^{1}

^{1}

^{1}

^{2}

^{3}

^{2}

^{3}

^{3}

^{1}

^{2}

^{3}

This paper presents a modular hydromechanical approach to assess the short- and long-term surface drainage behavior of arbitrarily deformable asphalt pavements. The modular approach consists of three steps. In the first step, the experimental characterization of the thermomechanical asphalt material behavior is performed. In the second step, information about the long-term material behavior of the asphalt mixtures is integrated on the structural scale via a finite element (FE) tire-pavement model for steady-state rolling conditions and time homogenization in order to achieve a computationally efficient long-term prediction of inelastic deformations of the pavement surface (rut formation). In the third step, information regarding the current pavement geometry (deformed pavement surface) is used to carry out a surface drainage analysis to predict, e.g., the thickness of the water film or the water depth in the pavement ruts as a function of several influencing quantities. For chosen numerical examples, the influence of road geometry (cross and longitudinal slope), road surface (mean texture depth and state of rut deformation), and rainfall properties (rain intensity and duration) on the pavement surface drainage capacity is assessed. These parameters are strongly interrelated, and general statements are not easy to find. Certain trends, however, have been identified and are discussed.

The ultimate goal of pavement design is to provide an optimal structure that meets the required service life while ensuring safety and minimizing costs and environmental impacts.

One major problem is the rapid increase in the volume of traffic with growing axle loads, which leads to amplified fatigue mechanisms and the premature failure of pavement structures. A further effect is intensified rut formation [

In this context, an increasing demand on design asphalt mixtures with a high resistance to permanent deformation at high temperatures is observable. To optimize the design of these materials, detailed analyses of the inelastic deformation are required. Hence, the results of laboratory tests can be used to determine the constitutive material relations between dynamic variables (i.e., stresses and forces) and kinematic quantities (i.e., strains and displacements). The most commonly used laboratory tests are the static creep-recovery test and the cyclic uniaxial compression (CUCT) test. Creep-relaxation tests are popular because of their simplicity. However, some authors have reported that the results do not correlate well with actual in-service pavement rutting [

By using the results of appropriate laboratory tests, numerical modeling techniques provide additional means for predicting the rutting behavior of asphalt pavements on the structural scale. These simulations employ material parameters which are obtained via the results of laboratory tests on the material scale. Material tests are also essential for the calibration and validation of any kind of numerical model in order to obtain realistic and reliable simulation results. Compared to the outcome of rutting tests on a laboratory scale, an advantage of modeling ruts (in combination with real tire models) is also that a more realistic geometrical form of the rut can be determined. Numerical models for rutting with different methodological approaches can be found in the literature (e.g., [

The combination of laboratory tests and numerical modeling is a promising way to make predictions regarding pavement rutting. The rut depth itself can be one of the results. However, further conclusions for the pavement performance can be drawn by coupling the results of the laboratory tests and the structural simulation with a model which deals with the surface drainage of the pavement.

Various methods to calculate or model pavement surface drainage exist in the literature. The still widely used Gallaway et al.’s equation was developed from empirical relationships and experimental data [

Furthermore, German guidelines use the simple value of rut depth (considering the specific cross slope situation as well) to describe the risk of aquaplaning. With a better knowledge of the surface drainage mechanism, this assessment could be improved. A 2D surface drainage model can provide additional information about the risk of critical water depths in the ruts leading to aquaplaning. Other important parameters, such as longitudinal slope and the geometrical form of the ruts—in interaction with the cross slope—can be integrated into the surface drainage model and improve information on the risk of aquaplaning in a specific case. Furthermore, it is possible to consider the effects of the surface texture in and next to the ruts or even their development over time. In this way, much more information about (future) surface drainage behavior can be determined than by merely using the value of rut depth to describe the risk of aquaplaning.

In this contribution, it is exemplarily shown that the experimental and numerical analysis of geometrical changes (deformations in the form of ruts) of arbitrary asphalt pavements can be combined and coupled with an analysis of functional properties addressing road users’ needs regarding safety and comfort. Therefore, a modular hydromechanical approach consisting of three steps (experimental material testing, long-term structural deformation finite element (FE) modeling, and surface drainage modeling) is proposed. This approach is also one of the first steps towards a more comprehensive, future vision of long-term pavement performance and its prediction with the help of in situ monitoring, data collection, data management, and modeling techniques—often called a “digital twin” in the context of industrial products or processes.

