Statistical Model for Dam-Settlement Prediction and Structural-Health Assessment
Publication: Journal of Geotechnical and Geoenvironmental Engineering
Volume 144, Issue 9
Abstract
This paper considers a hydrostatic–seasonal–time (HST) statistical model for predicting the settlement behavior of a concrete-faced rockfill dam (CFRD) during operation. The constituents of the model are innovatively related to viscoelastic–plastic material model components, with the novel addition of considering unloading–reloading behavior when determining the model parameters. The statistical model developed considers both load-related and time-related deformation behavior of rockfill and CFRDs. The discussion compares different functions from the literature describing time-dependent deformation of rockfill dams and CFRDs and emphasizes cautious selection of predictors to enhance any statistical model’s credibility. The model is applied to the case of the 200-m-high Kárahnjúkar CFRD. The aim of the statistical analysis is first to create a prediction model for short- and long-term settlements of the dam, and second to extract behavioral patterns for structural-health monitoring applications. This is important for the operational safety of large dams, and thus the resilience and sustainability of related civil infrastructure systems.
Introduction
Monitoring and surveillance of important infrastructures is an integral part of civil engineering practice. This has evolved along with technology and is referred to, in its most advanced form, as structural-health monitoring (SHM). SHM is the process of implementing a monitoring system along with a strategy to identify unfavorable change (damage) in the normal condition of a structure (Worden et al. 2007). An SHM management system embraces, first, damage diagnosis, integrating statistical prediction models with numerical mechanics models [e.g., using finite-element (FE) models], and second, damage prognosis for making decisions on maintenance or giving alarms for emergency preparedness (e.g., Farrar and Lieven 2007). SHM is important for the operational safety of large dams.
Fig. 1 demonstrates the use of statistical and mechanical models in dam engineering considering dam operation. This relates to an SHM procedure for prognosis of the health of a dam or its parts. However, judging from scientific publications on this topic for dams, statistical models have mostly been used in analyzing monitoring data from concrete dams. Thus, there is a need to establish this field in the case of geotechnical structures like embankment dams.
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The focus of this paper is on statistical prediction modeling of monitoring data from measurements in a concrete-faced rockfill dam (CFRD). In recent years, large dams of this type have generally been monitored extensively, some with more than 300 measuring locations. Each measuring location results in a time series of monitoring data for the dam surveillance to consider. This large amount of data theoretically enables comprehensive application of an SHM. There are reports on the usage of monitoring data from such dams for calibrating mechanics models. However, reports on statistical prediction models relating to SHM of a CFRD or other geotechnical structures are, by the authors’ experience, rare and incomplete.
The deformations of CFRDs govern their behavior during normal operation and induce strains and stresses in the concrete face. Thus, it is logical to initiate the SHM process by analyzing monitored deformations, such as the settlements. Hence, this paper proposes a multivariate linear regression (MVLR) statistical model to simulate the settlement of a large CFRD based on records of full-scale settlement data.
The statistical model proposed considers the settlement behavior of rockfill and relates the components of the model to a viscoelastic–plastic (VEP) material model. Additionally, the calculation procedure considers the unloading–reloading behavior of the dam material, which is a novel addition in such models.
This paper briefly reviews general statistical models for estimating dam response as well as multivariate linear regression methodology for obtaining the statistical parameters. Subsequently, rockfill material behavior is reviewed and the statistical model for the settlement behavior of a CFRD introduced, followed by relation of its components to a VEP material model. Discussion on the different parameters of the proposed model follows and includes a comparison of different functions from the literature on time-dependent deformation of rockfill dams and CFRDs. Finally, the test case involving the Kárahnjúkar CFRD is introduced, and the application of the proposed model is demonstrated.
The aim of the study is first to create a prediction model for the short- and long-term settlements, and second to extract behavioral patterns for structural health monitoring applications, for example, to distinguish between normal and abnormal behavior. Additionally, the objective is to demonstrate the possibility and advantages of considering physical factors and their relation to dam response in the development of a statistical model for a rockfill dam.
General Multivariate Hydrostatic–Seasonal–Time Regression Model
The general statistical model for the recorded response of a monitoring instrument in a dam is often called a hydrostatic–seasonal–time (HST) model (Chouinard et al. 1995) after the key components generally defined for the statistical model. Such models have been widely applied to concrete dams.
