1. bookVolume 43 (2021): Issue 2 (June 2021)
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Geostatistical analysis of spatial variability of the liquefaction potential – Case study of a site located in Algiers (Algeria)

Online veröffentlicht: 30 Jun 2021
Seitenbereich: 155 - 168
Eingereicht: 26 Nov 2020
Akzeptiert: 29 Mar 2021
Zeitschriftendaten
License
Format
Zeitschrift
Erstveröffentlichung
09 Nov 2012
Erscheinungsweise
4 Hefte pro Jahr
Sprachen
Englisch
Introduction

Soil liquefaction is a phenomenon of instability or loss of strength that can take place on a generally saturated or partially saturated granular medium. This event is characterized by a higher pore pressure linked to the contracting behavior of soil during the application of a rapid loading (earthquake, shocks, tidal wave, etc.). The occurrence of liquefaction in soils is often assessed using the originally simplified method proposed by Seed and Idriss (1971).[27] This method is based on the Standard Penetration Test (SPT), Cone Penetration Test (CPT), Marchetti Dilatometer Test (DMT), Shear Wave Velocity Measurement Test, and other laboratory tests, such as the cyclic triaxial test. These methods are based on the calculation of the safety factor. The in-situ sand liquefaction behavior depends on several characteristics, namely the grain size, water table, average density of soil, layer thickness of loose sand, as well as the history of static and dynamic stresses.[1]

Using the probabilistic approach (reliability methods) for the calculation of the risk of soil liquefaction made it possible to assess the effect of variability and uncertainty existing for each parameter of the CSR (cyclic stress ratio) and CRR (cyclic resistance ratio) functions on the estimation of liquefaction risk.[13,14,2] However, the geostatistical approach makes it possible to analyze the spatial structure of soil Liquefaction Potential Index (LPI) for the zoning of the risk potential in a site.[21,22,29]

Geostatistics, which was invented in 1962 by Georges Matheron, an engineer from the Corps des Mines in France, represents the set of probabilistic methods intended for the study of regionalized phenomena.[18] They have been successfully applied to issues related to mining geology, hydrogeology, environment, and so on. Geostatistical analysis, which takes into account two important issues, differs from the rest of the classical interpolation methods. The first issue lies in identifying the spatial structure of the variable studied using the variogram, which is considered a basic tool of geostatistics; the second one concerns the use of this spatial structure with known measured values, for the optimal prediction at unmeasured points, using the Kriging technique. Another advantage of the geostatistical analysis, over traditional interpolation methods, is the production of standard deviation maps of the predicted values. This allows engineers to identify areas of uncertainty.[3] Several studies on the use of geostatistical techniques (interpolation or simulation) in geotechnical applications have previously been carried out.[3,15,21]

On the other hand, it is interesting to recall that the Geographic Information System (GIS) was developed by the Environmental Systems Research Institute (ESRI). It is a tool that is primarily intended to manage, visualize, map, query, and analyze all data possessing a spatial component.[10] The geostatistical analysis extension of the GIS software package offers statistical tools for analyzing and representing continuous data on a map in order to generate surfaces. Exploratory spatial data analysis tools in the GIS software offer different perspectives on the data, such as their distribution, overall trends, and level of spatial autocorrelation.[8]

The present study aims to calculate the soil Liquefaction Potential Index (LPI), which represents a measure of the total risk of soil liquefaction over a strip 20 m deep. For this, a series of 62 static penetration tests (Cone Penetration Tests) were carried out in the study area. In addition, a geostatistical analysis was performed to study the spatial variation of the liquefaction potential index and to map the areas at risk of liquefaction using the Geographic Information System (GIS). The soil susceptibility to liquefaction was checked for a magnitude of 6.8 (magnitude of Boumerdes earthquake in 2003).[2] This value was used in the calculations, while taking as the Peak Ground Acceleration (PGA) value on the surface as amax = 0.30 g (value provided by the Algerian Parasismic Regulations).[26] The possible acceleration of the ground taken into account during earthquakes is between 0.2 g and 0.25 g

Presentation of the study site

The site under study is located in the Commune of Dar El Beïda, 16 km east of Algiers, in the Great Plain of Mitidja that stretches between the mountains of the Tellian Atlas and the Mediterranean Sea. Figure 1 gives an overview of the location on the studied site.

Figure 1

Geographical location of the study region and location of the CPT surveys.

The data used in this study originate from surveys carried out with a static penetrometer of Gouda type. The region under study is located in an area of high seismicity; it is classified in the seismic category Zone III, according to the Algerian earthquake regulation in force.[26] The behavior of the water table under the site study ground exhibited significant variations of up to 5 m, depending on the season. It was therefore advisable to consider that the water table was flush.

