Determination of the relative macroporosity and the effective aggregate width for different macropore geometries with disk infiltrometers

The relative macroporosity (wf) and the effective aggregate width (dag) are input parameters for several dual‐permeability models. As wf is geometrically related to dag, any improvement in its determination is directly extended to dag. The wf, as estimated by disk infiltrometers, applies only under the assumption that macropores are cylindrically shaped. We generalize the determination of wf for ring, hexagon, brick, and rectangular slab macropore‐matrix shapes using a transformation factor, ξ, obtained from pore‐scale modeling. The ξ was computed by dividing the relative macroporosity for noncylindrical shapes, wf_nc, over the relative macroporosity for cylindrical shapes, wf_c. The computation of ξ accounts for differences in the macropore area and macropore water flow between noncylindrical and cylindrical shapes. A total of 15 combinations of macropore width and effective aggregate width were used to construct the geometrical figures and compute both wf_nc and wf_c. For the cylindrical, ring, and rectangular slab shapes, the macropore water flow was solved using analytical solutions. For the hexagonal and brick shapes, the macropore water flow was solved numerically using COMSOL Multiphysics. Remarkably, the computed ξ was constant and equal to 1.5 for all four noncylindrical shapes under analysis. We show that the solution is exact for laminar flow under saturated conditions in the macropores with a rigid and wettable matrix. This methodology enables the derivation of a better estimate of wf and dag from disk infiltrometer data that include different macropore geometries. This information is crucial for the setup of dual‐permeability models in risk assessments and detailed studies.

over the relative macroporosity for cylindrical shapes, w f_c . The computation of ξ accounts for differences in the macropore area and macropore water flow between noncylindrical and cylindrical shapes. A total of 15 combinations of macropore width and effective aggregate width were used to construct the geometrical figures and compute both w f_nc and w f_c . For the cylindrical, ring, and rectangular slab shapes, the macropore water flow was solved using analytical solutions. For the hexagonal and brick shapes, the macropore water flow was solved numerically using COMSOL Multiphysics. Remarkably, the computed ξ was constant and equal to 1.5 for all four noncylindrical shapes under analysis.
We show that the solution is exact for laminar flow under saturated conditions in the macropores with a rigid and wettable matrix. This methodology enables the derivation of a better estimate of w f and d ag from disk infiltrometer data that include different macropore geometries. This information is crucial for the setup of dual-permeability models in risk assessments and detailed studies. INTRODUCTION been used for simulating water flow and chemical transport in drainage systems (Gardenas, Simunek, Jarvis, & van Genuchten, 2006;Scorza Júnior, Jarvis, Boesten, van der Zee, & Roulier, 2007;Tiktak, Hendriks, Boesten, & van der Linden, 2012) and water redistribution in semiarid environments (van Schaik, Hendriks, & van Dam, 2010). Therefore, dual-permeability models are an essential component of agricultural and environmental studies involving the computation of water and chemical mass balances. A large number of parameters, however, may discourage the use of dual-permeability models, especially because some parameters are difficult to measure in the field.
Parameterization of dual-permeability models is a difficult task, and it is considered relevant for further research (Van den Berg, Gottesbüren, Hammel, Jarvis, & Poot, 2014). A complete parameterization of dual-permeability models is essential for both local-and regional-scale studies. One possible approach is to obtain high-quality data to be used in parameter estimation by inverse methods (e.g., Andelst study; Scorza Júnior, Smelt, Boesten, Hendriks, & van der Zee, 2004). This methodology provides sufficient estimates for local studies, but the large data requirements and computational burden render it impractical for regional studies. Furthermore, inverse methods require reasonable initial estimates of dual-permeability model parameters and uncertainty bounds for computational efficiency (Arora et al., 2011;Köhne, Köhne, & Gerke, 2002). Therefore, the ability to roughly estimate dual-permeability model parameters is essential, regardless of whether more precise estimations are to be obtained through inverse methods. Additionally, it is pertinent for risk assessment or regional studies, where the availability of data for calibration is scarce.
The effective aggregate width (d ag ) and the relative macroporosity (w f ) are common input parameters for dualpermeability models. The d ag denotes the characteristic distance from the macropore wall to the center of the matrix. The distance is measured parallel to the soil surface in different cross-sectional planes over depth. The w f refers to the total area (or volume) of macropores over a reference area (or volume) of soil. The d ag is mathematically related to w f (see Section 2). Therefore, any improvement in the determination of w f is directly passed on to d ag . The last point is essential because d ag has typically been estimated by calibration or pedotransfer functions under field conditions (Scorza Júnior et al., 2007;. Therefore, alternative methods for an independent determination of d ag and w f under field conditions are useful. Accurate estimations of d ag have been performed under controlled laboratory conditions using cylindrical macropores surrounded by a cylindrical matrix mantle (Arora et al., 2011;Urbina et al., 2019). These conditions allow