The components of a general HST model represent the loading and environmental conditions influencing the recorded response, , as well as irreversible time-dependent effects such as creep. The HST model assumes that the various response properties can be studied separately, and can be formulated as follows (Chouinard et al. 1995; Léger and Leclerc 2007; De Sortis and Paoliani 2007; Mata 2011):where = recorded response; = effect of hydrostatic thrust; = reservoir elevation; = time; = seasonal environmental effect; = seasonal parameter; = irreversible time-dependent effects; and = residual error.
(1)
The parameters of statistical behavior models as presented in Eq. (1) can be obtained using different regression techniques. Herein, MVLR methods were used. MVLR methods were applied to a set of simultaneous observations to determine the relationship between the predictor (independent) variables and the response (dependent) variables. Predictions can then be made from the established relationship between the response variables or samples, , versus one or more predictor variables . The general relationship can be expressed as (Rencher 2002)
(2)
Here the predictors are represented with the matrix of variables, where is a column vector of ones and each variable has dimensions [] or observations ( is transpose). The matrix contains the regression coefficients, and is a matrix of residuals. Conversely, the matrix of variables represents the response variables, where each variable has dimensions []. Thus, the matrix is by () [ rows by () columns], and the matrix is by ( rows by columns). Expansion of Eq. (2) results in
(3)
In the case where there is only one response variable, the regression methodology is called multiple linear regression (Rencher 2002) and the matrix reduces to the vector . In that case is a vector of the regression coefficients, , and reduces to a vector containing the error terms. This method assumes that the errors are uncorrelated, and the expected value of the error terms is zero, i.e., ; but the variance is , where is the standard deviation.
The general HST model, first proposed by Ferry and Willm (1958) and Willm and Beaujoint (1967), has been used for analyzing the deformation monitoring data of dams, mainly concrete dams (Chouinard and Roy 2006; Léger and Leclerc 2007; De Sortis and Paoliani 2007). The same model approach has also been adopted for analyzing settlements of earth-rockfill dams (Hu et al. 2011) as well as for a CFRD (Wu et al. 2009).
This statistical model approach applies to the normal operational conditions and is generally valid for prediction within the limits of the hydrostatic loading and environmental conditions defined by a set of prediction or design variables used for the regression (Chouinard et al. 1995).
Rockfill Material Behavior
The HST model has mainly been used for concrete dams. In the case of rockfill dam settlements, an alternative to the philosophy behind the traditional presentation of the HST statistical model is to relate the different components of a statistical model to the deformation behavior, which for rockfill material generally depends on both loading and time.
Load-related deformation, plotted as stress against strain, provides information on material stiffness and its dependence on the stress-strain level induced in the material. Triaxial testing of granular soils has shown that the unloading and reloading modules are higher than the modulus on primary loading (Stewart 1986; Byrne et al. 1987). It has also been shown that the unloading and reloading moduli may differ slightly (Duncan et al. 1980; Saboya and Byrne 1993). Furthermore, the unloading process provides only partial recovery of the initial deformation.
Considering time-dependent deformation, several theories have been set forth to explain the time-dependent behavior of soils under load. Bjerrum (1967) proposed the concept of delayed compression under constant effective stress, and Janbu (1969, 1985) introduced the time resistance concept. Similar phenomena are observed for granular materials, such as rockfill (e.g., Augustesen et al. 2004; Lade et al. 2009; Justo and Durand 2000). Plotting deformation versus time for a rockfill sample subjected to stepwise loading up to a certain maximum has been shown to contain initial instantaneous deformation (compaction) and transient or delayed deformation for each load increment (the viscoelastic response), as well as steady-state (long-term) creep (e.g., Dolezalova and Hladik 2011).
Similarly, the deformations induced by the water pressure on the first filling of a CFRD include (1) the near-instantaneous response to the primary loading represented by the added water pressure, and (2) the delayed response. The instantaneous response is largely irrecoverable, e.g., due to particle breakdown as well as rearrangement of the rock particles during compression. During operation of a CFRD the particle breakdown continues, and creep dictates the behavior, with decelerated settlements approaching a horizontal asymptote during the operational lifespan. Additionally, variations in the reservoir elevation are likely to affect the settlement behavior. Unloading during a dam’s operation (from lowering the reservoir) results in partial recovery, mainly of the viscoelastic response, whereas reloading (raising the reservoir) induces this again. The stress condition depends on the location within the dam. The settlement response at the upper levels may be mainly due to viscoelastic response, while at the lower levels, where the state of stress is higher, plastic deformations may have occurred. Reservoir elevations within the limits already experienced by the dam are likely to result in viscoelastic response, mainly in the upper region of the dam. Conversely, hydrostatic pressure above the limits already experienced by the dam may induce further permanent settlement.