Methodology
Method for assessing the liquefaction potential
Determination of the cyclic stress ratio

The cyclic stress ratio (CSR) is calculated according to the following formula[27]: CSR=τcycσV0=0.65(σV0σV0)(amaxg)rd(1MSF) {\rm{CSR}} = {{{\tau _{{\rm{cyc}}}}} \over {{{\sigma_{{\rm{v0}}}^\prime}}}} = 0.65\left( {{{{\sigma _{{\rm{v}0}}}} \over {{{\sigma_{{\rm{v}0}}^\prime}}}}} \right)\left( {{{{{\rm{a}}_{\max }}} \over {\rm{g}}}} \right){{\rm{r}}_{\rm{d}}}\left( {{1 \over {{MSF}}}} \right) where amax is the maximum horizontal acceleration at the ground surface, g is the acceleration of gravity, σv0 and σv0 are the total and effective vertical stresses, respectively, rd the stress reduction factor, and MSF is the Magnitude Scaling Factor.

To calculate the factor rd, [31] proposed the following relationship: rd=(10.4113z0.5+0.04052z+0.001753z1.5)(10.4177z0.5+0.05729z0.0062z1.5+0.00121z2) {r_d} = {{\left( {1 - 0.4113{z^{0.5}} + 0.04052z + 0.001753{z^{1.5}}} \right)} \over {\left( {1 - 0.4177{z^{0.5}} + 0.05729z - 0.0062{z^{1.5}} + 0.00121{z^2}} \right)}} where z is depth below ground surface (m).

With regard to the Magnitude Scaling Factor (MSF), the lower bound equation, which was suggested by [31], was used as follows: MSF=102.24Mw2.56 {MSF} = {{{{10}^{2.24}}} \over {M_w^{2.56}}} where Mw is the moment magnitude of the earthquake.

Determination of the cyclic resistance ratio for the cone penetrometer tests

The cyclic resistance ratio (CRR) is estimated from CPT data following the procedure in [23], [24] as: CRR={0.833(qc1N,cs1000)+0.05ifqc1N,cs<5093(qc1N,cs1000)3+0.08if50qc1N,cs<160 {\rm{CRR}} = \left\{ {\matrix{ {0.833\left( {{{{{\rm{q}}_{{\rm{c}}1{\rm{N}},{\rm{cs}}}}} \over {1000}}} \right) + 0.05\,{\rm{if}}\,{{\rm{q}}_{{\rm{c}}1{\rm{N}},{\rm{cs}}}} < 50} \cr {93({{{{{{\rm{q}}_{{\rm{c}}1{\rm{N}},{\rm{cs}}}}}}} \over {1000}})^3 + 0.08\,\,{\rm{if}}\,50 \le {{\rm{q}}_{{\rm{c}}1{\rm{N}},{\rm{cs}}}} < 160} \cr } } \right.

The resistance to penetration of a standard cone may be calculated using the following equation: (qc1N)cs=Kc×qc1N {\left( {{q_{c1N}}} \right)_{cs}} = {K_c} \times {q_{c1N}} Kc is a correction factor that depends on Ic and defined by the equation of [23] as: {Kc=1.0ifIc1.64Kc=0.403Ic4+5.58Ic321.63Ic2+33.75Ic17.88ifIc>1.64 \left\{ {\matrix{ {\,{{\rm{K}}_{\rm{c}}} = 1.0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,{{\rm{I}}_{\rm{c}}} \le 1.64} \hfill \cr {{{\rm{K}}_{\rm{c}}} = - 0.403\,{\rm{I}}_{\rm{c}}^4 + 5.58\,{\rm{I}}_{\rm{c}}^3 - 21.63\,{\rm{I}}_{\rm{c}}^2 + 33.75{{\rm{I}}_{\rm{c}}} - 17.88\,{\rm{if}}\,{{\rm{I}}_{\rm{c}}} > 1.64} \hfill \cr } } \right. On the other hand, the corrected peak resistance qc1N (CPT) was determined using the following formula: qc1N=(qcσvoPat)(Patσv0)n {q_{c1N}} = \left( {{{{q_c} - {\sigma _{v0}}} \over {{P_{at}}}}} \right){\left( {{{{P_{at}}} \over {{{\sigma_{v0}^\prime}}}}} \right)^n} The exponent n is defined using the formula of [24] as: n=0.381(Ic)+0.05(σv0Pat)0.15wheren1 n = 0.381\left( {{I_c}} \right) + 0.05\left( {{{{{\sigma_{v0}^\prime}}} \over {{P_{at}}}}} \right) - 0.15\,{where}\,n \le 1 Note that Pat is the reference pressure (100 kpa), qc is the peak resistance, and σv0 and σv0 are the total and effective vertical stresses, respectively.

