Estimating Earthquake-Induced Pore Water Pressure: Hierarchy of Existing Methods

Abstract

The prediction of earthquake-induced excess pore water pressure is becoming a crucial step for liquefaction-related problems, such as the prediction of settlements, assessment of bearing capacity of shallow foundations, and efficiency of liquefaction mitigation measurements. The goal of this review is to provide a state of the art of the available methods for estimating excess pore water pressure from the most simplified approaches, based on simple hand-made calculations, to the most simplified models for pore pressure prediction, and finally to the most advanced models implemented in computer codes for seismic site response analysis. The hierarchy of the methods is enriched with exemplificative applications from real cases to help understand the classification provided and the usability of each method according to the needs of the professional practice.

1. Introduction

Seismically induced excess pore water pressure within saturated loose sandy soils is fundamentally triggered by cyclic loadings under undrained conditions. This reduces the soil effective stress state by activating a mechanism that can culminate in either catastrophic flow failure (liquefaction) or progressive strain degradation (cyclic softening).

Experimental research assessed that the generation of excess pore water pressure depends strongly on ground motion parameters (e.g., peak ground acceleration—PGA, duration, frequency content), with high correlation with PGA and Arias Intensity [1]. Kamura et al. [2] show results from continuously measuring ground acceleration and pore water pressure at a liquefaction-prone site in Japan. Based on the records, they established a threshold acceleration value that initiates the triggering of excess pore pressure and defined the “contribution duration” as the temporal interval during which the recorded ground motion accelerations surpassed the established threshold.

An investigation performed in a laboratory on layered sand columns showed that soil type, density, and layering/permeability contrasts significantly affect the dissipation rates and attained values of the pore pressure build-up, where lower permeability interlayers prolong high excess pore water pressure [3]. Large-scale shaking table model tests and centrifuge tests also confirm that initial liquefaction onset and pore water pressure evolution depend on seismic intensity and soil stratigraphy [4,5].

Experiments investigating the mechanisms of excess pore water pressure generation with a structure settled on soil deposits and without (free-field) show the complexity of the problem depending on the type and geometry of the foundations and soil properties and the layering configurations [6,7,8,9,10,11]. Due to their complexity, the effects of buildings on pore pressure generation are beyond the scope of this review. In addition, after excess pore pressure build-up, the dissipation and reconsolidation process controls post-earthquake deformation and settlements, with permeability and void ratio evolution playing a key role in reconsolidation behavior [12,13,14,15,16,17].

Cubrinovski et al. [18] demonstrated via effective stress dynamic analysis that the simple metrics (like CPT-based assessment) for the critical layers were essentially identical between sites that experienced liquefaction and sites that experienced no liquefaction after the 2010–2011 Canterbury seismic events, implying the urgency to consider the entire soil profile behavior (i.e., the “system response”) rather than just the properties of the individual weakest layer.

For the above reason, the prediction of the seismically induced excess pore water pressure is of utmost importance.

Machine learning approaches are emerging to predict excess pore water pressure response from stress histories and soil properties [19]. Since this is an emerging research area, it was also beyond the scope of this review.

Biondi [20] reviews in detail several pore water pressure relationships by classifying them into two different groups: the first encompasses empirical relationships which are formulated as analytical functions to replicate the outcomes of experimental testing, e.g., [21,22,23,24,25,26,27]; the second one includes relations derived from theoretical models of the cyclic soil behavior, with experimental data used for parameter calibration, e.g., [28,29,30].

One of the most popular analytical relationships belonging to the first group has been proposed by Booker et al. [31] as follows:

$${\mathrm{r}}_{\mathrm{u}}={\displaystyle \frac{2}{\mathsf{\pi }}}{\arcsin \left({\displaystyle \frac{\mathrm{N}}{{\mathrm{N}}_{\mathrm{L}}}}\right)}^{1/2\mathsf{\beta }}$$

(1)

where r_u is the pore pressure ratio defined as the ratio between the seismically induced pore pressure increment, Δu, and the initial effective overburden stress, σ’_v0, N is the number of cycles, N_L is the number of cycles to reach liquefaction, and β is a coefficient depending on and physical and mechanical soil properties.

Several modified versions of the above Equation (1) were proposed to better fit the test data; such as in a recent publication [32].

In such a way, N/N_L can be considered a factor of safety because it correlates with the seismic demand, N, and the soil resistance to liquefaction, N_L.

The use of the first group identified by Biondi [20] typically necessitates a comparison between the irregular loading due to an earthquake and the soil resistance as determined from laboratory investigations, which commonly employ uniform loads. To make this comparison possible, the irregular shaking must first be converted into an equivalent series of uniform cycles. Following the pioneering studies by Seed et al. [33], a shear stress history variable at any given time due to the earthquake can be reduced into an equivalent number of cycles, N_eq, with amplitude equal to 0.65 of the maximum shear stress, τ_max, which would cumulatively produce the same excess pore pressure buildup as the actual motion. A significant number of methodologies have been developed to estimate the equivalent number of uniform cycles of an earthquake (e.g., [34,35,36,37]). However, these approaches are often not easy to apply, and the results are highly dependent on the conversion criterion and the techniques used for selecting and counting the cycles that contribute to pore pressure accumulation [38].

Hashash et al. [39] classified the pore pressure generation models into three categories: stress-based, strain-based, and energy-based models, based on the variable used for predicting the pore pressure build-up (i.e., shear stress, shear strain, or dissipated energy, respectively). Hashash et al. [39] observed that if the cyclic stress-controlled tests inspired the stress-based approach, other subsequent research proposed a correlation with the shear strain [40,41] or the energy dissipated within the soil deposit [42].

In the strain-based pore pressure models, Chen et al. [43] proposed a model where residual excess pore water pressure is quantified by the volumetric strain changes owing to cyclic shearing, accounting for the concept of threshold shear strain, i.e., the strain below which excess pressure is not generated.

