Aleatory Uncertainty

In geotechnical engineering, natural variability can be used interchangeably with inherent or spatial variability, stemming from a combination of various geologic, environmental and physio-chemical processes (e.g. Lumb 1966; Phoon and Kulhawy 1999a). Various geological processes that take place with time will lead to changes of the geotechnical properties (Bjerrum 1967). As a result of the glacio-tectonic disturbance, for example, Pliocene deposits exhibit an extremely pronounced spatial variability (Jamiolkowski 2014). Natural variability is routinely discussed in standard texts (see Chapter 10 in Look

• 2014) and design guidelines (e.g. DNV 2017; JCSS 2006). For the simplest example of a test profile varying with depth within a single soil layer, it is customary to separate this profile into a trend function and a fluctuating component (Phoon and Kulhawy 1999a). In the literature, this fluctuating component or natural variability is modelled, whenever possible, as a statistically homogeneous random field (Vanmarcke 1983). All soil/rock properties in situ will vary vertically and horizontally, most often with more pronounced variability in the vertical direction. While a random field model is easy to describe probabilistically, it is noteworthy that a full statistical characterization of this model in 3D based on actual sparse and incomplete site investigation data is very difficult and a practical solution only emerged recently (Ching et al. 2020a,
• 2021). The characterization of site stratigraphy is another major missing feature of random field studies until quite recently (e.g. Wang et al. 2013, 2016, 2017, 2018, 2019a, b; Ching et al. 2015; Li et al. 2016; Qi et al. 2016; Cao et al. 2019; Shuku et al. 2020; Zhao et al. 2020). Phoon et al. (2021) proposed three challenges in data-driven site characterization.

As suggested in Annex D of ISO 2394 (ISO 2015), the characterization of natural variability is the most important element to develop risk-informed decision making in the form of RBD (e.g. Nadim 2017; Phoon and Ching 2019; Phoon et al. 2019). Notwithstanding the evolving research on this issue, it is useful and practical to present statistics that are (1) based on actual databases and (2) aligned with existing design practice. The key features are as follows: (1) COV of a geotechnical parameter found in Phoon and Kulhawy (1999a, b), who undertook a comprehensive compilation of soil data, does not take a unique value; (2) multivariate nature of geotechnical data can be exploited to reduce the COV; and (3) spatial variability affects the limit state beyond the reduction in COV arising from Vanmarcke’s spatial averaging (Phoon et al. 2016a). An extensive characterization of COV for specific sites can be found elsewhere (e.g. Soulie et al. 1990; Rehfeldt et al. 1992; Cherubini et al. 2007; Chiasson and Wang 2007; Jaksa 2007; Uzielli et al. 2007; Stuedlein 2011; Sruedlein et al. 2012a). Tables 2.1, 2.2 and 2.3 summarize typical natural variability of soil strength properties, index parameters and field measurements, respectively. Table 2.4 summarizes the statistics of index, strength and deformation properties of igneous, sedimentary and metamorphic rocks that are taken from Aladejare and Wang

(2017). The number of tests is presented as an indicator of the degree of statistical uncertainty associated with the mean and the COV. Based on the compiled soil data, approximate guidelines for inherent soil variability were proposed in Table 2.5 by Phoon and Kulhawy (1999a). Similar guidelines are available for properties of rocks (e.g. Phoon et al. 2016a; Aladejare and

Table 2.1 Summary of the natural variability of soil strength parameters

Property	Soil type	n	N		Property mean		Property COV (%)
			Range	Mean	Range	Mean	Range	Mean
	Fine grained	38	2-S38	101	6-412	100	6-56	33
	Clay, silt	13	14-82	33	15-363	276	1 1-49	22
	Clay	10	12-86	47	130-713	405	18-42	32
	Clay	42	24-124	48	8-638	112	6-80	32
	Sand	7	29-136	62	35-41	37.6	5-1 1	9
	Clay, silt	12	5-51	16	9-33	15.3	10-50	21
	Clay, silt	9	—	—	17-41	33.3	4-12	9
	Clay, silt	4	—	—	0.24- 0.69	0.51	6-46	20
	Clay, silt	3	—	—	—	0.62	6-46	23
	Sand	13	6-1 1 1	45	0.65- 0.92	0.74	5-14	9

(Source: Table I in Phoon and Kulhawy 1999a)

Note: s_u = undrained shear strength, ф' = effective friction angle, UC = unconfined compression test, UU = unconsolidated-undrained triaxial compression test, CIUC = consolidated-isotropic-undrained compression test.TC = triaxial compression test, DS = direct shear test, n = no. of data groups and N = no. of tests per group. For the two greyed rows, laboratory test type was not reported.