The soil behavior type index Ic is defined by the expression proposed by [23]: Ic=(3.47logQ)2+(1.22+logF)2 {I_c} = \sqrt {{{\left( {3.47 - {log}Q} \right)}^2} + {{\left( {1.22 + {log} F} \right)}^2}} With Q=[(qcσV0)σV0] Q = \left[ {{{\left( {{q_c} - {\sigma _{V0}}} \right)} \over {{{\sigma_{V0}^\prime}}}}} \right] F=fs(qcqV0)×100% F = \left\lceil {{{{f_s}} \over {\left( {{q_c} - {q_{V0}}} \right)}}} \right\rceil \times 100\% The quantity Q is known as the normalized tip resistance and F is the normalized friction ratio of the penetrometer sleeve.

Determination of the factor of safety

The local factor of safety (FS) coefficient for the soil liquefaction is such that[12]: Fs=CRRCSR {F_s} = {{{CRR}} \over {{CSR}}} Note that liquefaction is predicted to occur if FS < 1; however, no liquefaction is expected to take place if FS > 1.[12] It is worth mentioning that the soil becomes more resistant to liquefaction as the safety factor increases.

Determination of the liquefaction potential index

The first step consists of determining the safety factor (FS), and the second one is to find the liquefaction potential index (LPI), which is a measure of the total risk of liquefaction of soil to a depth of 20 m. The liquefaction potential index is defined as follows[11,12]: LPI=020F(z)w(z)dz {LPI} = \int_0^{20} {F\left( z \right)} w\left( z \right)dz According to [17] and [16], the liquefaction potential index can be calculated using the following expression: LPI=i=1NLwiFLiHi {LPI} = \sum\limits_{i = 1}^{NL} {{w_i}{F_{Li}}{H_i}} where wi(z) = 10 - 0.5z, and z denotes the depth in meters; Hi is the thickness of the discretized soil layers, NL is number of soil layers. Then, FLi may be defined as [17]: FLi={0forFs(z)1,01FsforFs(z)1 {F_{Li}} = \left\{ {\matrix{ {\,\,\,\,\,\,\,\,\,\,\,0} \hfill & {{for}} \hfill & {\,{F_s}\left( z \right) \ge 1,0} \hfill \cr {1 - {F_s}} \hfill & {} \hfill & {{for}\,\,\,{F_s}\left( z \right) \le 1} \hfill \cr } } \right. Furthermore, [28] classifies a site, according to the importance of the soil liquefaction phenomenon, into five categories: Very low for LPI = 0, Low for 0 < LPI ≤ 2, Moderate for 2 < LPI ≤ 5, High for 5 < LPI ≤ 15, and Very high for LPI > 15.

Table 1 presents an example about the evaluation of the liquefaction potential index for a typical borehole (BH34). Figure 2 shows some illustrative graphs to show the major steps involved in calculating the liquefaction potential at one CPT sounding location. The liquefaction potential index was calculated for each CPT profile for an earthquake scenario with Mw = 6.8 and having a peak horizontal ground acceleration amax = 0.3 g (Table 2). The CPT positions were defined by the Universal Transverse Mercator (UTM 31 Zone 31N) coordinate system.

Computation of LPI for peak horizontal ground acceleration of 0.3 g corresponding to Mw = 6.8.

Depth (m) Average Density (KN/m3) Overburden Stress σV0 (Kpa) Effective stress σV0 (Kpa) qc (Kpa) fs (Kpa) CRR rd CSR Fs z (m) H (m) w(z) F w(z)*F*H
2.50 19.58 48.95 23.95 2550.00 125.00 0.59 0.98 0.30 1.95 1.25 2.50 9.38 0.00 0.00
3.50 19.58 68.53 33.53 1140.00 41.00 0.19 0.98 0.30 0.62 3.00 1.00 8.50 0.38 3.21
8.50 19.58 166.43 81.43 2430.00 130.00 0.33 0.93 0.29 1.15 6.00 5.00 7.00 0.00 0.00
10.50 19.58 205.59 100.59 1450.00 51.00 0.13 0.89 0.28 0.48 9.50 2.00 5.25 0.52 5.43
14.50 19.58 283.91 138.91 2860.00 112.00 0.18 0.78 0.24 0.73 12.50 4.00 3.75 0.27 4.02
LPI 12.66

Figure 2

Example plots showing key steps in calculating liquefaction potential at one CPT (BH34): (a) tip resistance (qc); (b) side friction (fs); (c) soil type index (Ic); (d) corrected equivalent tip resistant (qc1N)cs; (e) cyclic stress ratio (CSR) and cyclic resistance ratio (CRR); (f) factor of safety (FS).