Core Ideas
• The relative macroporosity and effective aggregate width are mathematically related. • The transformation factor is constant for noncylindrical shapes. • Computation of the relative macroporosity for mixed geometries is included. • HYDRUS, MACRO, and SWAP models can be parameterized with this methodology. • The methodology can be applied to existing disk infiltrometer databases.
precise knowledge of w f , the effective macropore width, b e , and the macropore-matrix shape over depth. The b e is defined here as the half-radius of a cylindrical macropore or, for other shapes, some characteristic macropore halfwidth. Accurate determinations of d ag , w f , and b e under field conditions can be performed by computed tomography (Hu, Feng, Yang, & Wang, 2014;Zhang, Xu, Li, Hou, & Ren, 2017). However, this technology is not widely available to institutions worldwide. Alternatively, w f and b e can be estimated by disk infiltrometers under field conditions. Those values, along with the macropore-matrix shape, can be used to approximate d ag . Disk infiltrometers allow for the determination of d ag , w f, and b e only under the assumption of cylindrical macropores and quasi-steady-state one-domain flow conditions (Watson & Luxmoore, 1986). The use of disk infiltrometers to obtain d ag , w f, and b e in noncylindrical macroporematrix shapes is challenging and is a limitation of the methodology. Noncylindrical macropore-matrix shapes include rings, hexagons, bricks, and rectangular slabs. Those shapes have already been incorporated into various dual-permeability models (e.g., Gerke & van Genuchten, 1996). Therefore, the challenge is to generalize the computation of d ag and w f for noncylindrical macropore-matrix geometries with disk infiltrometers.
The main objective of this research was to compute a transformation factor (ξ) for the estimation of the relative macroporosity, w f , in the ring, hexagon, brick, and rectangular slab macropore-matrix shapes under laminar flow conditions. A mathematical and geometrical relation between d ag and w f for different macropore-matrix shapes is explicitly shown in Section 2. Implications of ξ in the computation of d ag are also presented in Sections 2 and 5. The outcomes of this research will allow for the generation of a theoretical estimate of the effective aggregate width, d ag, and the relative macroporosity, w f , for noncylindrical and cylindrical geometries, using disk infiltrometers. F I G U R E 1 Schematic representation of a disk infiltrometer measurement for cylinders (I), rings (II), hexagons (III), bricks (IV), and rectangular slabs (V). The homogenization and regularization steps are applied to obtain d ag and w f for noncylindrical shapes. b e is the effective macropore width; d ag is the effective aggregate width; w f_c, and w f_nc are the relative macroporosities for cylindrical and noncylindrical shapes, respectively; and ξ is the transformation factor

THEORETICAL FRAMEWORK
In this section, relevant physical concepts and the introduction of the novel transformation factor is included. The concept of "porous block" is commonly used in this manuscript; hence, it is crucial to know its definition. A porous block is an idealized geometrical figure formed by a macropore and its corresponding matrix. This term was used similarly in Gerke and van Genuchten (1996). An example of a cylindrical porous block is found in Figure 1, "Homogenization." The cylindrical porous block is formed by a cylindrical macropore of radius = 2b e and its matrix mantle of width = d ag . The matrix mantle is the soil matrix between the macropore wall and the limit of the porous block. Four types of noncylindrical porous blocks are found in Figure 1, "Regularization." For example, the brick shape porous block is formed by the half-width of the macropore (b e ) and its inner matrix, including a halfwidth = d ag .
The computation of the transformation factor, ξ, compares the four noncylindrical porous blocks against the cylindrical porous block, as is depicted in Figure 1, "Regularization." It is worth to mention that in Gerke and van Genuchten (1996), the cylindrical macropore surrounded by a cylindrical matrix mantle is denoted as a "hollow cylinder," whereas the ring porous block is denoted as "solid cylinder."

Young-Laplace capillarity theory
For water at 20 • C and a contact angle of zero between the liquid-vapor-solid interface, the equivalent macropore radius (r m , cm) for a given pressure head (h) is where h (cm) is the pressure head imposed at the base of the disk infiltrometer. For macropore shapes other than cylindrical and spherical, the following solution of

Vadose Zone Journal
Young-Laplace for parallel plates can be used: where d T (cm) is the width between two parallel plates (or matrix walls). The fixed pressure head employed at the base of the disk infiltrometer determines the effective macropore radius (Equation 1) or effective macropore width (Equation 2), above which a macropore is excluded from the water flow process. We advise using Equation 2 instead of Equation 1 for noncylindrical geometries as an approximation of the macropore width.