Settlement Prediction Model Proposed for CFRDs
This section explains the calculation procedure for the prediction model proposed here for a CFRD settlement during operation, i.e., from the end of the first filling. The inclusion of material behavior as described previously was an essential part of the model development, as further explained in the next section. The prediction model was developed by considering material behavior and stress state, but is expressed in Eq. (4) in the form of an HST model, with as the recorded response, as the load-related deformation considering effect of hydrostatic thrust, as the time-dependent settlement, and as the residual error. The seasonal component from temperature variations is generally insignificant for rockfill dam settlements and thus is not included. However, the model developed considers unloading–reloading () from variations in the reservoir elevation as well as the effect of the first filling (). The statistical model is presented as follows:
(4)
The symbols used in Eq. (4) are described in detail subsequently; however, for an overview, are regression coefficients; the symbols , , , and all relate to the water head; is the current stress, while is the previously experienced maximum stress; represents accumulated total days () divided by 100 (); and is a corresponding base value. The symbols and are indexes of summation, while , , and are the upper limit of summation.
The left-hand side of Eq. (4) represents the response on each observation day, i.e., at one point in time. Furthermore, the predictors on the right-hand side of the equation are calculated relating to that same observation day. The subsequent discussion of Eq. (4) assumes the resulting time series of settlement () since the end of the first filling. For proper consideration of delayed response to hydrostatic loading, the time series of the reservoir elevation must extend from days ( for the test case) prior to the end of the first filling. The time series should have time steps of 1 day. The regression assumes a sample set of size for the responses and predictors. However, the calculation requires time series of the reservoir elevation with time steps. The calculation procedure starts by calculating all the predictors on the left-hand side for each observation day, at time step (). The reservoir elevation at the observation day is at time step of the reservoir time series. The routines within the calculation procedure use the observed values of reservoir elevation and the time in days as well as the constraints described for each predictor. Subsequently the predictors are regressed to the observed settlements to obtain the regression coefficients.
In Eq. (4), the subscript on the first term refers to first filling and denotes that this term accounts for the settlements induced by the first filling. The parameters , , and are regression coefficients for the hydrostatic component, is a regression coefficient for the time effect component, and is a constant. The subscript for regression coefficient denotes that different coefficients are calculated, depending on whether the settlement occurs under unloading () or reloading () conditions. Similarly, the subscript for regression coefficients denotes that different coefficients are calculated, depending on whether the settlement occurs under loading () or unloading () conditions. The component accompanying is only used if the reservoir loading exceeds previous loadings.
In the model, is the dead water level and is the water head calculated as the difference of the water level at the relevant observation day and the dead water level. The different factors defined for the hydrostatic component account for delayed response to the hydrostatic loading. The predictor accounts for delayed response to the first reservoir impounding. The delay depends on location of the instrument within the dam, with the delay somewhat increasing toward the downstream side. At time step of this predictor, is the average value of the water heads observed at time step to [] of the reservoir time series (i.e., at to [] days prior to the observation day). The number is the number of observations included in the average. Thus, . For the test case, was generally used, and had a value of 110–160 days. Similarly, is a corresponding average of the water heads considering the first days of the reservoir elevation time series. The summation in Eq. (4) denotes that more than one predictor of this type may be required to capture this effect. In the case study presented herein, was used.
In the summation associated with , is a value that should be defined for each case studied. For instance, the test case uses . In the ratio , is the average water head observed on Days 1–3, 4–10, 11–30, 31–60, and 61–110 before the observation day, whereas is the average of water head observed on Days 1–3, 4–10, 11–30, 31–60, and 61–110 before the first date of settlement observation series.
The factor accounts for the loading induced when the reservoir elevation rises above an elevation to which the dam has previously been subjected, and thus the current stress () is larger than previously experienced maximum stress (). This component was not activated in the model because this loading condition was not relevant for the particular case studied. However, it is included in the model description for the sake of completeness. The parameter in the last summation of Eq. (4) is a value that should be defined for each case studied. Additional components may be required to describe the actual behavior under different loading conditions.
Other symbols used in Eq. (4) are , where represents the number of days passed since the end of construction (or when the embankment elevation surpassed the level of the instrument location) until the observation day; and , where is the number of days from the end of construction to the end of the first filling. Thus corresponds to the end of dam construction, or, as may apply, when the embankment elevation reached a level that was above the instrument location.