The present approach consists of several unidirectionally coupled modules (material, structure, and drainage) to describe the long-term behavior of arbitrary asphalt pavements from the material scale to the structural scale and to derive the surface drainage features over the long term. A graphical overview of the methodologies is provided in Figure

Overview of the modular unidirectional hydromechanical investigation for deformable asphalt pavements subjected to rut formation during their service life: pavement surface runoff model (PSRM) and arbitrary Lagrangian–Eulerian finite element method (ALE FEM).

The flowchart of the modular analysis is depicted in Figure

Flowchart of the modular analysis.

In the following subsections, more information on each module is provided, with a focus on the geometry-dependent model for surface drainage in this contribution.

The objective of the experimental material characterization is to reveal the material’s short- and long-term response to cyclic loading. Particularly for the prediction of plastic deformations, the quantification of accumulated inelastic deformations is important. The existence of different testing facilities and the general need of a test methodology for accurate inelastic strain measurements during the experimental testing of asphalt materials have already been briefly discussed in the introduction. To assess the thermomechanical material behavior of asphalt mixtures, i.e., to also take into account the influence of temperature, the short- and long-term tests are performed at different testing temperatures. The outcome of these force-controlled tests (strain-time plots) is then used to identify sets of material model parameters for each testing temperature. Besides the temperature-dependent mechanical response of the material, other thermal characteristic values such as heat capacity, thermal conductivity, and mass density also have to be identified from experiments. These additional quantities are used as the input parameters of the thermomechanical tire-pavement model (see [

The experimentally assessed asphalt behavior is numerically represented via a continuum mechanical model for asphalt [

The temperature dependency of the asphalt is captured via parametrization via sets of model parameters identified for discrete testing temperatures (see Section

To integrate and evaluate the short- and long-term material response on the structural scale (pavement structure), different numerical methods are available. Often, the FEM is used to carry out numerical simulations of pavement structures (see, e.g., [

In this contribution, an FE discretized configuration of tire and pavement is considered (see Figure

FE tire-pavement model for the structural analysis of the tire-pavement interaction using an ALE framework for both tire and pavement.

For a numerically efficient but still detailed analysis of the tire-pavement interaction, a stationary state of the tire in motion is numerically represented with the help of an ALE steady-state transport simulation. The idea of the steady-state analysis is based on the special description and decomposition of the motion of the tire and the pavement with respect to several configurations (initial configuration with Lagrangian frame

Due to the steady-state motion of tire and pavement, time derivatives (of quantities belonging to the current configuration) with respect to the moving reference configuration are redundant and the computational cost can be significantly decreased. In this context, the rotation of the tire is represented by the flow of the tire material through the fixed FE mesh of the tire, and no fine FE resolution of the whole circumferential direction of the tire is required (see Figure

The long-term prediction of the pavement’s rutting performance requires the use of a numerically efficient strategy. In this context, time homogenization—see, e.g., [

Information regarding the geometry of the deformed pavement surface (cross-sectional direction) as a function of the number of load cycles (tire overruns) are then transferred to the surface drainage analysis.

In the past, models for the surface drainage of pavements have often been developed solely from experimental data and were not based on physical modeling. Descriptive parameters or equations were derived from experimental results to gain a reduced model of pavement drainage, as in [

In this contribution, a 2D numerical model called the Pavement Surface Runoff Model (PSRM), developed at the University of Stuttgart [

The PSRM was designed to model pavement surface runoff and can, therefore, only simulate infiltration processes into porous pavements in a simplified way. A number of models dealing with flow and drainage in porous pavements have previously been proposed (e.g., [

The numerical calculations of the PSRM, which are based on fundamental hydromechanical modeling, use the DuMu^{X} software toolbox [

In the PSRM, standard geometries (e.g., superelevated transition sections with different cross slopes and longitudinal slopes or different pavement widths) can be generated for ideal even surfaces. More realistic and complex (uneven) topographies and geometries of pavement surfaces can be imported using 3D surface data. This data can either be measured or—as done in the present modular hydromechanical investigation—calculated based on a structural tire-pavement model (

In the PSRM, rainfall is simplified to a spatially uniform event, which is constant in time, as it is also implemented in a simplified way in other (pavement) drainage models. Furthermore, the rainfall intensity

The Navier–Stokes equations are a set of fundamental equations concerning fluid mechanics and characterize the conservation of mass, momentum, and energy in hydrosystems with a set of nonlinear partial differential equations.

If water is treated as an incompressible fluid and its motion as a temperature-independent flow, the conservation of energy can be neglected. For laminar flow conditions in combination with an ideal frictionless fluid and vertically acting gravity, the so-called incompressible Euler equations can be derived from the Navier–Stokes equations.

