## Abstract

This study first reviews the influence of grain size on the permeability of porous granular media in comparison to other factors, especially the sorting of grain size distribution, in order to improve the physical knowledge of permeability. The aim of this research is to counter the widespread misconception that the characteristics of water flow in granular porous media can be associated exclusively with an area regarding grain size. This review involves two different aspects. First, the dependence of the intrinsic permeability on the particle size distribution is highlighted, independently of the other internal factors such as porosity and average grain size, by simply reviewing the main existing formulas. Second, the historical literature on the influence of the average grain size in porosity is analyzed, and it is compared with the influence of the granulometric sorting. The most recognized data show that the influence of each of these two factors is of the same order, but it was not expressed in mathematical form, so a relationship of porosity versus average grain size and sorting is established. The two aforementioned steps conclude that the factors influencing permeability do not advise the use of area dimensions because it leads to only link permeability with the average grain size, especially when nonspecialists come into contact with earth sciences. Finally, after a review of the historical evolution of the permeability units, they are redefined to avoid the common misconception that occurs when the established unit leads to only a partial understanding of the key parameters influencing permeability.

## 1. Introduction

Historically, permeability characterizes all types of porous solid media, focusing this work on granular media, where permeability is a key petrophysical parameter due to its application in many fields of Earth sciences, such as hydrogeology, oil field, geological environment, geotechnics, and soil science in agronomy.

As initially established by Darcy [1], permeability is the ease with which water can move through pore spaces in porous media. The flow velocity of water depends on the internal characteristics of the medium and the pressure gradient to which it is subjected; however, considering the rigidity of the solid phase, in granular media, this flow velocity depends on the pressure gradient to which the water is subjected [2].

Over time, the concept of permeability long included the influence of temperature, which was first recognized at the same time as the concept of fluid viscosity [3, 4]. In this way, by including the fluid viscosity in the relationships to obtain permeability, a process to generalize the Darcy equation to the flow of any fluid was initiated. However, this process implied the separation of two very distinct aspects affecting the measurable flow velocity, the internal characteristics of the medium, and the characteristics of the fluid. Despite this, the term permeability was maintained up to 1940 to characterize flow in porous media.

In the study of the physical properties of rocks, quantifying permeability as the ability of fluid to flow through granular media must be performed using a reference fluid with unit viscosity (water); otherwise, the permeability of a medium cannot be referred to. For other common fluids, the flow velocity changes as a function of their viscosity. However, the universal concept of permeability cannot be applied to certain special fluids (supercritical fluids), and this has led to the adoption of specific definitions such as relative permeability. The same can be said about certain flows that do not characterize the medium but represent atypical cases that lead to specific relationships for each case. Cases with nonconventional fluids or flows were not analyzed in this study for the reasons presented in the Discussion.

The permeability of unconsolidated granular media depends on grain features (average size, shape, and size sorting), porosity [5, 6], and factors characterizing the fluid path, such as tortuosity and constrictivity [7]. The influence of tortuosity and constrictivity, which is another way to understand flow in granular media, is analyzed in the Discussion. For cemented media, permeability also depends on the degree of cementation; however, the presence of cement was not considered in this study.

The influence of average grain size has been widely quantified in the literature; however, the importance of grain size sorting was already stressed by Masch and Denny [8], and the computation of the grain size sorting influence was not adequately solved, which was analyzed by Díaz-Curiel et al. [9]. There have been proposals such as that from Berg [5], in which a deviation coefficient was related to the granulometric curve for normal distribution and 0.3 < Ø < 0.4, and Van Baaren [10], which included a variable sorting coefficient ranging from 0.7 to 1.0. Other researchers included the coefficient of granulometric uniformity *C*_{U} established by Hazen (*C*_{U} = *d*_{60}/*d*_{10}, where *d*_{60} and *d*_{10} being the mesh sizes of the sieve through which 60% and 10% of the sample passes, respectively) in relationships to estimate permeability: *κ* = 0.1·*C*_{U}^{1/3}·Ø^{5}/(1 − Ø)^{4}·*d*_{10}^{2} proposed by Mbonimpa et al. [11]; the “Shahabi correlation” *κ* = 1.2·*C*_{U}^{0.735}·*d*_{10}^{0.89}·Ø^{3}/(1 − Ø)^{2} referred to by Chapuis [12]; the “Pavchich relation” *κ* = 0.455·*C*_{U}^{1/6}·Ø/(1 − Ø)·*d*_{17} referred to by Říha et al. [13]; and the “Amer and Awad model” *κ* = 3.5 × 10^{−6}/*ν*·*C*_{U}^{0.6}·Ø^{3}/(1 − Ø)^{2}·*d*_{10}^{2.32} referred to by Arshad et al. [14]. These relationships were not appropriate in this study because the permeability values increased with grain sorting, which is contrary to the natural behavior, and the authors disagree with this positive correlation. Neither was considered appropriate in this study the “Beyer relationship” referred to by Wang et al. [15], written as *κ* = 6 × 10^{−4}·981·log(500/*C*_{U})·*d*_{10}^{2} because, for a range of *C*_{U} of natural sands (between 1 and 10), the resulting permeability variation is very low (<60%).

Regarding the influence of grain shape factors, here, it considers shape factor α as the “specific surface ratio” of grains to spheres of the same volume [16], the “ratio to spherical factor” [17], or the “angularity factor” [18]. With shape factor α, the factor 6 coming from the specific surface of spheres (*S*_{S} = 6/*d*) is included inside the constant factor in relationships to estimate permeability. Values of shape factor α are 1.1, 1.25, and 1.4 for rounded, medium angularity, and angular sands, respectively [18], or 1.0 (spherical), 1.02 (rounded), 1.07 (worn), 1.17 (sharp), and 1.29 (angular) [17]. New values from μ-imaging techniques and analytical solutions are being proposed, although a range from 1 to 1.5 is remaining for most natural sands [19]. Thus, the influence of the grain shape has not been considered in this study because it was substantially less than the other factors, as will be shown throughout this study.