Relative macroporosity
The relative macroporosity is obtained by dividing the total macropore area (m 2 ) over the total surface area of infiltration (A R , m 2 ) as follows: where N is the number of macropores, and A m (m 2 ) is the average macropore cross-sectional area. Equation 3 can be modified as follows: where A R over N is the average area of a porous block; therefore, the union of the macropore and its matrix area. Equations 3 and 4 can be used for any geometry. Practitioners should be aware that A R can be the total surface area of infiltration of a disk infiltrometer or the total surface area of infiltration of a noncylindrical porous block (see the computation of the transformation factor in Section 3).

Number of macropores
The number of macropores of a given size (i.e., r m or d T ) can be obtained by = where Q (m 3 s −1 ) and q (m 3 s −1 ) are water flow rate, and the meaning of those water flow rates is context dependent, because Equation 5 is used for different situations in this paper. Q may be the water flow rate from a disk permeameter under a given pressure head (see Equations 1 and 2), and q may be the water flow rate computed for the macropore by pore-scale simulation.

The effective aggregate width
The effective aggregate width is computed from a cylindrical or noncylindrical porous block. The w f and the effective macropore width (b e ) are input parameters for obtaining the effective aggregate width, d ag , for different porous blocks, as follows: where w f ]0,1[ (rings, hexagons, and bricks); where w f ]0,1] (rectangular slabs); where d ag is the width of the cylindrical matrix mantle in Equation 6, the radius of the ring matrix or the half-width for the brick matrix or the apothem for the regular hexagon matrix in Equation 7, and the half-width of the rectangular slab matrix in Equation 8. The b e is the half-radius of the macropore in Equation 6 and is the macropore halfwidth in Equations 7 and 8, and w f_c is the relative macroporosity for cylindrical shapes. The ξ was previously defined. Some of these equations were derived from Gerke and van Genuchten (1996). Equations 6-8 demonstrates the mathematical relation between d ag and w f for different geometries. Next, we will show how to approximate w f and b e for noncylindrical shapes with a disk infiltrometer, to subsequently approximate d ag using Equations 7 and 8.

Disk infiltrometer
The disk infiltrometer is a device that allows for the measurement of infiltration rates at different pressure heads imposed at the base of a disk (Perroux & White, 1988). These pressure heads can be transformed into a macropore radius (Equation 1) or macropore width (Equation 2) using the Young-Laplace capillarity theory. Therefore, the above equations (Equation 3-8) can be applied to different pressure heads if a threshold for differentiating between the macropores and the matrix system is set. Based on experimental evidence, Jarvis (2007) suggested the threshold value be −10 or −6 cm pressure head for use in Equations 1 and 2. Two example cases are presented below. In Case 1, we explain how d ag and w f could be computed from disk infiltrometers for cylindrical shapes. In Case 2, we show the limitations of the use of disk infiltrometer for the obtainment of d ag and w f for noncylindrical shapes.