It is assumed that laminar flow is the only drawback of the PSRM compared to surface drainage situations in reality. In measurements, water flow has been observed in the full range between laminar and fully turbulent conditions (see [

In pavement surface runoff, the horizontal spread of the water film is much larger than the water depth. For cases like these, the depth-averaged shallow water equations are ideal [

Integrating the incompressible Euler equations over the flow depth, the depth-averaged shallow water equations in their general form are obtained for a simplified model of the pavement surface (see Figure

Model of water flow with velocity

So far, (

The equation has to be further adapted for the modeling of pavement surface runoff. In surface drainage, the discharge

Furthermore, assuming a smooth pavement surface is not practical. Flow resistance has a large effect on flow velocities. In free flow cases, the roughness of the surface acts as shear stress against the gravitational force and decelerates the flow [

Assuming a steady state, an expression for the slope friction

For this case, the Darcy–Weisbach coefficient

Equation (

The inundation ratio

The coefficient

The free parameters are identified via the least squares method [

The experiments [^{2}) of three different asphalt types (SMA 11, SMA 8, and a mastic asphalt) and two concrete surfaces obtained by different texturing methods. The variation of MTD is between 0.4 mm and 1.8 mm. The flow conditions in the experiments can be assumed as 1D (no cross slope), with a flow path length of 2.5 m. An inflow construction at the top of the pavement sample ensures steady-state flow conditions. In the experiments, a variation of longitudinal slopes and runoff rates (simulating different rain intensities at certain flow path lengths) are tested. Water film thicknesses are measured using an ultrasonic device, which measures distances on the dry surfaces and on the surfaces with water flow, respectively. The measured difference between the distances, repeated at 27 fixed points on the surface, results in an average water depth. The experiments are described in more detail in [

More experimental data might improve the accuracy and applicability of the model. Although the model is only valid and applicable for five discrete values of MTD, these are, in fact, within a sufficiently wide range compared to realistic pavements.

Including the discharge

The PSRM uses the finite volume method (FVM) on a Cartesian grid to gain numerical solutions to the previously derived shallow water equations. A 2D grid is used and the conserved variables are stored as cell-averaged values at the center of each cell. The pavement is implemented as an impervious boundary, and no-flow conditions are imposed on the right and left sides of the model domain.

The numerical approach was implemented in DuMu^{X} (as explained in Section

Most drainage models for porous (e.g., [

In the following, the surface drainage model (Section

To demonstrate the fundamental features of the hydromechanical approach, the simple but illustrative example of a tire running on an artificial test section (thin asphalt layer on deformable subbase material), as sketched in Figure

Geometry and FE discretization of the benchmark example (cross section of the near field: thin asphalt layer on subbase material of the test track, both with elastoplastic material behavior) and FE model of the test tire [

As a result of the structural analysis, the permanent surface deformation (given in Figure

Deformed surface of the test track (cross-sectional direction) plotted for a global time of 5 years, 10 years, 20 years, and 30 years (number of load cycles = about 7·10^{7} per year) [

Information regarding the permanent surface deformation (2D) of a representative part of the cross section subjected to rut formation is used to generate a 3D geometrical model of the road surface, including the longitudinal and cross slopes via inter- and extrapolation of the geometrical data. Figure

Generated deformed pavement surface. (a) Unscaled. (b) 10x vertically scaled.

For the numerical examples of surface drainage, a variation of rain intensities

The chosen rain intensities represent (short) heavy rainfall events (e.g., “showers” during a thunderstorm) as they usually appear in Germany. The intensities of heavy rainfall events vary widely within Germany. Therefore, the determination of an “average” value for a certain duration of heavy rainfall events has a great uncertainty; the rain intensities which are chosen in the following examples have been varied accordingly. Statistical analyses of rain intensities in Germany [

Table

Simulation parameters for numerical examples of surface drainage (influencing quantities highlighted in bold).

Example | Duration (s) | MTD (mm) | ||||
---|---|---|---|---|---|---|

1 | 0.75 | 900 | 0.4 | 30 | ||

2 | 900 | 0.4 | 2.5 | 1.0 | 30 | |

3 | 0.75 | 0.4 | 2.5 | 2.0 | 30 | |

4 | 0.75 | 900 | 0.4 | 2.5 | 2.0 | |

5 | 0.75 | 900 | 30 |

The simulation results show a 2D distribution of water film depths (WFD) in different colors plotted over the pavement surface. The overall water depth is related to the texture depth (MTD), so the resulting water depth given in the figures represents the water height above the texture peaks, which is assumed to be relevant to aquaplaning aspects.

