Sea Sapporo

5 km

N

Figure 8 Investigation locations

0

1

2

3

4

5

Dep

th fr

om w

ater

surfa

cez

(m)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

Middlesection

Finalsection

(a)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

section

Finalsection

Dep

th fr

om w

ater

surfa

ce (m

)

8

9

10

11

12

(b)

Figure 9 Investigation results (a) at location A1 (b) at location A2

the strata with a soft interlayer The actual 120588 profile of stratawith a soft interlayer is determined when the calculated 120588cprofile matches the measured 120588c profile well

For bed sediments as shown in Figure 7(c) the measured120588c profile can be divided into three sections namely the firstconstant section the second curved section and the thirdlinear section The measured 120588c values are different fromthe actual 120588 values near the upper and lower boundariesof the fluid mud From the 120588c profile the upper and lowerboundaries of the fluid mud can be determined that is thedepth of the point from which density increases plus b andthe depth of the point from which density slightly increaseslinearly minus b The actual 120588 of the water layer is assumed tobe the measured 120588c of the first constant section The slope ofthe actual 120588 profile of the bed mud layer is assumed to be theslope of themeasured120588c profile of the third linear sectionTheactual 120588 distribution of the fluidmud layer can be assumed to

be a parabolic linear or square root function The 120588c profileis back-calculated assuming that the ND-CP passes throughbed sediments The actual 120588 profile of bed sediments can bedetermined when the calculated 120588c profile fits the measured120588c profile5 An Example of Practical Application

51 ND-CP Measured 120588119888 Profile The density profiles of bedsediments at Isahaya Bay Japan were investigated using theND-CP The details on the equipment and methods used inthis investigation can be found in a prior publication [23]TheND-CP measured density profiles at two locations (Figure 8)are used to illustrate how to use the proposed method todetermine the actual 120588 profile from the measured 120588c profile

The measured BCR and DCR and the calculated 120588cvalues at locations A1 and A2 are shown in Figure 9 BCR

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

Tokyo

JAPANA1

A2 Osaka

Kyushu

Isahaya Bay

Ariake Sea Sapporo

5 km

N

Figure 8 Investigation locations

0

1

2

3

4

5

Dep

th fr

om w

ater

surfa

cez

(m)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

Middlesection

Finalsection

(a)

0 50 0 1000 08 12 16

BCR (cps) DCR (cps) Density (gcm3)

Initialsection

section

Finalsection

Dep

th fr

om w

ater

surfa

ce (m

)

8

9

10

11

12

(b)

Figure 9 Investigation results (a) at location A1 (b) at location A2

the strata with a soft interlayer The actual 120588 profile of stratawith a soft interlayer is determined when the calculated 120588cprofile matches the measured 120588c profile well

For bed sediments as shown in Figure 7(c) the measured120588c profile can be divided into three sections namely the firstconstant section the second curved section and the thirdlinear section The measured 120588c values are different fromthe actual 120588 values near the upper and lower boundariesof the fluid mud From the 120588c profile the upper and lowerboundaries of the fluid mud can be determined that is thedepth of the point from which density increases plus b andthe depth of the point from which density slightly increaseslinearly minus b The actual 120588 of the water layer is assumed tobe the measured 120588c of the first constant section The slope ofthe actual 120588 profile of the bed mud layer is assumed to be theslope of themeasured120588c profile of the third linear sectionTheactual 120588 distribution of the fluidmud layer can be assumed to

be a parabolic linear or square root function The 120588c profileis back-calculated assuming that the ND-CP passes throughbed sediments The actual 120588 profile of bed sediments can bedetermined when the calculated 120588c profile fits the measured120588c profile5 An Example of Practical Application

51 ND-CP Measured 120588119888 Profile The density profiles of bedsediments at Isahaya Bay Japan were investigated using theND-CP The details on the equipment and methods used inthis investigation can be found in a prior publication [23]TheND-CP measured density profiles at two locations (Figure 8)are used to illustrate how to use the proposed method todetermine the actual 120588 profile from the measured 120588c profile

The measured BCR and DCR and the calculated 120588cvalues at locations A1 and A2 are shown in Figure 9 BCR

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

09 12 15

26

24

22

20

18

16D

epth

from

wat

er su

rface

z (m

) ＝ (gcＧ3)

B(1287231)

A(1000183)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

Parabolic Linear Square root

(gcＧ3)

D(128621775)

C(100019625)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

＝ (gcＧ3)

Parabolic Linear Square root

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12 15

26

24

22

20

18

16

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 10 Determination of the actual density profile of bed sediments at location A1 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

＝ (gcＧ3)

B(12331083)

A(10251045)

(a)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

(gcＧ3)