The existence of other factors that do not depend on grain size influenced the ultimate goal of this study: that is, not to recommend area units for permeability, which leads to the idea that it only depends on an area that characterizes the medium, and other factors do not receive enough consideration. Moreover, using permeability units with area dimensions can lead to the misconception that permeability is a static parameter.

This study first ascertains works showing the dependence of permeability on average grain size, separate from the dependence on porosity, and other works where this dependence is only related to porosity. Many of these models to estimate permeability are based on concepts that are more restrictive or more specific than those presented by granular media in nature.

Next, the well-known joint influence of porosity and average grain size on permeability, which later became known as intrinsic permeability, is analyzed through the relationships established by Slichter [20] and Kozeny [2], to which the influence of granulometric sorting given by the relationship developed by Díaz-Curiel et al. [9] has been added. This analysis shows that except for very fine silt grain sizes (<10 μm) or porosities lower than 0.05, the influence of granulometric sorting on permeability is of the same order of magnitude as the other factors.

Subsequently, the influence of grain size factors on porosity is analyzed, revealing that the dependence of porosity on average grain size is opposite to the positive correlation shown by permeability. After developing a new relationship for porosity as a function of grain size and size sorting, it is also concluded that the influence of grain size sorting on porosity is of the same order as that of average grain size. Therefore, the factors influencing permeability would not be reduced to a single dependence on grain size, as suggested by the current use of permeability units with area dimensions.

Finally, this study redefines the permeability units and the so-called hydraulic conductivity. The aim of this redefinition, which takes up the original concept of permeability, is to avoid the use of area dimensions for permeability because the idea is that the average grain size of a medium is the only characteristic that quantifies its permeability value. If appropriate units are established, they will no longer lead to the widespread misconception that permeability is a static (geometric) property. Moreover, the arguments that will be shown against the current equivalence between both units justify the need for this redefinition. Based on these findings, we recommend the use of the historical term “intrinsic permeability” to consider the granulometric characteristics and return to the term “permeability” to what today is generally called “hydraulic conductivity.”

## 2. Joined Influence of Grain Size, Size Sorting, and Porosity on Permeability

Over time, different relationships have been developed to estimate permeability from the cited factors, starting with grain size (when accessible). These studies built on the work of Hazen [21] for media with porosity values of approximately 36% [17, 22-26].

In parallel, many studies have been published in which grain size is not considered in the relationships developed to estimate permeability, instead limiting its dependence on the porosity of the medium [27-33]. For these authors, the developed relationships provided satisfactory estimates of permeability because either the change in grain size was implicit in the developed expressions or a uniform grain size was studied. Thus, in these cases, no widespread expression was developed, including grain size, but rather empirical expressions that required different coefficients in each area.

Also, from the beginning, some researchers applied both grain size and porosity, which is currently considered the best option for estimating the permeability of unconsolidated media. In most publications that adopted this criterion, the relationships use an effective grain diameter (in any type of sedimentation environment), Slichter’s equation [20] for determining the velocity of flow through a porous medium from the grain diameter, *d*, and porosity, Ø, can be expressed as follows [9]:

where υ is the flow velocity, Δ*P* is the pressure difference at the sample ends, *ℓ* is the length of the sample, and *η* is the dynamic viscosity of water. As in equation (1), *d* is expressed in cm, *ΔP* and *ℓ* are expressed in the same units, and the result of the Slichter equation is expressed in cm/s.

A more commonly used expression is the equation of Kozeny [2], which denotes *κ* as the intrinsic permeability:

where *C*_{K} ≈ 1/5, *γ* is the specific weight, *μ* is the kinematic viscosity, and *ℓ*_{FL} is the length traveled by the fluid in the direction of the pressure gradient (to *ℓ*_{FL}/*ℓ* was assigned an overall value of 1.41) so that $\u2207P/(lFL/l)$ is the effective pressure gradient. Kozeny’s equation remains the reference model for evaluating permeability as a function of grain size and porosity [34].

If equations (1) and (2) are compared, the intrinsic permeability in Slichter’s equation results in *κ* = *d*^{2}·Ø^{3.3}, and the dependence of both equations on porosity is similar (Ø^{3}/(1−Ø)^{2} ≈ 2.7·Ø^{3.3}). The resulting values of this expression vary between 1 × 10^{−6} and 0.027 for porosities between 0.01 and 0.25 and show a much lower variation range between 0.027 and ~0.4 for porosities between 0.25 and 0.5. The reason for differentiating these two porosity ranges is due to the factor having a positive correlation with porosity in the first range but a negative correlation in the second range (see Reference 35 and references therein).

The joint use of the average grain size and porosity for estimating permeability has continued to be developed [5, 30, 36-38], with certain studies [10, 39, 40] including the formation factor (formation resistivity to fluid resistivity ratio) and the power of porosity in Archie’s first law [41]. All these studies conclude that the best relationship κ(*d*,Ø) is that from Kozeny’s equation, although different constants have been used depending on the geological environment. However, to date, different publications have claimed that Kozeny’s equation deviates from the experimental values for low average grain sizes by underestimating [42] and overestimating [43].

Because the Kozeny equation was developed for homometric grains, some relationships were subsequently developed, including a coefficient related to grain size sorting. Some of these relationships can be found in Vienken and Dietrich [44], Devlin [45], Rosas et al. [46], and Říha et al. [13]; however, as mentioned in the Introduction, these do not adequately address the inclusion of sorting. Unlike the above relationships, in that, developed by Díaz-Curiel et al. [9], the intrinsic permeability decreased with sorting, adding a variable factor from 1 for very homometric grains to 0.05 for very poor sorting of certain detrital lithologies:

where $Sc=d75/d25$ is the sorting coefficient established by Trask [47], *d*_{75} and *d*_{25} being the mesh sizes of the sieve through which 75% and 25% of the sample passes, respectively. Díaz-Curiel et al. [9] also generalized the equation (3) to include the shape factor α: *κ*=(5.42/α)·*d*^{2}/*S*_{C}^{2}·Ø^{3}/(1 − Ø)^{2}; however, as it has been shown in the Introduction section, its influence is much smaller than the other factors due to its low range of variation.