In the framework of the stress-based pore pressure models, excess pore water pressure can be estimated in a simplified manner based on the damage parameter as proposed by Chiaradonna et al. [44,45]. This model utilizes a defined damage parameter, κ, which is an incrementing variable throughout the loading history and establishes a direct correlation with the pore pressure accumulation according to the following relationship:

$${\mathrm{r}}_{\mathrm{u}}=a{\left({\displaystyle \frac{\mathsf{\kappa }}{{\mathsf{\kappa }}_{\mathrm{L}}}}\right)}^b+c{\left({\displaystyle \frac{\mathsf{\kappa }}{{\mathsf{\kappa }}_{\mathrm{L}}}}\right)}^d$$

(2)

where a, b, c, and d are fitted model parameters, and κ_L is the damage parameter at liquefaction, defined as follows:

$${\mathsf{\kappa }}_{\mathrm{L}}=4\hspace{0.33em}{\mathrm{N}}_{\mathrm{r}}{\left({\mathrm{CSR}}_{\mathrm{r}}-{\mathrm{CSR}}_{\mathrm{t}}\right)}^{\mathsf{\alpha }}$$

(3)

where (N_r, CSR_r) identifies a reference point, CSR_t is the threshold below which there is no pore pressure accumulation, and the parameter α defines the dependency of the soil resistance on the number of cycles. This latter is analytically described as a relationship proposed by Park et al. [46]:

$${\displaystyle \frac{\left(\mathrm{CRR}-{\mathrm{CSR}}_t\right)}{\left({\mathrm{CSR}}_r-{\mathrm{CSR}}_t\right)}}={\left({\displaystyle \frac{N_r}{N_L}}\right)}^{{\displaystyle \frac{1}{\alpha }}}$$

(4)

Due to the proportionality between the damage parameter and the number of cycles for uniform cyclic loading [45], Equation (2) can be rewritten as follows:

$${\mathrm{r}}_{\mathrm{u}}=a{\left({\displaystyle \frac{\mathrm{N}}{{\mathrm{N}}_{\mathrm{L}}}}\right)}^b+\left(0.95-a\right){\left({\displaystyle \frac{\mathrm{N}}{{\mathrm{N}}_{\mathrm{L}}}}\right)}^d$$

(5)

where N/N_L is the same normalized number of cycles as Equation (1).

For a generic history of shear stress, this is normalized to the initial vertical effective stress as follows:

$${\mathsf{\tau }}^{*}(\mathrm{t})={\displaystyle \frac{\left|\mathsf{\tau }(\mathrm{t})\right|}{{\mathsf{\sigma }}_{\mathrm{v}0}^{\prime }}}$$

(6)

For a time instant, the damage parameter is computed as follows:

$$\mathsf{\kappa }(\mathrm{t})={\mathsf{\kappa }}_0+\mathrm{d}\mathsf{\kappa }$$

(7)

where κ₀ is the damage cumulated at the last reversal point of the function (τ*, CSR_t) reached at the time instant, t. The damage κ₀ is defined as follows:

$${\mathsf{\kappa }}_0=\left\{\begin{array}{ccc}\mathsf{\kappa }(\mathrm{t}-\mathrm{d}\mathrm{t})& \mathrm{if}{\dot{\mathsf{\tau }}}^{*}(\mathrm{t})=0& \mathrm{or}{\mathsf{\tau }}^{*}(\mathrm{t})={\mathrm{CSR}}_{\mathrm{t}}\\ {\mathsf{\kappa }}_0(\mathrm{t}-\mathrm{d}\mathrm{t})& \mathrm{if}{\dot{\mathsf{\tau }}}^{*}(\mathrm{t})\ne 0& \mathrm{or}{\mathsf{\tau }}^{*}(\mathrm{t})\ne {\mathrm{CSR}}_{\mathrm{t}}\end{array}\right.$$

(8)

i.e., κ₀ is a stepwise function assuming the value of the damage parameter gained at the time step (t − dt) every time the stress ratio is a local maximum or when τ* = CSR_t.

The increment of the damage parameter, dκ, in the time interval dt is as follows:

$$\mathrm{d}\mathsf{\kappa }=\left\{\begin{array}{cc}0& \mathrm{if}{\mathsf{\tau }}^{*}(\mathrm{t})<{\mathrm{CSR}}_{\mathrm{t}}\\ [{\mathsf{\tau }}_0^{*}(\mathrm{t})-\mathsf{\tau }(\mathrm{t}){]}^{\mathsf{\alpha }}& \mathrm{if}{\mathsf{\tau }}^{*}(\mathrm{t})\ge {\mathrm{CSR}}_{\mathrm{t}}\end{array}\right.$$

(9)

The accumulation of damage occurs every time τ* overtakes CSR_t.

This simplified model avoids converting the irregular stress history into an equivalent number of cycles and it can be used to predict the pore pressure ratio time history for any type of loading after the calibration of the pore water pressure—PWP—model parameters (a, b, c, and d of Equation (5) and CSR_t, N_r, CSR_r, and α of Equation (4)). Further details on the PWP model and calibration procedure can be found in Chiaradonna et al. [45,47].

If the pore pressure generation models can only predict the residual component of the excess pore pressure generated in saturated soil during cyclic loading, advanced constitutive models working in effective stress (e.g., [48,49,50,51]) can correctly predict both transient and residual components of the excess pore water pressure [52,53]. The first component has little influence on soil behavior under saturated conditions because it equalizes the variation of the applied mean normal stresses arising from the cyclic loading [54,55]. The residual component has a predominant impact on the soil strength and stiffness because it is the result of plastic strain of the soil skeleton [42,56].