D(1222106975)

C(1025105825)

(b)

Dep

th fr

om w

ater

surfa

ce z

(m)

09 12

112

110

108

106

104

102

Parabolic Linear Square root

＝ (gcＧ3)

(c)

Dep

th fr

om w

ater

surfa

ce z

(m)

104

10209 12

112

110

108

106

and ＝ (gcＧ3)

Actual Predicted ＝

＝Measured

(d)

Figure 11 Determination of the actual density profile of bed sediments at location A2 (a) measured 120588c profile (b) assumed 120588 profile (c)calculated 120588c profile (d) deduced 120588 profile

is approximately zero in water and approximately 50 cpsin bed mud DCR in bed sediments is lower than thatin water and decreases with increasing density 119877120588 can becalculated according to the measured BCR and DCR byusing (1) and then the density can be calculated by using(25) The density profiles show similar transitions and varysignificantly around the watermud interface The measured120588c profiles can be divided into three sections namely aninitial constant section a middle curved section and a finallinear section

52 Determination of the Actual 120588 Profile There is an obviousdifference between the actual 120588 and the measured 120588c profilesnear the upper and lower boundaries of the fluid mud layeras shown in Figure 7(c) The theoretical model proposed inSection 3 and the method proposed in Section 4 were used todeduce the actual 120588 profiles from the measured 120588c profiles asfollows

(1) For the measured 120588c profile at location A1 (Fig-ure 10(a)) the 120588c at and depth of the intersection of theinitial constant and middle curved sections (point A) are1000 gcm3 and 183 m respectively The 120588c at and depth ofthe intersection of themiddle curved and final linear sections(point B) are 1287 gcm3 and 231 m respectively For themeasured 120588c profile at locationA2 (Figure 11(a)) 120588c and depthat point A are 1025 gcm3 and 1045 m respectively 120588c anddepth at point B are 1233 gcm3 and 1083 m respectively(2) For the assumed 120588 profile at locationA1 (Figure 10(b))the depth of the upper boundary of the fluidmud at pointC is19625mwhich is equal to the depth at pointA (183m) plus b(01325 m)The depth of the lower boundary of the fluid mudat point D is 21775 m which is equal to the depth at point B(231m)minus b (01325m) 120588 at pointC is 1000 gcm3 whichis equal to 120588c at point A 120588 at point D is 1286 gcm3 which isequal to 120588c at point B minus kb (where k is the slope of thefinal linear section) For the assumed 120588 profile at location A2

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

(Figure 11(b)) the depth at pointC is 105825m and the depthat point D is 106975 m 120588 at point C is 1025 gcm3 and 120588 atpoint D is 1222 gcm3(3) The 120588 function of the first section (water layer) isassumed to be constant the 120588 function of the second section(fluid mud) is assumed to be parabolic linear or square rootand the 120588 function of the third section (bed mud) is assumedto be linear The 120588 of the first section is assumed to be 1000gcm3 (freshwater) and 1025 gcm3 (seawater) for 120588 profilesat locations A1 andA2 respectivelyThe parabolic linear andsquare root density functions for the second section can bedetermined according to the density and depth values at pointC and point D by (11) to (16) The slope of the third section isassumed to be 0005 and 008 respectively for the 120588 profilesat locations A1 and A2 based on the measured 120588c values at thefinal linear sections(4)The 120588c profiles are calculated by (21) to (25) assumingthe ND-CP passes through the strata with the assumed 120588profiles in Figures 10(b) and 11(b) The calculated 120588c profilesat locations A1 and A2 are shown in Figures 10(c) and 11(c)respectively The calculated 120588c profiles clearly vary when theassumed 120588 function of the second section varies (amongparabolic linear or square root functions) Therefore thecalculated 120588c profile can be changed by adjusting the assumed120588 profile(5) The calculated 120588c profiles for the different assumed120588 profiles are compared with the measured 120588c profiles Thecalculated 120588c profiles match the measured 120588c profiles wellif the density function of the second section is assumedto be a square root function [24] The deduced 120588 profilesare also shown in Figures 10(d) and 11(d) The 120588 valuesare clearly different from the 120588c values around the fluidmud layer Therefore it is suggested that the deduced 120588profile from the measured 120588c profile should be adopted topredict sediment transport determine nautical depth andplan dredging projects