Therefore, the value of intrinsic permeability depends, with a similar degree of importance, on the average grain size, porosity, and size sorting coefficient of the particles. This fact indicates that the influence of average grain size on permeability is only one of three relevant factors unless the porosity is considered to depend exclusively on that grain size. For this reason, the following review of factors influencing porosity and a new approach to it is presented.

## 3. Computing the Influence of Granulometric Factors on Porosity: A New Approach

Considering the concept of granular porosity, that is, the ratio between the inter-granular volume of pores among the grains and the total volume of the medium, the media in which the intergranular space does not contain clays or cement (consolidated media) were analyzed. For evaluating the influence of grain characteristics, this analysis overlooks the obvious influence of packing on porosity because its effect on permeability is included in the relationships mentioned in the previous section.

According to historical and subsequent research on the dependence of porosity on grain size [18, 20, 22, 48, 49] in media with spherical grains of the same size, porosity seems to be independent of grain size. With these artificial (unnatural) characteristics, porosity is a function of packing, whereby Ø ~ 26% for rhombohedral packing and Ø ~ 48% for cubic packing [20]. Furthermore, when the medium contains two-grain sizes, there is a clear dependence of porosity with the ratio between the proportions of these two diameters, although the porosities used are again in a range from approximately 25% to 50%.

However, for geological granular media, Fraser [22] concluded that the increase in the specific surface of pores (ratio of surface area to pore volume) that occur as grain size decreases results in an increase in the interstitial fraction between grains. This fact has also been explained by the lower ratio between diameters presented by media with a smaller grain size [31]. Thus, although porosity values may range from approximately 25% to 50% in the gravel/sand/silt series, in the absence of considerations regarding their sorting, the porosity of gravels is typically closer to 25%, whereas that of sands and silts is gradually near to 50%. Therefore, it is generally accepted that porosity increases with decreasing grain size.

Another particle size characteristic that affects porosity is grain size sorting, the influence of which has previously been considered using the size uniformity coefficient [13, 26, 50-53]. In some of these studies, the “Vukovic and Soro” relationship referred to by Cheng and Chen [51] is used to estimate porosity as the only sorting function: Ø = 0.255·(1 + 0.83^{C}^{u}), where *C*_{U} is the uniformity coefficient of grain size described above. Although the fact that this expression does not consider the average grain size, it supports our viewpoint on the influence of other factors that do not relate to an area characterizing the medium, following the aim of this study to compare the influence of average grain against grain size sorting, a review of both factors is necessary.

To analyze the dependence of porosity on grain factors, the dispersion shown by porosity measurements must be discussed. In granular porous media, classical methods [54, 55] for determining the porosity of clean sands were related to the samples, Ø = (*δ*_{GR} − *δ*_{T})/(*δ*_{GR} − *δ*_{FL}) [56], where *δ*_{T} and *δ*_{FL} refer to the density of the sample and the density of the fluid filling the pores, respectively. The other method involves volume measurements, Ø = (*V*_{T} − *V*_{GR})/*V*_{T} [57], where *V*_{T} and *V*_{GR} are the volume of the sample and grains, respectively, and *V*_{GR} = *M*_{GR}/*δ*_{GR}, where *M*_{GR} is the mass of all grains, and *δ*_{GR} is the density. The method of Krumbein and Monk [57] was applied to study the dependence of porosity on grain size and sorting, revealing a wide range of porosity values compared with those expected from gradual behavior. In other words, the data exhibited substantial differences from the family of curves, which can be represented as a continuous function of grain diameter and porosity. This dispersion has also been reported in previous studies [58, 59], although their possible sources are not analyzed in this study. Therefore, it is convenient to develop a relationship that includes average grain size and size sorting to determine the extent to which they affect porosity.

For the joint quantification of grain size and sorting, data from Rogers and Head [58] and Beard and Weyl [59] were selected because, besides being two renowned works, in this study, these data are considered the best, showing the influence of grain size and sorting on porosity. Later studies did not show significant changes in their results. Both studies showed that the porosity and grain size were negatively correlated in unconsolidated media. After comparing the results of Rogers and Head with the dry-loose and wet-packed samples of Beard and Weyl, the second set was selected because of the similarity in values. To obtain values corresponding to the same grain diameter, thereby enabling a comparison, both data sets were fitted to a family of curves given by the relationship:

where *C*_{Sc} is a sorting-dependent coefficient that has been fitted to a function of the type a + b·ln(*S*_{C}) obtaining the coefficients a and b for each data set. The resulting relationships for *C*_{Sc} are:

Averaging, the sorting-dependent coefficient would be:

The mean relative deviation from the Rogers and Head data set is 1.2% and from the Beard and Weyl data set is 3.9%. Figure 1 shows both data sets, together with the curves resulting from equation (4).

In summary, the influence of grain size sorting on porosity is of the same order as that of average grain size. This result reinforces that it is not advisable to use only the average grain size to characterize the permeability of natural media.

## 4. Permeability Units

### 4.1. Historical Evolution of Permeability Units

This section reviews the units of permeability and so-called hydraulic conductivity because different concepts and units have been reported for these parameters. In addition to being respectful of the original units, the ground of this review shows that changing the primary units of velocity to area units was unnecessary and led to an arbitrary equivalence between the permeability and intrinsic permeability units. As mentioned in the Introduction, this review aimed to address the misconception of permeability as a static characteristic that results from assigning area dimensions to permeability.

