Modeling and Simulation of Flow and Formation Damage of Asphalt-Paved Roads

Abstract

Porous asphalt is a standard asphalt built on aggregate storage bed which allows water to drain through it and reduces stormwater runoff. However, porosity of the porous asphalt and the storage bed may be effectively reduced due to trapping suspended solids from the water or from the asphalt damage. In this paper, we present mathematical modeling and numerical simulation of flow and damage of porous asphalt-paved roads. A mathematical model to describe the fine-particles transport carried by a two-phase flow in a porous medium is presented. The buoyancy, capillarity, and mixed relative permeabilities correlations to fit with the mixed-wet system are considered. Throughout this investigation, we monitor the changing of the fluids properties such as water saturation and solid properties such as porosity and permeability due to trapping the fine-particles.

1. Introduction

The impervious asphalt-paved roads may store significant amounts of thermal energy during summer. The stored thermal energy may be transferred to stream waters during runoff events. Moreover, as a result of fluctuations in ambient air temperatures—diurnal and seasonal, intensity of solar radiation of the asphalt-paved, a significant deformation of asphalt-paved may occur due to the heat and mass transfer between asphalt and water. On the other hand porous (pervious, permeable, or open-graded) asphalt is standard asphalt with reduced sand or fines that allows water to drain through it. Pervious asphalt built on aggregate storage bed reduces stormwater runoff. In addition to reducing runoff, this effectively traps suspended solids and filters pollutants from the water. The stormwater flows through the asphalt to the layer of crushed stone aggregate bedding and base that supports the asphalt while providing storage and runoff treatment. The use of porous asphalt can potentially reduce additional expenditures and land consumption for conventional collection, conveyance, and detention stormwater infrastructure. Compared to the lifetime of dense graded asphalt concrete roads, the lifetime of porous asphalt concrete is less [1]. The loss of stones from the road surface, called raveling, is mostly reported as the dominant defect in porous asphalt wearing course [2, 3]. Raveling is the start of major defects like potholes, because once a stone is gone, the surrounding stones will follow for lacking support in at least one direction [4]. Raveling also has negative influence on the noise reduction function and skid resistance of porous asphalt pavement. During service life, the pores tend to be clogged by dirt, dust, or other clogging agents. The formulation of fine-particles transport in two-phase flow in porous media has been studied experimentally and numerically in [5–7]. Formation damage is a common problem in reservoir development. The particles migrate through the porous media, deposit on the pore surfaces, and become trapped at pore constrictions to reduce the rock porosity and permeability.

Researchers have done many attempts to predict these phenomena experimentally and numerically; however, there is no study that considers the possible porosity and permeability reduction. So, in order to keep porous asphalt and its storage bed efficient, considering these kinds of effects may lead to interesting results. In the current work, we introduce modeling and numerical simulation of fine-particle transport in two-phase flow in porous asphalt-paved roads.

