This study provides experimental evidence of Forchheimer flow and the transition
between different flow regimes from the perspective of the pore size of
permeable stone. We first carry out seepage experiments on four
kinds of permeable stones with mesh sizes of 24, 46, 60 and 80, corresponding to mean particle sizes (50 % by weight)
of 0.71, 0.36, 0.25 and 0.18 mm, respectively. The seepage experiments show that an
obvious deviation from Darcy flow regime is visible. In addition, the
critical specific discharge corresponding to the transition between flow regimes
(from pre-Darcy to post-Darcy) increases with increasing particle
size. When the “pseudo” hydraulic conductivity (

Darcy (1857) conducted a steady-state flow experiment in
porous media and concluded that the specific discharge was proportional to the
hydraulic gradient, which is Darcy's law:

From the Forchheimer equation, we can see that when the specific discharge is sufficiently small, the inertial force can be ignored, and the equation transforms into the form of Darcy's law. On the other hand, when the specific discharge is sufficiently large, the viscous force can be ignored, and the equation transforms into fully developed turbulent flow.

In addition to a polynomial function such as the Forchheimer equation,
several power-law functions have also been proposed to describe
non-Darcy flow. One of the most commonly used power-law equations is the
Izbash equation (Izbash, 1931), which is written as

Because of its applicability for a wide range of velocity spectra and its sound physics, many scholars have adopted the Forchheimer equation (among many different types of equations) to explore non-Darcy flow. The theoretical background of the Forchheimer equation has also been discussed in detail (Panfilov and Fourar, 2006). Numerous experimental data have confirmed the validity of the Forchheimer equation for a variety of nonlinear flow phenomena (Geertsma, 1974; Scheidegger, 1958; Wright, 1968). The quadratic Forchheimer law has also been revealed as a result of numerical modeling in which the Navier–Stokes flow in corrugated channels was simulated (Koch and Ladd, 1996; Skjetne et al., 1999; Souto and Moyne, 1997). Thus, the Forchheimer equation will be selected as the representative equation to describe non-Darcy flow in this study.

Since the transition between Darcy flow and non-Darcy flow is important and
difficult to quantify, different scholars have carried out experiments using
a wide range of porous media, including homogeneous and heterogeneous porous
media. Most of the experimental studies have focused on the influence of the
mean particle size on the flow state transition when using homogeneous porous media.
In fact, in earlier studies, it was believed that the nonlinear (or non-Darcy) flow behavior in
porous media was due to the turbulent effect of the flow, and the
Reynolds number (

It is evident that most of the numerous experimental studies reviewed above on the transition from
Darcy flow to non-Darcy flow focused on the
effect of the mean particle size rather than the particle size distribution.
Recently, a few investigators have recognized the importance of particle size
heterogeneity in understanding the transition between flow regimes, and have
carried out a series of experiments to address the issue. For instance,
Van Lopik et al. (2017) provided new experimental data
on nonlinear flow behavior in 20 samples of various uniformly graded granular materials ranging from medium sands (

The purpose of this study is to provide a quantitative analysis of the effects of pore size on the transition of flow regimes between Darcy and non-Darcy flows, based on a series of laboratory experiments. To meet the objectives, we first carry out seepage experiments on permeable stones with four different particle sizes. After that, we conduct mercury injection experiments on permeable stones with four different particle sizes, obtaining the pore size distributions for different particle sizes. Finally, the effects of pore size on the transition between flow regimes and the Forchheimer coefficients are discussed, based on the experimental results.

The experimental device is mainly composed of three parts: a water supply
device, a seepage experimental device and a measuring device. A schematic
diagram of the experimental apparatus is shown in Fig. 1. The water supply
device consists of a tank, a centrifugal pump and a flow-regulating valve.
The seepage experimental device consists of a permeable stone and a
plexiglass column. The measurement device monitors the real-time water
temperature and pressure. The water temperature is measured using a
thermometer with a precision of measurement of 0.1

Schematic diagram of the experimental apparatus.