The mathematical expression for determining the value of permeability was established by Darcy [1], which calculates the flow of water, *Q*, that occurs at an environmental temperature (20°C, for which *η* = 1 cP) in a soil sample of cross-sectional area *A*, subjected to a pressure gradient $\u2207PH=\Delta h/l$:

where υ is the flow velocity, Δ*h* is the difference in water height at the sample ends, *ℓ* is the length of the sample, and *k*_{1} is a coefficient that Darcy [1] termed the “permeability-dependent coefficient.” In essence, the flow rate of a porous medium according to Darcy depends on the pressure gradient to which it is subjected and is given by υ = *Q*/*A*, where *Q* and *A* are directly measurable parameters. In some points of this study concerning units, the expression_{$\u2207PH=\Delta hH2O(m)/lSAND(m)\u2261mH2O/mSAND=(L\u22121/L\u22121)$} will be used (thereafter, $\u2207PH\u2261mH2O/mSAND$) to recognize the dimensionless pressure gradient used by Darcy. Because Darcy did not consider the characteristics of the fluid and considered the nondimensional pressure gradient $\u2207PH$, *k*_{1} resulted in velocity dimensions ([*k*_{1}] = (L^{3}·*T*^{−1})·(L^{−2})·(L^{−1}/L^{−1})= L·*T*^{−1}), whose unit of common use was cm/s.

In the case studied by Darcy, the grain size of the sand was 58% < 0.77 mm, 13% < 1.10 mm, 12% < 2.00 mm, and 17% > 2 mm (estimated average = 0.58 mm in log-normal distribution and sorting coefficient of 2.29), the measured porosity value was Ø = 0.38, the internal diameter of the pipe was 0.35 m, and the thickness of the sand layer was 1.10 m. The average value between flow velocities and pressure gradients was 0.03 cm/s, with an average relative deviation of 3.1%. With these data, the permeability-dependent coefficient value in Darcy’s experiment is:

This result is invariant for any units of volume and area, as long as both are in the same system of units (*Q* in cm^{3}/s and *A* in cm^{2} or *Q* in m^{3}/s and *A* in m^{2}).

Subsequently, different methods of generalizing Darcy’s law were developed in order to include the fluid characteristics and employ standard units of pressure (force per unit area).

Regarding the fluid characteristics, in their study on the influence of grain diameter, Slichter [20] introduced the viscosity coefficient *μ* into the relationship between υ = *Q*/*A* and the pressure gradient. Subsequently, without considering Darcy’s law, Nutting [54] also included the viscosity in their study on capillary conduction, which resulted in *υ* = *R*^{2}/8·(d*P*/d*x*)/*η*, where *R* is the capillary radius (the term conductivity was used approximately interchangeably with the permeability). Note that the viscosity, *η* = *μ*/*δ*, includes the dependence of permeability on fluid density and temperature.

The standard pressure units were included by the relation Δ*P* = *δ*·g·Δ*h*, where *δ* is the density of the fluid, and g is the value of the gravitational field on the Earth’s surface. Thus, *$\u2207P=\delta \u22c5g\u22c5\Delta h/l$* is the gravitational potential gradient, and *δ*·g is the conversion factor from $\u2207PCF$ to the pressure gradient in units of pressure. Subsequently, it was widely accepted that the flow velocity of any fluid through a sample subjected to a pressure gradient in standard units of pressure is the result of multiplying three factors: a *k*_{2} coefficient as an intrinsic parameter of the medium that is independent of the fluid, the opposite of the viscosity that reflects the characteristics of the fluid, and the pressure gradient to which the sample is subjected, that is:

The flow relationship related to the pressure gradient and viscosity in equation (7) is equivalent to that given by Nutting [54]. From equation (7), it follows that the *k*_{2} coefficient has the dimensions of the area, that is, [*k*_{2}] = (L^{3}·*T*^{−1})·(L^{−2})·(*M*·L^{−1}·*T*^{−1})·(L^{2}·T^{2}·*M*^{−1}) = L^{2}. Unlike *k*_{1}, which provides the flow velocity, *k*_{2} does not represent any surface of the medium traversed but is simply a parameter whose dimensions match those of the area. As will be seen later, the correspondence between *k*_{2} and some surfaces obtained from the geometric characteristics of the medium may lead to incorrect results.

In terms of the relationship between *k*_{1} and *k*_{2}, the post-Nutting development involved a complexity that the authors consider much more straightforward to resolve. Let $\u2207PCF$ be the conversion factor of the pressure gradient $\u2207P/\u2207PH$, which has pressure gradient dimensions since $\u2207PH$ is dimensionless. According to equations (5) and (7), the relationship between *k*_{1} and *k*_{2} is given using the following:

Therefore, before 1930, it was already considered that the relationship between velocity for water given by Darcy equation and the permeability for any fluid in standard units of pressure was the ratio $\u2207PCF/\eta $, whose dimensions are $[\u2207PCF/\eta ]=(L\u22121\u22c5T\u22121)$. Equation (3), or the equation obtained by substituting $\u2207P$with *δ*·g, has since been the most widely used relationship [12, 40, 45, 60-62].

By including viscosity and setting the pressure gradient in standard units (force per unit area), it was necessary to choose which of the two coefficients is most suitable for denoting permeability. Through analogies with classic physics terminology, the result obtained with the measurement could have been called hydraulic conductance, and the intrinsic coefficient *k*_{2} could have been called hydraulic conductivity. It must be noted that, although the term “hydraulic” is mostly used to refer to incompressible fluids, in some cases, this term led to consider it a characteristic of the medium for the flow of water instead of for the flow of any fluid. However, the historical development of the permeability unit did not follow this terminology, and the terms intrinsic permeability and hydraulic conductivity for *k*_{2} and *k*_{1}, respectively, have prevailed.

The *k*_{2} coefficient is called the intrinsic permeability in many previous studies [6, 25, 33, 63-66]. However, in many other publications on empirical formulas for estimating permeability from the grain diameter and porosity, the intrinsic coefficient *k*_{2} is called the hydraulic conductivity, which is assigned velocity units despite having units of area [14, 44, 46, 67-69]. The key publications that led to the establishment of these terms are Wyckoff et al. [70] and Hubbert [71].

Wyckoff et al. [70] presented a definition of permeability in accordance with Darcy’s concept, that is, considering water as the reference fluid, but quantitatively assigned the result of using a pressure gradient ∇$\u2207Patm$ in atm/cm because atm (atmosphere) was the common unit of pressure at that time. The established Darcy permeability unit was “a permeability of 2.00 Darcy means that a flow rate of 2 cm^{3}/s is obtained through a cross-section of 1 cm^{2} and a length of 1 cm under a pressure differential of 1 atm (76.0 cm Hg) for a fluid of 1 centipoise viscosity.” As in the previous concept of permeability, this definition implies that 1 Darcy = 1 cm/s. However, when considering the pressure gradient $\u2207P$ in atm/cm, the resulting permeability values are ~10^{3} times lower than those obtained by Darcy’s law, in which the pressure gradient is 1 m_{H2O}/m_{SAND} because 1 atm/cm is equivalent to 10 m_{H2O}/0.01 m_{SAND} = 10^{3} m_{H2O}/m_{SAND}.