Calculated values of liquefaction potential index (LPI).

Borehole Coordinates LPI Borehole Coordinates LPI


Easting (m) Northing (m) Easting (m) Northing (m)
BH1 519039.1 4062223 5.57 BH32 517267.3 4060744 24.11
BH2 519247.8 4061957 0.00 BH33 517582.2 4061421 16.97
BH3 519471.7 4061669 0.73 BH34 516748.6 4060166 12.66
BH4 519697.4 4061414 0.00 BH35 516205.9 4060353 26.85
BH5 519931.9 4061204 0.00 BH36 516494 4060620 26.54
BH6 519925.8 4060761 0.00 BH37 515785.8 4060475 21.12
BH7 519655.4 4061015 0.00 BH38 515446.7 4061284 23.90
BH8 519438.6 4061314 0.00 BH39 515676.1 4061850 30.48
BH9 519187.1 4061546 2.34 BH40 515196.3 4061971 32.72
BH10 518939.2 4061757 5.73 BH41 517075.7 4059634 27.74
BH11 518750.8 4062111 8.60 BH42 516106.9 4061729 29.61
BH12 519674.6 4060860 8.75 BH43 515283.1 4059841 13.44
BH13 519369.7 4060970 3.80 BH44 515430 4060130 12.45
BH14 519124.5 4061169 0.00 BH45 515829.7 4059942 23.93
BH15 518443.2 4061889 9.97 BH46 518340.9 4061656 19.64
BH16 519199.3 4060493 8.61 BH47 517981.9 4061644 6.18
BH17 518900 4060902 5.38 BH48 517343.3 4058514 0.93
BH18 518591 4061257 16.25 BH49 517268.7 4058736 0.00
BH19 518848 4060181 3.69 BH50 517249.5 4058958 0.00
BH20 518474.9 4060857 19.59 BH51 517536.8 4059125 0.21
BH21 518374.6 4060557 24.41 BH52 517465.8 4058892 0.00
BH22 518288 4060102 26.39 BH53 517511 4058648 3.24
BH23 518125.3 4059736 13.89 BH54 517506.3 4058348 0.00
BH24 517678.6 4059724 24.30 BH55 517628.5 4058870 1.83
BH25 517901.1 4060124 28.98 BH56 517669.1 4058681 5.21
BH26 518152.4 4060413 25.75 BH57 517711.5 4058493 1.61
BH27 518130.8 4060901 24.28 BH58 515780.4 4058544 6.00
BH28 517660.5 4060689 13.15 BH59 515724.8 4059143 3.29
BH29 517854.7 4060945 11.10 BH60 515743.1 4059410 4.85
BH30 517444.4 4059335 0.00 BH61 515869.5 4059687 11.92
BH31 517097.9 4061032 20.12 BH62 515898.42 4059487.6 11.66
Geostatistical procedure
Variogram modeling

The variogram is a tool that allows measuring the dissimilarity between values based on their separations. It describes the spatial continuity of the regionalized variable.[8] The theoretical variogram is defined as: γ(h)=12Var[Z(x)Z(x+h)]=12E[(Z(x)Z(x+h))2] \gamma\left( h \right) = {1 \over 2}{\rm{Var}}\left[ {Z\left( x \right) - Z\left( {x + h} \right)} \right] = {1 \over 2}{\rm{E}}[ {{{\left( {Z\left( x \right) - Z\left( {x + h} \right)} \right)}^2}}] where Z(x) is an observed value at a particular location, h is the distance between ordered data, Var is the variance of a random variable, E is the expectation operator of a random variable.

The experimental variogram, proposed by [18], is defined as follows[8]: γe(h,θ)=12N(h,θ)i=1N(h,θ)[Z(xi)Z(xi+h)]2 {\gamma_e}\left( {h,\theta } \right) = {1 \over {2N\left( {h,\theta } \right)}}\sum\limits_{i = 1}^{N\left( {h,\theta } \right)} {{{\left[ {Z\left( {{x_i}} \right) - Z\left( {{x_i} + h} \right)} \right]}^2}} Note that xi is an experimental point; (xi + h) is an experimental point located at distance h from xi, in direction θ; Z (xi) is the value measured at point xi; Z (xi + h) is the value measured at point (xi + h); N (h, θ) is the number of pairs of observations in the direction considered θ and separated by distance h.