Effects of cross slope and longitudinal slope in combination with rut depth

Rain intensity of 0.75 mm/min with a duration of 15 min and an MTD value of 0.4 mm were chosen for the simulation. The cross slope

The results in Figure

The bulges at the edge of the ruts result in a backwater effect, which occurs next to the (higher edges of the) ruts in all cases. This effect is more significant with cross slopes with a lower gradient, and thus higher water depths appear here. The effect of the longitudinal slopes on the backwater effect is ambiguous. The area of increased water depths is smaller with a cross slope of

Example

Comparison of different rain intensities

For this example, rain intensity is set to 0.75 mm/min and 2.0 mm/min, respectively, with a duration of 15 min in each case. An MTD value of 0.4 mm was chosen. The cross slope

In Figure

Example

Water depths during heavy rainfall

The time required to reach a critical depth of water in the ruts in the case of a heavy rainfall is investigated in this example. As in Example

The simulations are based on a rain intensity of 0.75 mm/min with a duration of 15 min, an MTD value of 0.4 mm, a cross slope of

Critical states regarding aquaplaning tend to be reached earlier within the ruts than outside of the ruts. With the help of calculations, as shown in Figure

It can also be seen that critical values of water depth seem to appear in the ruts earlier than outside the ruts.

Example

Development of drainage with increasing rutting during the pavement’s service life

In this example, simulations have been carried out for a rain intensity of 0.75 mm/min with a duration of 15 min, an MTD value of 0.4 mm,

In Figure

Example

Effects of texture on the drainage of a pavement surface with ruts

In this example, the calculations are carried out with a rain intensity of 0.75 mm/min with a duration of 15 min. Two MTD values with different cross slopes

Different roughness properties of pavement surfaces—numerically expressed via the MTD—have a significant effect on surface drainage. In general, higher MTD values offer a certain water drainage capability in texture cavities, but also increase the flow resistance. Both effects coexist, but one of them can be predominant in different geometric situations (e.g., depending on the slope or flow path length) and, in consequence, can have a major influence on water depth. This contradictory influence of surface roughness on pavement drainage has not yet been sufficiently discussed either qualitatively or quantitatively.

These different (contradictory) effects can be seen in the results depicted in Figure

Example

This paper presents a modular hydromechanical approach to assess the short- and long-term surface drainage behavior of arbitrary, deformable asphalt pavements.

Firstly, the results of the laboratory tests of the materials used in the pavement provide information about the material’s mechanical short- and long-term characteristics at different temperatures.

Secondly, an FE model dealing with tire-pavement interaction is used to predict the long-term permanent deformation performance of arbitrary asphalt pavements on the structural scale. The model assumes steady-state rolling conditions in order to carry out computationally efficient analyses. In addition, the efficiency is increased by computing the long-term structural response using a time homogenization technique. Based on material parameters from experimental material characterization tests, the structural model provides rut depths and the rut geometry for arbitrary tire-pavement configurations as a function of the number of load cycles.

Thirdly, the results of the long-term pavement modeling (rut depths and rut geometry as a function of the load cycles) are used in a unidirectional coupling to predict the surface drainage behavior of the rutted pavement surface, especially the water depth within the ruts. The basic hydromechanical modeling is based on depth-averaged shallow water equations. This modeling technique can consider ruts (and uneven areas in general) in a pavement because of its 2D runoff simulation. Modeling water runoff instead of using the rut depth as a benchmark for the risk of aquaplaning offers an additional benefit. Important pavement design and surface parameters can be included in the surface drainage model, e.g., cross slope, longitudinal slope, or even transition areas, which have a major influence on the surface drainage behavior. It can be shown that the surface texture (represented by MTD values) also affects the runoff behavior and the resulting water depths. Varying these parameters enables the pavement’s drainage performance and related aquaplaning risks to be assessed. Thus, a prediction of the functional property of surface drainage as well as its implications on road safety could be improved by the present modular coupling, which delivers more precise results by including information about material and simulated surface geometries as a function of the pavement’s service life.

Regarding this interdisciplinary hydromechanical modular approach, it is obvious that the combination and coupling of experimental and numerical methods (material, structure, and drainage) can fundamentally improve prediction methods for the long-term, multiphysical behavior of asphalt pavements and their performance. Future research work will concentrate on systematic parameter analyses using the hydromechanical modular approach for application-oriented studies regarding varying material characteristics, existing pavement construction types as well as varying traffic, wheel loads, pavement slopes, etc.

Parts of the raw data supporting the modular analysis are from previously reported studies and datasets, which have been cited in the text. The processed drainage data are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) under Grants KA 1163/30, RE 1620/4, WE 1642/11, and LE 3649/2 within the DFG Research Group FOR 2089.

^{X}: A multi-scale multi-physics toolbox for flow and transport processes in porous media