Later, Hubbert [71] employed a different constant of proportionality *K*_{H} to represent the historical concept of permeability and defined the specific fluid conductivity *σ = −N·d*^{2}*·δ/η*, obtaining *K*_{H} = *N*·*d*^{2}·(*δ*·g/*η*), where *N* is a constant of proportionality (whose value depends on the geometrical shape of the internal structure of the medium). Then, using *K*_{H} to refer to the coefficient *k*_{1} according to the permeability in Darcy’s law, Hubbert established that *K*_{H} was the product: *K*_{H} = *N*·*d*^{2} and decided to refer to the permeability as *K*_{H}, which has the dimensions of the area, instead of the intrinsic permeability. Subsequently, *K*_{H} was renamed as hydraulic conductivity by Richards [72]. Regarding the units, Hubbert made different proposals: In 1940, he stated that the unit of permeability in cm^{2} was arbitrary (“=10^{n} [cm^{2}], the choice of n is entirely arbitrary”) and recommended a range of values between 10^{−15} and 10^{−10} cm^{2}. In 1957, Hubbert [73] stated that *N* was approximately 6 × 10^{–4} for randomly arranged spheres, finally concluding that the equivalent of 1 Darcy was 10^{−8} cm^{2}. Since Hubbert’s publications, the use of area dimensions for permeability and equivalence has become the most widespread practice today [5, 74-81]. Despite this fact, some authors maintain the use of Darcy units [82-87], which shows the reluctance to accept surface units.

### 4.2. Redefinition of Permeability Units

Permeability, as obtained by Darcy’s law, is partly an intrinsic parameter of the medium, in the sense that the concept of flow itself and the inclusion of the sample length across the pressure gradient make it independent of its volume. However, as this law does not consider the influence of the fluid characteristics, that is, viscosity (and temperature within it), the intrinsic permeability more effectively characterizes the internal properties of the medium. Despite this, the authors believe that the use of water to determine the permeability of a medium is still an appropriate criterion because water is a reference fluid for many aspects of fluid mechanics and for the Celsius temperature scale. Also, water is a fluid that facilitates the comparison of experimental measurements, which is one of the objectives of the international conventions of units. In other words, the flow capacity of a porous medium is a doubly intrinsic parameter, covering both the characteristics of the fluid and those of the medium. In this sense, the concept of permeability is intrinsic since it refers to a fluid of unitary characteristics, and the most accurate way to qualify the second feature and that of the internal structure is the use of intrinsic permeability.

The proposal in this study takes up to the original concept of permeability to designate the flow velocity of water in a porous medium when subjected to a certain pressure gradient. Therefore, assigning the term permeability to the *k*_{1} coefficient of Darcy’s law is considered more appropriate than the proposal of Hubbert [71], who assigned it to the *k*_{2} coefficient in equation (7). Moreover, the coefficient of permeability *k*_{2} used in the equation (7), which has area dimensions, should have been assigned to the intrinsic permeability as indeed many authors have adopted (see references marked with # above).

Thus, the permeability unit would be defined by the following statement: “a permeability of 1 Darcy is that for which a medium under a pressure gradient of 1 cm_{H2O}/cm_{SAND} allows a water flow velocity of 1 cm/s” (note that the pressure gradient of 1 cm_{H2O}/cm_{SAND} = 1 m_{H2O}/m_{SAND} used by Darcy). In contrast to some earlier definitions, in this definition, the sample cross-sectional area and flow rate are expressly omitted, unlike some conventional definitions, because the concept of flow velocity is considered to be universal, and the same for pressure gradient.

The proposal for intrinsic permeability remains as a property of a medium that relates the flow velocity of any fluid to the pressure gradient to which it is subjected. Thus, the unit of intrinsic permeability is defined as follows: “an intrinsic permeability of 1 cm^{2} is that for which a medium under a pressure gradient of 1 cm_{H2O}/cm_{SAND} allows an uncompressible fluid of 0.01 poise viscosity to have a flow velocity of 1 cm/s.” This is an indirect definition, as it refers to a property, intrinsic permeability, as a function of the result it leads to in another property, that is, the flow velocity, resulting in area dimensions.

Note that the above two definitions cannot be considered to be included in the so-called centimeter-gram-second system of units (CGS) since pressure is not.

Considering the inclusion of a viscosity of 0.01 poise and the same pressure gradient, the correspondence between the unit of permeability of 1 Darcy and the unit of intrinsic permeability in cm^{2} is:

This equivalence contrasts very strongly (six orders of magnitude) with that established since Hubbert [73] as 1 Darcy ≡ 10^{−8} cm^{2}. This is the other reason why a redefinition of units needs to be considered, in addition to the fact that area units lead to the misconception that permeability is a static property.

It should be noted that different units of a given property may give different values simply due to differences in the conversion factor. Thus, the maximum values of the permeability of granular geological media found in the literature differ depending on the chosen unit convention, although they are the same if the corresponding conversion factor is applied. The permeability values shown in most studies are somewhat lower than 10 Darcy, while in others the maximum value is around 10,000 Darcy [88, 89]. The difference probably lies in whether one considers the gradient of pressures used for the definition of 1 Darcy or that following the definition by Wyckoff et al. [70] is considered, which was established for a pressure gradient approximately 1000 times greater than that considered in Darcy’s law.