2. Mathematical Modeling

The basic equations that govern the flow of the two-phase flow in porous media are mass conservation equation and constitutive equation (Darcy’s law). The two-dimensional governing equations may be written as𝜕𝜑𝜌_𝑤𝑆_𝑤𝜌𝜕𝑡=−∇⋅_𝑤𝑣_𝑤,𝜕𝜑𝜌_𝑎𝑆_𝑎𝜌𝜕𝑡=−∇⋅_𝑎𝑣_𝑎, (1a)(1b)𝑣_𝑤=−𝐾𝑘_𝑟𝑤𝜇_𝑤∇𝑝_𝑤−𝜌_𝑤𝐠,𝑣_𝑎=−𝐾𝑘_𝑟𝑎𝜇_𝑎∇𝑝_𝑎−𝜌_𝑎𝐠, (2a)(2b)where 𝑆 is the saturation and 𝑣 [m/s] is the velocity. 𝑤 stands for the wetting phase (water), and 𝑎 stands for the nonwetting phase (air). 𝜑 is the porosity of the medium, and ∇=(𝜕/𝜕𝑥,𝜕/𝜕𝑧) is the divergence operator. 𝐾 [m²] is the absolute permeability, 𝑘_𝑟𝛼 [dimensionless] is the relative permeability, 𝜌 [kg m³] is the density, 𝑝 [Pa] is the pressure, and 𝐠=(0,−𝑔)^𝑇 is the gravitational acceleration. 𝜇 is the viscosity. The fluid saturations for the two-phase flow are interrelated by 𝑆_𝑤+𝑆_𝑎=1. (3) The total velocity is 𝑣_𝑡=𝑣_𝑤+𝑣_𝑎. (4) Summing the saturation equation for water phase and the oil phase, one obtains ∇⋅𝑣_𝑡=0. (5) Moreover, adding the constitutive equations for each phase, (1a)–(2b), and substituting into (5), we end up with 𝑣_𝑡=−𝐾𝜆_𝑡(𝑆)∇𝑝_𝑤+𝜆_𝑎(𝑆)∇𝑝_𝑐+𝐾𝜒(𝑆)𝑔, (6) where 𝜆_𝛼(𝑆)=𝑘_𝑟𝛼(𝑆)/𝜇_𝛼 is the mobility, 𝜆_𝑡(𝑆)=𝜆_𝑎(𝑆)+𝜆_𝑤(𝑆) is the total mobility, and 𝜒=𝜆_𝑎(𝑆)𝜌_𝑎+𝜆_𝑤(𝑆)𝜌_𝑤. In order to derive the pressure equation, substitute (6) into (5). We obtain ∇⋅𝜆_𝑡(𝑆)∇𝑝_𝑤+𝜆_𝑎(𝑆)∇𝑝_𝑐(𝑆)−𝜒(𝑆)𝑔=0. (7) Substituting the constitutive equation of the water phase, (2a) and (2b), into (1a) and (1b) gives 𝜕𝜑𝑆_𝑤𝜕𝑡+∇⋅𝑣_𝑤=0. (8) Therefore, water velocity may be written as 𝑣_𝑤=𝑓_𝑎𝑣_𝑡+𝜆_𝑤𝐾𝑓_𝑎∇𝑝_𝑐+𝜆_𝑤𝑓_𝑎𝐾Δ𝜌𝑔, (9) where Δ𝜌=𝜌_𝑤−𝜌_𝑎.