Table 2.2 Summary of the natural variability of soil index parameters

Index	Soil type0	n	N		Index mean		Index COV (%)
			Range	Mean	Range	Mean	Range	Mean
	Fine grained	40	17-439	252	13-105	29	7-46	18
	Fine grained	38	15-299	129	27-89	51	7-39	18
	Fine grained	23	32-299	201	14-27	22	6-34	16
	Fine grained	33	15-299	120	12-44	25	9-57	29
	Clay, silt	2	32-118	75	—	0.094	60-88	74
	Fine grained	6	5-3200	564	14-20	17.5	3-20	9
	Fine grained	8	4-315	122	13-18	15.7	2-13	7
	Sand	5	—	—	30-70	50	1 1-36	19
	Sand	5	—	—	30-70	50	49-74	61

(Source: Table 2 in Phoon and Kulhawy 1999a)

Note: w_n = natural water content, w_L = liquid limit, w_p = plastic limit, PI = plasticity index, LI = liquidity index, у = total unit weight, y_d = dry unit weight and D_r = relative density.‘Fine-grained materials derived from a variety of geologic origins (e.g. glacial deposits, tropical soils and loess). ^bTotal variability for the direct method of determination. ‘Total variability for the indirect determination using SPT values.

Wang 2017) and cemenr-mixed soils (e.g. Liu et al. 2015; Pan et al. 2018,

2019). They are more empirically grounded than the indicative standard deviations of some soil strength and stiffness properties noted in Table 2.6, which are given in Section 3.7 - Soil Properties of the JCSS Probabilistic Model Code (JCSS 2006). Phoon et al. (2016a) made the following observations:

The highest COV values seem to be associated with measurements of soil stiffness.

With respect to field measurements, the COV values of natural variability for sand are higher than that for clay.

The COV values of natural variability for index parameters are the lowest, with the possible exception of the relative density and liquidity index.

Intuitively, the more information an engineer has that can provide different insights into the properties of the geomaterials and the behaviour of geotechnical structures, the more reliable judgement he or she can make, such as the selection of design parameters and calculation methods. Ching et al. (2010) utilized a Bayesian analysis to reduce the uncertainties in the estimation of undrained shear strength by incorporating multivariate

Table 2.3 Summary of the natural variability of field measurements

Test type	Property	Soil type	n	N		Property value		Property COV (%)
				Range	Mean	Range	Mean	Range	Mean
CPT		Sand	57	10-2039	1 15	0.4-29.2	4.1	10-81	38
CPT		Silty clay	12	30-53	43	0.5-2.1	1.59	5-40	27
CPT		Clay	9	—	—	0.4-2.6	1.32	2-17	8
VST		Clay	31	4-31	16	6-375	105	4-44	24
SPT		Sand	22	2-300	123	7-74	35	19-62	54
SPT		Clay, loam	2	2-61	32	7-63	32	37-57	44
DMT		Sand to clayey sand	15	12-25	17	64-1335	512	20-53	33
DMT		Clay	13	10-20	17	119-455	358	12-32	20
DMT		Sand to clayey sand	15	12-25	17	346-2435	1337	13-59	37
DMT		Clay	13	10-20	17	502-876	690	12-38	20
DMT		Sand to clayey sand	15	10-25	15	9.4-46.1	25.4	9-92	50
DMT		Sand, silt	16	—	—	10.4-53.4	21.6	7-67	36
DMT		Sand to clayey sand	15	10-25	15	0.8-8.4	2.85	16-130	■ 53
DMT		Sand, silt	16	—	—	2.1-5.4	3.89	8-48	30
DMT		Sand to clayey sand	15	10-25	15	1.9-28.3	15.1	20-99	44
DMT		Sand, silt	16	—	—	1.3-9.3	4.1	17-67	38
PMT		Sand	4	—	17	1617-3566	2284	23-50	40
PMT		Cohesive	5	10-25	—	428-2779	1084	10-32	15
PMT		Sand	4			5.2-15.6	8.97	28-68	42