However, this ambiguity is eliminated if the equations relating intrinsic permeability to grain size and porosity, such as the Kozeny equation and its various modifications, are considered to be valid. Taking into account the data from Darcy’s experiment, with an average diameter of 0.58 mm and a granulometric sorting coefficient of 2.29, the resulting intrinsic permeability value obtained by the modified Kozeny’s equation given by equation (3) is *k*_{2}= 0.00039 cm^{2} (=0.00031 if an average shape factor of 1.26 is adopted for natural sands). Comparison of this result for water viscosity of 0.01 poise with that obtained in equation (6), *k*_{1}= 0.03 cm/s, indicates that Kozeny’s equation corresponds to the same pressure gradient as Darcy’s equation; therefore, the relationship in equation (9) establishes the correct equivalence.

It should be noted that the correspondence between permeability and intrinsic permeability is not the same as the transformation between velocity dimensions and area dimensions but the result of converting a coefficient derived from two ratios reflecting different magnitudes and measurement conditions. Thus, considering that:

the exclusively dimensional equivalence of the intrinsic permeability with a permeability of 1 Darcy would be:

The discordance between equations (11) and (9) is because the dimensional equivalence of equation (11) is not an adequate process.

In this sense, as the unit of intrinsic permeability has area dimensions, but is not an area, the use of their multiples that affect the correspondence with permeability should not be done as a transformation of units of area. See, for example, the difference between 1 μ[m^{2}] = 10^{−6}[m^{2}] ≠ 1 µm^{2} = 10^{−12} m^{2}. Each increase of 10 in the scale of permeability must correspond with an equal increase in the scale of intrinsic permeability, that is, without considering the power of 2.

Several studies employ the equivalence established by Hubbert [73] but refer to the International System of Units (SI) [79]. However, the authors have not found any report in which the International Committee for Weights and Measures (Paris, France) establishes it as an accepted derived unit. The alternatives proposed in this study are: “a hydraulic conductance of 1 m/s is that for which a medium under a pressure gradient of 1 Pa/m allows a water flow velocity of 1 m/s” and “a hydraulic conductivity of 1 m^{2} is that for which a medium under a pressure gradient of 1 Pa/m allows an uncompressible fluid of 1 Pa·s viscosity to have a flow velocity of 1 m/s.” These definitions complete and differentiate all the units used to date, although it should be noted that a hydraulic conductivity of 1 [m^{2}] does not generally occur in natural porous media.

Once the four units have been redefined, it is also important to propose the recommended symbols for each term, as the difference between hydraulic conductance and permeability, or between hydraulic conductivity and intrinsic permeability, is not limited to a matter of units. Considering the usual practice with other physical units mentioned above, the following symbols are proposed: *C*_{H} for hydraulic conductance, *k* for permeability, *κ* for intrinsic permeability, and *ς*_{H} for hydraulic conductivity.

Regarding the relationships between the different properties, the correspondence between the hydraulic conductance unit and the hydraulic conductivity unit (conversion of the media behavior for water to the media behavior for any fluid) is:

The factor 10^{−3} is the result of including the viscosity of the water (0.001 Pa·s).

The rest of the relationships between the units are:

The factor 10^{6} is derived from 10^{2} for the passage from m to cm and 10^{4} for the different pressures. That is, considering the pressure gradient ∇*P*_{H} in m_{H2O}/m_{SAND} of Darcy’s law, the resulting value for *C*_{H} is ~10^{4} times lower than that obtained under a pressure gradient of 1 Pa/m because 1m_{H2O}/m_{SAND} ≈ 10^{4} Pa/m.

The factor 10^{–2} results from including the viscosity of the water (0.01 poise).

The factor 10^{–7} is the result of considering 10^{–4} for the different pressures, 10^{–1} for the passage from poise to Pa·s (1 Pa·s = 10 poise), and 10^{–2} for the conversion from cm/s to *m*/s, in the expression of the three terms involved *υ·η*/∇*P*, resulting in units of [m^{2}].

In short, according to equations (12) to (15), the equivalence between the defined units is given by equation (16):

## 5. Qualitative Correlation Between Permeability and Grain Size

In this study, it was considered inappropriate to make a univocal assignment between the internal surface of the medium and intrinsic permeability or hydraulic conductivity. This is because such an assignment leads to the innate consideration that the grain size *d* (or its square) of a sample provides a sufficient approximation of its permeability. However, the large difference between the grain size of natural porous media and the proportionally smaller range of other factors allows a representation of the qualitative correlation between the grain size and permeability. In Díaz-Curiel et al. [35], the different signs of the permeability behavior in cemented media (positive) versus unconsolidated media (negative) were presented in a single relationship.

Figure 3 shows the bands representing the normal value range of the units and terms defined in this study versus the corresponding values of the grain diameter for specific lithologies with different porosities and grain size sorting. The central values of these bands corresponded to the results obtained using the Kozeny equation (equation 2) for a gradual distribution of porosities and average grain sizes between gravels (Ø~0.25–0.35 and *d* > 2 mm) and clays (Ø~0.50 and *d* < 0.004 mm); a normal distribution was selected as the gradual function, although any other gradual function would produce similar curves. The existence of these bands with respect to the curves resulting curves from the Kozeny equation is due to the possibility of different porosities and sorting of each average grain size. Figures 3(a) and 3(b) show that the relationship between the flow properties in porous media and the grain size is not univocal because, for a given grain size, the corresponding permeability varies by approximately two orders of magnitude. For example, sands with an average grain diameter of 1 mm have permeability values between 2 × 10^{−3} and 6 × 10^{−1} Darcy, which corresponds to an approximate range of intrinsic permeability between 2 × 10^{−5} and 2 × 10^{−3} cm^{2}.

## 6. Discussion

As this work deals with factors influencing permeability, the numerous indirect relationships for estimating permeability are not discussed in detail. One such relationship is established using nuclear magnetic resonance (NMR) techniques, with direct empirical relationships being observed between total resonance signal relaxation time and permeability if the transverse relaxation time is correctly decomposed [90]. NMR techniques additionally allow certain internal factors to be obtained, such as the distribution of grain sizes.