On the other hand, a mathematical model is developed to describe the fine-particles transport carried by two-phase flow in porous media. Assuming that we have a number 𝑚 of size intervals of particles in the water phase, the transport equation for each size interval 𝑖 of the particles in the water phase can be written as 𝜕𝜙𝑆_𝑤𝐶_𝑖𝜕𝑡+𝑣_𝑤⋅∇𝐶_𝑖=∇⋅𝜑𝑆_𝑤𝐷_𝑖∇𝐶_𝑖+𝑅_𝑖, (10) where 𝑖=1,2,…,𝑚. 𝐷_𝑖 is the dispersion coefficients of particles in size interval 𝑖 in the water phase. 𝑅_𝑖 is the net rate of loss of particles in size interval 𝑖 in the water phase. The net rate of loss of particles may be written as [5–7] follows: 𝑅_𝑖=𝜕𝛿𝜙_𝑖,𝜕𝑡 (11) where (𝛿𝜙)_𝑖=𝑣_𝑖+𝑣^∗_𝑖 is the porosity change due to release or retention of particles of interval 𝑖 in the water phase. 𝑣_𝑖 is the volume of the particles of interval size 𝑖 in contact with the water phase available on the pore surfaces per unit bulk volume of sandstone. 𝑣^∗_𝑖 is the volume of the particles of interval size 𝑖 entrapped in pore throats from the water phase per unit bulk volume of sandstone due to plugging and bridging. At the critical velocity of the surface deposition only particle retention occurs while above it retention and entrainment of the particles take place simultaneously [8]. A modified Gruesbeck and Collins’s model for the surface deposition is expressed by [5–7] the following: 𝜕𝑣_𝑖=𝛼𝜕𝑡^𝑑,𝑖𝑣_𝑤𝐶_𝑖when𝑣_𝑤≤𝑣_𝑐,𝛼_𝑑,𝑖𝑣_𝑤𝐶_𝑖−𝛼_𝑒,𝑖𝑣_𝑖𝑣_𝑤−𝑣_𝑐when𝑣_𝑤>𝑣_𝑐, (12)  where 𝛼_𝑑,𝑖 is the rate coefficients for surface retention of the particles in interval 𝑖. 𝛼_𝑒,𝑖 is the rate coefficients for entrainment of the particles in interval 𝑖. 𝑣_𝑐 is the critical velocity. Similarly, the rate of entrapment of the particles in interval 𝑖 is 𝜕𝑣^∗_𝑖𝜕𝑡=𝛼_{𝑝𝑡,𝑖}𝑣_𝑤𝐶_𝑖, (13) where 𝛼_{𝑝𝑡,𝑖} is the pore throat blocking constants. Porosity may be changed because of particles deposition on the pore surfaces or blocking of pore throats. The porosity variation may be expressed by [5–7] the following: 𝜙=𝜙₀−𝛿𝜙_𝑖, (14) where 𝜙₀ is the initial porosity. Also, the permeability variation due to particles deposition on the pore surfaces or blocking of pore throats may be expressed as [5–7] follows: 𝐾=𝐾₀1−𝑓𝑘_𝑓+𝑓𝜙𝜙₀^𝑙, (15) where 𝐾₀ is the initial permeability, 𝑘_𝑓 is a constant for fluid seepage allowed by the plugged pores, 𝑓=1−∑_𝑖𝛼_𝑓,𝑖𝑣^∗_𝑖 is the fraction of the original cross-sectional area open to flow, and the value of the exponent 𝑙 has range from 2.5 to 3.5. For the particles transport carried by fluid stream in the porous media, deposition on pore surfaces and blockage in pore throats may occur. The retained particles on pore surfaces may desorb for hydrodynamic forces and then possibly adsorb on other sites of the pore bodies or get entrapped at other pore throats.

3. Results and Discussion

In order to get physical insights for the problem under consideration, we consider that typical cross-section for pervious pavement system consists of six porous layers (Figure 1) that can be simplified to one-dimensional system for calculations. The porous asphalt layer taken is 15 cm, the choker course layer is 20 cm, the filter course is 30 cm, the filter blanket is 8 cm, the reservoir course is 10 cm, and finally the native material is optional that we consider it 20 cm. We consider the top boundary as inlet of water-particles suspension. So water imbibes into the porous asphalt and replaces the air that leaves the medium from the same boundary. This type of flow in porous media is called countercurrent imbibition [9–12]. In countercurrent imbibition both wetting and nonwetting phases flow through one inflow-outflow boundary. Therefore, the total velocity becomes zero, 𝑣_𝑡=0. Therefore, the flow equations may be written as 𝜕𝑃_𝑤𝜕𝑧=−𝑓_𝑤(𝑆)𝜕𝑃_𝑐(𝑆)(𝑆)𝑣𝜕𝑧−𝜒𝑔,_𝑤=𝐾𝜆_𝑤𝑓_𝑎𝜕𝑃_𝑐.𝜕𝑧+Δ𝜌𝑔 (16) Therefore, 𝜕𝜑𝑆_𝑤𝜕𝜕𝑡+𝐾𝜆𝜕𝑧^𝑤𝑓_𝑎𝜕𝑃_𝑐𝜕𝑧−Δ𝜌𝑔=0. (17) The relative permeabilities are given in terms of the normalized wetting phase saturation 𝑆, given as 𝑘_𝑟𝑤=𝑘⁰_𝑟𝑤𝑆^𝑎,𝑘_𝑟𝑎=𝑘⁰_𝑟𝑎(1−𝑆)^𝑏, (18) where 𝑎 and 𝑏 are positive numbers, 𝑘⁰_𝑟𝑤=𝑘_𝑟𝑤(𝑆=1) is the endpoint relative permeability to wetting phase, and 𝑘⁰_𝑟𝑎=𝑘_𝑟𝑎(𝑆=0) is the endpoint relative permeability to nonwetting phase. The normalized wetting phase saturation is given by 𝑆𝑆=^𝑤−𝑆_𝑖𝑤1−𝑆_𝑟𝑎−𝑆_𝑖𝑤,0≤𝑆≤1, (19) where 𝑆_𝑖𝑤 is the irreducible water saturation and 𝑆_𝑟𝑎 is the residual air saturation.