Case 1: Disk infiltrometers and cylindrical shapes
Let us assume a field condition where four cylindrical macropores of a radius ≥0.06 cm are found (Figure 1, I).
Placing the disk at 0 and −3 cm pressure head allows us to measure the macropore water flux in that pressure head range as follows: -3] is the water flux displaced by the disk infiltrometer (m s −1 ) for the pressure head range [0, −3], Q 0 is the water flow displaced by the disk infiltrometer at 0 pressure head (m 3 s −1 ), and Q 3 is the water flow displaced by the disk infiltrometer at −3 cm pressure head (m 3 s −1 ). The minimum macropore radius that transports water in q D[0,-3] is obtained by substituting −3 cm in Equation 1, which results in r m = 0.05 cm.
The number of cylindrical macropores, N c , can be obtained by applying Darcy's law (the numerator in Equation 10) and the Hagen-Poiseuille equation (the denominator in Equation 10), assuming quasi-steady-state laminar flow conditions (Dunn & Phillips, 1991;Watson & Luxmoore, 1986), the outcome of which was where η is the dynamic viscosity of water (kg m −1 s −1 ), ρ is the density of water (kg m −3 ), g is the gravitational constant (m s −2 ), and r m, −3 , is 0.05 cm for h = −3 cm. The derivation of Equation 10 can be found in the supplemental material. The primary assumption in Equation 10 is one-domain flow conditions that produce a unitary gradient. Hence, the saturated hydraulic conductivity from Darcy (K) is equal to q D .
The average macropore cross-sectional area of the cylindrical porous block (A m in Equations 3 and 4) is computed by setting r m = 0.05 cm as the radius. The computation of d ag requires choosing the shape of the porous block. Recall that in this case, the macropore shape is cylindrical, and for practical applications, we set a cylindrical matrix mantle. Therefore, the equations below for w f (Equation 3) and d ag (Equation 6) are valid for a cylindrical porous block: where the subscript [0, −3] means that the variable was computed for that pressure head range, and the subscript −3 (in A m and b e ) means that those variables were computed with the corresponding maximum macropore radius involved in the infiltration process (0.05 cm). Recall that b e = 0.025 cm in Equation 12 (half-radius). Practitioners should notice that the assumption of a cylindrical porous block for the computation of d ag is just a conceptualization. Hence, the matrix surrounding the cylindrical macropore can be different (e.g., squared matrix). However, cylindrical macropores and cylindrical matrix mantle have been applied in previous research with dual-permeability models for the obtainment of d ag and w f (Arora et al., 2011;Urbina et al., 2019). Therefore, it is convenient to keep that conceptualization. Previous settings imply that the four cylindrical porous block areas do not fit below the disk infiltrometer area. That issue occurs because the conceptualization performed here looks to keep the relative macroporosity constant between the porous blocks. This conceptualization is one of the characteristics of homogenization ( Figure 1).
Homogenization is based on Equation 11 for w f (Figure 1, "Homogenization"). We observed that Equation 11 (see Equation 4) implies that the total surface area of infiltration (A R ) is being divided by the number of macropores (N). Therefore, using the previous example, four individual porous blocks are generated by homogenization. Every porous block contains a cylindrical macropore of radius 0.05 cm, and they contain the same matrix area, which is conceptualized as a cylindrical matrix mantle. Thus, the four porous blocks have the same d ag and w f . A result of homogenization is that the w f computed using the four macropores over the disk infiltrometer area by Equation 11 must be the same as that computed for an individual porous block using Equation 4. Additionally, the water flow transported by all the porous blocks is the same as the displaced water by the disk infiltrometer (see Equation 10). Regularization (Figure 1) is not applied to cylindrical macropores shapes.
Previous computations in Equations 11 and 12 indicate that both d ag and w f must be computed for more pressure head ranges until the threshold of −10 cm pressure head is achieved. Additionally, including more pressure head ranges reduces the error associated with homogenization (Figure 1, I). Understanding homogenization is critical because it is the base of the estimation of d ag and w f for noncylindrical geometries.

Case 2: Disk infiltrometers and noncylindrical shapes
The average macropore cross-sectional area (and thus w f ) for noncylindrical shapes (Figure 1, II-V) cannot be computed using disk infiltrometers because more than one dimension is required (in cylindrical cases, only the radius is needed to compute the average macropore crosssectional area). We generate a three-step process that includes the homogenization and regularization steps in order to compute the relative macroporosity for noncylindrical shapes, w f_nc (Figure 1).
Let us consider infiltrated water measured by disk infiltrometer in a pressure head range of [0, −3], and that the macropore width, d T , is computed by Equation 2 setting h = −3 cm. The first step is to observe the macroporematrix shape under field conditions where the disk infiltrometer is to be applied. The second step is to presume that the infiltrated water is transported by a bundle of cylindrical porous blocks (Figure 1, II-V, "Homogenization"), where each cylindrical macropore in a cylindrical porous block has a radius equal to the macropore width or r m = d T = 2b e = 0.05 cm (see Equations 1 and 2). From the bundle of cylindrical porous blocks, we compute an incorrect w f following the methodology of Case 1. To rectify w f , we apply the regularization step, which is the third step. Regularization implies the use of the relative macroporosity of a cylindrical porous block (w f_c ) multiplied by a transformation factor, ξ, to obtain the relative macroporosity for the noncylindrical porous block (w f_nc ) that was initially observed under field conditions (Figure 1, I-IV, "Regularization"). Therefore, the transformation factor, ξ, is defined as the ratio between w f_nc and w f_c , and is only applied in the regularization step for noncylindrical shapes (Figure 1). Computing the transformation factor, ξ, is the objective of this research.
The effective aggregate width, d ag , can now be computed for different macropore-matrix shapes because we can compute the relative macroporosity of noncylindrical shapes following the abovementioned methodology. The relative macroporosity for cylindrical shapes w f_c along with ξ can be set in Equations 7 or 8 to compute d ag .
Care should be taken to use the half-width of the macro-pore in Equations 7 or 8. In the previous example where d T = 0.05 cm (macropore width), b e = 0.025 cm (macropore half-width for noncylindrical shapes).
To simplify the analysis, a regular hexagon represents the irregular hexagon after regularization, where the apothem of the regular hexagon is d ag . A square shape represents bricks and rectangular slab shapes, to have a unique d ag after regularization (Figure 1, IV and V, "Regularization").