The effective stress models necessitate several input data to accurately characterize the response of granular soils subjected to elastic or elastoplastic deformations. Furthermore, they must capture common features observed in soils susceptible to liquefaction, specifically the following: the volumetric strain accumulation under cyclic loading, dilatancy at large shear strains, and the effect of fabric history on dilatancy [57].

This review aims to pragmatically provide a classification from the point of view of the possible applicative methods in professional practice. The novelty of the proposed hierarchy is the categorization based on both the complexity and the level of accuracy of the performed analysis in view of practical applications. The following Section 2 describes the predictive approaches graded with an increasing level of complexity (synthetic, simplified, approximate, and rigorous), while the following sections focus on each level. Then, all the methods are applied to an Italian case study and critically discussed in Section 7. Further exemplificative applications are reported in Appendix A. Supplementary Materials also include example spreadsheets to support the application of the most popular methods. Limits and benefits are listed for each method, finally giving a guide for selecting the most suitable one according to the relevance of the problem and requested resources.

2. Hierarchy of Approaches for Excess Pore Water Pressure Prediction

The prediction of the accumulation of excess pore water pressure induced by seismic loading can be assessed with two possible approaches (Figure 1):

• ‘Decoupled’ approach, where the computation of excess pore pressure is performed after the end of the calculations, i.e., hand-made calculations of the safety factor against liquefaction or ground response analysis in total stress (e.g., [58]). In both cases, the seismic demand used in the calculations does not account for the generation of pore water pressure.
• ‘Coupled’ approach, which consists of non-linear dynamic analyses in effective stress by using either a simplified or an advanced soil constitutive model [39,59]. In this second case, the seismic demand used in the calculation of the excess pore water time history is progressively affected by pore pressure generation.

Within the framework of the ‘decoupled’ approach, two distinct methods with increasing complexity can be identified (Figure 1):

1.1.

‘Synthetic’ methods can provide an overall estimate of the expected pore pressure build-up, independently of time;

1.2.

‘Simplified’ methods are able to provide the history of the excess pore pressure by adopting one of the pore pressure generation models, such as reported by Hashash et al. [39]. This method can be adopted after the execution of total stress ground response analysis.

Within the framework of the ‘coupled’ approach, two distinctions can be drawn in terms of complexity of the method and analysis level, respectively (Figure 1):

2.1.

‘Approximate’ methods or ‘loosely coupled’ analyses that can predict pore pressures by utilizing pore pressure generation models in conjunction with cyclic response models formulated in total stress [60,61]. The crucial aspect that guarantees the coupling between the previous two ingredients of the analysis is the adoption of a degradation model, i.e., how the generated pore pressure models affect the cyclic soil response. The most common degradation models have been proposed by [62].

2.2.

‘Rigorous’ methods or ‘fully coupled’ analyses that employ advanced soil constitutive models that are intrinsically formulated within the effective stress framework. These models simultaneously and fully predict both the stress–strain response and the pore pressure response of the soil, addressing the coupled nature of the soil skeleton and pore–fluid interaction [63,64,65].

Figure 1 graphically summarizes the discussed classification.

Figure 1. Hierarchy of existing approaches for pore water pressure accumulation.

Figure 1. Hierarchy of existing approaches for pore water pressure accumulation.

3. Synthetic Methods

The ‘synthetic’ methods provide an overall estimate of the maximum expected pore pressure build-up, i.e., just a single value of r_u, independent of the time of attainment (Figure 1). This type of prediction can be adopted after the calculation of the safety factor against liquefaction. Theoretically, the safety factor against liquefaction can be computed according to different approaches, again classified as stress-based, strain-based, and energy-based depending on whether the seismic demand and soil resistance are quantified in terms of normalized stress, strain, or dissipated energy, respectively. However, since the stress-based approach is the most popular and universally accepted one, it will be referred to in the following.

According to the stress-based procedure [66], the factor of safety (FS) is calculated as the ratio between the cyclic resistance ratio (CRR) of the soil and the seismic demand induced by the earthquake, called the cyclic stress ratio (CSR), both referring to the same conditions (e.g., effective stress confinement, magnitude of the seismic scenario, and so on). All the details can be easily found in the literature [67,68,69,70].

Although easy to use, Olson et al. [71] identified three main limitations in this approach: (1) application of several correction factors, e.g., the depth reduction, magnitude scaling, and overburden correction factors; (2) the use of a total stress framework to describe the seismic demand, even though it is part of an effective stress phenomenon; and (3) because it is based on a database of observed surface manifestations, there are uncertainties when it is applied outside of database (e.g., depths > 10 m).

Figure 2 shows the possible combinations from which the safety factor against liquefaction can be determined. Indeed, both CSR and CRR can be defined by using two possible levels of accuracy:

• Level A or a, respectively, for CSR and CRR: where the minimum effort is requested in terms of both experimentation and calculation to define FS. This is the most immediate and largely adopted level in codes and guidelines;
• Level B or b, respectively, for CSR and CRR: where a bigger effort is required compared to level A/a to define FS in terms of both experimentation and calculation.

Figure 2. Calculation of the safety factor according to different analysis levels.

Figure 2. Calculation of the safety factor according to different analysis levels.

In detail, from the perspective of the seismic demand, the CSR can be calculated as follows:

• Level A: hand-made calculation according to the simplified expression:

$$\mathrm{CSR}=0.65{\displaystyle \frac{{\mathsf{\tau }}_{\max }}{\mathsf{\sigma }{\prime }_{\mathrm{v}}}}=0.65\cdot {\displaystyle \frac{{\mathrm{a}}_{\max }}{\mathrm{g}}}\cdot {\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{v}}}{\mathsf{\sigma }{\prime }_{\mathrm{v}}}}\cdot {\mathrm{r}}_{\mathrm{d}}$$

(10)

where σ_v is the total vertical stress and σ’_v is the vertical effective stress at a depth z, a_max is the maximum horizontal acceleration expected at the ground surface, 0.65 is the coefficient introduced to have an equivalent constant shear stress amplitude starting from the maximum shear stress, τ_max, g is the gravity, and r_d is a reduction factor accounting for soil deformability, usually a function of the depth and moment magnitude, M_w, of the seismic event [68].