Figure 1 

Typical cross-section for pervious pavement system.

For the capillary pressure we used the general correlation 𝑝_𝑐=𝑐_𝑤𝑆_𝑤−𝑆_𝑤𝑟1−𝑆_𝑤𝑟^−𝑎𝑤+𝑐_𝑎𝑆_𝑎−𝑆_𝑎𝑟1−𝑆_𝑎𝑟^−𝑎𝑎, (20) where 𝑐_𝑤 and 𝑐_𝑎 are constants that represent the entry pressure for imbibition and drainage, respectively. The constants 1/𝑎_𝑤 and 1/𝑎_𝑎 are the pore size distribution indexex for imbibition and drainage, respectively.

The transport equation for the interval 𝑖 of the particles can be written as 𝜕𝜑𝑆_𝑤𝐶_𝑖𝜕𝑡+𝑣_𝑤𝜕𝐶_𝑖=𝜕𝜕𝑧𝜕𝑧𝜑𝑆_𝑤𝐷_𝑖𝜕𝐶_𝑖𝜕𝑧+𝑅_𝑖,𝑅_𝑖=𝛼_𝑑,𝑖+𝛼_{𝑝𝑡,𝑖}𝑣_𝑤𝐶_𝑖,𝑢_𝑤≤𝑢_𝑐𝛼_𝑑,𝑖+𝛼_{𝑝𝑡,𝑖}𝑣_𝑤𝐶_𝑖−𝛼_𝑒,𝑖𝑣_𝑖𝑣_𝑤−𝑣_𝑐𝐶_𝑖,𝑢_𝑤>𝑢_𝑐. (21) The surface deposition of the particles in the interval 𝑖 is 𝜕𝑣_𝑖=𝛼𝜕𝑡^𝑑,𝑖𝑣_𝑤𝐶_𝑖,𝑣_𝑤≤𝑢_𝑐𝛼_𝑑,𝑖𝑣_𝑤𝐶_𝑖−𝛼_𝑒,𝑖𝑣_𝑖𝑣_𝑤−𝑣_𝑐𝐶_𝑖,𝑣_𝑤>𝑢_𝑐. (22) The rate of entrapment of the particles in interval 𝑖 is 𝜕𝑣^∗_𝑖𝜕𝑡=𝛼_{𝑝𝑡,𝑖}𝑣_𝑤𝐶_𝑖. (23) The initial conditions are 𝑆_𝑤=𝑆_0𝑤,𝐶_𝑖=𝑣_𝑖=𝑣^∗_𝑖=0,𝑡=0,0≤𝑧≤𝐻, (24) where 𝐻 is the depth and 𝑆_0𝑤 is the initial water saturation. The boundary conditions are 𝑆_𝑤=1−𝑆_𝑜𝑟,𝐶_𝑖=𝐶_𝑖,0,𝑣_𝑖=𝑣^∗_𝑖=0,𝑡≥0,𝑧=0,𝑑𝑆_𝑤=𝑑𝑧𝑑𝐶_𝑖=𝑑𝑧𝑑𝑣_𝑖=𝑑𝑧𝑑𝑣^∗_𝑖𝑑𝑧=0,𝑡≥0,𝑧=𝐻, (25) where 𝐶_𝑖,0 is the concentration of particles in the particles suspension at the inlet boundary. The governing equations (17)–(23) are solved numerically along with the initial and boundary conditions, (24)–(25). An efficient algorithm is used to solve the above high-nonlinear parabolic partial differential equation in one space variable 𝑧 and time 𝑡. The Galerkin method is used for spatial discretization [13], while the time integration for the resulting ordinary differential equation is done with an adaptive time step. 100 points of the spatial grid were used during calculations and were enough to provide an acceptable accuracy. Now, we consider one-size particles suspension in the water phase at the inlet, with the following parameter [6, 7], 𝛼_𝑑,𝑖=8 m⁻¹, 𝛼_{𝑝𝑡,𝑖}=17 m⁻¹, 𝛼_𝑒,𝑖=400 m⁻¹, 𝑣_𝑐=4.6×10⁻³ m/s, and 𝐷_𝑖=5.6×10⁻³ m²/s. The inlet particles concentration values are 𝐶_𝑖,0=0.0 (without particles), 0.0005, and 0.01. The remaining model parameters are 𝑆_𝑤𝑟=𝑆_𝑜𝑟=0.001, 𝑎_𝑤=𝑎_𝑎=0.5, 𝑐_𝑤=500, 𝑐_𝑎=−500, 𝜑₀=0.2, 𝑙=3, 𝑘_𝑓=0.015, 𝛾_𝑓=0.01, 𝑘_𝑟𝑤0=𝑘_𝑟𝑎0=1, and 𝑎=𝑏=4.

