## 1 Introduction  
Several permeable friction courses (PFCs) have been widely used in permeable pavement systems to control rainwater quantity and quality during urbanization [1; 2; 3]. A PFC is a porous asphalt layer that is laid on the surface layer of a conventional impermeable asphalt pavement [4]. Porous asphalt is an asphalt material that is designed to have a high porosity so that water can infiltrate its pores and drain out vertically, laterally, or both. Hence, a PFC is used as a tool to drain out water and has the potential to reduce the peak flow of pollutants in rainwater [5; 6; 7]. Literature studies have reported that the properties of porous asphalt material (such as porosity and permeability) have a strong effect on the drainage of a PFC. In addition, the geometric design of the pavement, such as the longitudinal slope (\(S_{y}\)), cross slope (\(S_{x}\)), length (\(L\)), width (\(W\)), and thickness (\(T\)), also has remarkable drainage effects [8; 9]. When using PFCs for stormwater management, the rainfall intensity (\(I\)) is an important factor that should be considered. To date, several studies have investigated the effects of these factors on the drainage of a PFC pavement. To achieve this, three methods have been used: the rainfall simulator experiment, laboratory experiment, and finite element analysis. The finite element analysis is a key method for observation because of its obvious benefits such as saving cost, saving time, and the high accuracy of the results [10]. [PERSON] et al. [8] utilized a finite element program called Seep3D to observe the effect of pavement geometric design, including cross slope, longitudinal slope, thickness, width (\(S_{x}\), \(S_{y}\), \(T\), and \(W\)), and rainfall intensity (\(I\)), on the drainage performance of a pavement. In their study, the permeability (\(k\)) of the porous asphalt was assumed to be 20 mm/s. The allowable rainfall intensity, which was the maximum rainfall intensity that could be applied over the pavement without surface ponding, was extracted according to the different values of the pavement geometric design. From the results of the study, a curve consisting of the ratio of the pavement thickness to the width (\(T/W\)) and the maximum allowable rainfall intensity was plotted. The outcome indicated that at a high value of cross slope (\(S_{x}\)), the longitudinal slope (\(S_{y}\)) did not seem to have any effect on the drainage of the pavement. In addition, the study found that a higher ratio of thickness to width (\(T/W\)) resulted in a lower effect on the longitudinal slope (\(S_{y}\)) of the drainage performance. In another study, [PERSON] et al. [11] proposed a computational finite element model to predict the flow of water in a PFC during rainfall. The flow of water in the body of the PFC was estimated by a two-dimensional flow using the equation of diffusion, whereas the flow of water over the surface of the PFC was estimated by a three-dimensional flow using the equation of Richard. The study highlighted that the thickness (\(T\)) of the PFC pavement played an important role on the effect of the time for surface ponding. However, the thickness (\(T\)) of the PFC had a less significant effect on the phreatic line in its body. In the permeable pavement design guide, the time at which surface ponding is initiated, known as the time for surface ponding (\(t_{p}\)), is a key factor in evaluating the peak flow of drainage in urban areas [12; 13]. In low impact development techniques, \(t_{p}\) is called the delay time of peak flow, one of the typical factors for permeable pavement system design [14]. For transportation construction, it is also a critical factor for safety movements [15]. To date, there have not been many studies focused on the time for surface ponding of a PFC subjected to rainfall by finite element analysis. [PERSON] et al. [15] investigated the time for surface ponding of a PFC using a finite element analysis program called Flow 3D.