In terms of other more classical relationships for indirect permeability estimation, such as the formation factor *F* referred to in section 2 (see Reference 35), it must be indicated that resistivity values have sometimes been inappropriately used instead of the formation factor [62]. It should also be noted that the relationship developed by Scheidegger [91] for a model of multiple straight and parallel conduits with a constant cross-section (*k* = *R*^{2}/[8·*F*] where *R* is the radius of the conduits) uses a coefficient that produces results far removed from the empirical values. Scheidegger’s relation and subsequent similar relations with a given “length” *λ* (*k = λ*^{2}/*a·F*) [75, 92-94] are invalid in unconsolidated natural media since the permeability values they provide increase as *F* decreases; this is contrary to the empirical data on unconsolidated media [35]. In these relationships, the use of the formation factor, which is a dimensionless characteristic, has also created the impression that permeability depends only on the square of the length. Furthermore, if conductive solids such as clays are present between the grains, the actual formation factor in this case is the equation given by Díaz-Curiel et al. [35].

Other static techniques for indirectly estimating the permeability of samples, such as X-ray tomography reconstruction models [95-98], where in situ application is somewhat limited, have not been considered. In relation to images taken using microcomputed tomography and scanning electron microscopy techniques, the process of estimating permeability first requires image segmentation and subsequent utilization of the relationship *k*_{PRED} = *f*(Ø*,r,σ*[*r*]) as a function of porosity Ø, mean pore size *r*, and standard deviation of the pore size distribution *σ*(*r*) [85]. Computational methods are also being applied mainly machine-learning techniques, including artificial neural networks, genetic algorithms, adaptive neuro-fuzzy inference systems, and their hybrids [56, 99-105]. Their application to permeability prediction, particularly in the study of reservoirs, requires a specific relationship (*k*_{PRED}) and data sets for training, which is a decisive aspect of the results, conditioned in several cases by the scarcity of experimental data [102, 106-108]. Other problems that have arisen are the difference in method performance depending on the site where they have been applied and the high degree of ambiguity (several models may correspond to the same permeability) [109-112]. They have provided promising results and are positioned as powerful tools for recognizing possible patterns because they allow the prediction of correlations in large data sets and characterize highly nonlinear relationships [104, 113, 114]. These methods typically rely on models generated using the aforementioned 2D and 3D image scanning methods, flow modeling (lattice Boltzmann methods) methods [30, 115-118], and the pore network method [119-122].

A factor affecting permeability that was not included in this study is the tortuosity of the fluid path through the granular medium, whose values are generally low (in Kozeny’s equation, an average value of 1.41 was adopted; tortuosity ranges from 1.1 to 1.7 in Koponen et al. [123]; from 1.0 to 1.6 in Barrande et al. [124]; and from 1.05 to 1.65 in Matyka et al. [125] and depend on factors such as average grain size and sorting, grain shape, and porosity). Analysis of different tortuosity values can be performed using internal characteristics [87] and image simulations [126]. The first reason for not considering tortuosity is that its conventional definition does not adequately characterize the tortuosity of a medium [127]. The second reason is that, as conventionally defined, tortuosity can only be obtained from indirect estimation equations, and the new definition of tortuosity proposed by Díaz-Curiel et al. [127] is very recent. Therefore, it is worth noting the limitations of different relations to estimate permeability derived from different electrical tortuosity values (*T*_{E} = (*F*·Ø)^{p} with *p* = 1 or 2) when only the power *p* = 0.5 should have been used [127].

Another factor related to the complexity of the fluid flow is the constriction of the paths through the pores (widening and narrowing of the paths in the pores). Although this characteristic was first analyzed for electrical resistivity [128, 129], the most widely used relationships are based on the relationship between the molecular diffusion coefficient in an open space (*D*_{OPEN}) and the resulting diffusion coefficient in porous media (*D*_{POR}). Thus, Petersen [130] starting from the Weissberg [131] equation *D*_{POR}/*D*_{OPEN} = Ø/*T*_{T} (where *T*_{T} is the tortuosity) and Van Brakel and Heertjes [132] (who studied diffusion in partly saturated homogeneous isotropic monodisperse sphere packing) arrived at the ratio of diffusion coefficients defined as *D*_{POR}/*D*_{OPEN} = Ø·ξ/*T*_{T}, where ξ denotes the constrictivity. This relationship has limitations such as the use of expressions for mixtures of water and solid media and the equivalence used with different electrical tortuosity values [127].

Therefore, the tortuosity can encompass some of the internal factors considered in Kozeny’s equation, resulting in an expression similar to Panda and Lake [133] that is not restricted to the capillary model. Thus, more recent relationships that include constrictivity in estimating permeability [7, 134] are equivalent to Kozeny’s equation.

In terms of the interest of researchers in the redefinition of units, apart from the misconception that the assignment of surface units leads to permeability, the main progress of the present redefinition of the units is the new equivalence between the surface units of intrinsic permeability and the units of permeability. If the development of the first part of section 4.1 (equation (9)) is not considered, it could be said that the new system of units established does not represent a great step since the unit of permeability is again the one established from Darcy’s law. However, in the authors’ opinion, the fact that this development (which synthesizes how flow is derived from a potential) was not carried out at the time is the reason for the change in the existing permeability units up to 1940. Additionally, the return to the original permeability units is a clear contrast of this study with respect to the current units, given that the velocity units for permeability were replaced by area units. The new units of conductance and hydraulic conductivity are in themselves an approach to the existing ones.

Next, we discuss some limitations that could arise if (contrary to common practice in petrophysics) it is considered that permeability units should be modified in the presence of some special flows or fluids:

A possible limitation would be to consider that the permeability values obtained for high flows or for gases lead to different definitions of permeability. However, the general practice in petrophysics to determine the permeability of a medium is to use a reference fluid with unit viscosity (such as water) and to include the viscosity value to consider different fluids. The possible combinations of fluid characteristics and particular flows are what justify the use of permeability as the value that a medium presents for the flow of water (as the reference fluid), as well as the use of intrinsic permeability to quantify the internal properties of the medium independent of the fluid. To obtain the permeability for gases that are not considered in the fields of hydrogeology, geotechnics, or soil science in agronomy, corrections such as the Klinkenberg [135] relationship are used without changing the permeability concept. In terms of the relative permeability for mixed gas/water flows, which occurs in the vadose zone (partial saturation of the pore space), it is not analyzed here because it comprises a different parameter than the worldwide concept of permeability.