Table 1 summarizes the coefficients used for the test case.
| Residuals () | Hydrostatic | Seasonal | Time | |||||
|---|---|---|---|---|---|---|---|---|
| Response prior to the end of first filling (other than transient response, see ) | Transient response (viscoelastic) to first filling | Transient response (viscoelastic) to subsequent unloading and reloading | Loading above previous maximum loading and subsequent unloading and reloading | Not included | Time dependent creep | |||
| Unloading | Reloading | Loading | Unloading | Reloading | ||||
| , , , , | , , , , | Not relevant for the test case () | Not included | |||||
Components of the Proposed Statistical Model
The fitting accuracy of a statistical model and its representation of actual behavior depends on the choice of influence factors (predictors) and their physical relation to the recorded data. In Eq. (4), the predictors of the proposed model are grouped as components of a HST statistical model, whereas this section introduces a different grouping basing on material behavior as introduced in Table 1. This entails relating the predictors to the components of a viscoelastic–plastic material model. This approach was important in the development of the model to ensure that each of the traditional HST components actually spans and captures the settlement behavior of a CFRD. For this purpose, this section first discusses rockfill deformation and then the different components of the statistical model from the consideration of material properties and deformation behavior.
Relation to a Viscoelastic–Plastic Material Model
VEP constitutive model components (e.g., Oyen and Cook 2003) can be used to describe rockfill deformation behavior, and they have been used in a finite-element analysis of CFRDs (e.g., Gan et al. 2014). In Fig. 2, the components of the statistical model proposed through Eq. (4) are related to elements of a VEP material model. The relation considers that the true material behavior is inherent in the recorded data used for the regression.
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In Fig. 2, is a deformation modulus describing the instantaneous compression of CFRDs when loaded (which replaces the elastic modulus of the Maxwell model) and represents a time-dependent viscous function to describe the time-dependent creep of CFRDs (which replaces a constant viscous coefficient of the Maxwell model). Furthermore, and are respectively the elastic modulus and viscosity coefficients for the transient elastic creep, i.e., the viscoelastic behavior (Kelvin-Voigt model). Finally, and are respectively the plastic hardening modulus and plastic viscosity coefficient, and is the yield stress.
The VEP model in Fig. 2 is essentially as presented by Gan et al. (2014) for the rockfill in an FE analysis of a CFRD, with the exception of the different possible loading conditions indicated with the shaded boxes. For example, there is the possibility of a different settlement behavior if the reservoir loads the dam to a higher stress level. This may result in stresses exceeding the yield stress or in deformation behavior related to primary loading. However, once the full supply level is reached in reservoir impounding, any additional reservoir loading is generally limited to a few meters. When effects of the first filling no longer have to be accounted for, and while the loading (stress state ) does not exceed previous loading [and the stress state is within the yield stress state ()], the model reduces to a viscoelastic model.
The remainder of this section explains the selection of the functions used in the statistical model in Eq. (4), aiming at capturing the material behavior.
Time-Dependent Creep
The selection of a predictor for creep is important because this generally governs the overall settlement during normal operation of a dam. The long-term creep of CFRDs has been studied using recorded data and rockfill creep behavior from triaxial tests. For example, Alonso and Oldecop (2007) used experimental data to investigate the relationship between strain, stress, and time for long-term creep. They found that a linear strain-log(time) function gave the best fit and argued that this was a feature also supported by settlement records from rockfill dams. Hunter and Fell (2003) also found that for CFRDs constructed of well-compacted rockfill, a log(time) function reasonably describes the overall settlement. The results were established by considering the start of the postconstruction deformation at the end of construction of the main rockfill body.
Various researchers have presented other formulations of the long-term creep of CFRDs. Table 2 presents a summary of six different formulations.
| Function | Expression | Explanation | Reference |
|---|---|---|---|
| F1 | are regression coefficients and is obtained from triaxial tests | Wu et al. (2009) | |
| In Fig. 3, | |||
| F 1.1: is positive, and are negative | |||
| F 1.2: and are positive, is negative | |||
| F 1.3: is positive, , and is negative | |||
| F 1.4: , , and is negative | |||
| F2 | are regression coefficients | Hu et al. (2011) | |
| In Fig. 3, | |||
| F 2.1: and are positive | |||
| F 2.2: is negative, is positive | |||
| essentially results in Function F6 | |||
| F3 | and are obtained from creep tests | Haifanga and Yinqi (2012) [a general version of the initial Ohde’s equation (Ohde 1939; Janbu 1963)] | |
| is a point in time of the creep test duration | |||
| In Fig. 3, and | |||
| F4 | and are regression coefficients | Zhou et al. (2009) | |
| In Fig. 3, and | |||
| F5 | is obtained from triaxial tests | Zhou et al. (2011) | |
| is proportional to confining stress | |||
| In Fig. 3, and | |||
| F6 | is a regression coefficient | Various authors have regressed this formula with monitoring data | |
| In Fig. 3, | |||
| , where is a base value of accumulated total number of days from a selected reference date |
Note: , where is the accumulated total days (unit of time is days).
In Fig. 3, the functions of Table 2 are plotted against time to visualize potential shapes of the curves within a timeframe of 200 years. It is clear that some time constraints are required for Functions F1 and F2 since they do not provide a natural description of the settlement behavior. On the other hand, the relations numbered F3–F6 may be considered further. Of those functions, F3–F5 use regression coefficients from triaxial tests. In this respect, F3 is particularly sensitive to the parameters used since two parameters must be fitted simultaneously in addition to the linear regression coefficient. Function F5 assumes convergence to a finite value , which is dependent on the stress state. This stress dependence of the creep limit, , is an important feature of this expression considering that the state of stress at any point in a CFRD changes during its different life-cycle phases.
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At the initiation of prediction model development, it is desirable to start with a relation that describes the settlement behavior appropriately through a function that can be regressed linearly with other predictors. In this study, the functional form F6 was selected since it has been shown to regress reasonably well to the actual settlement behavior of CFRDs, particularly those of well-compacted rockfill. Function F6 is, in fact, the simplest version of the Janbu time resistance concept (Janbu 1969, 1985). Function F6 increases infinitely to an asymptote, with the increment decreasing at each time step, which is in agreement with the general settlement behavior of a CFRD.
Hydrostatic Effect
In HST models for concrete dam displacements, the hydrostatic effect is usually expressed as a third-degree polynomial function (Chouinard and Roy 2006), similar to what has been used for earth-rockfill dams (Hu et al. 2011). Some researches (Wu et al. 2009) have, through back and forth shifts between physical and empirical relationships, derived an expression for the hydrostatic loading, resulting in functions with the depth of water as a power of the elastic modulus number. However, the modulus number can take on a wide range of values from negative to positive (Soroush and Jannatiaghdam 2012), which makes the model impractical. Furthermore, it is important to maintain a basis for physical relationships and to bear in mind that a statistical model assumes that different response properties can be studied separately. The transparent relationship of actual behavior is advantageous for statistical model application and was preferred in the development of the proposed model.
Fitzpatrick et al. (1985) presented simplified schemes to estimate the rockfill modulus, during construction on one hand and during first filling on the other [see also Hunter and Fell (2003)]. These moduli are widely used in the empirical design of CFRDs. The modulus for the construction phase is calculated from the measured settlements of the rockfill dam body, whereas the first filling deformation modulus is calculated from the measured deflections of the face slab. The two moduli capture the overall deformation behavior, i.e., no distinction is made between the two different sources of deformation, the near-instantaneous response to loading, and the delayed contribution. Still, they represent a clear physical relation between deformation and loading as well as strains and stresses. The hydrostatic loading at the relevant location represents the stress increment and the strain is derived from dividing the measured deformation by the thickness of the dam layer underneath (with the same alignment as the deformation). A similar artifact modulus can be obtained considering only the vertical deformation, i.e., the dam settlements, during operation. This is shown in Fig. 4, explaining the expression for the artifact modulus, rewritten in Eq. (5) to extract the settlementwhere = artifact settlement modulus; = unit weight of water; = settlement at depth from the reservoir surface; and = depth of rockfill column.
(5)
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In a statistical model the settlements at different times are known at measuring locations within the dam. For each measuring location, the value of is known from the dam geometry, the value of is obtained for each time step from monitoring data of the reservoir elevation, and the density of water is a constant value. The settlement deformation during operation at a certain location within the dam body can thus be related in a simple way to the water pressure, i.e., the reservoir elevation , using Eq. (5).
It seems reasonable in a regression model to relate the deformation directly to and consider that the regression is in a way providing values representing relation to an artifact moduli at different measuring locations, although within the limitations of the regression model. For operational conditions, it is additionally possible to account for unloading and loading conditions arising from variations in the reservoir level.
This applies to both the instantaneous response as well as the delayed response, both of which have to be accounted for in the statistical model. Once the time lag between the loading and response has been identified from the acquired data, this can be incorporated into the statistical model to account for the delayed response.
Test Case
The case used for testing the prediction models studied in this paper was the 198-m-high and 700-m-long Kárahnjúkar CFRD in Iceland (Fig. 5). The reservoir elevation since the start of the first impounding is shown in Fig. 6. The dam is constructed of palagonite tuff rockfill with dam zoning based on the state of the practice at the time of design and construction.
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Instrumentation of the dam related to settlement monitoring includes benchmarks at the dam crest (CS) and on the downstream side (TS), as well as hydraulic settlement (HS) gauges within the dam body. The locations of the crest stations labeled CS-1 to CS-12 are shown in Fig. 5. The HS gauges are located within three main sections of the dam, as shown in Fig. 7 (labeled HSA-, HSB- or HSC-, with a number). The main benchmarks on the downstream side, labeled TS, serve as reference stations for the HS gauges.
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Fig. 8(a) presents all the available monitoring data series, with a starting point (all settlements set to 0) at the end of June 2007, i.e., during the first impounding. The starting point selection considers the CS data series, which have the latest starting point in time. Fig. 8(a) presents processed time series. The processing (Sigtryggsdóttir et al. 2013) included interpolation between readings taken at uneven time intervals to obtain synchronized time series with equal time steps. The actual readings were conducted frequently during construction and the first impounding. However, the interval between readings has gradually increased, resulting in some loss of detail in the monitored settlement behavior.
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In Fig. 8(b), all the time series shown in Fig. 8(a) are standardized. A standardized variable has a 0 mean and a standard deviation of 1 (Rencher 2002) and retains the shape properties of the original variable. Thus, standardization can be used to put data sets on the same scale for further analysis or comparison. The high correlation in the overall settlement behavior can be observed from Fig. 8(b). However, it is clear that there are details in the deformation history that require further attention.
Analysis
The stations HSA-5 to HSA-7 captured reasonably well the settlement response induced by the variations in the reservoir elevation. Similar details were also observed at other locations, including at the crest, where stations CS-7 to CS-9 are of special interest because they are located at the maximum dam section. Hence, these stations were selected for the analysis.
Fig. 9 displays the standardized settlement at stations HSA-5 to HSA-7, extending from the end of the first impounding. A delay in the response to the reservoir impounding was observed, with the gauges closer to the upstream side showing the earliest response. The delay in the response to the first filling was about 110 days for HSA-5, 160 days for HSA-6, and 190 days for HSA-7. This time delay was accounted for in the statistical model.
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Fig. 10(a) presents the settlement at station HSA-5. The settlement time series starts when the reservoir elevation rises above the station location at 585 m above sea level. Fig. 10(b) shows how these settlements plot against an estimate of a proportional value of the deviatoric stress. This proportional value can be expressed as follows:where = estimate of the deviatoric stress at time (the current monitoring time); and = estimate of the deviatoric stress at time (start of the settlement time series). The estimated deviatoric stress was calculated considering stresses within the dam at the instrument location. These comprise earth pressure as well as pressure from hydrostatic loading. The resulting stress-deformation relationship in Fig. 10(b) for HSA-5 resembles the general stress-strain relationship presented, e.g., by Byrne et al. (1987) for initial and repeated loading.
(6)
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The importance of incorporating the delayed response to the reservoir loading in the model is evident from Fig. 9, as well as from similar results from other locations within the dam. It is also of interest to alternate between the unloading and reloading behavior. The proposed model in Eq. (4) therefore accounts for these effects. The stress state was calculated for each time step. Different regression coefficients were calculated for the unloading versus the reloading conditions. This was determined from a loading function, as illustrated in Fig. 11 for station HSA-5, with 0 representing the reloading conditions, and 1 the unloading conditions. The stress level was also calculated and compared with previous stress levels. As mentioned previously, reloading above the previously experienced stress state has not been of significant influence for the dam during its operation.
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Multiple regression was performed on the time series from HSA-5, HSA-6, and HSA-7 using the processed settlement time series from the end of the first filling as the response variable. Subsequently, a multivariate multiple regression with the time series recorded at CS-7, CS-8, and CS-9 as response variables was conducted using both a training data subset as well as a prediction check data subset.