6 Summary and Conclusions

Thenuclear density cone penetrometer (ND-CP) can be usedfor in situ investigation on the soil density while it measuresthe composite density (120588c) of the soil within a spheroidcentered at the midpoint between the gamma ray source andthe detector A theoretical model for calculating 120588c and amethod for deducing the 120588 profile from the measured 120588cprofile were proposed in this studyThe following conclusionscan be drawn from this study(1)A theoretical model for predicting the count rate ratio(119877120588) of the ND-CP is proposed and equations for calculating119877120588 are derived for the case of the ND-CP penetratinginto strata with different density distribution functions Thecalculated 120588c profiles exhibit a good fit to the laboratory-measured 120588c profiles by the ND-CP indicating that theproposed theoretical model can be used to calculate the 120588cwithin the spheroid A comparison of the 120588 and 120588c valuesshows that the 120588c profiles are considerably different from theactual 120588 profiles at the boundaries of each stratum where thedensity changes suddenly

(2) A method for deducing the actual 120588 profile from themeasured 120588c profile is proposed according to the differencesbetween the 120588 and 120588c profilesThe depth of and density at theboundaries of each stratum are first determined based on thefeatures of the measured 120588c profile Then the 120588 profile of thestrata is assumed and the120588c profile is back-calculated Finallythe calculated 120588c profile is compared with the measured120588c profile and the actual 120588 profile is identified when thecalculated 120588c profile fits the measured 120588c profile(3) The proposed model and method are used to deter-mine the actual 120588 profiles from measured 120588c profiles ofbed sediments in the field The actual 120588 values are clearlydifferent from the 120588c values around the fluid mud layer Itis suggested that the deduced 120588 profile should be adoptedto predict sediment transport determine nautical depth andplan dredging projects The findings in this study would bebeneficial for the in situ determination of density profilesusing the ND-CP Further research should be conducted toverify the proposed model and method in determining theactual 120588 profiles from the measured 120588c profiles by the ND-CPin the field of geotechnical and pavement engineering

Acronyms119886 Equatorial radius of the spheroid2119887 Gamma ray source-detector separationdistance119860 Constant 06264119861 Constant 40954119862 Constant 68422119861119862119877 Background count rate119863119862119877 Density count rate119878119862119877 Standard count rate119896 Slope of the final linear region

ND-CP Nuclear density cone penetrometer119877120588 Count rate ratio119881 Volume of spheroid119911 Depth119911i Depth corresponding to the midpoint ofthe ellipse1199111 Depth at the top of stratum1199112 Depth at the bottom of stratum1205721 Fitting parameter for parabolic densitydistribution1205722 Fitting parameter for square root densitydistribution1205731 Fitting parameter for linear densitydistribution120588 Density120588c Composite density1205880 Density value of constant densitydistribution1205881 Density value at the top of stratum1205882 Density value at the bottom of stratumΔ119877120588 Count rate ratio from a small volume120575119911 Differential of 119911120597119881 Differential of 119881120597119877120588 Differential of count rate ratio

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

Data Availability

The data used to support the findings of this article areavailable from the authors upon request

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof the National Key Research and Development Program ofChina (Grant No 2017YFC0805402) the Major Program ofNational Natural Science Foundation of China (Grant No51890911) and the National Natural Science Foundation ofChina (Grant No 51509181)

References

[1] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley New York NY USA 3rd edition1996

[2] Y Wu H Lyu J Han and S Shen ldquoDewatering-inducedbuilding settlement around a deep excavation in soft deposit inTianjin Chinardquo Journal of Geotechnical and GeoenvironmentalEngineering vol 145 no 5 Article ID 05019003 pp 1ndash14 2019

[3] D Sarkar and A Haldar Physical and Chemical Methods inSoil Analysis- Fundamental Concepts of Analytical Chemistryand Instrumental Techniques New Age International Ltd NewDelhi India 2005

[4] S-J Zheng ldquoThe new method for engineering geologicinvestigation-Nuclear density cone penetrometerrdquoUndergroundEngineering and Tunnels vol 4 no 2 pp 16ndash21 1995(Chinese)

[5] S HorpibulsukW Katkan andA Apichatvullop ldquoAn approachfor assessment of compaction curves of fine grained soils atvarious energies using a one point testrdquo Soils and Foundationsvol 48 no 1 pp 115ndash125 2008

[6] WHMcAnally C FriedrichsDHamilton et al ldquoManagementof fluid mud in estuaries bays and lakes I Present state ofunderstanding on character and behaviorrdquo Journal of HydraulicEngineering vol 133 no 1 pp 9ndash22 2007

[7] D Campbell and J KennethHenshall ldquoBulk densityrdquo in Soil andEnvironmental Analysis-PhysicalMethods K A Smith and C EMullins Eds pp 315ndash348 Marcel Dekker Inc New York NYUSA 2nd edition 2000

[8] National Energy Administration of China DL 5270-2012 TestCode for Density and Moisture by Nuclear Methods ChinaElectric Power Press Beijing China 2012