(Source: Table 3 in Phoon and Kulhawy 1999a)

Note: CPT = cone penetration test,VST = van shear test, SPT = standard penetration test, DMT = dilatometer test, PMT = pressuremeter test, q_c = CPT tip resistance, q_t = corrected CPT tip resistance, N_SPT = SPT blow count (number of blows per 305 mm), A and В = DMT A and В readings, E_D = DMT modulus, l_D = DMT material index, K_D = DMT horizontal stress index, p_L = PMT limit stress and E_PMT = PMT modulus.

Rock type	Property		No. of data groups	No. of tests per group		Range of data	Property mean value		Property COV (%)
	Type	Parameter		Range	Mean		Range	Mean	Range	Mean
Igneous	Index		12	9-40	13	1.60-3.06	2.49-2.75	2.64	0.7-5.6	2.7
			6	5-172	38	1.45-3.04	2.65-2.88	2.73	0.7-1.7	1.2
			2	5-8	7	55.10-97.12	60.50-96.88	78.69	0.4-10.7	5.6
			13	5-517	84	10.90-30.30	21.60-28.34	23.39	0.9-13.7	4.9
			3	4-1 1	8	0.09-4.00	0.45-2.76	1.34	29.9-93.1	61.5
			19	9-172	26	0-42.50	0.20-16.56	4.82	35.7-107.1	69.7
			16	5-517	62	17.00-76.00	32.19 -75.75	46.28	0.8-48.9	16.8
			4	9-108	38	1 1.00-88.00	39.20-68.50	59.65	3.1-67.1	24.2
	Rock mass		2	57-70	64	37.00-77.00	52.1-53.0	52.55		17.5
			2	6-57	32	30.00-100.00	65.00-97.03	81.02		2.9
	Strength		42	5-164	55	0.75-380.02	24.78- 329.70	103.90	8.9-103.9	38.7
			16	5-517	60	0.26-34.14	4.39-26.05	10.70	20.7-87.5	36.0
			14	5-517	101	0.15-17.40	2.49-12.53	7.26	14.1-98.1	40.6
			3	7-29	7	0-176.00
			5	7-29	7	0-66.80
			2		3	8.00-35.00	13.00-32.00	18.83		22.9
	Deformation		25	5 -108	40	0.50-97.38	4.10-71.90	38.40	4.0-91.2	30.7
			18	5-172	20	0.04-0.44	0.17-0.33	0.27	9.4-23.5	14.1

Sedimentary	Index		27	6-120	28	1.35-3.61	1.73-3.00	2.38	0.4-13.0	5.6
			10	10-49	17	1.79-2.95	2.63 - 2.71	2.67	0.4-3.4	1.4
			4	19-49	35	1.30-99.00	71.10-92.30	85.17	5.4-84.4	38.4
			21	2-778	92	8.80-31.60	19.38-27.20	24.07	0.1-12.7	6.2
			7	1 1-121	28	0.02-16.00	0.37-7.00	3.13	48.0-188.0	97.0
			47	7-262	31	0-67.88	0.26-39.94	10.68	1.0-181.6	53.4
			29	6-510	76	9.00-76.00	15.00-70.00	40.26	0.8-33.7	18.7
			12	6-44	18	4.20-96.00	12.87-60.16	46.89	5.4-39.2	20.3
	Rock mass		4	17-120	41	15.00-89.00	22.60- 60.40	46.13	17.1- 27.0	21.5
			5	15-120	49	4.50-99.30	57.40-80.80	70.00	1 1.2-52.1	29.0
	Strength		73	6-470	50	0.63-345	4.40-264.00	62.80	0.4-109.6	42.8
			26	3-77	25	0.06-76.60	1.20-17.00	7.90	1.6-59.3	31.5
			37	6-1305	140	0.05-14.60	0.23-16.21	3.52	2.9-91.7	40.9
			9	13-58	30	0-96.00	2.57-31.82	21.23	15.7-79.0	42.8
			12	9-58	25	7.00-66.00	24.93-58.31	41.71	3.9-30.6	14.1
			5	8-58	25	2.00-41.00	4.00-21.00	17.94	14.2-27.5	20.9
	Deformation		50	6-121	30	0.06-196.26	0.59-73.17	23.7	7.0-128.1	43.0
			13	2-62	16	0.03-0.45	0.17-0.39	0.24	4.0-75.4	25.6