Ordinary kriging

Ordinary kriging is an interpolation procedure that provides, from the available data, an optimal, linear and unbiased estimate of the property under study and whose estimation error is minimized.[6] The interpolated value at the point x0, denoted Z* (x0), is given by: Z*(x0)=i=1NλiZi(x) {Z^*}\left( {{x_0}} \right) = \sum\limits_{i = 1}^N {{\lambda _i}{Z_i}\left( x \right)} The weights λi are chosen so as to minimize the estimation variance. This should lead to the following ordinary Kriging equations [19, 7, 29]: {i=1,nλiγ(xi,xj)+μ=γ(xj,x0)(i,j=1,2,...n)i=1Nλi=1 \left\{ {\matrix{ {\sum\limits_{i = 1,n} {{\lambda _i}\gamma \left( {{x_i},{x_j}} \right) + \mu = \gamma \left( {{x_j},{x_0}} \right)\left( {i,\,j = 1,2,...n} \right)} } \cr {\sum\nolimits_{i = 1}^N {{\lambda _i} = 1} } \cr } } \right. The kriging system of equation (19) will take the following matrix form: (γ11γ(x1,x2)γ(x1,xn)1γ(x2,x1)γ22γ(x2,xn)1γ(xn,x1)γ(xn,x2)γnn11110)(λ1λ2λnμ)=(γ(x1,x0)γ(x2,x0)γ(xn,x0)1) \left( {\matrix{ {{\gamma _{11}}} & {\gamma \left( {{{\rm{x}}_1},{{\rm{x}}_2}} \right)} & \cdot & {\gamma \left( {{{\rm{x}}_1},{{\rm{x}}_{\rm{n}}}} \right)} & 1 \cr {\gamma \left( {{{\rm{x}}_2},{{\rm{x}}_1}} \right)} & {{\gamma _{22}}} & \cdot & {\gamma \left( {{{\rm{x}}_2},{{\rm{x}}_{\rm{n}}}} \right)} & 1 \cr \cdot & \cdot & \cdot & \cdot & \cdot \cr {\gamma \left( {{{\rm{x}}_{\rm{n}}},{{\rm{x}}_1}} \right)} & {\gamma \left( {{{\rm{x}}_{\rm{n}}},{{\rm{x}}_2}} \right)} & \cdot & {{\gamma _{nn}}} & 1 \cr 1 & 1 & \cdot & 1 & 0 \cr } } \right)\left( {\matrix{ {{\lambda _1}} \cr {{\lambda _2}} \cr \cdot \cr {{\lambda _{\rm{n}}}} \cr \mu \cr } } \right) = \left( {\matrix{ {\gamma \left( {{{\rm{x}}_1},{x_0}} \right)} \cr {\gamma \left( {{{\rm{x}}_2},{x_0}} \right)} \cr \cdot \cr {\gamma \left( {{{\rm{x}}_{\rm{n}}},{x_0}} \right)} \cr 1 \cr } } \right) where n is the number of information available; xi and xj are measurement points; Z (xi) and Z (xj) are the values measured at points xi and xj; x0 is the point (or volume) to be estimated; µ is the Lagrange parameter; γ (xi, xj) is the value of the variogram γ (h) for the distance h between xi and xj; γ (xj, x0) is the value of the variogram γ (h) for the distance h separating xi and x0.

Kriging also makes it possible to evaluate the uncertainty associated with the estimator in the form of the kriging variance, which is given as: σe2=i=1,nλjγ(xi,x)γ¯(x,x)+μ \sigma _e^2 = \sum\limits_{i = 1,n} {{\lambda _j}\gamma \left( {{x_i},x} \right) - \bar \gamma \left( {x,x} \right) + \mu } where γ¯(x,x)γ¯(x,x) \bar \gamma \left( {x,x} \right)\bar \gamma \left( {x,x} \right) is the mean value of the variogram between two points belonging to the volume x. Note that if the volume is reduced to a point, then the distance between these points is zero.

Results and discussion
Distribution of the liquefaction potential index data

ArcGIS Geostatistical Analyst provides statistical tools that allow analyzing and mapping continuous data.[9] Figure (3) illustrates the location of the data and the spatial distribution of the liquefaction index on the site under study. It is easy to notice that the distribution of boreholes is not regular and does not cover the entire study area. The statistical distribution of the Liquefaction Potential Index (LPI) is illustrated by the histogram in Table 3. The Kurtosis coefficient (1.79) is greater than 0, which implies that the distribution is more flattened than a normal distribution. The Skewness coefficient (0.47) indicates that this distribution is not symmetrical; it is an asymmetric distribution, with a spread towards the low values. Therefore, a logarithmic transformation could certainly help to make the distribution of the data more symmetrical. However, this transformation is not recommended by several authors.[4,20,25,30]

Figure 3

Spatial distribution and histogram of liquefaction potential index.

Basic statistics of liquefaction potential index.