[9] B Pontecorvo ldquoNeutron well loggingrdquo Oil Gas Journal vol 40pp 32-33 1941

[10] J Homilius and S Lorch ldquoOn the theory of gamma rayscattering in boreholesrdquo Geophysical Prospecting vol 6 no 4pp 342ndash364 1958

[11] ACMeigh andBO Skipp ldquoGamma-ray andneutronmethodsof measuring soil density and moisturerdquo Geotechnique vol 10no 3 pp 110ndash126 1960

[12] P A Ruygrok ldquoEvaluation of the gamma and neutron radiationscattering and transmission methods for soil density andmoisture determinationrdquo Geotechnical Testing Journal vol 11no 1 pp 3ndash19 1988

[13] T F Fwa and S A Tan ldquoExperimental evaluation of a laboratorytwin-probe nuclear gage for specimen density measurementrdquoJournal of Testing and Evaluation vol 20 no 1 pp 59ndash65 1992

[14] ASTM ldquoStandard D 2922-05 Standard Test Methods for Den-sity of Soil and Soil-Aggregate in Place by Nuclear Methods(Shallow Depth)rdquo Tech Rep ASTM International West Con-shohocken Pa USA 2005

[15] T Lunne P K Robertson and J J M Powell Cone PenetrationTesting in Geotechnical Practice Blackie Academic amp Profes-sional London UK 1997

[16] S-L Shen J-P Wang H-N Wu Y-S Xu G-L Ye and Z-YYin ldquoEvaluation of hydraulic conductivity for both marine anddeltaic deposits based on piezocone testingrdquoOcean Engineeringvol 110 pp 174ndash182 2015

[17] J L Ledoux J Menard and P Soulard ldquoThe penetro-gammadensimeterrdquo in Proceedings of the 2nd European Sym-posium on Penetration Testing vol 2 pp 679ndash682 BalkemaAmsterdam Netherlands 1982

[18] T Shibata M Mimura and A K Shrivastava ldquoUse of RI conepenetrometer data in foundation engineeringrdquo in Proceedingsof the 13th International Conference on Soil Mechanics andFoundation Engineering vol 1 pp 147ndash150 New Delhi India1994

[19] M Mimura A K Shrivastava T Shibata and M NobuyamaldquoPerformance of RI cone penetrometers in sand depositsrdquo inProceedings of the International Symposium on Cone PenetrationTesting CPTrsquo95 vol 2 pp 55ndash60 Linkoping Sweden 1995

[20] M Mimura and A K Shrivastava ldquoRI-cone penetrometersexperience in naturally and artificially deposited sandrdquo inProceedings of the First International Conference on Site Char-acterization vol 1 pp 575ndash580 Atlanta Ga USA 1998

[21] G R Dasari M Karthikeyan T-S Tan M Mimura and K-KPhoon ldquoIn situ evaluation of radioisotope cone penetrometersin claysrdquo Geotechnical Testing Journal vol 29 no 1 pp 45ndash532006

[22] T Umezaki T Kawamura andM Yoshimura ldquoInvestigation ofsediment environments in closed water body using RI-densityLogrdquo in Proceedings of the 17th International Conference onSoil Mechanics and Geotechnical Engineering pp 1050ndash1053Alexandria Egypt 2009

[23] R Jia T Hino T Hamada J Chai andM Yoshimura ldquoDensityand undrained shear strength of bed sediment from ND-CPTrdquoOcean Dynamics vol 63 no 5 pp 507ndash517 2013

[24] R Jia THino J Chai THamada andMYoshimura ldquoInterpre-tation of density profile of seabed sediment fromnuclear densitycone penetration test resultsrdquo Soils and Foundations vol 53 no5 pp 671ndash679 2013

[25] G Schlieper ldquoPrinciples of gamma ray densitometryrdquo MetalPowder Report vol 55 no 12 pp 20ndash23 2000

[26] M Karthikeyan Application of radioisotope cone penetrometerto characterize a lumpy fill [Dissertation thesis] NationalUniversity of Singapore 2005

[27] W R Parker G C Sills and R E A Paske ldquoIn situ nucleardensity measurement in dredging practice and controlrdquo inProceedings of the First International Symposium on DredgingTechnology pp 25ndash42 England UK University of Kent 1975

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Mathematical Problems in Engineering

[28] K Preiss ldquoNon-destructive laboratory measurement of marinesediment density in a core barrel using gamma radiationrdquoDeep-Sea Research vol 15 pp 401ndash407 1968

[29] M Karthikeyan and T T Soon ldquoProfiling of heterogeneous soilusing nuclear-density cone penetrometerrdquo Geotechnical TestingJournal vol 31 no 6 pp 513ndash525 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom