Quantifying Durability and Failure Risk for Concrete Dam–Reservoir System by Using Digital Twin Technology

Abstract

This study presents a digital twin approach to quantifying the durability and failure risk of concrete gravity dams by integrating advanced numerical modelling with field monitoring data. Building on a previously developed finite element model for dam–reservoir interaction analysis, this research extends its application to the assessment of existing, fully operational dams by using digital twin technology. One such case study of a digital twin is given for the concrete gravity dam, Salakovac. The numerical model combines finite element formulations representing the dam as a nonisothermal saturated porous medium and the reservoir water as an acoustic fluid, ensuring realistic simulation results of their interactions. The selected finite element discrete approximations enable the detailed analysis of the dam failure mechanisms under varying extreme conditions, while simultaneously ensuring the consistent transfer of all fields (displacement, temperature, and pressure) at the dam–reservoir interface. A key aspect of this research is the calibration of the numerical model through the systematic definition of boundary conditions, external loads, and material parameters to ensure that the simulation results closely align with observed behaviour, thereby reflecting the current state of the ageing concrete dam. For the given case study of the Salakovac Dam, we illustrate the use of the digital twin to predict the failure mechanism of an ageing concrete dam for the chosen scenario of extreme loads.

1. Introduction

Dams are hydraulic structures built to retain and utilize stored water for energy production and other purposes, contributing to efficient water management. In Bosnia and Herzegovina, most of the dams are part of hydroelectric power plants, supplying approximately a quarter of the national electricity demand. Serving as a reliable and renewable energy source, they play a crucial role in efforts to achieve sustainability goals and support the global transition to clean energy. In addition to electricity generation, dams offer other benefits, including flood control and stable water supply.

Since dams are a key component of hydroelectric power plants, any damage to them can lead to reduced electricity production or even a complete shutdown of the plant. Furthermore, more severe damage or dam failure can have catastrophic consequences for downstream communities. Most of the dams in Bosnia and Herzegovina currently operating at full capacity are concrete gravity or arch dams built several decades ago, emphasizing the need for regular monitoring, maintenance, and modernisation efforts that can be structured within digital twin technology. Continuous monitoring of the dam’s behaviour, along with the development and use of numerical models, is essential for tracking the current condition and analysing the response to various extreme-load scenarios. This approach allows for the early identification of potential safety issues and the implementation of appropriate preventive measures to ensure the structure’s safety.

Numerical modelling of dam–reservoir interaction includes developing detailed numerical models for both the dam structure and the reservoir water, along with a representation of their interaction. To ensure a complete analysis of dam failure risk, the numerical model of the dam must be capable of capturing the failure modes, including crack initiation and propagation. This requires considering the complete list of long-term loads, such as self-weight, reservoir pressure, and seasonal temperature variations. Here, these loads (leading to dam ageing) are combined with short-duration extreme loads, like those induced by earthquake events. In addition, calibrating the numerical model by properly defining boundary conditions and identifying ageing material properties based on existing monitoring data is essential to ensure a realistic and precise simulation that reflects the current state of the dam. This, in turn, provides a foundation for further analyses of dam safety under extreme conditions. Numerous design studies of concrete dams are available in the literature, offering valuable insights into their behaviour under various loading scenarios and significantly contributing to the comprehensive understanding of dam performance. However, not many studies are available for combined conditions of ageing dams, which is the main originality of this work.

There are many previous studies on long-term dam loading. Numerical approaches to the thermal analysis of concrete dams have received considerable attention, as thermally induced displacements and stresses are major factors governing structural performance and long-term integrity. These studies have underscored the importance of accurately modelling heat transfer mechanisms, thermal loads, and boundary conditions, as these factors directly influence the temperature distribution within the dam. Within the finite element method framework, studies by Venturelli [1], Léger et al. [2], and Sheibany and Ghaemian [3] detailed how environmental conditions, such as ambient air, water, and foundation temperatures, affect temperature and stress distributions in concrete gravity dams. These studies investigated the relationship between seasonal temperature changes and the resulting stress profiles, contributing significantly to the understanding of the thermo-mechanical behaviour of concrete dams under varying environmental conditions. Complementing these efforts, Fairbairn et al. [4] provided a comprehensive review of the factors contributing to thermal cracking in massive concrete structures, such as dams, and outlined modelling strategies and methodologies for assessing cracking risks. Mirzabozorg et al. [5] and Shao et al. [6] developed models to simulate thermal distribution in concrete dams, explicitly accounting for the effects of solar radiation. Furthermore, Žvanut et al. [7] analysed the thermal behaviour of concrete dams while considering additional variables including insolation, shading, water levels, and spillover, which significantly influence the temperature field and stress distributions. Leitão and Oliveira [8] examined the effects of solar radiation and water temperature profiles on the thermal response of concrete dams, highlighting the importance of accurately modelling these spatio-temporal actions to enhance thermal analysis outcomes.

There are also many previous studies on the effect of extreme loads on dams. Here, seismic analysis of concrete dams represents a critical aspect of dam engineering, as it ensures their safety under earthquake loading. This remains a subject of extensive ongoing research. Rezaiee-Pajand et al. [9] provided an overview of various dynamic analysis techniques applied to concrete dams, highlighting recent advancements in methodologies used to assess their performance under earthquakes. Jianwen et al. [10] evaluated various methodologies for analysing seismic cracking in concrete dams. They systematically compared traditional and advanced analysis techniques, including the finite element method, shaking table tests, and other numerical simulation approaches, to assess their effectiveness in predicting seismic damage. Their study highlighted the advantages and limitations of each method, providing insights into their applicability to real-world scenarios. Haghani et al. [11] used the extended finite element framework to simulate crack propagation and evolving damage mechanisms within the dams during seismic events. In a recent study, Arici and Soysal [12] provided a comprehensive synthesis of seismic analysis techniques for concrete dams, covering numerical modelling, experimental methods, and analytical methods. They highlighted the complexities in predicting crack propagation and failure modes under seismic loads and discussed the importance of considering factors such as dam–reservoir interaction, material properties, and loading conditions in seismic assessments.

Here, we bring these two studies together to evaluate the failure safety of ageing concrete gravity dams, relying upon digital twin technology. A digital twin is a virtual representation of a physical system that is continuously updated with real-time monitoring data to reflect its current condition and performance. Digital twins have emerged as a powerful tool in dam safety management through the integration of monitoring data with numerical models to enhance predictive capabilities and structural assessment [13,14]. By combining monitoring data with physics-based numerical modelling, digital twins provide a dynamic platform for simulating various operational and extreme scenarios. This supports more accurate prediction of structural responses and informed decision making for dam operation and risk management. Aside from numerical modelling, digital twin technology also makes strong use of machine learning. Recent advances in machine learning have introduced powerful data-driven methods for structural health monitoring of dams, enabling the prediction of thermal distribution, displacements, and potential anomalies based on historical and real-time monitoring data [15,16,17]. However, physics-based numerical modelling remains essential for a full understanding of the underlying behaviour of dams and quantifying the risk of failure under extreme loads. Numerical models incorporate material properties, boundary conditions, and physical laws governing structural response, allowing for detailed scenario-based simulations of dam performance under extreme loads. As such, machine learning and numerical modelling are increasingly seen as complementary tools to be combined within the digital twin, with integrated approaches offering the potential to enhance predictive accuracy and deepen physical understanding in dam safety assessments [18].

The present study focuses on the use of numerical modelling in structural health monitoring and predictive analysis for ageing dams. Specifically, this paper presents a digital twin of the Salakovac Dam, developed by integrating a physics-based numerical model of dam–reservoir interaction with monitoring data. Here, the numerical model of dam–reservoir interaction previously developed by the authors [19,20] is applied for the first time in the context of an ageing dam. Using available monitoring data, a comprehensive model is developed that reflects the current condition of the dam. This process involves calibrating the model through the appropriate definition of boundary conditions and the identification of material parameters, ensuring that the numerical results closely align with observed data, and thereby providing a realistic representation of the behaviour of the dam under operating conditions, including self-weight, reservoir pressure and seasonal temperature variations. The case study further demonstrates the potential of the model to quantify the remaining structural resistance of the dam under dam–reservoir interaction and to capture the associated failure modes during extreme loading conditions, with earthquake loading serving as a representative scenario in this study. By relying on a numerical model calibrated with monitoring data to reflect the current condition of the dam and capable of capturing localised failure within the dam–reservoir system, the proposed framework makes a key contribution by enabling the assessment of the overall safety of the dam under extreme scenarios, while also accounting for the long-term effects of operational loads that contribute to dam ageing.

The outline of the paper is as follows: Section 2 provides an overview of the main features of the numerical model of dam–reservoir interaction used in this study. Section 3 details the calibration of the numerical model for the Salakovac Dam, including its geometry and the definition of boundary conditions. Section 3 further presents several numerical analyses aimed at identifying the ageing material parameters of the model. The results are discussed, focusing on the ability of the model to replicate the observed behaviour of the dam under operating conditions. This section also includes the analysis aimed at illustrating the capacity of the model to predict the response of the dam to progressively increasing lateral loads, including cracking and localised failure, and thereby its suitability for assessing the structural safety of the dam under extreme events. Section 4 summarises the key conclusions and outlines potential directions for future research.

2. Numerical Model of Dam–Reservoir Interaction

The complexities of numerical modelling of the dam–reservoir interaction arise from two main aspects. The first one is the complex behaviour of the cohesive materials in the dam structure interacting with the reservoir. Namely, for such dams, typically built from concrete, rock, or soil, all of which are cohesive and porous materials, the water in the reservoir acts not only as a source of loading on the dam but also as a source of pore saturation. Furthermore, aside from the self-weight and the reservoir pressure, the seasonal temperature variations are among the key loads on the dam. Hence, the numerical model of the dam ought to be capable of representing a nonisothermal saturated poroelastic medium. The assumption of elastic behaviour may be valid for the majority of dams under the long-term standard operational loads, due to their conservative design. However, to assess the safety of the dam under extreme loads, the numerical model must further be capable of capturing inelastic behaviour and potential failure mechanisms.

The second complexity in the numerical modelling of dam–reservoir interaction emerges due to the numerical approximation of the water in the reservoir. The simplest and still widely used method for modelling dam–reservoir interaction is the added mass or Westergaard approach [21]. In this approach, the influence of the reservoir water is accounted for by adding a fraction of its mass to the dam during vibrations. The added mass is determined under the assumption of an incompressible fluid, making it particularly effective for estimating hydrodynamic pressure in reservoirs [22]. The acoustic wave theory is commonly used in dam–reservoir interaction problems to describe the motion of water in the reservoir, as confined reservoir conditions result in small movements. Within the finite element framework, two widely used approaches for modelling reservoir water as an acoustic fluid are the Eulerian and Lagrangian formulations [23,24]. In the Eulerian formulation, displacements are the state variables in the structure, while pressures (or velocity potentials) are the state variables in the fluid. Due to the difference in state variables between the structure and the fluid, this formulation requires a special solution for the fluid–structure interface, referred to as the arbitrary Lagrangian–Eulerian (ALE) approach [25]. In contrast, the Lagrangian approach uses displacements as state variables for both the structure and the fluid, eliminating the need for special treatment of the fluid–structure interface. However, this approach suffers from numerical issues, such as spurious zero-energy modes [26], which require special numerical treatment [27,28]. To address the issue of spurious modes, a mixed displacement/pressure-based fluid finite element formulation is proposed, where nodal displacements and pressures both serve as the unknown variables [29].

Here, we build on the numerical model of dam–reservoir interaction previously proposed by the authors in [19,20]. This study extends its use to dams under full operating conditions. The main goal is to provide an approach that combines advanced numerical modelling with field monitoring data that allows for robust analysis of dam response under combined long-term and transient extreme conditions. In the proposed model, the dam is represented as a nonisothermal saturated porous medium, while the reservoir water is treated as an acoustic fluid, modelled using the Lagrangian formulation with a mixed displacement/pressure-based approximation. Therefore, the key feature of the proposed model pertains to its carefully selected numerical representations of both the dam and reservoir, enabling a direct solution for the dam–reservoir interaction through a global finite element assembly procedure. More precisely, the selected finite element approximations for the dam and reservoir water yield the same degrees of freedom at each finite element node, allowing for a seamless connection of structure and fluid elements at the shared nodes. Furthermore, the model includes the coupling of mechanical motion, pore fluid flow, and heat flow, and allows for the simulation of the fracture process zone and localised failure in the dam.

Next, the main components of the numerical model for dam–reservoir interaction are outlined. For more details, we refer to [19,20]. All numerical implementations and computations in this study were performed with the research version of the computer code FEAP (Finite Element Analysis Program), developed by R. L. Taylor [30].

2.1. Numerical Model of Dam Structure

2.1.1. Governing Equations

The response of a nonisothermal saturated porous medium is governed by three sets of equations [31]. The first set is the equations of motion defining the mechanical response, the second set is the continuity equation for the pore fluid flow, and the third set is the energy equation for the heat flow. The first two sets of equations are the equations of Biot’s porous media theory, which defines the coupling between the solid phase and the pore fluid [32]. The addition of the energy equation and thermal coupling further extends this theory to the nonisothermal case, which accounts for the influence of seasonal temperature variations on the behaviour of the dam. The governing equations are defined under the assumption that no phase change occurs and that thermal equilibrium is achieved between the solid phase and the pore fluid.

The equations of motion for a nonisothermal saturated porous medium are written as

$$\nabla ·\mathit{\sigma }+\mathbf{b}-\rho \mathbf{a}=\mathbf{0}$$

(1)

where $\mathit{\sigma }$ is the total stress tensor, $\mathbf{b}$ is the body force vector, $\mathbf{a}$ is the solid-phase acceleration vector, and $\rho$ is the mass density of the mixture, which is assumed constant.

According to Terzaghi’s principle of effective stresses, the total nominal stress is decomposed into the effective stress and the pore pressure, written as

$$\mathit{\sigma }={\mathit{\sigma }}^{\prime }-\mathbf{I}bp$$

(2)

where ${\mathit{\sigma }}^{\prime }$ is the effective stress tensor; $\mathbf{I}$ is the second-order identity tensor; p is the pore pressure, assumed positive in compression; and b is Biot’s constant.

The effective stress tensor, considering thermal coupling, consists of two parts. The first part is the mechanical part, ${\mathit{\sigma }}_u$, which arises from nonhomogeneous displacements, and the second part is the thermal part, ${\mathit{\sigma }}_T$, which results from temperature changes. The effective stress tensor is expressed as

$${\mathit{\sigma }}^{\prime }={\mathit{\sigma }}_u+{\mathit{\sigma }}_T$$

(3)

The thermal part is computed as

$${\mathit{\sigma }}_T={\mathit{\beta }}_T(T-T_0)$$

(4)

where ${\mathit{\beta }}_T$ is the thermal stress tensor for the isotropic case, defined as ${\mathit{\beta }}_T={\beta }_T\mathbf{I}$, and $T_0$ is the reference temperature.

The continuity equation for pore fluid flow through a nonisothermal saturated porous medium is written as

$${\displaystyle \frac{1}{M}}\dot{p}+b\nabla ·\dot{\mathbf{u}}-{\overline{\beta }}_{sf}\dot{T}-\nabla ·\left({\displaystyle \frac{k}{{\gamma }_f}}\nabla p\right)=0$$

(5)

where M is Biot’s modulus, k is the coefficient of permeability of the isotropic porous medium, ${\gamma }_f$ is the specific weight of the fluid, and ${\overline{\beta }}_{sf}$ is the thermal expansion coefficient of the mixture.

The energy equation for heat flow through a saturated porous medium under the assumption of thermal equilibrium between the solid phase and the pore fluid is written as

$$\rho c_T\dot{T}+\nabla ·{\mathbf{q}}_T-s=0$$

(6)

where $\rho c_T$ is the effective heat capacity of the mixture, ${\mathbf{q}}_T$ is the heat flux, and s is the heat source.

The heat flux ${\mathbf{q}}_T$ is defined by Fourier’s law of heat conduction, written as

$${\mathbf{q}}_T=-k_T\nabla T$$

(7)

where $k_T$ is the coefficient of thermal conductivity of the isotropic saturated porous medium.

In the formulation of the energy equation, the mechanical contributions to the energy balance are neglected, which is justified for brittle cohesive materials [33]. This assumption simplifies the equation, resulting in an uncoupled form where temperature changes are considered independently of displacements and pore pressures.

2.1.2. Finite Element Approximation

The response of a structure represented in terms of a nonisothermal saturated porous medium in the model is approximated using a discrete lattice framework that incorporates thermo-poro-mechanical coupling. The choice of the discrete beam lattice model lies in its ability to reproduce the linear elastic response of an equivalent continuum model. Furthermore, it allows for an efficient and robust implementation of highly complex nonlinear behaviour, including the fracture process zone and localised failure [34,35], as well as the coupling of different fields, necessary to achieve a realistic response [35,36,37].

The main idea of discrete lattice models is to approximate the structure as an assembly of Voronoi cells, with cohesive links connecting adjacent cells. As a result, the mechanical response of the structure is obtained on a mesh of 1D finite elements that effectively model the behaviour of the cohesive links. The discrete lattice model is constructed by exploiting the duality property between the Voronoi diagram and the Delaunay triangulation. Specifically, the Delaunay triangulation forms a mesh of triangles, where each edge represents a cohesive link. Thus, the Delaunay triangulation forms a mesh of cohesive links, modelled using 1D finite elements, while the Voronoi diagram defines the cross-sections for these finite elements (see Figure 1).

In this particular model, the behaviour of cohesive links is modelled with nonlinear Timoshenko beam finite elements with enhanced kinematics in terms of embedded discontinuities in the axial and transverse directions, allowing for more accurate simulation of crack formation and propagation in both mode I (opening mode) and mode II (sliding mode) [35]. The constitutive model adopted for the Timoshenko beam elements in this study includes a linear elastic part, followed by exponential softening after the ultimate stress is reached.

The pore pressure and temperature fields are approximated with a mesh of CST (constant strain triangle) finite elements [38], which corresponds to the mesh of Delaunay triangles (Figure 1). Here, the choice of the Hammer quadrature rule for numerical integration allows for integration points to be placed at Gauss integration points of Timoshenko beam finite elements; this is used to easily exchange field values for displacements, pore pressure, and temperature [19].

For compact presentation, only the final system of equations derived through the standard finite element approximation procedure is presented below. For a more detailed explanation, please refer to [19,20]. The resulting system of equations for the poro-thermo-mechanical coupled problem at the level of the Timoshenko beam element is therefore written as

$$\begin{array}{cc}\hfill {\mathbf{M}}_{uu}^e\ddot{\overline{\mathbf{u}}}+{\mathbf{f}}^{int,e}-{\mathbf{K}}_{up}^e\overline{\mathbf{p}}& ={\mathbf{f}}^{ext,e}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathbf{K}}_{up}^{e,\mathsf{T}}\dot{\overline{\mathbf{u}}}-{\mathbf{K}}_{pT}^e\dot{\overline{\mathbf{T}}}+{\mathbf{D}}_{pp}^e\dot{\overline{\mathbf{p}}}+{\mathbf{K}}_{pp}^e\overline{\mathbf{p}}& ={\mathbf{q}}^{ext,e}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathbf{D}}_{TT}^e\dot{\overline{\mathbf{T}}}+{\mathbf{K}}_{TT}^e\overline{\mathbf{T}}& ={\mathbf{s}}^{ext,e}\hfill \end{array}$$

(10)

where ${\mathbf{M}}_{uu}^e$ is the mass matrix, ${\mathbf{f}}^{int,e}$ is the internal load vector resulting from displacements and temperature changes, ${\mathbf{K}}_{up}^e$ and ${\mathbf{K}}_{pT}^e$ are the coupling matrices, ${\mathbf{D}}_{pp}^e$ is the compressibility matrix, ${\mathbf{K}}_{pp}^e$ is the permeability matrix, ${\mathbf{D}}_{TT}^e$ is the heat capacity matrix, ${\mathbf{K}}_{TT}^e$ is the conductivity matrix, and ${\mathbf{f}}^{ext,e}$, ${\mathbf{q}}^{ext,e}$, ${\mathbf{s}}^{ext,e}$ are the load vectors. The matrices ${\mathbf{K}}_{up}^e$, ${\mathbf{K}}_{pT}^e$, ${\mathbf{D}}_{pp}^e$, ${\mathbf{K}}_{pp}^e$, ${\mathbf{D}}_{TT}^e$, and ${\mathbf{K}}_{TT}^e$ are computed as

$$\begin{array}{cc}\hfill {\mathbf{K}}_{up}^e& ={\int }_0^{L^e}{\mathbf{B}}_{up}^{\mathsf{T}}b{\mathbf{N}}_{up}dx;{\mathbf{K}}_{pT}^e={\int }_{{\Omega }_{CST}^e}{\mathbf{N}}_p^{\mathsf{T}}{\overline{\beta }}_{sf}{\mathbf{N}}_{T}d\Omega \hfill \\ \hfill {\mathbf{D}}_{pp}^e& ={\int }_{{\Omega }_{CST}^e}{\mathbf{N}}_p^{\mathsf{T}}{\displaystyle \frac{1}{M}}{\mathbf{N}}_{p}d\Omega ;{\mathbf{K}}_{pp}^e={\int }_{{\Omega }_{CST}^e}{(\nabla {\mathbf{N}}_p)}^{\mathsf{T}}{\displaystyle \frac{k}{{\gamma }_f}}\nabla {\mathbf{N}}_{p}d\Omega \hfill \\ \hfill {\mathbf{D}}_{TT}^e& ={\int }_{{\Omega }_{CST}^e}{\mathbf{N}}_T^{\mathsf{T}}\rho c_T{\mathbf{N}}_{T}d\Omega ;{\mathbf{K}}_{TT}^e={\int }_{{\Omega }_{CST}^e}{(\nabla {\mathbf{N}}_T)}^{\mathsf{T}}{k}_T\nabla {\mathbf{N}}_{T}d\Omega \hfill \end{array}$$

(11)

where $\mathbf{N}$ denotes the matrix of the shape functions [38], and $\mathbf{B}$ the matrix of their derivatives. It is important to note that in Equations (9) and (10), the parts of matrices ${\mathbf{D}}_{pp}^e$, ${\mathbf{K}}_{pp}^e$, ${\mathbf{K}}_{pT}^e$, ${\mathbf{D}}_{TT}^e$, and ${\mathbf{K}}_{TT}^e$ in (11) are extracted to the nodes of the Timoshenko beam finite element.

The mass matrix for the Timoshenko beam is derived by distributing the total mass of the element to its nodes (Figure 2), leading to a diagonally lumped mass matrix, expressed as

$${\mathbf{M}}_{uu}^e={\displaystyle \frac{1}{2}}\rho diag\left(A_{tot},A_{tot},I^e,A_{tot},A_{tot},I^e\right)$$

(12)

where $A^e$ and $I^e$ are the cross-sectional properties of the Timoshenko beam—the area and second moment of inertia of the cross-section.

To ensure computational stability over the long-term simulation period required to account for seasonal temperature variations, the global system of equations is solved by using the mid-point time integration scheme and its energy-conserving variant.

2.2. Numerical Model of Reservoir

2.2.1. Governing Equations

The reservoir water is modelled as an inviscid fluid with small, irrotational motion, reflecting typical reservoir conditions. The governing momentum and continuity equations, formulated based on acoustic wave theory, are written as

$$\begin{array}{cc}\hfill \rho \dot{\mathbf{v}}+\nabla p& =0\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}+{\displaystyle \frac{\dot{p}}{\beta }}& =0\hfill \end{array}$$

(14)

where $\mathbf{v}$ is the velocity vector, p is the pressure, $\rho$ is the mass density, and $\beta$ is the bulk modulus.

For irrotational flows, the vorticity constraint is expressed as

$$\begin{array}{c}\hfill \nabla \times \mathbf{v}=0\end{array}$$

(15)

2.2.2. Finite Element Approximation

The motion of reservoir water is modelled by using the Lagrangian formulation and mixed displacement/pressure-based finite element approximation [29]. The main advantage of this approach lies in the resulting set of degrees of freedom, which includes both displacements and pressure, ensuring seamless integration with the formulation used for the dam [20]. Specifically, the selected approximations permit the direct exchange of both displacement and pressure degrees of freedom at the dam–reservoir interface, allowing all computations to be performed in a fully monolithic manner. Additionally, this framework allows the reservoir water to be considered both as an external load and as a source of pore saturation.

In the mixed approach, the strong form of the governing equations represents a penalized version of the previously introduced acoustic fluid equations, and is expressed as follows:

$$\begin{array}{cc}\hfill \nabla p+\nabla \times \mathbf{\Lambda }-{\mathbf{f}}^b& =0\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{u}+{\displaystyle \frac{p}{\beta }}& =0\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \nabla \times \mathbf{u}-{\displaystyle \frac{\mathbf{\Lambda }}{\vartheta }}& =0\hfill \end{array}$$

(18)

where $\mathbf{u}$ is the displacement vector, $\mathbf{\Lambda }$ is the ’vorticity moment’, $\vartheta$ is the penalty parameter, and ${\mathbf{f}}^b$ is the external load vector, which, in addition to body forces, also includes the inertia force $-\rho \ddot{\mathbf{u}}$.

The reservoir is discretized using a $\mathrm{Q}4$-$\mathrm{P}1$-$\Lambda 1$ finite element, which provides a linear interpolation for displacements and constant approximations for pressure and the ’vorticity moment’. Again, only the final system of equations derived through the standard finite element procedure is presented next. For more details, please refer to [19,20]. Hence, the resulting system of equations at the element level is written as

$${\left[\begin{array}{ccc}{\mathbf{A}}_{uu}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\right]}^e{\left\{\begin{array}{c}\ddot{\overline{\mathbf{u}}}\\ \ddot{\overline{\mathbf{p}}}\\ \ddot{\overline{\mathbf{\lambda }}}\end{array}\right\}}^e+{\left[\begin{array}{ccc}\mathbf{0}& {\mathbf{L}}_{up}& {\mathbf{L}}_{u\lambda }\\ {\mathbf{L}}_{up}^{\mathsf{T}}& {\mathbf{L}}_{pp}& \mathbf{0}\\ {\mathbf{L}}_{u\lambda }^{\mathsf{T}}& \mathbf{0}& {\mathbf{L}}_{\lambda \lambda }\end{array}\right]}^e{\left\{\begin{array}{c}\overline{\mathbf{u}}\\ \overline{\mathbf{p}}\\ \overline{\mathbf{\lambda }}\end{array}\right\}}^e={\left\{\begin{array}{c}{\mathbf{f}}^{\mathrm{b}}\\ \mathbf{0}\\ \mathbf{0}\end{array}\right\}}^e$$

(19)

where

$$\begin{array}{cc}\hfill {\mathbf{A}}_{uu}^e& ={\int }_{{\Omega }_f^e}\rho {\mathbf{N}}_u^{\mathsf{T}}{\mathbf{N}}_{u}d\Omega ;\hfill \\ \hfill {\mathbf{L}}_{up}^e& =-{\int }_{{\Omega }_f^e}{\mathbf{V}}_u^{\mathsf{T}}{\mathbf{N}}_{p}d\Omega ;{\mathbf{L}}_{u\lambda }^e={\int }_{{\Omega }_f^e}{\mathbf{D}}_u^{\mathsf{T}}{\mathbf{N}}_{\lambda }d\Omega \hfill \\ \hfill {\mathbf{L}}_{pp}^e& =-{\int }_{{\Omega }_f^e}{\displaystyle \frac{1}{\beta }}{\mathbf{N}}_p^{\mathsf{T}}{\mathbf{N}}_{p}d\Omega ;{\mathbf{L}}_{\lambda \lambda }^e=-{\int }_{{\Omega }_f^e}{\displaystyle \frac{1}{\vartheta }}{\mathbf{N}}_{\lambda }^{\mathsf{T}}{\mathbf{N}}_{\lambda }d\Omega \hfill \end{array}$$

(20)

with $\mathbf{N}$ as the matrix of shape functions, and $\mathbf{V}$ and $\mathbf{D}$ as the divergence and curl matrices of the shape functions, respectively.

The pressure and ’vorticity moment’ can be statically condensed at the element level, with their values derived from the computed displacements. However, the objective is to directly connect the dam and reservoir finite elements at common nodes, ensuring the exchange of both motion and pressure at their interface. This is achieved by extrapolating the pressure computed within an element to the nodes of the $\mathrm{Q}4$ finite element used for displacement approximation. Consequently, the pressure at each node is determined as the average of the pressures computed in all finite elements sharing that node. Through this post-processing procedure, a $\mathrm{Q}4$-$\mathrm{P}4$ finite element for the reservoir is constructed, enabling direct exchange at the dam–reservoir interface.

The temperature of the reservoir water acts as a boundary condition at the dam–reservoir interface, specifying the temperature along the dam surface in direct contact with the water. Therefore, the temperature degree of freedom is also included in the finite element formulation, resulting in a $\mathrm{Q}4$-$\mathrm{P}4$-$\mathrm{T}4$ finite element; however, temperature values at all nodes are assigned as predefined input parameters representing boundary conditions.

3. Digital Twin: Salakovac Dam

In this section, we seek to present different steps in constructing the digital twin for the chosen structure of the concrete gravity Salakovac Dam. First, we present the measurements of long-term loading effects and then the matching numerical model results used to complement the (rather) scarce measurements by adding physics-based results that can enhance the digital twin predictive abilities. We then illustrate the use of such a digital twin in predicting the failure of an ageing dam.

3.1. Geometry and Measurements of Long-Term Loading Effects

Salakovac Dam is located on the Neretva River, near the city of Mostar in Bosnia and Herzegovina. The construction of the dam began in early 1977 and was completed in December 1981, with the dam becoming fully operational in 1982. Since then, the dam has been consistently operating at full capacity. Salakovac Dam, Grabovica Dam, Jablanica Dam, and Mostar Dam form a chain of hydropower facilities on the Neretva River that significantly contribute to the electrical energy production in the country. The installed capacity of the Salakovac Dam is $208.5\mathrm{MW}$, with an average annual energy production of $356.6\mathrm{GWh}$.

The Salakovac Dam is a large concrete gravity dam with a maximum height of $70\mathrm{m}$. The crest length is $230.5\mathrm{m}$ and is positioned at an elevation of $127\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$. The reservoir level is relatively stable, with the maximum level at $124.7\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$, the normal level at $123\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$, and the minimum level at $118.5\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$. The Salakovac Dam is composed of 17 blocks, varying in length from 6 to $22.5\mathrm{m}$. The longitudinal view of the Salakovac Dam is depicted in Figure 3. The dam reaches its maximum height of $70\mathrm{m}$ in block 8, which has a length of $15\mathrm{m}$. The cross-section at the maximum height of the dam is shown in Figure 4.

Since the construction of the Salakovac Dam dates back to 1977, rehabilitation and modernisation of the monitoring system was undertaken in 2012, which included the installation of new measurement equipment. The monitoring system in Salakovac Dam includes thermometers for monitoring the concrete temperature; one thermometer for monitoring the water temperature in the reservoir installed upstream at an elevation of $102.33\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$; one thermometer for monitoring the air temperature; and automatic coordimeters, along with alignment points, for monitoring the horizontal displacements of the dam. The air temperature, water temperature, concrete temperature, and horizontal displacement of the dam at the coordimeters are recorded automatically every two hours, whereas the horizontal displacements of the dam at the alignment points are measured manually once a month. The positions of the thermometers for monitoring concrete temperature (T), automatic coordimeter (C), and the alignment point (A) in the cross-section of block 8 are shown in Figure 4.

For the subsequent numerical analyses, we use monitoring data from the five-year period from 1 January 2015 to 31 December 2019. The end date was selected due to frequent failures in measurement equipment starting in 2020, which led to significant gaps in the available data that would have to be interpolated, thereby introducing additional uncertainties. The data from 1 January 2014 to 31 December 2014 are used to determine the initial temperature distribution in the dam. The monitored temperature data are processed and used in the form of daily averages.

3.2. Numerical Simulations Providing the Corresponding ’Measurements’

The geometry of the Salakovac dam allows for the simplification of the numerical model by assuming plane strain conditions. Following this assumption, the problem can be treated as two-dimensional, reducing computational complexity and time while still capturing the key aspects of the dam’s behaviour. This approach is commonly used for concrete gravity dams, which are typically analysed using two-dimensional numerical models, in contrast to concrete arch dams that often require three-dimensional analysis. Nevertheless, in certain gravity dam configurations, three-dimensional effects can become significant. These effects can influence both the static and dynamic response of the dam, and in such cases, three-dimensional modelling may be necessary to ensure a more accurate structural assessment [1].

The plane strain numerical model of Salakovac Dam is built using the geometry of the cross-section at the maximum height of the dam, specifically through block 8 (Figure 4). Due to the relatively small changes in the reservoir level, the reservoir level in the numerical model is kept fixed at $123\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$. The finite element mesh of the Salakovac Dam is shown in Figure 5.

In order to completely define the numerical model, boundary conditions must be specified for the displacement, pore pressure, and temperature fields. At the base of the dam structure and the reservoir, displacements are constrained in both horizontal and vertical directions, while along the left edge of the reservoir, displacements are fixed in the horizontal direction. A zero-pore-pressure boundary condition is imposed on the exposed surfaces of the dam structure and the top surface of the reservoir. However, defining boundary conditions for heat flow remains a significant challenge in establishing a predictive numerical model of the Salakovac Dam. Proper specification of these conditions is essential, as they directly affect the temperature distribution within the dam body and consequently the overall structural response of the dam. These conditions must be carefully specified at the dam–reservoir interface, the dam–foundation interface, and the exposed surfaces of the dam, as these are sources of uncertainty that can significantly influence the reliability of the numerical model.

3.2.1. Dam–Reservoir Interface Temperature Measurements

At the dam–reservoir interface, the reservoir water temperature serves as a boundary condition. However, the accurate representation of the water temperature profile with depth is limited due to only a single thermometer being available for water temperature measurements, located upstream at an elevation of $102.33\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$. This lack of measurement data necessitates the use of alternative estimation methods to approximate the temperature profile. To address this, the water temperature profile was estimated by relying on the method proposed by Bofang [39], supplemented with parameter fitting. According to this method, the variation in water temperature with depth is defined by the following expression:

$$T(y,t)=T_m\left(y\right)+A\left(y\right)cos(\omega ·(t-t_0-\epsilon \left(y\right)))$$

(21)

where $y\left[\mathrm{m}\right]$ is the depth, $t\left[\mathrm{day}\right]$ is the time, $T_m[°\mathrm{C}]$ is the mean annual temperature of the water, $A[°\mathrm{C}]$ is the amplitude of the annual variations in the water temperature, $\omega ={\displaystyle \frac{2\pi }{365\left(366\right)}}$ is the frequency of temperature variations, $t_0\left[\mathrm{day}\right]$ is the time of the maximum air temperature, and $\epsilon \left[\mathrm{day}\right]$ is the phase difference of the maximum water and air temperatures.

The mean annual temperature of the water $T_m$, the amplitude of the annual variations in the water temperature A, and the phase difference of the maximum water and air temperatures $\epsilon$ are given by the following expressions:

$$T_m\left(y\right)=a_1+(T_s-a_1)e^{a_{2}y}$$

(22)

$$A\left(y\right)=a_3·e^{a_{4}y}$$

(23)

$$\epsilon \left(y\right)=a_5-a_6{e}^{a_{7}y}$$

(24)

where $T_s[°C]$ is the mean annual water temperature at the water surface, and $a_1\dots a_7$ are the unknown parameters that can be obtained through the monitored temperatures.

In order to obtain the unknown parameters, the measurement data from the single available thermometer for water temperature are insufficient. Therefore, these data are supplemented with concrete temperature measurements from the thermometers located closest to the dam–reservoir interface, specifically thermometers $\mathrm{T}1$, $\mathrm{T}4$, and $\mathrm{T}7$. Namely, the mean annual temperature of the concrete and the amplitude of the annual temperature variations in the concrete at these thermometers, due to their proximity to the reservoir, are assumed to be equal to the corresponding values of the water at the same elevation. The mean annual water temperature at the water surface $T_s$ is assumed to correspond to the mean annual concrete temperature at the thermometer $\mathrm{T}1$. Using these data, the values of parameters $a_1\dots a_4$ are determined by applying the least squares method. The unknown parameters $a_5$ and $a_6$ are determined from the known phase difference at the level of the thermometer for water temperature and the assumed phase difference of 3 days at the water surface. The value of parameter $a_7$ is taken as $-0.085$, as suggested by [39]. The monitored air and water temperatures that were used as input parameters were first smoothed using the Robust Locally Estimated Scatterplot Smoothing (RLOESS) method, with a window size of 90 days [40], as shown in Figure 6. The monitored concrete temperatures at thermometers $\mathrm{T}1$, $\mathrm{T}4$, and $\mathrm{T}7$ are shown in Figure 7.

The obtained values of the parameters $a_1\dots a_7$ are provided in

Table 1. Values of unknown parameters $a_1\dots a_7$.

	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$	${\mathit{a}}_7$
2014	$10.780$	$-0.124$	$7.861$	$-0.049$	$24.755$	$21.755$	$-0.085$
2015	$10.622$	$-0.139$	$10.459$	$-0.050$	$41.675$	$38.675$	$-0.085$
2016	$10.618$	$-0.145$	$9.036$	$-0.045$	$28.381$	$25.381$	$-0.085$
2017	$10.580$	$-0.110$	$11.650$	$-0.049$	$30.798$	$27.798$	$-0.085$
2018	$10.691$	$-0.138$	$11.023$	$-0.054$	$22.338$	$19.338$	$-0.085$
2019	$10.445$	$-0.137$	$10.645$	$-0.052$	$29.827$	$26.827$	$-0.085$

. The predicted and monitored water temperatures at the level of the thermometer for the selected period are shown in Figure 8. The smooth transition between the years is achieved through a linear blending. Given the significant lack of measurement data, it can be concluded that a reasonable match between the predicted and monitored water temperatures is achieved. Temperature peaks during the warmer months are more accurately predicted, while those in the colder months are somewhat overestimated. The water temperature profiles with depth at the beginning of each month for the year 2017 are shown in Figure 9.

It is important to note that the previous assumptions are approximate estimates. To more accurately determine the water temperature profile with depth in the reservoir, additional measurements of water temperature at various depths are necessary.

3.2.2. Dam–Foundation Interface Temperature Measurements

At the bottom of the dam, a constant temperature of 9.5 °C is assigned to simulate conditions at the dam–foundation interface. This value is based on the analysis of concrete temperature measurements from thermometer $\mathrm{T}8$, located in the central part of the bottom section of the dam. The recorded temperatures show minimal variation over time, stabilizing around the value of 10.3 °C, indicating that no significant heat inflow or outflow occurs at this interface (Figure 10).

This stabilized temperature represents the thermal equilibrium maintained at this location throughout the dam’s operational period. However, due to the position of this thermometer, a slight increase compared to the dam–foundation interface temperature must be accounted for when defining this boundary condition. This increase results from heat flow through the dam body due to seasonal temperature variations, which have also stabilized over time. By adjusting the dam–foundation interface temperature to align the computed and observed values at the location of $\mathrm{T}8$, a temperature of 9.5 °C at the dam–foundation interface is determined for this boundary condition. To eliminate the temperature difference at the corner node where the dam–reservoir and dam–foundation boundaries intersect, even though it is approximately 1 °C, no temperature boundary condition was applied along a 3 m length from the corner node on the dam–foundation interface, allowing for a smoother temperature transition.

3.2.3. Exposed Surfaces Temperature Measurements

The Salakovac Dam is a structural component of the Salakovac Hydropower Plant, an impoundment-type facility separated by an expansion joint from the rest of the construction up to an elevation of $88\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}$ (Figure 4). Importantly, no heat transfer occurs through the expansion joint, which means the surface up to an elevation of $88\mathrm{m}.\mathrm{a}.\mathrm{s}.\mathrm{l}.$ is treated as thermally insulated, with no boundary conditions applied in that region.

For the remaining exposed surfaces, heat inflow and outflow due to seasonal temperature variations are modelled using convective and radiative boundary conditions, while also accounting for solar radiation.

The convective boundary condition at the exposed surfaces of the dam is defined by Newton’s cooling law as

$$q_c=h_c(T-T_{air})$$

(25)

where $q_c[\mathrm{W}/{\mathrm{m}}^2]$ is the convective heat flux due to the temperature differences of the dam surface and the surrounding air, $h_c[\mathrm{W}/\mathrm{m}°\mathrm{C}]$ is the convective heat transfer coefficient, $T[°\mathrm{C}]$ is the temperature of the dam surface, and $T_{air}[°\mathrm{C}]$ is the temperature of the surrounding air.

The radiative boundary condition is defined with the Stefan–Boltzmann law as

$$q_r=e{C}_s(T^4-T_{air}^4)$$

(26)

where $q_r[\mathrm{W}/{\mathrm{m}}^2]$ is the radiative heat flux due to electromagnetic radiation, e is the emissivity of the surface, and $C_s=5.67·{10}^{-8}[\mathrm{W}/{\mathrm{m}}^2{\mathrm{K}}^4]$ is the Stefan–Boltzmann constant. Since the temperature field in the model is introduced in °C, for radiative boundary conditions where the temperature is raised to the power of 4, a conversion from $\mathrm{K}$ to °C must be carefully handled.

Solar radiation increases the temperature of the exposed surfaces of the dam, which impacts the overall temperature distribution throughout the dam body. The intensity of solar radiation depends on factors such as the time of day, season, latitude, and weather conditions. Accurate modelling of solar radiation is particularly important in relatively thin sections, where solar radiation can have a significant effect on thermal stresses and the overall structural response of the dam. In thicker cross-sections, solar radiation primarily affects the temperature at the surface, with no considerable influence on the interior sections [5].

Due to the lack of measurements that would enable more precise modelling, the effect of solar radiation is accounted for by adjusting the temperature of the surrounding air [1,41]. Considering the location of Salakovac Dam in Jablanica, which experiences a Mediterranean climate, the temperature of the surrounding air is obtained by increasing the monitored air temperatures by 1 °C in winter, 2 °C in spring and autumn, and 3 °C in summer. The increase is applied gradually over a 30-day period. The monitored air temperatures and adjusted temperatures of the surrounding air that serve as an input in the thermal analysis of the Salakovac Dam are shown in Figure 11.

3.3. Digital Twin Construction and Results

3.3.1. Thermal Analysis and Identification of Thermal Parameters: Thermal Conductivity and Specific Heat

Identifying the thermal parameters of the numerical model as part of the digital twin is essential to accurately simulate the effects of seasonal temperature variations on the structural response of the dam. Due to the uncoupled form of the energy equation governing the heat flow, the thermal analysis of Salakovac Dam can be performed independently from other effects. The primary objective of this analysis is to identify the thermal conductivity $k_T$, and the specific heat $c_T$. Given the small range of temperature variations during dam operation, these parameters are considered temperature-independent.

To perform a reliable thermal analysis, the initial temperature distribution in the body of the dam must be known. Since there are no concrete temperature recordings for the entire operational period of the Salakovac Dam, an iterative procedure is employed to establish the initial temperatures. In the first step, a constant temperature of 10 °C is applied to all finite element nodes of the dam. Subsequently, a transient thermal analysis is performed for the period from 1 January 2014 to 31 December 2014, with the previously defined boundary conditions for heat flow. The temperatures obtained on the nodes at the end of this analysis are applied to the model as the new initial temperatures, and the analysis is repeated. This process is iterated multiple times until the computed temperatures on the nodes at the end of two consecutively performed analyses closely align [5]. The criterion adopted in this study for the total number of iterations performed is that the consecutively computed average nodal temperature should not differ by more than $1\%$. The computed temperatures on the nodes at the end of the last iteration are applied as the initial temperatures in the thermal analysis of the Salakovac Dam for the period from 1 January 2015 to 31 December 2019.

The optimal values of $k_T$ and $c_T$ are determined by minimizing the root mean square error (RMSE) between the computed and monitored concrete temperatures at thermometers T1–T9, computed as

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(27)

where $y_i$ is the observed value at point i, ${\widehat{y}}_i$ is the predicted value at point i, and n is the total number of data points (observations).

An optimization procedure is performed on a cubic surface fitted using the RMSE values computed for various combinations of these parameters. Guided by literature recommendations [42,43], and an analysis of the computed RMSE values, constraints were included in the optimization process specifying that $k_T$ should range from $2.5$ to $4.0\mathrm{W}/\mathrm{m}°\mathrm{C}$ while $c_T$ should fall between 840 and $1000\mathrm{J}/\mathrm{kg}°\mathrm{C}$. The optimal values of thermal conductivity and specific heat that minimize RMSE were determined to be $k_T=3.4\mathrm{W}/\mathrm{m}°\mathrm{C}$ and $c_T=840\mathrm{J}/\mathrm{kg}°\mathrm{C}$. The identified thermal parameters result in an RMSE value of 1.03 °C, indicating reliable temperature predictions for the observed temperature range in the dam body, especially when considering the assumptions made in the definition of the boundary conditions for heat flow. Other parameters are selected as follows: mass density $\rho =2400\mathrm{kg}/{\mathrm{m}}^3$, convective heat transfer coefficient $h_c=25\mathrm{W}/\mathrm{m}°\mathrm{C}$, and emissivity $e=0.85$.

A comparison of the computed and monitored concrete temperatures at thermometers T1–T9 is shown in Figure 12. Given the limited amount of monitoring data used to define the boundary conditions for heat flow, a reasonably good match can be observed. The computed temperatures show a slight difference compared to the monitored values, whether in terms of computed amplitudes or the timing of the peaks. These discrepancies are largely attributed to the uncertainties in boundary conditions defined at the dam–reservoir interface and the exposed surfaces. A more accurate prediction of the temperature distribution in the dam body requires a more precise definition of all measurement data to define boundary conditions, which has to be supported by additional monitoring efforts. The computed temperature at thermometer T9 did not capture a monitored oscillation of approximately 1 °C, which is likely due to a local heat source that could not be identified from the available data and drawings. The computed temperature fields for 1 January 2017 and 1 August 2017 are shown in Figure 13.

To assess the mesh dependence of the model, the results are computed for two mesh densities: a coarse mesh with 1288 finite elements and a fine mesh with 3358 elements. As shown in Figure 14, the results are practically the same.

3.3.2. Thermo-Mechanical Analysis and Identification of Thermal Expansion Coefficient

Based on the measurement data, and taking into account that no visible cracks have been observed during the inspection of the dam, it can be assumed that the behaviour of the dam remains linear elastic under long-term loads, including self-weight, reservoir pressure, and seasonal temperature variations. Consequently, the thermal expansion coefficient of the Timoshenko beam finite element ${\overline{\beta }}_s$ is identified by comparing the computed and observed horizontal displacements recorded with the automatic coordimeter at point $\mathrm{C}1$ over the selected period from 2015 to 2019, under the assumption of linear elastic behaviour.

The monitored horizontal displacements are primarily influenced by reservoir pressure and seasonal temperature variations, with additional contributions from factors such as time-dependent deformations and foundation movements. To ensure a consistent parameter estimation process, only the horizontal displacements due to the temperature variations at point $\mathrm{C}1$ are considered for comparison. To achieve this, the thermal component of the monitored horizontal displacements is first extracted. This is done using a statistical model, as shown in [44]. Specifically, the statistical model predicts horizontal displacements u as a combination of three components: the first accounts for temperature changes $u^t$, the second is influenced by reservoir pressure $u^r$, and the third captures unexpected behaviour $u^o$. The horizontal displacements predicted with the statistical model are obtained as

$$\begin{array}{cc}\hfill u& =u^t+u^r+u^o\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill u^t& =a_{1}cos\left({\displaystyle \frac{2\pi t}{365}}\right)+a_{2}sin\left({\displaystyle \frac{2\pi t}{365}}\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill u^r& =b_1+b_{2}H+b_3{H}^2\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill u^o& =c_{1}t+c_2{t}^2\hfill \end{array}$$

(31)

where $t\left[day\right]$ is the time, and $H\left[m\right]$ is the reservoir level. The unknown parameters $a_1,a_2$, $b_1,b_2,b_3$, and $c_1,c_2$ are obtained by minimizing the RMSE between the predicted and monitored horizontal displacements at point $\mathrm{C}1$.

The horizontal displacements at point $\mathrm{C}1$ obtained using a statistical model (Figure 15) result in an RMSE value of $0.62\mathrm{mm}$, suggesting a good fit. These results are next used for the identification of the thermal expansion coefficient ${\overline{\beta }}_s$ in the numerical model.

In the numerical model, due to the slow variation in temperature in concrete caused by seasonal changes, the thermal displacements of the dam are assumed to be primarily governed by thermo-mechanical effects. The thermal displacements of the dam depend on the temperature changes over time, rather than the reference temperature $T_0$. The reference temperature is often called the closing temperature when the final segment is cast and the structure becomes restrained. For older dams, the reference temperature can be assumed to be equal to the mean annual concrete temperature of a sufficiently thick dam [45]. Therefore, the mean annual concrete temperature for the selected years 2015 to 2019 is obtained as a volume-weighted average of the temperatures computed within the finite elements. The computed values are 12.10 °C, 12.01 °C, 11.99 °C, 12.09 °C, and 12.03 °C, respectively. These results show that the computed mean annual concrete temperature does not vary significantly. Thus, a reference temperature of 12 °C is selected while acknowledging its secondary importance in the overall parameter identification analysis.

The coefficient of thermal expansion ${\overline{\beta }}_s$ is obtained by minimizing the RMSE between the computed and statistically obtained horizontal displacements at point $\mathrm{C}1$ caused by temperature variations. The statistically obtained horizontal displacements used in the identification process include both the thermal component and the component that accounts for unexpected behaviour since the thermal component alone does not capture the trend in the mean annual air or water temperature over the selected period (Figure 6). It should be noted that this is a rough estimate, as this part of the displacement over the selected period also accounts for other effects, such as creep or foundation movement, but in this particular case, these effects are considered to have a smaller influence.

To allow for the comparison of relative changes in displacements over time due to temperature variations, the horizontal displacement at the start of the selected period is subtracted from all subsequent values, effectively setting the initial displacement to zero. This adjustment is applied to both the numerically and statistically obtained displacements. The optimal value of ${\overline{\beta }}_s$ is determined to be $6·{10}^{-6}/°\mathrm{C}$, resulting in an RMSE of $1.06\mathrm{mm}$, indicating a moderate fit. The computed results are shown in Figure 16. By analysing the results, it can be concluded that the model predicts well the amplitude for the years 2017–2019. However, for 2015–2016, the computed amplitudes show some discrepancies. A closer examination of the monitored displacements in 2015 reveals an abrupt jump at the beginning of the year, as well as two additional jumps mid-year, indicating a potential error in the measurements. Additionally, the numerically obtained displacements show a slight lag compared to the reference displacements.

It is important to recognize that the computed thermal displacements are directly related to the computed temperature distribution within the dam, which is, in turn, controlled by the extent and quality of monitoring data through its role in the definition of boundary conditions. Additionally, the statistically obtained thermal displacements also contain a certain level of error and the measurements themselves may be affected by external factors. Most importantly, the overall discrepancies in computed total horizontal displacements are not significantly affected any further, as the horizontal displacement component due to reservoir pressure remains relatively constant, as shown in Figure 15 and discussed next.

The displacements of the dam due to seasonal changes are also computed for two mesh densities. A comparison of displacements at point $\mathrm{C}1$ indicates that the computed results remain nearly unaffected by mesh refinement, as shown in Figure 17.

3.3.3. Poro-Mechanical Analysis and Identification of Young’s Modulus of Elasticity

The Young’s modulus of elasticity for the Timoshenko beam finite element E is determined by comparing statistically derived and numerically computed horizontal displacements at point $\mathrm{C}1$ under reservoir hydrostatic pressure. Given that the reservoir level at Salakovac Dam is relatively stable, the horizontal displacements due to reservoir pressure show little variation, as shown in Figure 15, which features the reservoir-induced component of horizontal displacements obtained through a statistical model. As the water level drops, the displacements decrease from the baseline value of $9.71\mathrm{mm}$, which can be considered the steady state under normal conditions. Therefore, Young’s modulus E in the numerical model is determined by matching this displacement considering that the response of the dam under reservoir pressure remains linear elastic and is primarily governed by poro-mechanical coupling. The other parameters are set as follows: Biot’s constant $b=1$, Biot’s modulus $M={10}^6\mathrm{MPa}$, the coefficient of permeability $k={10}^{-10}\mathrm{m}/\mathrm{s}$, and the bulk modulus of the reservoir water $\beta =2070\mathrm{MPa}$.

The computed reservoir pressure and pore pressure fields are shown in Figure 18. The identified value of E is determined to be $18540\mathrm{MPa}$. In analysing this value, it is important to consider that the dam was constructed over 40 years ago, and due to the sustained loads over this time, creep has occurred in the concrete. Since creep is not explicitly included in the numerical model, its approximate effect on the overall dam displacements is commonly accounted for by using a reduced value of the instantaneous Young’s modulus [46]. This adjusted or effective modulus $E_{eff}$ can be expressed as

$$E_{eff}=E_{inst}·(1-r)$$

(32)

where $E_{inst}$ is the instantaneous Young’s modulus and r is the reduction factor representing long-term deformation.

Thus, the Young’s modulus E identified in the previously described manner actually reflects this reduced value, ensuring that long-term deformations caused by sustained loads, such as reservoir hydrostatic pressure, are represented in the model. It is also important to note that during the selected period from 2015 to 2019, a certain amount of creep occurred, and this is covered in the statistical model through the component that accounts for unexpected behaviour. However, due to the relatively short period for significant creep to take place, this component is completely attributed to the thermal part of the displacements, as it is considered to have a much greater influence.

The statistically computed displacements served primarily as a supporting tool for parameter identification. Accordingly, the numerically obtained horizontal displacements due to both reservoir pressure and temperature variations at point $\mathrm{C}1$ are directly compared with measured values and discussed, as shown in Figure 19. In addition to RMSE, which assigns greater weight to larger errors, the mean absolute error (MAE) is also evaluated to quantify the average error, computed as

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(33)

The obtained RMSE and MAE values are $1.40\mathrm{mm}$ and $1.02\mathrm{mm}$, indicating a good overall fit. As evident from Figure 19, the error is primarily driven by the lag in computed displacements, particularly their thermal part, as the numerical model captures the amplitudes and overall seasonal trend well but exhibits a phase shift, causing displacements to respond faster than observed data. These discrepancies are influenced by the model assumptions, which could be further refined with additional monitoring data to strengthen key factors, reduce uncertainties, and enhance the overall reliability of the simulation.

To improve the reliability of the identification process, displacement measurements at additional coordimeter points in the middle section of the dam would have also been beneficial. However, such measurements were not available in this case. The other available measurements of horizontal displacements are at alignment point $\mathrm{A}1$. However, the reliability of the parameter identification process using alignment point $\mathrm{A}1$ may be limited due to measurement errors and the sparse availability of horizontal displacement measurements, which are recorded only once a month. Therefore, the comparison of computed and monitored displacements at point $\mathrm{A}1$ is shown in Figure 20 solely to illustrate the trend in the results.

For other analyses, such as thermal stress analysis or earthquake analysis, it is necessary to know the value of the instantaneous Young’s modulus for the Timoshenko beam. Typically, the effective modulus is derived by reducing the instantaneous modulus by $20\%$ to $40\%$ [1]. Alternatively, the instantaneous value of Young’s modulus can be identified by matching the observed jump in statistically derived displacements corresponding to a known short-term change in reservoir level. In this study, a change in reservoir level of $1.3\mathrm{m}$ is selected as this aligns with the length of the finite element. The corresponding displacement jump to match is approximately $0.55\mathrm{mm}$. The identified value of the instantaneous modulus is determined to be $30340\mathrm{MPa}$, reflecting a $39\%$ reduction, aligning with the upper end of the range suggested in the literature. However, the identification of the instantaneous modulus should be further validated through additional, carefully planned and controlled in situ measurements of displacements due to the reservoir level change.

3.3.4. Lateral Overload as Extreme Load Condition

The numerical model of dam–reservoir interaction, calibrated under operating long-term conditions, serves as the foundation for subsequent analysis of the dam under extreme load scenarios. In addition to providing insight into the current behaviour of the dam, the proposed numerical framework is further used to demonstrate its potential to quantify the remaining structural resistance of the dam beyond standard operating loads. Consequently, this analysis allows Timoshenko beam finite elements to fail in both mode I and mode II, effectively simulating the crack opening and crack sliding failure modes, which, in turn, enables the modelling of localised failure within the dam.