• Level B: the results of total stress ground response analysis, processed according to the equation:

$$\mathrm{C}\mathrm{S}\mathrm{R}=0.65{\displaystyle \frac{{\mathsf{\tau }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathsf{\sigma }{\prime }_{\mathrm{v}}}}$$

(11)

where τ_max is the maximum shear stress coming from the seismic site response analysis in total stress at the depth z, 0.65 is the coefficient adopted to convert the irregular loading into an equivalent constant shear stress amplitude, and σ’_v is the vertical effective stress.

The earthquake scenario is defined only by the pair made by the maximum expected acceleration at the site and the moment magnitude of the seismic event (a_max, M_w) at the basic level A, while a selection of accelerograms, typically spectro-compatible with a target spectrum [72], is required for level B. In addition, in level B, for performing the total stress ground response analysis, even in the most simple one-dimensional configuration, it is necessary to define the depth of the seismic bedrock, the stratigraphy of the soil layering, the soil properties of each deformable soil deposits in terms of unit weight, shear wave velocity, and non-linear and dissipative soil behavior (usually expressed as normalized shear modulus and damping ratio vs. shear strain curves). Therefore, in level B, the better accuracy in the definition of the CSR is because the τ_max is accounting for the seismic soil response to the applied shaking, while, in level A, the local site effects are only accounted in the definition of a_max via the amplification factors [72].

From the perspective of cyclic soil resistance, the CRR can be calculated as follows:

• Level a: empirical equations that correlate the CRR to the results of normalized mechanical soil parameters as obtained from in situ tests, e.g., Cone Penetration Test—CPT [68,70], Standard Penetration Test—SPT [68], Flat Dilatometer Test—DMT [73], and shear wave velocity measurement [74,75]. An example of an empirical relationship between CRR for the reference conditions (M_w = 7.5; σ’_v0 = 1 atm) and the normalized corrected blow count, (N₁)_60cs, as measured during SPT, or the normalized and corrected cone tip resistance of CPT, q_c1NCS, is proposed by [68] as follows:

$$\mathrm{C}\mathrm{R}{\mathrm{R}}_{\mathrm{M}=7.5,\mathsf{\sigma }\prime =1}=\exp \left({\displaystyle \frac{({\mathrm{N}}_1{)}_{60\mathrm{c}\mathrm{s}}}{14.1}}+{\left({\displaystyle \frac{({\mathrm{N}}_1{)}_{60\mathrm{c}\mathrm{s}}}{126}}\right)}^2-{\left({\displaystyle \frac{({\mathrm{N}}_1{)}_{60\mathrm{c}\mathrm{s}}}{23.6}}\right)}^3+{\left({\displaystyle \frac{({\mathrm{N}}_1{)}_{60\mathrm{c}\mathrm{s}}}{25.4}}\right)}^4-2.8\right)$$

(12)

$$\mathrm{C}\mathrm{R}{\mathrm{R}}_{\mathrm{M}=7.5,{\mathsf{\sigma }}_{\mathrm{v}}\prime =1}=\exp \left({\displaystyle \frac{{\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}}{113}}+{\left({\displaystyle \frac{{\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}}{1000}}\right)}^2-{\left({\displaystyle \frac{{\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}}{140}}\right)}^3+{\left({\displaystyle \frac{{\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}}{137}}\right)}^4-2.8\right)$$

(13)

• Level b: direct estimation of the cyclic soil resistance via the cyclic resistance curve, i.e., the relationship between the cyclic resistance ratio and the number of cycles necessary to induce liquefaction as measured by cyclic simple shear or cyclic triaxial tests. However, laboratory tests have some limitations to reproduce the site conditions and consequently some corrections need to be applied prior to using the curve for engineering applications [69,76,77,78].

Level b is characterized by higher accuracy compared to Level a due to the fact that the relationship between CRR and N is directly investigated for the considered soil. This implies that the estimation of the excess pore water pressure can be performed both following a direct way via the normalized number of cycles, as for Equation (1) or Equation (5), or an indirect way via the relationship between r_u and the safety factor, FS. Further insights are provided in the next subsections on Ab and Bb levels.

The FS calculation according to the method Aa is the most popular and adopted in several building codes and guidelines [72], while the method Bb is less popular due to the difficulty of having undisturbed samples of liquefiable soils to be properly tested in a laboratory. The hybrid combination Ba and Ab, even though characterized by a different level of accuracy, is still possible if a site is well-documented enough to allow the execution of a ground response analysis in total stress but cyclic laboratory tests are missing (method Ba) or if the cyclic resistance curve is experimentally available for the liquefiable layer but other data for seismic response analysis are not available (method Ab). This last case is quite rare in practical applications.

More insights into the four combinations are provided in the following subsections, with special focus on level Aa and Bb.

3.1. Level Aa: Safety Factor Calculated by Using the Semi-Empirical Equation for CSR and Empirical Charts for CRR

According to this level, the factor of safety is as follows:

$$\mathrm{F}\mathrm{S}={\displaystyle \frac{\mathrm{C}\mathrm{R}\mathrm{R}}{\mathrm{C}\mathrm{S}\mathrm{R}}}=\left({\displaystyle \frac{\mathrm{C}\mathrm{R}{\mathrm{R}}_{\mathrm{M}=7.5,\mathsf{\sigma }{\prime }_{\mathrm{v}}=1}}{\mathrm{C}\mathrm{S}\mathrm{R}}}\right)\cdot \mathrm{M}\mathrm{S}\mathrm{F}\cdot {\mathrm{K}}_{\mathsf{\sigma }}\cdot {\mathrm{K}}_{\mathsf{\alpha }}$$

(14)

where CSR is calculated according to Equation (10), CRR_{M=7.5,σ’v=1} represents the cyclic resistance ratio (CRR) normalized to a seismic magnitude of M_w = 7.5 and to a reference effective overburden stress of σ’_v = 103 kPa, and Magnitude Scaling Factor (MSF) is incorporated to account for the influence of the earthquake duration. Finally, K_σ and K_α are correction factors accounting for the effects of the effective overburden stress and the initial static shear stress on the horizontal plane, respectively [67,68].