Number of values Mean Median Standard deviation Minimum Maximum Skewness Kurtosis
62 11.459 8.6759 10.456 0 32.718 0.472 1.793
Global trends

The trend analysis tool provides a three-dimensional perspective of the data.[9] The green line indicates the decreased value in the east-west direction, and the blue line shows a bell curve in the north-south direction. Liquefaction potential index values are higher in the northwest but there is not a sufficiently big trend in our data (Figure 4).

Figure 4

Trend analysis for the liquefaction potential index data.

Semivariogram modeling

The variogram is used to characterize the spatial continuity of the liquefaction potential index (LPI).[21,22] Figure 5 shows the experimental variogram, calculated for multiple distances, with a step of 300 meters, while assuming that this is an isotropic soil. It is worth noting that the fitting of several models, available in the GIS software (ArcGIS Geostatistical Analyst), was performed. The experimental values were mostly best fitted to the spherical model with R2 = 0.943 and minimum residual sum of squares (Table 4). The spherical model adopted in this analysis is defined by the following formula: γ(h)={c0+c(3h2ah32a3)for0<h<ac0+cforh>a \gamma \left( h \right) = \left\{ {\matrix{ {{c_0} + c\left( {{{3h} \over {2a}} - {{{h^3}} \over {2{a^3}}}} \right)} \hfill & {{for}\,0 < h < a} \hfill \cr {\,\,\,\,\,{c_0} + c} \hfill & {\,\,\,{for}\,h > a} \hfill \cr } } \right. where γ(h) is the semivariogram, h is the lag distance, a is the range, C0 is the nugget, and C is the sill.

Figure 5

Experimental and theoretical semivariograms for the liquefaction potential index.

Properties of the fitted semivariograms of the liquefaction potential index.

Model type Nugget C0 Partial Sill C+C0 Major range a (m) C/C+C0 RSS R2
Spherical 0 120 1800 1 1728 0.943
Exponential 0 135 1060 1 2633 0.885
Linear 38 146 3440 0.735 7272 0.629
Gaussian 5 125 1200 0.96 2205 0.900

Note: R2 is the coefficient of determination; RSS is the residual sum of squares, and c/c+c0 is nugget-sill ratio.

If the variogram actually depends only on the norm of h, it is said to be isotropic. On the other hand, if it depends on the h direction also, then it is anisotropic. Figure 6 shows directional variograms along four directions, that is, 0°, 45°, 90°, and 135°, with respect to the azimuthal direction. It is easy to see that the variograms have almost the same plateau, and the semivariogram in the 90° (east-west) direction grows more slowly than in the other directions. Therefore, one may say that there is a great spatial continuity of the liquefaction potential index in that direction, with a range of around 2600 meters. It can therefore be deduced that the geotechnical properties of the soil are more homogeneous in the east-west direction in comparison with the other directions. Semivariograms in the 0° (north-south) and 45° directions grow faster. Consequently, it can be said that there is a small spatial continuity of the liquefaction potential index along these 2 directions, with a range of about 1200 meters. It can hence be deduced that the geotechnical properties of the soil are rather heterogeneous.

Figure 6

Anisotropic semivariogram of liquefaction potential index variorum for directions 0°, 45°, 90°, and 135°.

Figure 7 depicts the theoretical model of a two-dimensional anisotropy, with a spherical model with a range of 2700 m, following the maximum direction of continuity of 90°, with a range of 1150 m. In the other minimum direction of 0° continuity, the plateau remains constant and equal to 120.

Figure 7

Anisotropic semivariogram model for the liquefaction potential index.

Cross validation

Cross-validation consists of temporarily eliminating a point from the data set, then estimating its value by kriging using the remaining data and the variogram model that is to be adjusted.[5] The clouds of correlations between the measured (LPI) values (on the abscissa) and the interpolated (LPI) values (on the ordinate) were compared for the two isotropic and anisotropic approaches. The results obtained showed a good value of the coefficient of determination R2, while taking into account the anisotropy of the variogram depicted in Figure (8).

Figure 8

Cross validation of the semivariogram model for the liquefaction potential index (isotropic analysis A, anisotropic analysis B).

Liquefaction potential index mapping

Ordinary kriging mapping of the soil liquefaction potential index over the entire study site was carried out. Maps for the ordinary kriging interpolation results as well as the corresponding kriging standard deviation maps for the two cases, isotropic and anisotropic, were presented. Figures 9a and 10a depict the spatial variations of the potential liquefaction index (LPI) of the soil in the study area. It was noted a large area was susceptible to liquefaction, particularly in the northwest of the region under study. Similarly, Figures 9b and 10b present the spatial variation of the kriging standard deviations. Consequently, regions of high uncertainty can therefore be visualized; the higher the standard deviation, the greater the uncertainty. The kriging standard deviation does not depend on the measured values of the liquefaction potential index (LPI); it depends only on the variogram model and the measurement point distributions. As for Figures 11a and 11b, they show the respective spatial variations of the potential index of liquefaction (LPI) of soil for maximum acceleration values of soil 0.2 g and 0.25 g, while assuming that this is an isotropic soil. It is worth mentioning the proportional relationship existing between the liquefaction risk and the increase or decrease in seismic acceleration.