METHODS
The computation of the transformation factor explained below is illustrated comparing one noncylindrical porous block against one cylindrical porous block. Therefore, we include homogenization and regularization from Figure 1, II-IV.

Transformation factor
The transformation factor, ξ, is the ration of the relative macroporosity of a noncylindrical porous block (w f_nc ) against the relative macroporosity of a cylindrical porous block (w f_c ) as follows: where the subscripts "nc" and "c" at each variable mean noncylindrical and cylindrical shapes, respectively, and q is macropore water flow. The left-hand side of Equation 14 was derived from Equation 3 for w f . The right-hand side of Equation 14 was obtained by setting the noncylindrical porous block as the reference shape, hence the number of noncylindrical porous blocks is one (N nc = 1). Therefore, in this setting, the total surface area of infiltration, A R , is equal to the area of the noncylindrical porous block (matrix and macropore area). The number of cylindrical shapes, N c , is different from one, and it is computed using Equation 5 where Q means the macropore water flow of the noncylindrical porous block (reference), q nc , and q means the macropore water flow of the cylindrical porous block, q c . In simple terms, the meaning of N c is the amount of cylindrical porous block necessary to transport the same amount of water as the reference noncylindrical porous block. Hence, the transformation factor ξ accounts for differences in the macropore area and macropore water flow between noncylindrical and cylindrical shapes. To quantify ξ, we generated 15 geometrical figures for rings, hexagons, bricks, and rectangular slabs combining three macropore widths, d T (0.075, 0.050, and 0.025 cm), and five effective aggregate widths, d ag (0.5, 1.25, 2.5, 3.75, and 5.0 cm). The radius of the transitional cylindrical porous block was set as d T = 2b e (homogenization and regularization step in Figure 1). Previous information is enough to compute w f_nc and w f_c and thus the transformation factor by Equations 13 and 14. An example of the construction of the macropore-matrix shapes is presented in the supplemental material.

Average macropore cross-sectional area
where A m_nc is the average macropore cross-sectional area of the noncylindrical porous block calculated from the previous 15 combinations of d T and d ag . The average macropore cross-sectional area of the cylindrical porous block, A m_c , is also needed in Equation 14 and is computed by setting r m = d T (Figure 1, "Regularization"). The A m_nc is demarcated from the matrix limit until the dashed red border (Figure 2). The noncylindrical porous block limits are denoted by the dashed red outer border in Figure 2, including the soil matrix.

Macropore water flow
Macropore water flow for the cylindrical and noncylindrical porous block is needed in Equation 14 for the computation of the transformation factor. The macropore water flow was computed in the vertical direction for cylindrical and noncylindrical porous blocks. The macropore water flow for the noncylindrical porous blocks was computed from the dashed green outer border to the macroporematrix boundary (Figure 2,_II-IV). The previously macropore water flow computed for all noncylindrical shapes was divided by two to represent the macropore water flow of the porous block (dashed red outer border to the macropore-matrix border). For more details, see the examples in the supplemental material.
The following conditions were imposed for macropore water flow: (a) laminar flow in the macropore, (b) the macropore has no interaction with the matrix (single-domain flow), and (c) the macropore is fully saturated.
The macropore water flow for the cylindrical porous block was computed analytically from the Hagen- Where the q c units are cubic meters per second, Δp is the pressure difference between the inlet and outlet boundary (Pa), θ is the angle between the horizontal and flow direction (degrees) where θ = −90 • applies to vertical flow, and Δx is the length of the macropore (m). The macropore water flow for noncylindrical porous blocks was computed analytically for the ring (Equation 20) (Papanastasiou, Georgiou, & Alexandrou, 1999), and rectangular slabs (Equation 21) (Spurk & Aksel, 2020) where q nc units are cubic meters per second, and Equation 21 was solved up to n = 21. The macropore water flow for hexagons and brick shapes was obtained through numerical simulation using COMSOL Multiphysics software in the vertical flow direction. Laminar flow conditions were imposed with no-slip boundary conditions at the walls for water at 20 • C. A no-flow condition was imposed in the macropore walls. Therefore, no interaction with the soil matrix was allowed (single-domain flow assumption). The following set of equations were solved using COMSOL: where υ is the fluid velocity field (m s −1 ), p is pressure (Pa), and I is the identity matrix. The macropore water flow for hexagon and brick shapes was finally obtained by integrating the fluid velocity field over the outlet area. A fine physics-controlled mesh was generated, and the numerical solutions were computed iteratively until stationary conditions were achieved. In the flow equations (both analytical and numerical), the length of the macropore Δx was set as 5 cm, and the pressure head difference Δp = 0 Pa.