After the definition of FS, a rough estimation of the seismic-induced excess pore pressure ratio (r_u) can be performed using one of the relationships between r_u and FS, such as reported by Iwasaki et al. [79], Marcuson et al. [80], and Chiaradonna and Flora [81]. The latter proposed the following expression:

$${\mathrm{r}}_{\mathrm{u}}={\displaystyle \frac{2}{\mathsf{\pi }}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left[{\mathrm{F}\mathrm{S}}^{-{\displaystyle \frac{1}{2\mathrm{b}\mathsf{\beta }}}}\right]\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{F}\mathrm{S}>1$$

(15)

where the exponents b and β can be estimated as a function of normalized SPT blow count, (N₁)_60cs, or as a function of normalized and corrected cone tip resistance of CPT, q_c1NCS, and estimated fines content, FC. The analytical expressions for b and β are the following:

$$\mathrm{b}=-1.487\cdot 10^{-8}\cdot {{\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}}^3+1.291\cdot 10^{-5}\cdot {{\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}}^2-5.722\cdot 10^{-4}\cdot {\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}+0.163$$

(16)

$$\mathrm{b}=-1.000\cdot 10^{-6}\cdot ({\mathrm{N}}_1{{)}_{60\mathrm{c}\mathrm{s}}}^3+2.216\cdot 10^{-4}\cdot ({\mathrm{N}}_1{{)}_{60\mathrm{c}\mathrm{s}}}^2+1.727\cdot 10^{-3}\cdot ({\mathrm{N}}_1{)}_{60\mathrm{c}\mathrm{s}}+0.1557$$

(17)

$$\mathsf{\beta }=0.01166\cdot \mathrm{F}\mathrm{C}+0.3536\cdot {\left({\mathrm{q}}_{\mathrm{c}1\mathrm{N}\mathrm{c}\mathrm{s}}-\mathsf{\Delta }{\mathrm{q}}_{\mathrm{c}1\mathrm{N}}\right)}^{0.264}-0.4979$$

(18)

$$\mathsf{\beta }=0.01166\cdot \mathrm{F}\mathrm{C}+0.1091\cdot {\left(({\mathrm{N}}_1{)}_{60\mathrm{c}\mathrm{s}}-\mathsf{\Delta }({\mathrm{N}}_1{)}_{60}\right)}^{0.5}+0.5058$$

(19)

where Δq_c1N and Δ(N₁)₆₀ are the increment components of the normalized soil parameters, q_c1NCS and (N₁)_60cs, due to the beneficial effects of the fines content on the soil cyclic resistance for CPT and SPT, respectively [68].

Figure 3 shows the r_u-FS charts for different FC values.

Example Application for Level Aa

During the 30 October 2020 earthquake, water eruptions were observed at the basement level of a residential building equipped with stone columns in Izmir city center [82]. This residential structure is located 2.6 km inland (from the shoreline), where the groundwater table was detected 1 m below ground level. Foundation and subsoil data are constrained, having been sourced from the original geological–geotechnical investigation report prepared during the building’s construction. To mitigate soil liquefaction potential, an array of 199 vertical drains was installed beneath the foundation. After the 2020 earthquake, a water eruption was observed from the vertical drains/columns, with water visible on the basement floor. The absence of finishes in this embedded story allowed for clear visibility of the effusive event. Eyewitness accounts confirm that the floor remained saturated for several days post-event. No external liquefaction manifestations (e.g., sand boils) or ground bulging was observed around the structure or in the vicinity.

Representative soil column consists primarily of sandy soils with variable content of silty or clay fines in the first 10 m below the ground level. Quantitative data on soil properties were derived from SPT performed at four locations beneath the building’s corners, with a mean SPT N-value of 10 observed within the upper 10 m (Figure 4).

According to level Aa, a two-step procedure was implemented for the analysis of soil response under seismic loading: the initial step involved estimating the liquefaction potential of the underlying soil layers. This was achieved by calculating FS using the SPT chart according to [68]; the second step focused on a preliminary evaluation of r_u through Equation (15). The expected r_u ranges in the interval 0.1–0.4, with a maximum of 0.8 at 7 m (Figure 4). A hydraulic gradient of 3–6 m of the water column is associated to this result, which is enough to activate the vertical drains.

3.2. Level Ba: Safety Factor Calculated by Using Dynamic Total Stress Analysis for CSR and Empirical Charts for CRR

The estimation of r_u for level Ba follows the same procedure of level Aa, i.e., a two-step procedure made up of an FS calculation via Equation (14) and a r_u calculation via Equation (15). The only difference between level Ba and Aa is the fact that CSR comes from a seismic site response analysis in total stress via Equation (11), and not from the application of Equation (10).

3.3. Level Bb: Safety Factor Calculated by Using Dynamic Total Stress Analysis for CSR and Cyclic Resistance Curve for CRR

The estimation of r_u for level Bb follows a three-step procedure consisting of the following: (1) performing a total stress ground response analysis; (2) defining for each depth of the liquefiable layer the pair (CSR, N_eq), where CSR is calculated according to Equation (11) and N_eq is calculated with one of the available relationships for estimating an equivalent number of cycles, e.g., Biondi et al. [38]; (3) estimating r_u in a direct way (d in Figure 1) as a function of the normalized number of cycles, e.g., by applying Equation (1) or Equation (5), or in an indirect way (i in Figure 1) as a function of the safety factor, e.g., by applying Equation (15).