Analysis Results
The results from the multiple regression analysis on the time series HSA-5 are presented in Fig. 12. Fig. 12(a) compares the model results (MLR) with the processed time series (HSA-5), Fig. 12(b) highlights the details of the settlement behavior due to annual cycles in the reservoir elevation, and Fig. 12(c) extracts the time-dependent settlement in the period considered from the model. Residuals from comparison of the model and the processed response time series are plotted in Fig. 13(a). The shape of the residual plot resembles a normal probability distribution, and the expected value of the residuals is zero [. Fig. 13(b) plots the residual for station HSA-5 from a regression model that does not take the stress state into account. Comparison of the distribution of the residuals in Figs. 13(a and b) indicates that a better fit is achieved when the stress state is considered.
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In Fig. 14(a) the model is compared with the actual readings from station HSA-5. The figure demonstrates how intervals between readings have gradually increased since the first impounding, as previously mentioned. It is thus to be expected that the accuracy of the processed time series used for the regression had simultaneously gradually decreased. Figs. 14(b and c) show similar analysis results for stations HSA-6 and HSA-7, respectively. However, the estimate of the deviatoric stress is less accurate for stations close to the middle of the dam or at the downstream side than for those at the upstream face.
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Fig. 15 presents model results from multivariate multiple linear regression, simultaneously considering three response variables at the crest, i.e., at stations CS-7, CS-8, and CS-9. These response variables were obtained from geodetic surveys on benchmarks, and there were not as many measurements available as there were for the HS stations. The response variable data set was divided into a training sample subset and a prediction check sample subset. In Fig. 15, the model results considering the training sample subset are presented with an unbroken line, while the model results considering the prediction check sample subset are presented with a dotted line. The actual survey readings are also presented in the figure with a circle denoting a survey data point. It can be seen that the prediction check sample subset agrees reasonably well with the actual response. Furthermore, the prediction model complements the readings and brings out details that are lost in the actual data since there are only two surveys conducted annually in this period.
GT.1943-5606.0001916/asset/bc902103-1139-4e8a-a111-037ab19e7ade/assets/images/large/figure15.jpg)
The residuals for station CS-9 are plotted in Fig. 16. Two residual plots are provided: (1) one for the regression results shown in Fig. 15 from a model that considers stress state from unloading–reloading conditions, and (2) another for a regression model that does not consider the stress state. Comparison of the distribution of the residual in Figs. 16(a and b) indicates a better fit when the stress state is considered, which corresponds to the results previously seen for data from HSA-5. The correlation of the training data subset with the model results was very high in both cases (both ), but slightly higher for the model considering the stress state, indicting a better fit. Similar results were obtained for other locations. This is in agreement with observations from triaxial tests (Stewart 1986; Byrne et al. 1987), which indicated a slightly different modulus for unloading and reloading.
GT.1943-5606.0001916/asset/152688d1-ff1a-40b9-be4f-bb7d00389531/assets/images/large/figure16.jpg)
Fig. 17 presents a prediction of time-dependent settlement from the multivariate multiple regressions of variables for stations CS-7, CS-8, and CS-9. The survey readings at these stations are plotted on the figure and start at the end of the first filling.
GT.1943-5606.0001916/asset/01459e3a-3713-4157-9c3a-29ade5c806e5/assets/images/large/figure17.jpg)
Discussion
Application of the proposed model in Eq. (4) to the different settlement time series of the case study demonstrates that the model adequately captures the response and gives a credible comparison to the response variables used. The prediction of the crest settlements in the validation period (Fig. 17) is also convincing. The settlement prediction will improve as the model is updated as new data are collected.
Required Performance History
The methodology used in developing the prediction model requires information on both the hydrostatic loading, i.e., the reservoir elevation, and the response variable, i.e., the settlements.
The proposed model requires daily values of the reservoir elevation and actual observations for the period used in the regression. The application of the model for prediction with detailed pattern identification due to changes in reservoir filling is mainly dependent on the hydrostatic component, and thus also on accurate prediction of the hydrostatic loading from the reservoir, e.g., based on reservoir inflow and outflow for reservoir elevation forecasting.
Conversely, monthly values of the settlement can be considered to provide a minimum amount of data points for processing to a data set suitable for obtaining the regression parameters of the prediction model. More closely spaced observations in time will enhance the regression and improve the reliability of the model, particularly for identification of details in the behavioral pattern of the settlement. Preferably, additional observations should be collected during periods of sudden and large changes in the reservoir elevation. The number of available data points for the case studied herein and the varying time interval between observations for the crest stations at the top of the test case dam is portrayed by the readings shown in Fig. 15. The original settlement time series were processed to produce daily values (Sigtryggsdóttir et al. 2013).