Four different particle sizes of permeable stones are selected to carry out the seepage experiment in this study. It is necessary to provide a brief overview of the preparation process of permeable stone, which is a type of artificially made tight porous medium formed by sand grains and a cementing compound. In the process of preparing permeable stones, sand of a certain particle size and the cementing compound are put in a mold, which is consolidated at room temperature. Permeable stone is widely used in daily life. At present, the most commonly used permeable base materials in urban road construction, “sponge” city construction and ecological restoration research are large-pore cement-stabilized gravel, a large-diameter permeable asphalt mixture, and so on (Guan et al., 2021; Q. Li et al., 2019; Suo et al., 2021; Yu et al., 2021). The discharge capacities of various permeable stones differ. However, increasing the pore space will lead to decreased pavement performance and mechanical strength (Han et al., 2016; Wang et al., 2021). Therefore, many scholars have carried out a lot of research on controlling the proper pore space of permeable stone (Alvarez et al., 2010; Prowell et al., 2002; Xie and Watson, 2004).

We carry out seepage experiments on four kinds of permeable stones with different mesh sizes of 24, 46, 60 and 80, where the mesh size is defined as the number of mesh elements (all with square shapes) in a 1 inch by 1 inch square, which means that a greater mesh size implies a smaller particle size. For instance, we can convert the above four different mesh sizes of permeable stones into corresponding particle sizes of 0.71, 0.36, 0.25 and 0.18 mm, respectively. In respect to pore composition, the pore distribution is concentrated over a narrow pore size range; the proportion of large pores and small pores is very small. The average particle size can reflect the overall permeability of the porous medium. The pore structure of permeable rock will not change during the process performed in the seepage experiment under room temperature. The photographs of the four kinds of permeable stones with different particle sizes are shown in Figs. 2 and 3.

The photograph of permeable stones with four different particle sizes.

Permeable stones with different particle sizes:

It is worth mentioning that the contact surface between the sample and the plexiglass column is sealed to prevent any preferential flow through the wall of the plexiglass column. After the permeable stone has been inserted into the plexiglass column, both ends are sealed with silicone glue. Water passing through the permeable stone is then collected by a cylindrical tank. Moreover, the ratio of the internal diameter of the column to the particle size of permeable stone is greater than 12, which, according to Beavers et al. (1972), can eliminate any possible wall effect on the seepage. When carrying out the experiment, it usually takes about 2 h to saturate the permeable stone. For each packed sample, more than 25 tests with different constant inlet pressures were conducted under steady-state flow conditions. In addition, for each group of permeable stones, repeated tests under the same experimental conditions were carried out 3–4 times to ensure the accuracy of the results.

In this study, the mean particle size corresponds to 50 % by weight. Such a definition of mean particle size may be different from those used in
some other studies, such as Fetter (2001), which used
10 % by weight as the mean particle size. The relationship between the
specific discharge (

Variation of

Figure 4 shows that when

Variation of

We can divide the

The particle size, grain size distribution and degree of sorting are the main factors that determine the size and shape of pores. The shape of the pores determines the tortuosity and distribution of flow paths, which are related to the viscous and inertial flow resistances. It was generally accepted in previous studies that the pore sizes of porous media have an impact on the seepage law (Maalal et al., 2021; Zhou et al., 2019). However, the structures of natural porous media are very complex, and it is difficult to quantify the effects of the arrangement of the particles on the seepage law. The characteristics of the pore size distribution contain critical information for quantifying the flow regimes. Mercury intrusion porosimetry and the nitrogen adsorption isotherm are two commonly used methods to characterize the pore sizes and their distribution (Rijfkogel et al., 2019). Other techniques can also be used to derive the pore size distribution, such as small-angle neutron and X-ray scattering measurements, CT images and nuclear magnetic resonance (Anovitz and Cole, 2015; Hall et al., 1986; Kate and Gokhale, 2006; Lindquist et al., 2000). In this study, we will use the mercury injection technique to measure the pore size distributions of the four permeable stones with different particle sizes and then use this information to describe the flow regimes.

To quantitatively study the pore size and pore throat distribution, we need
to envisage a physically based conceptual model to describe the pore
structures of permeable stones. The most commonly used model is the so-called
capillary model (Pittman, 1992; Rezaee et al.,
2012; Schmitt et al., 2013), which approximates the connected
pores as many parallel capillaries. Capillary forces are generated at
the phase interface due to the surface tension between the solid and liquid
phases when liquid flows in a capillary. The capillary force is directed
toward the concave liquid level. It was shown by Washburn (1921) that

Since mercury is a nonwetting phase to solids, to get mercury into the pores of the permeable stone, an external force (or displacement pressure) must be applied to overcome the capillary force. When a greater pressure is applied, the mercury can enter smaller pores. When a certain pressure is applied, the injection pressure is equivalent to the capillary pressure in the corresponding pore. Therefore, we can calculate the corresponding capillary radius according to Eq. (4), where the volume of mercury injected is the pore volume.