Numerical solution error and Reynolds number
The numerical error of COMSOL simulations is computed by comparing COMSOL flow with the two available analytical solutions (Equations 20 and 21). The error was computed as a percentage: where x is the flow computed by the analytical solution (Equations 20 and 21), and y is the flow computed by COM-SOL numerically.
The assumption of laminar flow conditions was tested with the Reynolds number, Re, computed for rings and rectangular slabs as follows: where D H = 4(A R /wet perimeter), v avg is the average flow velocity (m s −1 ) or the flow divided by the cross-sectional area of the macropore, and D H is the hydraulic diameter (m).

Transformation factor
The relative macroporosity obtained from presuming cylindrical shapes (w f_c ) was always lower than the actual value for noncylindrical shapes (w f_nc ) (Figure 3). The transformation factor, ξ, was almost constant with some small deviations in rings, hexagons, and bricks porous blocks (Figure 3). Recall that macropore water flow for both hexagons and bricks was obtained by numerical simulation, whereas analytical solutions were used for rings and rectangular slabs. The ξ appears to vary between 1.45 and 1.67 for all noncylindrical porous blocks. The deviations were mainly in the combinations generated with larger macropore widths and smaller effective aggregate width (d ag ).

F I G U R E 3
The transformation factor, ξ (black line), computed dividing the relative macroporosity for noncylindrical shapes, w f_nc , by the relative macroporosity of cylindrical shapes, w f_c . The lowermost x axis label is shape type, macropore width (cm), and effective aggregate width (cm)

F I G U R E 4
The left y axis shows the percentage error between the analytical and numerical computation of water flow for rings and rectangular slab shapes. The right y axis shows the Reynolds number (Re). The x axis shows (from bottom to top) shape type, macropore width (cm), and effective aggregate width (cm)

Numerical solution and Reynolds number
The percentage error was higher for the smallest effective aggregate width for both rings and rectangular slabs shapes (Figure 4). Generally, the percentage error increases when the macropore width increases (Figure 4). The Reynolds number was lower than 100, and the correlation between the Reynolds number and the percentage error was 0.23, which is not significant.

Transformation factor
The transformation factor, ξ, is almost constant across noncylindrical porous blocks under different geometrical arrangements (Figure 3). Using the analytical solutions available for rings and rectangular slabs, we demonstrate that ξ is approximately equal to 1.5 (see the supplemental material). However, using the numerical solution for hexagons and bricks, the solution for ξ seems to vary between 1.49 and 1.67 (Figure 3). This variation was produced by numerical error, which we computed for the ring and rectangular slabs using both analytical and numerical solutions (Figure 4). The most significant percentage numerical error occurs for the smallest effective aggregate width and the largest effective macropore width (Figure 4). For hexagonal and brick shapes, the largest deviations of ξ are of similar magnitudes as the percentage numerical errors previously mentioned for rings and rectangular slabs.
No turbulent flow conditions were observed in this analysis. The maximum Reynolds number computed was <100 for both rings and rectangular slabs shapes, whereas turbulent flow occurs at Reynolds number >1,000 (Jarvis, 2007). Consequently, ξ with a value of 1.5 applies to all the geometries analyzed under laminar flow conditions. The ξ of 1.5 applies for any macropore orientation (e.g., slope macropores). This conclusion can be seen by solving Equation 14 directly for rings or rectangular slabs where the pressure component [ΔP -ρgΔxsin(θ)] is canceled (see the supplemental material).

5.1.1
The effective macropore width The effective macropore width (b e ) for field conditions may show a range of values. Only the minimum b e in a pressure head range is found with the disk infiltrometer. Therefore, the actual value of the effective macropore width is a source of uncertainty. In Watson and Luxmoore (1986) database, the minimum radius (2b e ) was set to each pressure head range for the computation of w f . The reason for using the minimum radius at each pressure head range was explained in Dunn and Phillips (1991). They concluded that for a macropore composed of different radius over depth, the narrowest part in the tube regulates the flow. The b e can also be estimated by using the representative mean pore radius (Moret & Arrúe, 2007).

