On Infiltration and Infiltration Characteristic Times

In his seminal paper on the solution of the infiltration equation, Philip (1969), https://doi.org/10.1016/b978-1-4831-9936-8.50010-6 proposed a gravity time, tgrav, to estimate practical convergence time and the time domain validity of his infinite time series expansion, TSE, for describing the transient state. The parameter tgrav refers to a point in time where infiltration is dominated equally by capillarity and gravity as derived from the first two (dominant) terms of the TSE. Evidence suggests that applicability of the truncated two‐term equation of Philip has a time limit requiring higher‐order TSE terms to better describe the infiltration process for times exceeding that limit. Since the conceptual definition of tgrav is valid regardless of the infiltration model used, we opted to reformulate tgrav using the analytic implicit model proposed by Parlange et al. (1982), https://doi.org/10.1097/00010694-198206000-00001 valid for all times and related TSE. Our derived gravity times ensure a given accuracy of the approximations describing transient states, while also providing insight about the times needed to reach steady state. In addition to the roles of soil sorptivity (S) and the saturated (Ks) and initial (Ki) hydraulic conductivities, we explored the effects of a soil specific shape parameter β, involved in Parlange's model and related to the type of soil, on the behavior of tgrav. We show that the reformulated tgrav (notably tgrav=F(β)S2/Ks−Ki2, ${t}_{\text{grav}}=\,F(\beta ){S}^{2}/{\left({K}_{s}-{K}_{i}\right)}^{2},$ where F(β) is a β‐dependent function) is about three times larger than the classical tgrav given by tgrav,Philip=S2/Ks−Ki2 $\,{t}_{\text{grav},\text{Philip}}={S}^{2}/{\left({K}_{s}-{K}_{i}\right)}^{2}$ . The differences between the classical tgrav,Philip and the reformulated tgrav increase for fine‐textured soils, attributed to the time needed to attain steady‐state infiltration and thus i + nfiltration for inferring soil hydraulic properties. Results show that the proposed tgrav is a better indicator of time domain validity than tgrav,Philip. For the attainment of steady‐state infiltration, the reformulated tgrav is suitable for coarse‐textured soils. Still neither the reformulated tgrav nor the classical tgrav,Philip are suitable for fine‐textured soils for which tgrav is too conservative and tgrav,Philip too short. Using tgrav will improve predictions of the soil hydraulic parameters (particularly Ks) from infiltration data compared to tgrav,Philip.

review of Vereecken et al. (2019) highlights the importance of infiltration processes at scales ranging from the pedon to global.
The infiltration process can generally be described by solving the Richards (1931) equation when the relationships between soil water content, matric potential, and hydraulic conductivity are defined explicitly. Alternatively, direct measurement of actual infiltration provides a convenient field approach to determining soil hydraulic parameters such as the sorptivity and saturated hydraulic conductivity (Ross et al., 1996). Sorptivity (S) expresses the capacity of a soil to absorb and release water by capillarity. In contrast, the saturated hydraulic conductivity (K s ) reflects the ability of a soil to transmit water under the influence of gravity.
As a unique solution of the Richards equation, the time series expansion (TSE) introduced by Philip ( , 1969 remains a widely used model for 1D ponded infiltration. Although, the TSE comprises an infinite series of different components, generally only the first two terms are being used in practice, with the higher-order terms in the infinite time series requiring a more systematic analysis. To analyze the convergence and time domain validity of the TSE, Philip (1969) introduced a characteristic time termed gravity time, t grav , at which gravity begins to dominate infiltration over capillarity. Philip (1969) assumed t grav to be a practical measure for the time range of useful convergence of the TSE to real data, thus allowing use of the two-term TSE. Hereby, t grav was simply formulated in terms of S and ΔK = K s -K i , where K i is the hydraulic conductivity at the initial (prior to infiltration) soil water content, θ i . Although not mentioned explicitly by Philip (1969), one may infer that only the two first terms of the TSE are used to formulate t grav . The definition of t grav is then explicit since the first term of the infiltration equation represents the effects of capillarity, and the second term the effects of gravity.
The parameter t grav has been used for a range of applications. For example, Ross et al. (1996) used t grav to scale the time and cumulative infiltration to obtain a dimensionless implicit analytical equation for infiltration. In another study on K s data obtained from positive-head tension and single-ring pressure infiltrometer data with classical undisturbed soil core measurements, Reynolds et al. (2000) used t grav as an index of the time at which steady-state infiltration will be reached. While designing a Beerkan Estimation of Soil Transfer (BEST) method, Lassabatere et al. (2006) further used t grav to determine the maximum time for which transient expressions should be fitted to experimental data to properly estimate S and K s . In that case, the authors used the t grav concept to compute the time validity of TSE for other types of analytical models. This last example shows that the concept of t grav developed regarding Philip's TSE may be generalized to other analytical models.
Whether or not t grav is a correct indicator for the time domain validity of Philip's TSE and its truncated forms accuracy, and/or if it is an accurate indicator for attaining steady-state infiltration, a precise determination of t grav itself is still of great importance. A limitation of the two-term infiltration approximation, as used to formulate t grav , is that it cannot be used for long infiltration times. Several studies indicate that the higher-order TSE terms can describe the infiltration process much better the two-term equation for some soils (Kutílek & Krejca, 1987;Rahmati et al., 2019Rahmati et al., , 2020. On the other hand, conceptually, t grav that marks when 50% of the cumulative infiltration is dominated by capillarity alone offers a generalizable alternative concept that it is valid irrespective of the number of terms included in the infiltration equation as well as regardless of the invoked model. A possible improvement when reformulating t grav could be the use of an infiltration model without any time constraints. An attractive approach for this is the analytic approximation provided by Parlange et al. (1982), further denoted as AAP (Analytic Approximation by Parlange). The analytic approximation of Parlange et al. (1982) has been referred to as a "quasi-exact implicit formulation of Haverkamp et al. (1994) for 1D flow" by several authors. While Haverkamp et al. (1994) introduced useful refinements to the original expression of Parlange et al. (1982), in line with scientific convention, the expression (and modification thereof) should be referred to as the Parlange et al. (1982) equation as suggested by Haverkamp et al. (1994). We will use AAP because this formulation is (a) widely adopted due to its physical basis; (b) valid for the entire infiltration process (Haverkamp et al., 1994;Parlange et al., 1982); and (c) also admits its own TSE by analogy with Philip's TSE (Moret-Fernández et al., 2020;Rahmati et al., 2019Rahmati et al., , 2020, thus enabling users to determine which component reflects the capillary or gravity effects on infiltration (Rahmati et al., 2020). In addition to S and ΔK, the AAP formulation includes a soil specific shape parameter β that also impacts infiltration and depends on the soil hydraulic parameters. We therefore seek to broaden the formulation of t grav by considering all variables S, ΔK = K s − K i , and β as used in the AAP formulation so that the general features of the cumulative infiltration curve can be captured for different soils.