By continuously increasing the injection pressure, one can obtain the curve of injection pressure versus the volume of injected mercury, from which one can also obtain the pore-throat distribution curve and the capillary pressure curve. Based on the amount of mercury injected at different injection pressures, the relation between the injection pressure and the injection saturation can be derived, and is shown in Fig. 6.

Schematic diagram of the change in pressure with the saturation: the initial stage (A–B), the intermediate mercury entry stage (B–C) and the end stage (C–D) are shown.

Figure 6 shows that the mercury injection curve can be divided into three
stages. Firstly, during the initial stage (A–B), which has a very mild slope,
the intake pressure is very small and the intake saturation is also very
low. With increasing injection pressure, the intake saturation
slowly increases. Secondly, during the intermediate mercury entry stage
(B–C), which has a steep slope, a small pressure change will lead to a
significant saturation change. This means that the pores are relatively
uniform and the differences in pore size are small. It is well known that
for mercury injection experiments, as the injection pressure increases, the
injection saturation will gradually increase, and eventually all the pores
will be filled with mercury. As can be seen from Fig. 7, with the continuous
injection of mercury, the pressures in permeable stones with different
particle sizes vary with the saturation, which is reflected in the different
pressures

Variation of the pressure with the saturation for four permeable stones with different particle sizes.

We can make a number of interesting observations based on Fig. 7. Firstly,
the pressure at the starting point (when the saturation begins to increase),
denoted as

Pressure characteristic values of four permeable stones with different particle sizes.

To observe the pore size distributions of the four permeable stones with different particle sizes in more detail, we can calculate the percentages of pores with particular sizes in the permeable stones according to the mercury injection curves, as shown in Figs. 8–11.

Histogram of the pore size distribution of the permeable stone with a particle diameter of 0.71 mm.

Histogram of the pore size distribution of the permeable stone with a particle diameter of 0.36 mm.

Histogram of the pore size distribution of the permeable stone with a particle diameter of 0.25 mm.

Histogram of the pore size distribution of the permeable stone with a particle diameter of 0.18 mm.

From Figs. 8 to 11, we find that the pore sizes of the four permeable
stones are uniform and fall within narrow ranges. The pore size
distributions for the four different particle sizes show a skewed normal
distribution. Also, the most common pore size (the peaks of the curves,
see Figs. 8–11) of permeable stones with different particle sizes are
different: 124, 99, 83 and 59

Gaussian function characteristic values for four permeable stones with different particle sizes.

The pore size distribution falls within an ever-narrower range as the mesh size becomes larger. Moreover, the cumulative percentage frequency curves of the pore size distributions with different particle sizes are exhibited in Fig. 12, and the results are shown in Table 3.

The cumulative frequency curves of the pore size distributions.

Pore size characteristic values for four permeable stones with different particle sizes.

Note:

Figure 12 shows that

The analysis of non-Darcy coefficients has always been of interest to many researchers working in different disciplines of porous media flow (Moutsopoulos et al., 2009; Sedghi-Asl et al., 2014; Shi et al., 2020). Different scholars have obtained a large amount of data through different experimental and simulation methods. They have performed quadratic fitting of the specific discharge and hydraulic gradient curves and developed numerous expressions for the Forchheimer coefficients. We obtained the coefficients of different fitting equations, which are shown in Table 4.

The Forchheimer coefficients for empirical relations.

Sidiropoulou et al. (2007) focused on the Forchheimer coefficients of porous media and evaluated the Forchheimer equation above. The validity of these equations has been verified using different experimental data. In addition, the root mean square error (RMSE) was used as a criterion to quantitatively evaluate the coefficients (Moutsopoulos et al., 2009). The different forms of the Forchheimer coefficients described in Table 4 are based on different assumptions and simplifications of the pore structure. Consequently, these coefficients are applicable under specific conditions with different degrees of accuracy.