It could also be argued that since this study does not analyze other fluids flowing through the granular media such as supercritical fluids, the new units are limited to the areas mentioned above. However, in addition to the fact that the current units do not differentiate these cases either, it would be relevant to special conditions that are rare in nature. The same applies to the fact that this study does not refer to Knudsen flow [136] because, if the pore size is larger than 1 µm, there is only bulk diffusion [137]; thus, it would only affect very low permeability media such as clays or marls.

Another possible exception would be that the new units are limited to Darcian behavior without considering the changes introduced by the Forchheimer equation [138]. Non-Darcy flows are not considered because they do not characterize the medium (they do not affect C

_{H}or ς_{H}), but it is a particular case of high velocity that occurs for certain gas flows and not for water flow where*k*or κ are determined. Except for when the fluid’s behavior becomes chaotic, this nonlinear flow with respect to the pressure gradient occurring at very high velocities is commonly used in the field of hydrogeology, for example, by Jacob’s equation [139]. However, such velocities do not occur for groundwater flow in nature; as an example of a very high natural groundwater flow, if a pressure increment of 5 atm and an average pore diameter reaching 2 mm are considered, the Reynolds number obtained for groundwater flow is <100, which does not correspond to turbulent flow [140]. Again, it should be noted that the current units do not differentiate between these cases either.

In summary, these possible limitations reinforce the initial need to redefine the unit system where the permeability of a medium is best characterized by the value it takes for a reference fluid of unit viscosity.

## 7. Conclusions

The analysis of the relative magnitudes of the three major factors affecting permeability, that is, average grain size, porosity, and grain size sorting, highlights that an exclusive link between the permeability of granular porous media and grain size is confusing, concluding that the use of area dimensions for permeability should be rejected.

The redefinition of units made in this study does not entail any change in their applicability and validity but only in the resulting values, which can be summarized as a change in scale. Thus, with regard to the possibility of applying the permeability unit in the CGS and the hydraulic conductance unit in the SI, these have not entailed any change in the operating procedures described in the standard test method for measuring conventional hydraulic conductivity (ASTM D2434-22 or ASTM D5856-15). In addition, although most published studies used surface units for permeability, certain studies [75, 78, 96] maintained the assignment of velocity dimensions for that parameter, presenting empirical data that validated the results that would be obtained with the redefined units. Nevertheless, the most important aspect to be verified is the validity of the equivalence between permeability and intrinsic permeability in this study. As discussed in section 4.2, after presenting equation (9), the results of the main equations characterizing the flow capacity in porous media, that is, Darcy’s law [1] and Kozeny’s equation [2], demonstrate the validity of the equivalence: 1 Darcy of permeability ≡ 10^{-2} [cm^{2}] of intrinsic permeability. Therefore, the new terms assigned and the redefinitions of the units related to flow in porous media given in this study are respectful of the original concepts, comply with the necessary scientific rigor, and exhibit total correspondence with these commonly used relationships [1, 2]. Furthermore, these terms comprise a new set of clearly structured terms, unlike those established in 1940. The difference with existing terms is the return to the original concept of permeability, thereby discarding the changes suggested by Hubbert [71], who used the term “fluid conductivity” to reflect the permeability and “permeability” to reflect what is typically termed the intrinsic permeability. The term “hydraulic conductance” and the new meaning of “hydraulic conductivity” have been established to most closely resemble the definitions of the SI, giving a new meaning to the term hydraulic conductivity that is currently used for permeability.

Moreover, the key difference between the new set of units considering the conventional units is that the original unit for the permeability of 1 Darcy = 1 cm/s for a dimensionless pressure gradient is employed, instead of the definition of 1 Darcy given by Wyckoff et al. [70] for a pressure gradient of 1 atm/cm. Likewise, the equivalence 1 cm/s ≡ 10^{−2} cm^{2} derived from the equations of Kozeny [2] and others is employed, thereby discarding the assignment of 1 cm/s ≡ 10^{−8} cm^{2} established by Hubbert [73], which represents a drastic difference to the equivalence currently used.

In contrast to the units of permeability established in communications later than Darcy’s work, the derivative assignment of velocity units for permeability from Darcy’s equation makes it an easily measurable parameter that provides values directly related to the flow in granular porous media. The same can be said for the hydraulic conductance. Moreover, handling multiples of velocity units is straightforward, as opposed to understanding the conceptual complexity of using area units. There is a possible disadvantage, which here is considered as an advantage: the obtained value corresponds to the flow of water as a unitary fluid.

The intrinsic permeability (or hydraulic conductivity) simultaneously provides values independent of the fluid type; however, it does not elucidate the actual flow in porous media because it does not differentiate water from other fluids such as oil. Moreover, it provides values with area dimensions that do not correspond to any existing surface of the medium. Therefore, the link between permeability and grain (or pore) size in granular media leads to a misconception regarding the concept of permeability.

Our redefinition matches the historical definition of the permeability coefficient (since Darcy [1]) and intrinsic permeability (since Kozeny [2]). However, the equivalence between them was demonstrated in this study. The new concept of hydraulic conductance and the redefinition of hydraulic conductivity are based on the general division of the SI; the flow velocity of a fluid with a viscosity different from that of water in a granular porous medium depends on the permeability and pressure gradient (as in Darcy’s equation [1]), to which the fluid viscosity is added in the simplest manner (equation 8), and the structural characteristics of the medium are reflected by the intrinsic permeability and hydraulic conductivity as the units used for the pressure gradient.

Acceptance of the proposed units will help advance dynamic permeability estimation techniques (i.e., those that are more related to fluid mobility in granular media) from a cross-sectional view of the analyzed media (which is closely related to the free area) to analyze the extent to which this area is obstructed in the perpendicular direction.

## Data Availability

The necessary data are included in the manuscript.

## Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper

## Acknowledgments

Part of this work was supported by the Regional Government of Madrid (CARESOIL-CM project grant number P2018/EMT-4317).