Errors in the determination of the relative macroporosity and the number of macropores
The w f obtained by presuming cylindrical porous block was, for all combinations, lower than the actual w f for noncylindrical porous blocks. Therefore, the number of macropores is miscalculated if cylindrical shapes are always presumed, as is considered in the methodology of Watson and Luxmoore (1986). These outcomes reveal that for a more accurate determination of w f by disk infiltrometers, the shape of the macropores should be con-sidered. The macropore-matrix shape is usually not mentioned in reported disk infiltrometer studies (Nachabe, 1995;Schwärzel et al., 2011).
Because ξ was constant and equal to 1.5 between noncylindrical shapes, the actual noncylindrical shape is not relevant for computing w f . The user needs to differentiate only between cylindrical and noncylindrical shapes under field conditions. In the case that the macroporematrix shape cannot be recognized over the total surface area of infiltration, a cylindrical shape should be assumed.

Errors in the determination of the effective aggregate width
The miscalculation of w f presuming cylindrical shapes propagates as an error into the effective aggregate width, d ag . The ratio between the actual value for d ag and the miscalculated value is proportional to 1/ξ (see Equations 7 and 8). Two primary sources of error can be produced when computing d ag . The first error relates to choosing the correct macropore-matrix shape under field conditions but without applying the transformation factor. The second error is mismatching the shape under field conditions. In the second case, it is assumed that the transformation factor was applied correctly.
Let us consider a field condition with rectangular slab macropores, where b e = 0.05 cm, w f_c = 0.004, ξ = 1.5, and, using Equation 8, d ag = 8.28 cm. The first error (not applying ξ in Equation 8) overestimates the final value of d ag by 50% (d ag = 12.45 cm). The second error of choosing a wrong macropore-matrix shape underestimates d ag by 82% (d ag = 1.48 cm) when cylindrical macropores (Equation 6) are used instead of rectangular slabs (Equation 8). In the case of choosing rings, hexagons, or bricks instead of the actual rectangular slab shape, the value of d ag is overestimated a 100% (d ag = 16.59 cm). The overestimation of d ag is related to larger macropore spacings. Smaller distances between the macropore wall and the center of the matrix imply less preferential flow. Distances of 0.1 cm are considered close to equilibrium (Gerke & van Genuchten, 1993a) or one domain flow (Larsson & Jarvis, 1999). Therefore, it is relevant to choose the correct macropore-matrix shape under field conditions for d ag. However, if we denoted rings, hexagons, and bricks shapes as even-sided polygons or "closed shapes," practitioners need to differentiate only between cylindrical, "closed shapes," and rectangular slabs for d ag . Hence, it is not necessary to know the actual "closed shape" under field conditions, which is convenient. Previous conclusions are different from the one for w f , where the user only needs to differentiate between cylindrical and noncylindrical shapes.

Physical assumptions in the estimation of the transformation factor
The ξ computed for noncylindrical porous blocks is based on simplifying assumptions of the physical processes for macropore water flow, which could deviate from the actual flow at field conditions. Quasi-steady-state and onedimensional flow conditions are a requirement for the developed methodology. Quasi-steady-state conditions are commonly challenging to achieve, especially for smaller pressure heads values at the base of the disk (Šimůnek, Wendroth, & van Genuchten, 1999). Watson and Luxmoore (1986) proposed to saturate the soil with a double ring initially and then place the disk infiltrometer to reduce the time it takes to reach a quasi-steady state. A onedimensional flow condition can be achieved by increasing the diameter of the disk infiltrometer. A larger diameter will increase the vertical hydraulic head gradient relative to the horizontal hydraulic head gradient (Šimůnek, Angulo-Jaramillo, Schaap, Vandervaere, & van Genuchten, 1998;Wooding, 1968).