Figure 9

Liquefaction potential hazard map predicted by ordinary kriging (a) and corresponding standard deviation map (b) for earthquake magnitude of 6.8 and peak ground acceleration of 0.3 g (isotropic analysis).

Figure 10

Liquefaction potential hazard map predicted by ordinary kriging (a) and corresponding standard deviation map (b) for earthquake magnitude of 6.8 and peak ground acceleration of 0.3 g (Anisotropic analysis).

Figure 11

Liquefaction potential hazard map for peak ground acceleration of 0.2 g (a) and 0.25 g (b) (Isotropic analysis).

Conclusions

The coupling of GIS geographic information system techniques and geostatistical analysis made it possible to carry out a spatial modeling of the risk of soil liquefaction over the entire study site, using digital maps for the estimate of the probable spatial distribution of the liquefaction potential index (LPI) of soil. Moreover, the kriging standard deviation maps of the (LPI) predicted values are quite important tools for geostatistical analysis. These maps can be used to identify areas of uncertainty. The results obtained showed that the higher the standard deviation, the greater the uncertainty. The presence of a large zone susceptible to liquefaction, in particular in the northwestern part of the study area, was confirmed by the results obtained. Note that the kriging standard deviation did not depend on the measured (LPI) values, but depended only on the variogram model and the distributions of the measurement points. Furthermore, the findings indicated that there is a proportional relationship between the liquefaction risk and the increase or decrease in seismic acceleration.

Figure 1

Geographical location of the study region and location of the CPT surveys.
Geographical location of the study region and location of the CPT surveys.

Figure 2

Example plots showing key steps in calculating liquefaction potential at one CPT (BH34): (a) tip resistance (qc); (b) side friction (fs); (c) soil type index (Ic); (d) corrected equivalent tip resistant (qc1N)cs; (e) cyclic stress ratio (CSR) and cyclic resistance ratio (CRR); (f) factor of safety (FS).
Example plots showing key steps in calculating liquefaction potential at one CPT (BH34): (a) tip resistance (qc); (b) side friction (fs); (c) soil type index (Ic); (d) corrected equivalent tip resistant (qc1N)cs; (e) cyclic stress ratio (CSR) and cyclic resistance ratio (CRR); (f) factor of safety (FS).

Figure 3

Spatial distribution and histogram of liquefaction potential index.
Spatial distribution and histogram of liquefaction potential index.

Figure 4

Trend analysis for the liquefaction potential index data.
Trend analysis for the liquefaction potential index data.

Figure 5

Experimental and theoretical semivariograms for the liquefaction potential index.
Experimental and theoretical semivariograms for the liquefaction potential index.

Figure 6

Anisotropic semivariogram of liquefaction potential index variorum for directions 0°, 45°, 90°, and 135°.
Anisotropic semivariogram of liquefaction potential index variorum for directions 0°, 45°, 90°, and 135°.

Figure 7

Anisotropic semivariogram model for the liquefaction potential index.
Anisotropic semivariogram model for the liquefaction potential index.

Figure 8

Cross validation of the semivariogram model for the liquefaction potential index (isotropic analysis A, anisotropic analysis B).
Cross validation of the semivariogram model for the liquefaction potential index (isotropic analysis A, anisotropic analysis B).

Figure 9

Liquefaction potential hazard map predicted by ordinary kriging (a) and corresponding standard deviation map (b) for earthquake magnitude of 6.8 and peak ground acceleration of 0.3 g (isotropic analysis).
Liquefaction potential hazard map predicted by ordinary kriging (a) and corresponding standard deviation map (b) for earthquake magnitude of 6.8 and peak ground acceleration of 0.3 g (isotropic analysis).

Figure 10

Liquefaction potential hazard map predicted by ordinary kriging (a) and corresponding standard deviation map (b) for earthquake magnitude of 6.8 and peak ground acceleration of 0.3 g (Anisotropic analysis).
Liquefaction potential hazard map predicted by ordinary kriging (a) and corresponding standard deviation map (b) for earthquake magnitude of 6.8 and peak ground acceleration of 0.3 g (Anisotropic analysis).