According to Eq. (2), the hydraulic gradient (

We can see from Eq. (6) that

The uniform-diameter cubic arrangement of porous media is a
rather ideal medium. The shape and arrangement of particles in natural pore
aquifers are usually irregular. Therefore, the above-mentioned linear
correlations between

Variation of

Variation of

We can see from Fig. 13 that the coefficient

Experimental fitting coefficients of different homogeneous particles sizes.

In the above sections, we analyzed the influence of the particle size on the
seepage coefficient. Furthermore, the pore size and pore specific surface
area are also related to the arrangement and sorting degree of the particles;
that is, to the porosity of the porous medium. To explore the effect of the sorting
degree on the seepage coefficient, we draw a schematic diagram of different
sorting degrees of particles, as shown in Fig. 15a and b. The degree of
particle sorting is one of the most important factors affecting the pore size. In
porous media with a poor sorting degree, the pore size is usually determined
by the diameter of the smallest particle. We can see from Fig. 15 that the
pores between the larger particles are filled by smaller particles,
resulting in even smaller pores. In addition, a poorer sorting degree of
particles leads to a larger pore specific surface area and a stronger
viscous force of flow, which can lead to a larger

Schematic diagrams of different particle sizes and
arrangements:

Furthermore, we have also provided schematic diagrams of spherical
particles with equal sizes in two simple arrangements, namely a cubic
arrangement and a hexahedron arrangement, as shown in Fig. 15a and c. The
cubic arrangement is the less compact arrangement, with a pore diameter of
0.414

Characteristic values of the pore structure for different arrangements with the same particle size.

However, the structure of natural porous media is much more complex and heterogeneous than what is shown in Fig. 15, so it is difficult to quantitatively describe the effect of the sorting degree and arrangement on the seepage law.

In view of this, we can use a macro parameter, porosity (

Variation of

Variation of

This study presents experimental results for the Forchheimer flow in four different permeable stones with different mesh sizes, including 24 mesh size (0.71 mm), 46 mesh size (0.36 mm), 60 mesh size (0.25 mm) and 80 mesh size (0.18 mm). The effects of the mean pore size and pore size distribution on the transition of the flow regime from pre-Darcy to post-Darcy are discussed. In addition, the mercury injection technique is proposed as a method to investigate the pore distributions of the permeable stones. Beyond that, the Forchheimer coefficients are specifically discussed. The main conclusions can be summarized as follows:

The relationships between the specific discharge (

When the specific discharge is small, only a small fraction of the pore
water flows through the pores. The rest of the pore water adheres to the
surfaces of the solid particles (it is immobile), partially blocking the flow
pathways. As the specific discharge increases, more pore water becomes
mobile and participates in the flow. Hence, the pseudo hydraulic conductivity
increases with increasing specific discharge. When the specific
discharge increases to the critical specific discharge (

The mercury injection experiment results show that the mercury injection curve can be divided into three segments. The beginning and end segments of the mercury injection curves for the four permeable stones with different particle sizes are very gentle, while the main (or intermediate) mercury injection curve is steep, indicating that the pore size distribution falls within a narrow range and the proportions of large pores and small pores are relatively small.

The porosity decreases as the mean particle size of the permeable stone
increases, while the mean pore diameter increases. The porosity faithfully reflects
the influences of the particle diameter, sorting degree and arrangement mode of the
porous medium on the seepage parameters. A larger porosity leads to smaller
coefficients

The coefficient

The data can be made available by contacting the first author or the corresponding authors.

ZL wrote the paper and carried out the experiment, JW and HZ revised this paper, TX and LH analyzed the experimental data and drew diagrams, and KH designed the experimental scheme.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Experiments in Hydrology and Hydraulics”. It is not associated with a conference.

This study was supported by the National Natural Science Foundation of China (grant no. 41402204), the National Key Research and Development Program of China (no. 2018YFC0604202) and the Fundamental Research Funds for National Universities, China University of Geosciences (Wuhan). The authors thank Zhongzhi Shen of China University of Geosciences for his great help in developing the experimental setup. The authors want to express their sincere appreciation of the constructive comments made by the two anonymous reviewers and the associate editor for improving the quality of the manuscript.

This research has been supported by the National Natural Science Foundation of China (grant no. 41402204), the National Key Research and Development Program of China (grant no. 2018YFC0604202) and the “CUG Scholar” Scientific Research Funds at China University of Geosciences (Wuhan) (grant no. 2020106).

This paper was edited by Jorge Isidoro and reviewed by two anonymous referees.