Figure 5 shows a scheme with the three steps described above.

3.4. Level Ab: Safety Factor Calculated via the Semi-Empirical Equation for CSR and the Cyclic Resistance Curve for CRR

The estimation of r_u for level Ab follows a two-step procedure consisting of the following: (1) defining for each depth of the liquefiable layer the pair (CSR, N_eq), where CSR is calculated according to Equation (10) and N_eq is calculated with one of the available relationships for estimating an equivalent number of cycles, e.g., Biondi et al. [38]; (2) estimating r_u in a direct way (d in Figure 1) as a function of the normalized number of cycles, e.g., by applying Equation (1) or Equation (5), or in an indirect way (i in Figure 1) as a function of the safety factor, e.g., by applying Equation (15).

4. Simplified Methods

Simplified methods can be considered an upgrade of the B-level methods as described in the previous sections, where the time history of a reference variable (e.g., shear strain, shear stress or dissipated energy) is extracted from the results of the seismic site response analysis in total stress and is processed in a pore pressure generation model according to a decoupled approach. In the stress-based approach, instead of considering the maximum shear stress only as for the B-level of synthetic methods, the entire time history of the shear stress is processed in a stress-based pore pressure model, such as the PWP model based on the damage parameter described above. An exemplificative application is reported for a real case study in the following Section 7.2. A further application for an ideal soil column is reported in Appendix A.1.

5. Approximate Methods

The approximate methods include all the codes for seismic response analysis in effective stress, where a loosely coupled approach is adopted. This level of analysis enables the incorporation of the dissipation of the generated excess water pressure into the calculations depending on the hydraulic boundary conditions. This is not possible in the decoupled approach. In the following Section 7.3, an example is provided for the simplified PWP model based on the damage parameter, which is implemented in the one-dimensional code SCOSSA [84]. A further example for an ideal soil column is reported in Appendix A.2 This latter aims to highlight the difference between the decoupled (simplified) and coupled (approximate) analysis.

6. Rigorous Methods

The rigorous methods include all the effective stress ground response analysis, where a fully coupled approach is adopted. An application is reported for a real case study in the following Section 7.4. A further example aiming to show a comparison between the approximate and the rigorous method is reported in Appendix A.3.

7. Application of Hierarchical Methods to a Case Study

The described methods are applied in this section to a liquefaction case study: a site in Pieve di Cento (Bologna province) located in an area that experienced liquefaction manifestations after the 2012 Emilia Earthquake [85] and was the setting of a real-scale liquefaction experiment in October 2018 [86,87].

Chiaradonna et al. [88] numerically investigate the dynamic response of the site for several seismic scenarios with a return period, T_R, ranging from 50 to 2475 years, associated with a probability of exceedance, P_VR in 50 years [72], ranging between 63% and 2%. For the proposed study, the earthquake scenario with a T_R = 140 years (P_VR = 50%) is considered, which is the most severe among those analyzed in [88] that do not induce the full liquefaction of the liquefiable deposits. For the sake of simplicity and making the logical flow more understandable, only one of the seven selected acceleration records will be considered in the following, i.e., the East–West component of the M_w = 5.2 12 October 1997 Central Italy earthquake recorded at the FORC recording station [89]. Figure 6 reports the time history of acceleration used in dynamic analyses. Further details about the definition of the input motion adopted in the analyses can be found in Chiaradonna et al. [88] and are omitted here for the sake of brevity.

An extensive in situ campaign [86] provided the data for soil characterization. The stratigraphy is composed of several distinct units (Figure 7a), beginning with a 0.9 m thick surficial crust of silty sand (SS1), which transitions into a sandy silt layer extending to approximately 2.9 m depth, partially above (SS2) and below (SS3) the groundwater table, which is located at 1.8 m depth. Below this, a grey silty sand layer (GSS) is encountered between 2.9 m and 6.0 m, interrupted by a thin intermediate clayey layer (Clay1) situated between 4.4 m and 4.8 m. From the 6.0 m mark to the maximum investigated depth, a thick silty clayey deposit (Clay2) dominates the profile. The upper sandy silty layer (SS) is characterized by heterogeneity (well-graded, with low-plasticity fine content ranging from 60% to 85%), whereas the deeper GSS is more homogeneous, exhibiting a narrow fine content range of 5% to 12% (Figure 8b). The clay deposit has a plasticity index, PI, equal to 0.55 and a percentage of clay ranging from 50 to 80%. The Soil Behavior Type index [90] resulting from in-situ tests is illustrated in Figure 8a. Based on susceptibility analyses, the GSS deposit has been identified as the liquefiable critical layer (Figure 7e and Figure 8a). This finding is supported by a combination of factors, including full saturation, loose state (Dr = 40%), low effective stress state, and an average SBT < 2.0 (Figure 8a).

7.1. Synthetic Methods Application

According to level Aa, soil capacity (CRR) was estimated on CPT-based chart reported by [68], while the seismic demand (CSR) was calculated by Equation (10) where a_max was equal to the stratigraphic amplification factor multiplying the maximum expected acceleration value on rock formation defined for a T_R = 140 years and the mean magnitude defined from the de-aggregation map of the selected site as reported in the Italian seismic hazard model [91,92], i.e., PGA = 0.094; M_w = 4.93.

For the application of charts in Figure 3, it is of outmost importance to have a reliable estimation of the fines content of soils. For this case study, an estimate of the fines content has been carried out as a function of the SBT index according to the relationship proposed by [68]:

FC = 80(I_c + C_FC) − 137

(20)

where the error coefficient C_FC has been calibrated specifically for this site on the basis of the grain size distribution curves and I_c profiles (Figure 8), and it is equal to −0.0775.