This study indicates that monitoring data from three to five cycles of seasonal variations in the reservoir elevation after the initial impounding is required to achieve reasonable prediction of the overall settlement behavior during normal operation. A part of the available data should be used for a validation check of the model. The model parameters can be updated, if required, based on past performance as additional data are gathered from repeated reloading cycles. As the time span of the training data set increases, the prediction of the long-term settlement will become more reliable.
Measured versus Predicted Performance
The benefits of the predicted response from a statistical model as presented here relate to both the long- and short-term prediction.
If the predicted long-term response indicates unreasonably large settlements compared with design estimates, there are mainly two focus points for further evaluation. First, the influence on the required freeboard of the dam, considering that rockfill dams, including CFRDs, are vulnerable to overtopping. Second, the increase in settlement-induced stresses in the concrete face slab and the potential for excess cracking leading to unacceptable leakage. The stresses induced in the slab due to the long-term settlements can be estimated, e.g., in an FE model.
Under normal operation, the actual reservoir elevation is often recorded automatically with daily values available for large CFRDs, while a survey of the settlement is often manual and only conducted a few times a year. Thus, the actual reservoir elevation can be used to predict the short-term settlements in between actual observations as well as at the observation day. In this case, one relevant application of the prediction model would be to compare the surveying measurements to the predicted response. If the variation between the two is larger than a prespecified deviation, the measurement should be repeated. This could reduce or even eliminate outliers from the surveying data and enhance the accuracy of the observations collected. This suggestion is based on the fact that strange outliers were observed in the surveying data used in this study. However, if a repeated measurement still does not follow the prediction, more frequent observations should be conducted along with a site inspection, followed by a detailed evaluation of the change in settlement behavior and analysis of the effects on structural integrity.
The patterns observed in the settlement behavior can be interpreted to signify the dam’s structural health and normal state of operation. It is difficult to envision causes for changes in this behavior, unless possibly if the dam would be subjected to extreme loading. Still, any changes in these details, as well as sudden deviations from the long-term settlement behavior, should call for further investigation and analysis to assess the dam’s structural health and safety.
Concluding Remarks
In this paper the development of statistical models of a rockfill dam deformation behavior has been described and the importance of a cautious selection of predictors for the credibility of a statistical model demonstrated. A statistical model was proposed and tested for the case of Kárahnjúkar CFRD. Such models are required for SHM management, as well as being consistent with International Commission on Large Dams–recommended (ICOLD 2012) dam engineer’s practice on the usage of numerical models. Thus, the paper presents important information, considering the limited and incomplete reports hitherto available for rockfill dam settlements.
The proposed statistical model [Eq. (4)] has elements of a general HST model, but without the seasonal component [], which is irrelevant. At the same time, components of the statistical model are innovatively related to a viscoelastic–plastic constitutive material model, with the novel addition of considering the unloading–reloading behavior of the dam settlement. For rockfill dams, this consideration is clearly an option and aids in defining predictors that relate to actual physical factors influencing the response variable. The algorithm developed for the regression has aspects related to FE analysis in terms of stress state and varying loading functions. The incorporation of a more accurate stress state information, e.g., from an FE model, would enhance the resulting statistical model, and vice versa, the data regression would provide valuable reference data for FE modeling. Such joint consideration of a statistical model and mechanics model is inherent in a comprehensive SHM process.
The statistical analysis of the settlement monitoring data has brought forth behavioral patterns, including details in the dam settlement induced by a viscoelastic response to variations in the reservoir hydrostatic loading. The behavioral pattern can be interpreted to signify the dam’s structural health and normal state of operation. The statistical model is well suited for an online prediction process, with regular updates based on past performance. Comprehensive SHM management incorporating statistical models as discussed herein is important for the operational safety of large dams, and thus the resilience and sustainability of the related civil infrastructure systems.
Acknowledgments
The support and input by the late Professor Ragnar Sigbjörnsson is gratefully acknowledged. The authors thank Landsvirkjun for access to the monitoring data.
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Journal of Geotechnical and Geoenvironmental Engineering
Volume 144 • Issue 9 • September 2018
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Received: Jul 1, 2017
Accepted: Feb 7, 2018
Published online: Jun 27, 2018
Published in print: Sep 1, 2018
Discussion open until: Nov 27, 2018
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