Figure 2 shows the water saturation against the dimensionless distance with various imbibition times and inlet particles concentrations. It is notable that the concentration of particles increases the water saturation in particular after significant time of imbibition. Moreover, the saturation of water increases with the time imbibition.

Figure 2 

Water saturation against the height with various imbibition times and inlet concentrations.

The normalized particles concentration is plotted in Figure 3 against the dimensionless distance with various inlet concentrations and imbibition times. From this figure it can be seen that the concentration of particles increases as the concentration at the inlet increases for positive 𝑅_𝑖. An interesting phenomenon can be observed from Figure 3 that the concentration increases with time but after a certain time of imbibition concentration of particles start to decrease with time. This may be interpreted by increasing the rate of particles precipitation on the pore surface of the porous medium. The opposite is true for minus 𝑅_𝑖. It is noteworthy that, in Figure 3, the dimensionless concentration goes larger than 1 because the concentration of particles increases by erosion of more particles from the medium (−𝑅_𝑖). On the other hand, the reference concentration was the inlet concentration of the particles coming with water from outside.

Figure 3 

Normalized particles concentration against height with positive/minus 𝑅𝑖 and various imbibition times.

Figures 4 and 5 illustrate the ratios of the permeability and porosity against the dimensionless distance with various imbibition times, respectively. Both the permeability and the porosity are reduced due to the precipitation of the particles on the pores walls. It is interesting to note that the reduction rates of the permeability and porosity depend on the layer permeability of the medium.

Figure 4 

Permeability ratio against the height with various imbibition times.

Figure 5 

Porosity ratio against the height with various imbibition times.

4. Conclusions

In this pape, we presented numerical modeling and simulation of particle transport in two-phase flow in porous asphalt-paved roads. Numerical experiments have been performed to explore these phenomena and to study the possible porosity and permeability variations. We found that both the permeability and the porosity are reduced due to the precipitation of the particles on the pores walls. These results may help engineers to keep porous asphalt and its storage bed efficient by looking for cleaning methodologies to avoid media blocking.

Acknowledgment

The authors would like to thank the Institute of Scientific Research and Revival of Islamic Heritage, Umm Al-Qura University, for supporting the Project no. 43208015.