Figure 11

Liquefaction potential hazard map for peak ground acceleration of 0.2 g (a) and 0.25 g (b) (Isotropic analysis).
Liquefaction potential hazard map for peak ground acceleration of 0.2 g (a) and 0.25 g (b) (Isotropic analysis).

Computation of LPI for peak horizontal ground acceleration of 0.3 g corresponding to Mw = 6.8.

Depth (m) Average Density (KN/m3) Overburden Stress σV0 (Kpa) Effective stress σV0 (Kpa) qc (Kpa) fs (Kpa) CRR rd CSR Fs z (m) H (m) w(z) F w(z)*F*H
2.50 19.58 48.95 23.95 2550.00 125.00 0.59 0.98 0.30 1.95 1.25 2.50 9.38 0.00 0.00
3.50 19.58 68.53 33.53 1140.00 41.00 0.19 0.98 0.30 0.62 3.00 1.00 8.50 0.38 3.21
8.50 19.58 166.43 81.43 2430.00 130.00 0.33 0.93 0.29 1.15 6.00 5.00 7.00 0.00 0.00
10.50 19.58 205.59 100.59 1450.00 51.00 0.13 0.89 0.28 0.48 9.50 2.00 5.25 0.52 5.43
14.50 19.58 283.91 138.91 2860.00 112.00 0.18 0.78 0.24 0.73 12.50 4.00 3.75 0.27 4.02
LPI 12.66

Basic statistics of liquefaction potential index.

Number of values Mean Median Standard deviation Minimum Maximum Skewness Kurtosis
62 11.459 8.6759 10.456 0 32.718 0.472 1.793

Properties of the fitted semivariograms of the liquefaction potential index.

Model type Nugget C0 Partial Sill C+C0 Major range a (m) C/C+C0 RSS R2
Spherical 0 120 1800 1 1728 0.943
Exponential 0 135 1060 1 2633 0.885
Linear 38 146 3440 0.735 7272 0.629
Gaussian 5 125 1200 0.96 2205 0.900

Calculated values of liquefaction potential index (LPI).

Borehole Coordinates LPI Borehole Coordinates LPI


Easting (m) Northing (m) Easting (m) Northing (m)
BH1 519039.1 4062223 5.57 BH32 517267.3 4060744 24.11
BH2 519247.8 4061957 0.00 BH33 517582.2 4061421 16.97
BH3 519471.7 4061669 0.73 BH34 516748.6 4060166 12.66
BH4 519697.4 4061414 0.00 BH35 516205.9 4060353 26.85
BH5 519931.9 4061204 0.00 BH36 516494 4060620 26.54
BH6 519925.8 4060761 0.00 BH37 515785.8 4060475 21.12
BH7 519655.4 4061015 0.00 BH38 515446.7 4061284 23.90
BH8 519438.6 4061314 0.00 BH39 515676.1 4061850 30.48
BH9 519187.1 4061546 2.34 BH40 515196.3 4061971 32.72
BH10 518939.2 4061757 5.73 BH41 517075.7 4059634 27.74
BH11 518750.8 4062111 8.60 BH42 516106.9 4061729 29.61
BH12 519674.6 4060860 8.75 BH43 515283.1 4059841 13.44
BH13 519369.7 4060970 3.80 BH44 515430 4060130 12.45
BH14 519124.5 4061169 0.00 BH45 515829.7 4059942 23.93
BH15 518443.2 4061889 9.97 BH46 518340.9 4061656 19.64
BH16 519199.3 4060493 8.61 BH47 517981.9 4061644 6.18
BH17 518900 4060902 5.38 BH48 517343.3 4058514 0.93
BH18 518591 4061257 16.25 BH49 517268.7 4058736 0.00
BH19 518848 4060181 3.69 BH50 517249.5 4058958 0.00
BH20 518474.9 4060857 19.59 BH51 517536.8 4059125 0.21
BH21 518374.6 4060557 24.41 BH52 517465.8 4058892 0.00
BH22 518288 4060102 26.39 BH53 517511 4058648 3.24
BH23 518125.3 4059736 13.89 BH54 517506.3 4058348 0.00
BH24 517678.6 4059724 24.30 BH55 517628.5 4058870 1.83
BH25 517901.1 4060124 28.98 BH56 517669.1 4058681 5.21
BH26 518152.4 4060413 25.75 BH57 517711.5 4058493 1.61
BH27 518130.8 4060901 24.28 BH58 515780.4 4058544 6.00
BH28 517660.5 4060689 13.15 BH59 515724.8 4059143 3.29
BH29 517854.7 4060945 11.10 BH60 515743.1 4059410 4.85
BH30 517444.4 4059335 0.00 BH61 515869.5 4059687 11.92
BH31 517097.9 4061032 20.12 BH62 515898.42 4059487.6 11.66

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