(Continued)

Rock type	Property		No. of data groups	No. of tests per group		Range of data	Property mean value		Property COV (%)
	Type	Parameter		Range Mean			Range	Mean	Range	Mean
Metamorphic	Index		9	9-24	16	2.58-3.1 1	2.58-2.82	2.70	0.3-2.5	1.2
			3	10-25	16	2.18-2.92	2.66-2.72	2.71	1.7-3.0	2.4
			2	10-1 1	1 1	95.87-99.33	98.12-98.44	98.28	1.0-1.1	1.0
			4	5-92	43	24.71-28.15	25.50-26.56	26.10	1.5-3.1	1.8
			4	9-13	1 1	0-6.60	0.05-3.85	1.39	49.8- 130.4	1 18.4
			1 1	2-32	16	0.06-22.40	0.32-1.88	0.78	12.5- 1 13.6	61.8
			9	2-92	16	31.66-63.00	37.47-56.50	51.63	2.7-16.3	10.0
			3	2-9	7	46.00-82.00	57.83-64.00	61.35	9.0-39.8	20.8
	Strength		23	2-151	25	4.62-320.00	32.81 — 150.00	81.45	7.9-70.4	43.1
			10	2-151	53	0.55-19.00	3.18 -12.10	10.54	9.2-59.0	31.5
			14	2-92	25	0.52-13.30	2.79-6.56	4.61	6.9-88.0	51.2
			3			0-76.00
			3	13-20	1 1	15.00-60.60		40.87		3.4
			1			3.00-33.00	3.00-29.00
	Deformation		9	2-41	8	1.00-88.40	1 1.57-38.50	24.33	22.6- 82.1	43.9
			3	4-37	14	0.02-0.40		0.24		86.1

(Source: data taken from Tables 2-9 in Aladejare and Wang 2017)

Note: p = bulk density, G_s = specific gravity, l_d2 = slake durability index, у = unit weight, w_n = water content, n = porosity R = Schmidt hammer hardness (R_L = L-type Schmidt hammer hardness), S„ = Shore scleroscope hardness, GSI = geological strength index, RQD = rock quality designation, o_c = uniaxial compressive strength, ct_m = Brazilian tensile strength, l_s = point load strength index (l_s50 = l_s for diameter SO mm),c = cohesion, ф = friction angle, mi = Hoek-Brown constant, E = Young’s modulus, and о = Poisson ratio.

Test type	Property	Soil type	Mean	COV (%)
Lab strength		Clay	10-400 kPa	20-55
		Clay	10-350 kPa	10-30
		Clay	150-700 kPa	20-40
		Clay and sand	20°-40°	5-15
Lab index		Clay and silt	13%-100%	8-30
		Clay and silt	30%-90%	6-30
		Clay and silt	15%-25%	6-30
		Clay and silt	10%-40%	_a
		Clay and silt	10%	_a
		Clay and silt	13-20 kN/m3	< 10
		Sand	30%-70%	10-40; 50-70b
CPT		Clay	0.5-2.5 MPa	< 20
		Clay	0.5-2.0 MPa	20-40
		Sand	0.5-30 MPa	20-60
VST		Clay	5-400 kPa	10-40
SPT		Clay and sand	10-70	25-50
DMT		Clay	100-450 kPa	10-35
		Sand	60-1300 kPa	20-50
		Clay	500-880 kPa	10-35
		Sand	350-2400 kPa	20-50
		Sand	1-8	20-60
		Sand	2-30	20-60
		Sand	10-50 MPa	15-65
PMT		Clay	400-2,800 kPa	10-35
		Sand	1,600- 3,500 kPa	20-50
		Sand	5-15 MPa	15-65

(Source: Table 7 in Phoon and Kulhawy 1999a)

Note: ^aCOV = (3%-l2%)/mean. ^bThe first range corresponds to the total variability of the direct method of determination, and the second range is the total variability of the indirect determination by SPT values.

information. Ching and Phoon (2015) used a multivariate database CLAY/7/6310 to estimate the mobilized undrained shear strength from a variety of undrained shear strengths obtained using different test procedures (e.g. direct shear, unconfined compression, field vane, consolidated undrained compression and extension and isotropically consolidated undrained compression). Muller et al. (2014) extended the method of Ching et al.