The excess pore pressure ratio profile for level Aa is reported in Figure 9g. Liquefaction is attained in the GSS2 layer, while it is only locally attained in the above GSS deposit.

According to level Ba, seismic site response analysis in total stress has been performed with the code STRATA [90] for the estimation of the seismic demand (CSR), while soil capacity (CRR) was quantified on the empirical chart proposed by [68], as in the previous level.

For the execution of the dynamic analysis in total stress, it was necessary to define the geotechnical model of the site, i.e., the shear wave velocity profile and the shear modulus reduction curves, and damping ratio curves vs. shear strain for each soil unit. The shear wave velocity profile was established based on the results obtained from a Cross-Hole (CH) test conducted in situ [86]. Figure 7b shows the experimental profile of the shear wave and the one adopted in the model. The investigation into the non-linear and dissipative characteristics of the soils was obtained by conducting cyclic torsional shear tests specifically on the grey silty sand (GSS) deposits (Figure 7f). For the remaining soil layers (Figure 7f), namely the silty sand (SS) and clay formations, the constitutive behavior was characterized by utilizing experimental curves derived from tests performed on Scortichino deposits, which are located approximately 20 km away. This approach was adopted due to the substantial geological similarity between the Scortichino and Pieve di Cento sites [93,94].

Figure 9 compares the obtained pore water pressure ratio profile for the Ba level with that from the Aa level. The seismic demand defined from the dynamic analysis is slightly higher than the CSR from Equation (10) at the depth of the liquefiable layers (Figure 9e), implying a slightly lower FS to which is associated with a higher r_u profile.

According to level Ab, the seismic demand, CSR, is the same as the previous Aa level based on Equation (10), while the cyclic strength of soils is directly defined by the cyclic resistance curves of the three liquefiable units (Figure 7g). In this case, the cyclic resistance curve is required to define an equivalent number of cycles for the considered motion. Due to the simplicity of the CSR definition, the relationship proposed by Idriss [95], which is only a function of magnitude, was adopted. For a magnitude of approximately 5, the Idriss [95] relationship leads to an equivalent number of cycles of 2 (Figure 10). In this case, the number of cycles is independent of the depth (Figure 11a). The pair at each depth (N_eq, CSR) is used to examine the cyclic resistance curve of each soil unit and define CRR and N_L, i.e., the soil resistance in correspondence with N_eq and the number of cycles to induce liquefaction for a CSR loading, respectively (Figure 10). The ratio N_eq/N_L is used to estimate the r_u via Equation (5), calibrated as reported in Figure 7h according to the direct way (Figure 11d—continuous black line). Alternatively, according to the indirect way, the FS is first calculated as CRR/CSR (Figure 10) and then the r_u calculated according to Equation (15), where the coefficients b and β are estimates as a function of the CPTU results (Equations (16) and (18), respectively). The vertical profiles of r_u are reported in Figure 11d. The variability between the direct and indirect ways is quite limited.

According to level Bb, the seismic demand was estimated from the total stress analysis in terms of maximum shear stress (Equation (11)), while the cyclic strength of soils is directly defined by the cyclic resistance curves of the three liquefiable units (Figure 7g). With respect to the previous case, at each depth a time history of acceleration derived from dynamic analysis is available. As a consequence, it was possible to apply a more refined relationship for the quantification of the equivalent number of cycles, such as that proposed by Biondi et al. [38], where N_eq is a function of the PGA and Arias intensity of the accelerogram. The vertical profiles of N_eq are reported in Figure 11a. It is slightly higher than that adopted in the Ab level (see also Figure 10 for a specific depth). Consequently, this implies that the CRR is lower (Figure 10) as well as the FS, resulting in remarkably higher r_u profiles in both direct and indirect ways (Figure 11d). The b level for the definition of the CRR, although characterized by better accuracy in the definition of the soil cyclic resistance, is affected by a large variability in the results due to the uncertainties connected to the definition of the equivalent number of cycles.

7.2. Simplified Method Application

The simplified method described in Section 4 can be considered as an upgraded version of the Bb level, because the input data are substantially the same: the total stress dynamic analysis for the seismic demand and the cyclic resistance curve for the soil resistance. However, the conversion process to a uniform loading of the irregular shaking is bypassed thanks to the application of the PWP model, according to a decoupled approach. The PWP model, which requires determination of the cyclic resistance and excess pore water pressure curves (Figure 7g,h), was calibrated on the results of CPTU for GSS layers, according to [47]. Because the fines content exceeds 30%, the r_u-N/N_L relationship defined in the laboratory on Scortichino sandy silt [94] was adopted for the SS unit due to the similarity in the grain size distribution.

The histories of shear stress from the total stress analysis at all the available depths were processed with the PWP model to predict the time histories or r_u. Figure 12 shows the PWP model application at a representative depth of 2.7 m.

The damage parameter accumulates only when the normalized stress, τ*, overcomes the threshold, CSR_t (dashed black line in Figure 12b).

Equation (2) is used to predict the excess pore pressure (Figure 12c,d). Higher values of r_u are attained in the first 10 s, where the intensity of the loading is the highest.

For each time history along the vertical depth gained from the total stress analysis, the process described in the above Figure 12 was applied. The resulting r_u profile is shown in Figure 13a,b (labeled as STRATA + PWP).

7.3. Approximate Method Application

Loosely coupled effective stress analysis in a one-dimensional condition has been performed with the code SCOSSA [44].

The only additional parameter that is new compared to the simplified method is the degradation parameter of the Matasovic and Vucetic [62] degradation model, which was set to the average of 3.5, consistent with the author’s suggestions.

The remaining data for the analysis are the same used in the total stress analysis reported in Figure 7. The resulting r_u profile is shown in Figure 13a,b (labeled as SCOSSA).