Given that the assumption of linear elastic behaviour for Salakovac Dam is supported by monitoring data and regular inspections, with no visible cracks or other indications of structural damage, the fracture limits and fracture energies for the Timoshenko beam elements cannot be directly identified for this specific dam. Therefore, this analysis is intended solely to illustrate the capability of the model to predict the maximum overload capacity and associated failure mechanisms. Hence, the fracture parameters for Timoshenko beams have been selected to ensure that no cracks develop under self-weight, reservoir pressure and seasonal temperature variations. Specifically, the tensile fracture limit is determined as the threshold at which initial cracks would form under these operational loads. The same limit is applied for shear, while the compressive limit is set to seven times the tensile limit, reflecting a typical strength ratio observed in ageing concrete. The corresponding values are as follows: the ultimate stresses in tension, compression, and shear are ${\sigma }_{f,t}=1\mathrm{MPa}$, ${\sigma }_{f,c}=7\mathrm{MPa}$, and ${\sigma }_{f,s}=1\mathrm{MPa}$, respectively. The fracture energies in tension, compression, and shear are $G_{f,t}=0.002\mathrm{MN}/\mathrm{m}$, $G_{f,c}=0.02\mathrm{MN}/\mathrm{m}$, and $G_{f,s}=0.02\mathrm{MN}/\mathrm{m}$, respectively. The Young’s modulus E is set to 30340 $\mathrm{MPa}$.

The analysis follows a sequential loading program. Initially, the self-weight, reservoir pressure, and a one-year cycle of seasonal temperature variations for the year 2015 are applied to establish the deformed state of the dam under operational loads. Subsequently, horizontal ground acceleration is applied incrementally, inducing inertial forces and hydrodynamic pressures, thereby simulating a progressively increasing lateral overload. This approach serves as a simplified representation of seismic extreme load applied in the spirit of a pushover. Specifically, the analysis assumes progressive inertial loading to determine the critical acceleration capacity of the dam and the associated failure modes. Unlike a classical pushover analysis, this approach does not rely on predefined lateral force distributions. Instead, inertial forces develop naturally according to the mass distribution, while hydrodynamic pressures develop directly through the dam–reservoir interaction. Furthermore, by adopting an inertial loading framework, the load path evolves with the mass and stiffness of the system, allowing crack initiation and propagation to occur without predefined constraints and enabling the simulation of localised failure with efficient convergence rates and computational efforts.

The computed horizontal displacements of the dam crest relative to the applied horizontal ground acceleration are presented in Figure 21a. The critical acceleration capacity of the dam is found to be $1.5\mathrm{g}$. The corresponding lateral load at this critical acceleration is $120\mathrm{MN}$, as shown in Figure 21b. Beyond this threshold, progressive and localised failure initiates within the dam. The results are compared for two mesh densities, showing a slight influence of mesh refinement only on the computed post-peak response.

To analyse the structural performance of the dam, a capacity curve is compared with the Eurocode 8 seismic demand spectrum in ADRS format [47], considering soil category A (rock) and a site-specific peak ground acceleration (PGA) of $0.18\mathrm{g}$ (Figure 21c). The acceleration capacity considerably exceeds the seismic demand, demonstrating a significant safety margin of approximately $3.3$ relative to the limit state. This margin decreases as the seismic demand increases with higher PGA levels; for instance, it reduces to $1.5$ for a PGA of $0.4\mathrm{g}$, suggesting that a stronger seismic event would be required for the dam to reach this critical state. Additionally, Figure 21c includes a line corresponding to the first natural period of the dam–reservoir system, $T=0.257\mathrm{s}$. This line aligns closely with the initial slope of the capacity curve, indicating that the response of the dam is predominantly governed by its first mode of vibration.

This inertial approach triggers the failure mode shown in Figure 22 and Figure 23, where cracking initiates at the heel and the downstream face, propagating inward into the body of the dam. Upon reaching the critical acceleration, a large macro-crack forms, leading to a reduction in the load-bearing capacity of the dam. This type of failure mode is a plausible scenario for concrete gravity dams, as documented by prior studies [48,49,50]. Moreover, it poses a considerable risk to downstream areas due to the loss of water containment, underscoring the importance of more detailed analysis and the efforts to build a digital twin.

The proposed numerical model of the dam–reservoir interaction offers a unified approach to solving the dam–reservoir interface, allowing for a straightforward computation of hydrostatic and hydrodynamic pressures exerted on the dam due to ground acceleration [51]. Figure 24 presents the computed total pressures at the upstream face of the dam under a horizontal ground acceleration of $1\mathrm{g}$, compared with analytical solutions where the hydrodynamic pressure distribution is computed according to Westergaard [21] and Von Kármán [52]. The computed pressure distribution aligns closely with the Von Kármán model, which is derived using the linear momentum balance principle. The Westergaard approach yields higher hydrodynamic pressure values, which are considered a conservative estimate often used for safety considerations. The computed pressure distributions are also compared for two different mesh densities, showing no significant difference in the results.

The reliability of the presented safety analysis in practical applications depends on the accurate identification of fracture parameters, which are essential for capturing realistic failure behaviour. Nevertheless, with proper calibration, the model shows potential in capturing failure thresholds and corresponding failure modes, supporting its use in seismic safety assessments.

4. Conclusions

This study presents a comprehensive numerical analysis of the Salakovac Dam, integrating advanced finite element modelling and monitoring data to construct its digital twin that can evaluate the response of the dam under both long-term operational and transient extreme conditions. For successful predictions, we here account for the influence of all fields: mechanical, thermal, and pore pressure. Namely, the numerical model represents the dam as a nonisothermal saturated porous medium and the reservoir water as an acoustic fluid, utilizing carefully selected finite element approximations that facilitate the seamless transfer of both motion and pressure at the dam–reservoir interface. By relying on the monitoring data, the numerical model of dam–reservoir interaction is calibrated to reflect the observed behaviour of the Salakovac Dam under long-term operating loads, including reservoir pressure and seasonal temperature variations. Despite the limited availability of monitoring data, the obtained error metrics confirm a good overall agreement between the modelled and observed responses, reinforcing the reliability of the model. Furthermore, the ability of the model to evaluate the failure threshold and capture the associated failure mode under increasing lateral loads is also demonstrated, illustrating its potential for safety assessments under earthquake events.

The approach presented in this study can be adapted and applied to other dams, serving as a learning tool for further refinement and enhancement of the proposed approach based on the specific characteristics and requirements of each case. This adaptability ensures that the numerical framework can be continuously improved, making it a versatile tool for assessing current condition and predictive analysis across various dam structures and conditions. For instance, future research will aim to apply the model to analyse the performance of the dam under various representative earthquake ground motions, accounting for cyclic behaviour, high-frequency excitation, and potential resonance effects. Moreover, this study assumes material homogeneity and applies a simplified approach to long-term deformations by reducing the instantaneous Young’s modulus. Future work will aim to enhance the numerical model by incorporating rheological formulations to more accurately capture time-dependent effects such as creep. In addition, it is intended to integrate probabilistic methods to account for uncertainties associated with material heterogeneity and parameter variability in large-scale structures like dams.

Overall, this study provides a robust numerical framework for dam–reservoir interaction modelling for application in structural health monitoring, safety assessment, and predictive analysis of ageing dams. The findings of this study underscore the importance of precise boundary condition data, showing that prediction accuracy is highly dependent on the extent and quality of monitoring. This highlights the necessity of incorporating key numerical modelling requirements into the design and optimization of the monitoring system. By ensuring that the monitoring setup provides a comprehensive and high-quality dataset aligned with numerical modelling needs, the accuracy and reliability of structural assessment can be significantly improved.

In this particular case, it is necessary to at least include measurements of the water temperature at two more points across the depth and displacements of the dam at two additional points in the middle section of the dam, to refine the boundary condition and improve the reliability of the identification process. Furthermore, it would be beneficial to have recordings of concrete surface temperature at certain points, and if possible of the foundation, as well. Additionally, targeted tests should be designed to obtain more accurate data for determining the values of the instantaneous Young’s modulus and fracture parameters. This can be achieved by using comparative material testing based on concrete specimens taken at the time of dam construction and submitted to a comparable loading program in order to quantify the potential long-term degradation, as proposed in our recent works [53,54].