(2010) to reduce the total uncertainty of spatially averaged values of undrained shear strength. This extended multivariate approach was then utilized by Muller et al. (2016) to assess the representative average values and reduce the associated uncertainties of undrained shear strength for staging the construction of embankments on soft clay.

Compared to the mean and COV value of a single quantity, the characterization of correlation can be more challenging. Correlation exists between different soil/rock parameters, such as between the undrained shear strength

Table 2.6 Indicative standard deviations of soil properties (Source Table 3.7.4.2 in JCSS 2006)

and the OCR. This is called cross-correlation. It is most commonly reported in association with transformation models (Kulhawy and Mayne 1990). Correlation also exists between the same parameter measured at different spatial locations. This is called spatial correlation. It is commonly reported as an autocorrelation function parameterized by a value called the scale of fluctuation. The scale of fluctuation describes the distance over which the

Table 2.1 Summary of the range of the scale of fluctuation (SOF) values for natural soils

(Source: data taken from Table 8 in Cami et al. 2020) Note: N = number of studies collated.

(Source: data taken from Table 4 in Pan et al. 2018 and Table 9 in Cami et al. 2020)

Data source: Liu et al. (201S) and Pan et al. (2018,2019).

Note: Small-scale SOF is the fluctuation due to insufficient mixing or positioning error, while large-scale SOF is the fluctuation due to natural variation of water content.

soil/rock parameters are similar or correlated and therefore, it is also known as correlation length. Recently, Cami et al. (2020) collated a comprehensive database of the horizontal and vertical scale of fluctuation values. The results are summarized in Table 2.7 for natural soils and Table 2.8 for cement-mixed soils to provide engineers with a sense of the probable range of the scale of fluctuation values. For natural soils, the horizontal scale of fluctuation ranges from nearly 0 to 100 m with the most probable range of 0 to 60 m, while the vertical scale of fluctuation ranges from nearly 0 to 9 m with the most probable range of 0 to 5 m.

Typically, soil-structure interaction occurs over a finite volume of soil that is called the influence zone. For example, this can be identified from the shearing surface around a pile tip observed by Vesic (1977). The properties of soil mass beyond this zone have an insignificant effect on soil-structure interaction behaviour. It is the mobilized values in this zone that one should consider in design. In EN 1997-1:2004, Clause 2.4.5.2 - Characteristic values of geotechnical parameters - described the influence zone with the following two application rules:

• 1. The influence zone governing the behaviour of a geotechnical structure at a limit state is usually much larger than a test sample or in an in situ test. The characteristic value of the governing parameter is often the mean of a range of values covering a large surface or volume of the ground.
• 2. The influence zone at this limit state may depend on the behaviour of the supported structure. For instance, when considering the ultimate limit state of a building on several footings, the governing parameter should be the mean strength over each individual zone of ground beneath a footing if the building is unable to resist a local failure. If, however, the building is stiff and strong enough, the governing parameter should be the mean of these mean values over the entire zone or part of the zone of ground under the building.

In the consideration of the natural or spatial variability of a soil mass, the problem becomes more complicated because the formation of the influence zone is complex in spatially heterogeneous soils. It can depart significantly from the classical symmetrical failure mechanisms developed for homogeneous or simple linearly varying soils. It is advantageous in practice to convert the governing parameter (spatially variable) into a homogeneous spatial average (e.g. Vanmarcke 1977,1983). This is similar to the concept of homogenization (e.g. Arwade and Deodatis 2011; Arwade et al. 2016). Several studies on the characterization of a spatially variable soil mass using the spatial average have been implemented for shear strength (e.g. Ching et al. 2016a), Young’s modulus (e.g. Griffiths et al. 2012; Paiboon et al. 2013; Ching et al. 2017b), bearing capacity of a footing (e.g. Soubra et al. 2010; Honjo and Otake 2013), settlement of a footing (e.g. Fenton and Griffiths 2005) and active lateral force (e.g. Hu and Ching2015).Tabarroki et al. (2021) proposed a mobilization-based characteristic value that accounts for spatial variability, failure mechanism, and their interactions correctly.