7.4. Rigorous Methods Application

A fully coupled effective stress non-linear dynamic analysis has been carried out with the Finite Difference Method bidimensional code FLAC [96].

The liquefiable units (SS, GSS type) were modeled using the constitutive model PM4Sand version 3.1 [97], with the model parameters listed in

Table 1. Calibrated parameters for PM4Sand and PM4Silt.

Parameter	SS1	SS2	SS3	GSS1	Clay1	GSS2	Clay2
Apparent relative density DR	0.38	0.38	0.38	0.35	-	0.35	-
Undrained shear strength su/σ’v0	-	-	-	-	0.25	-	0.25
Shear modulus coefficient, Go	1769	1021	791	548	511	534	473
Contraction rate parameter, hpo	0.7	0.7	0.7	0.82	26	0.85	27

.

Calibrations were developed for representative mean values of q_c1Ncs, with values of 66 and 60 for SS and GSS layers, respectively.

The shear modulus coefficient (Go) was calibrated on the in situ shear wave velocity, Vs, at the middle of the stratum, the apparent relative density (D_R) was estimated as 38% and 35% for SS and GSS layers, and the contraction rate parameter (h_po) was iteratively set to match the cyclic strength in 15 uniform cycles as predicted by the cyclic resistance curves of Figure 7g. All the secondary parameters were set to their default values.

The clay deposits were modeled using the constitutive model PM4Silt [98] with the parameters listed in Table 1. The Go was defined on the measurement of Vs at the middle of the layer. The h_po was adjusted to produce reasonable slopes for the cyclic strength curves of clay soils (obtained from CPTU), as reported by [69]. All the secondary parameters were set to their default values.

The resulting r_u profile is shown in Figure 13a,b (labeled as FLAC).

Figure 13a compares the r_u profiles obtained from the time histories of the three dynamic analyses. It is possible to observe that by increasing the level of complexity of the methods (simplified to approximate to rigorous level), the degree of conservatism of the analysis significantly decreases. This means that a major effort in the definition of the geotechnical model and complexity of the constitutive model, and also in the computational effort, is translated into benefits in cost saving and efficiency of the design process.

Figure 13b shows the comparison of the previous profiles with the synthetic methods. In this last case, there is no clear tendency but a large variability of the four levels. However, the basic Aa level is undoubtedly characterized by a great level of robustness compared to the other synthetic levels, due to the fact that it is based on the traditional safety factor calculation which is widespread throughout the world.

8. Utility of the Excess Pore Water Pressure Predictions

In addition to the identification of the liquefaction triggering (by definition), excess pore water pressure prediction can be used to verify the safety and serviceability of structures even in conditions far from failure or to correctly reproduce the system soil response recognized as the discriminating factor in observed or not-observed damage after strong earthquake sequences, just to mention two prominent problems.

In detail, the predicted excess pore water pressure can be used to contribute to the assessment of the following problems, although it is not limited to just these:

• Quantify the severity of sand ejecta manifestations [99,100,101];
• Estimation of the free-field consolidation settlements [102];
• Degraded bearing capacity of shallow foundations [103];
• Practical design of liquefaction mitigation interventions [104,105];
• Evaluating the influence of excess pore water pressure on permanent slope displacement during seismic action [106];
• Assessment of the uplift response of underground structures [107];
• Mapping vulnerability indexes of the liquefaction potential [108].

9. Discussion and Conclusions

This review aimed to provide a systematic classification of the existing methods for predicting excess pore water pressure induced by seismic loading.

Table 2. Summary of input data and the required effort of the hierarchical methods.

Method		Type of Computation	Type of Dynamic Analysis	Data for Seismic Demand	Data for Cyclic Soil Resistance	Calibration Effort	Computational Effort
Synthetic	Level Aa	Analytic	-	Max acceleration, magnitude	In situ test	-	Very low
	Level Ab	Analytic	-	Max acceleration, magnitude	Cyclic Resistance Curve	-	Very low
	Level Ba	Numerical	Total stress	Time history of the input motion	In situ tests	-	Low/ Medium
	Level Bb	Numerical	Total stress	Time history of the input motion	Cyclic Resistance curve	-	Low/ Medium
Simplified		Numerical	Total stress	Time history of the input motion	PWP model	Low	Low/ Medium
Approximate		Numerical	Loosely coupled effective stress	Time history of the input motion	Cyclic response + PWP model	Low/ Medium	Low/ Medium
Rigorous		Numerical	Fully coupled effective stress	Time history of the input motion	Advanced soil constitutive model	High	High

summarizes the required input data and the computational effort (i.e., the required resources) for each method of the hierarchy to guide a cost–benefit analysis for the choice of the most appropriate level.

The basic level for estimating r_u from the safety factor against liquefaction (method Aa) is still not very common in engineering practice and should become much more popular in the future, in accordance with the performance-based design of structure and infrastructure.

The decoupled approaches provide conservative estimation of the pore water pressure compared to the coupled ones; however, their practical use is not always justified or convenient, especially for the simplified methods, because the same efforts of resources can easily lead to an approximate level.

The most accurate method we have at the moment is the rigorous one, which implies the use of advanced effective stress constitutive models; these are currently more accessible than in the past, thanks to guided calibration procedures provided by researchers [109,110].

The stress-based PWP model proposed by [45] can be applied transversally to all the different levels of complexity, and therefore, it seems a promising tool for predicting seismic-induced pore water pressure build-up in professional practice.

Future paths for the research can explore the applicability of hierarchical methods to several case studies with different seismic intensities to explore the range of variability of the results. It is worth mentioning that the findings of the present study should be further consolidated by analyzing a larger dataset of sites and seismic scenarios. This type of work has particular relevance for medium/low seismicity areas where remarkable excess pore pressure is expected more than complete liquefaction [111].