Mixed shapes
Macropore-matrix shapes are likely mixed under field conditions. In that case, we advise that the same procedure should be followed for each shape independently. However, the final w f value to be used in the models should be multiplied by a weight factor corresponding to their relative abundance over the total surface area of infiltration and the differences in water flow: where ϕ c is the proportion of the relative macroporosity corresponding to cylinders, the subscripts "n" and "nc" are cylindrical and noncylindrical shapes, ′ f _c is the relative macroporosity presuming that cylindrical macropores transport all the infiltrated water by disk infiltrometer (computed following Case 1 in Section 2), and ϕ Ac is the proportion of cylindrical porous blocks over the total number of porous blocks (including cylindrical and noncylindrical shapes). The value of ϕ Ac is subjectively chosen by the practitioner under field conditions and is an open subject for future research.
Each relative macroporosity can be used to compute d ag for mixed shapes in a similar way. Equation 26 is for two macropore-matrix shapes, including a cylindrical macropore and a noncylindrical macropore. Equation 26 can be expanded to more macropore combinations. More details about the construction of Equations 26 and 27 are given in the supplemental material.
In Equations 27, the macropore water flow for the cylindrical (q c ) and noncylindrical (q nc ) shape is needed. This computation can be done by using the analytical solutions for cylinder (Equation 19), ring (Equation 20), and rectangular slab (Equation 21). In the case that hexagon or brick shapes are mixed with cylinders, the user might use numerical solutions. However, following the same construction of shapes performed in this research for the transformation factor, the macropore water flow through hexagons can be approximated as 3.5 times the one for rectangular slabs. In the case of bricks, the transformation factor is 4.0 times the one for rectangular slabs.
Follow-up studies can be conducted to implement a more precise pore-scale model of the macropore water flow for non-cylindrical porous blocks. Under field conditions, the flow can be in rivulets or films (Germann, Helbling, & Vadilonga, 2007;Nimmo, 2010) or even turbulent (Beven & Germann, 2013;Chen & Wagenet, 1992;Jarvis, 2007), especially for bigger diameters (or width) of macropores (Germann, 1987). We anticipate that if the macropore water flow for noncylindrical shapes is different from laminar, the transformation factor should increase from 1.5. In follow-up studies, we advise that the macropore water flow for the transitional cylindrical porous block should be kept as a fully saturated cylindrical macropore shape under laminar flow conditions, to maintain the physical meaning of ξ.
The user should be aware that this methodology is an initial approximation for d ag and w f presuming nearly quasi-steady-state and one-dimensional flow conditions. Commonly, three-dimensional flow is expected under field conditions when the disk infiltrometer is applied (Stewart, Abou Najm, Rupp, & Selker, 2016).

CONCLUSIONS
Disk infiltrometers have been typically used to obtain the relative macroporosity (w f ) and the number of macropores in the field, under the presumption of cylindrical macropore shapes. We generalize the computation of w f with disk infiltrometers by introducing a transformation factor, ξ, derived from pore-scale modeling. The ξ was computed, accounting for differences in macropore area and water flow between the actual shape (ring, hexagon, brick, and rectangular slab) and the transitional cylindrical shape (cylindrical macropores). Macropore water flow was solved analytically for the cylindrical, ring, and rectangular slab shapes. Macropore water flow for hexagon and brick shapes was solved numerically using COMSOL Multiphysics software. The ξ appeared to be constant and equal to 1.5 for all the noncylindrical shapes analyzed. An explicit mathematical and geometrical relation was presented for w f, and the effective aggregate width (d ag ) for all the studied macropore-matrix shapes. The computation of ξ for w f allows the computation of d ag from disk infiltrometer data for noncylindrical geometries. The use of ξ in obtaining d ag and w f is an exact solution under the physical assumptions used for macropore water flow in this study. We derived an equation for computing w f for soils containing mixed macropore-matrix shapes. The user needs to know the relative abundance of cylindrical porous blocks over the total number of porous blocks below the surface area of infiltration.
The current methodology can be applied to previous databases of disk infiltrometer data. Three conditions are required for the use of this methodology: (a) there should be quasi-steady-state conditions during the disk infiltrometer measurement, (b) the macropore-matrix shape should be known, and (c) the water displaced from the disk infiltrometer reservoir should be measured for at least two pressure head ranges, always including a measurement at zero pressure head.
Both w f and d ag are input parameters for several dual-permeability models, such as HYDRUS, SWAP, and MACRO. The d ag is used in both water flow and chemical transport components of currently available models and is a very sensitive parameter to water flow, water content over depth, and pesticide leaching (Gerke & van Genuchten, 1993a;Jarvis & Larsbo, 2012;Larsbo, Roulier, Stenemo, Kasteel, & Jarvis, 2005;. To our knowledge, this is the first study that has developed an independent estimate for d ag under field conditions for different geometries with disk infiltrometers. Therefore, the findings of this investigation may have substantial implications on the modeling of water flow and solute transport in soil profiles, and the leaching of agrochemicals and other pollutants to groundwater and surface water. This methodology can be applied to find an initial estimate of d ag and w f for risk assessment problems and regional studies, or for use as an initial value for fitting parameters in inverse models.