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For both the Philip and AAP TSE models, adding terms to the TSE-based t grav formulation reflects the persistent effects of capillarity at relatively long infiltration times when infiltration is dominated by gravity. If the two-term TSE infiltration equation is used, the first term is purely capillary driven and the second term is driven by gravity (Lassabatere et al., 2006;Philip & Farrell, 1964, among others). However, the role of higher-order TSE terms, whether capillary-or gravity-driven, remains controversial, primarily because they contain ratios of K s n−1 /S n−2 (with n = 1, 2, …, ∞). For example, Rahmati et al. (2020Rahmati et al. ( , 2021 analyzed the contributions of the higher order TSE terms of the AAP model under the premise that the role of gravity increases with increasing infiltration time. They concluded that although the higher-order terms still include the capillary parameter S, the terms remain largely controlled by gravity. A characteristic of the TSE is that all higher terms (those beyond the first two) have negligible impact on the onset of infiltration, whereas their contribution increases with increasing infiltration time. Since S is a capillary parameter, an important question arises about the role of S in the higher-order TSE terms, and whether these terms mainly reflect gravity. A simple answer to this question can be found in the mathematical form of the infinite series in terms of θ as derived by . The series contains the variables S and K s for all infiltration times, but with significantly diminishing contributions of S beyond the first (absorption) term. Also, Philip's solution of the Richards (1931) equation involves a perturbation of the exact absorption case without gravity, while also considering uniform soil properties and initial soil water contents. These premises are often violated since the advancing infiltration front may reach layers with different water retention and hydraulic conductivities, thereby physically and mathematically affecting the infiltration rate at the surface. From a physical perspective, gravity will dominate the flow process at late infiltration times, and hence one may expect then vertically downward gravity-driven saturated flow (Waechter & Philip, 1985). However, by neglecting capillary, no information on the transition from near-dryness to near-saturation in regions below the infiltration front will be obtained. As correctly stated by Waechter and Philip (1985), this is a "physically interesting and practically important limit of flows strongly dominated by gravity, with capillary effects weak but nonzero".
Because of the above ambiguities in the expression of t grav , this study aims to clarify the definition of the characteristic infiltration time. Specific objectives are to: (a) reformulate the infiltration characteristic time t grav using the AAP formulation of (Parlange et al., 1982), (b) study its relation to the classical t grav introduced by Philip (1969), hereafter denoted as t grav,Philip , and (c) discuss potential applications of this soil property regarding our understanding and modeling of water infiltration into soils.

Philip's Infiltration Theory
Based on an exact solution of the Richards (1931) equation, Philip ( , 1969 derived the following expression for 1D ponded infiltration, known as Philip's TSE: where I(t) denote cumulative infiltration [L] at a given time t [T] and A 1 to A ∞ [L/T n/2 ] are coefficients of the infinite time series: Philip (1969) showed that A 1 is equal to the sorptivity S, while A 2 is proportional to K s , suggesting that 2 = ( s − i) , where c is a constant equal to 1 / 2 , 2 / 3 , and 0.38 depending upon the selected diffusivity model (i.e., linearized, a δ -function, and/or nonlinear). Philip (1969) also determined the time domain validity of his TSE formulation. By comparing time and geometric series expansions, he found that the series solution should converge to the infiltration curve described by the Richard equation for ( ∕ 2) 2 . By noting that in most cases K i /K s << 1 and 2 ≈ 1∕2 ( − ) , Philip (1969) concluded that the convergence time or time domain validity could be defined as 4 . Using S, K s , and K i , he approximated t grav,Philip as: For initially dry soil conditions such that K i can be neglected, the above equation simplifies to: Philip (1969) furthermore showed that the condition 4 was too conservative and that the factor 4 could be replaced by a smaller value. He then stated that, nominally, ≤ expresses the practical time domain validity for the TSE. We note that prior to Philip (1969), Philip and Farrell (1964) had defined the infiltration characteristic time, t grav,P&F , as well as the infiltration characteristic length, I grav,P&F , as:

Parlange's Analytical Solution of the Infiltration Equation
Equation 3 holds subject to the approximate time constraint that an attractive approach to overcome the time constraint is through comparisons with AAP. The AAP solution, which is valid for all times and gives an accurate estimate of the cumulative infiltration, can also be expanded in two-term (2T), three-term (3T) or higher approximations, via the same approach of TSEs (e.g., Lassabatere et al., 2009). These models may be used to provide a more reliable estimate of t grav . We first provide a background of the AAP as summarized by Rahmati et al. (2020).
The AAP cumulative infiltration solution (Parlange et al., 1982), which was redefined by Haverkamp et al. (1994), is given by: where β is a soil-dependent dimensionless integral shape parameter, usually fixed at 0.6 (default value). Fuentes et al. (1992) demonstrated that the shape parameter β is related to the soil hydraulic functions by: where θ s is the saturated volumetric water content (L 3 L −3 ), θ i is the initial water content (L 3 L −3 ), and D(θ) is the soil water diffusivity. For initially dry soils (with K i ∼ 0), the AAP solution reduces to: A simplified two-term (2T) approximate expansion of Equation 8 was proposed by Haverkamp et al. (1994) to describe the transient state valid for short to intermediate infiltration times. The expansion is identical to  two-term equation: where (1) = (2) = 2 − 3 (10) Additional expansions were proposed by Rahmati et al. (2019) using three terms (3T), by Moret-Fernández et al. (2020) considering four terms (4T), and Rahmati et al. (2020) using five terms (5T), that is, where c(3) to c(5) are defined as: These expressions for c(3) to c(5) assume that the initial hydraulic conductivity is negligible. We adopt this hypothesis to simplify the derivations of t grav . We aim at simplifying t grav by (a) neglecting the initial hydraulic conductivity and thus neglecting the dependency of t grav on initial conditions and (b) using the approximate expansions instead of the implicit AAP model to ease computations.

Exact Implicit Formulation of the Characteristic Time t grav
According to the basic definition of t grav , one can rewrite any TSE at time equal to t grav as (Rahmati et al., 2020): where I grav is the cumulative infiltration at time t grav , also known as the infiltration characteristic length. I grav in Equation 15 can be computed immediately by evaluating any analytical model at time t grav . Since Philip's TSE is not clearly defined when using a limited number of terms, we consider it more robust to reformulate t grav by using the AAP solution, which is valid for all infiltration times. To do this, Equation 6 is redefined in terms of t = t grav by substituting I grav from Equation 15 into Equation 6 to give: If K s , K i , ΔK (=K s -K i ), S, and β are known, this equation can be solved analytically for t grav using MATLAB, Mathematica, R, Python, or some other appropriate software. Those approaches can be combined with scaling procedures to separate the effects of scale and the hydraulic shape parameters. A computationally more effective scaling procedure as proposed by Varado et al. (2006) and further developed by Lassabatere et al. (2009), also for the sorptivity  can be used. For this purpose, we scale cumulative infiltration, I(t), and time, t, by: Rewriting the above equations for t = t grav yields: where * grav is the scaled t grav parameter, and * ( * grav ) corresponds to the scaled cumulative infiltration at * grav . Substitution of the above expressions into Equation 16 leads to: Several functions for the hydraulic conductivity could be used in Equation 19. For this study we used the Mualemvan Genuchten (MvG) model (Mualem, 1976;van Genuchten, 1980) for K i given by: where K(S e ) is soil hydraulic conductivity at effective saturation (S e ), or alternatively at a given soil water content θ, m is a model parameter, and θ s and θ r are the saturated and residual water contents (L 3 L −3 ), respectively.

Substitution of Equation 20 into Equation 19 gives:
where δ is a coefficient accounting for the effects of initial soil water content and the parameter m on * grav : Equation 21 is identical to Equation 16, except that K s , K i , and S are eliminated to lead to a more generalized dimensionless definition of * grav from where we can view * grav as a function of the infiltration-related soil variable β and the functional coefficient δ given by: The function c(β, δ) is to be obtained from Equation 21 by numerical resolution using a root-finding algorithm. Finally, the dimensional t grav can be recovered immediately from Equation 17b to yield: where F(β, δ) defines a functional relationship depending on β and δ: For an initially dry soil (lim →0 = 0 ), one can simply set δ = 0. Equation 21 then simplifies to: In this case, * grav may be viewed as a function of β only. Equation 24 can then be simplified also by setting K i = 0: By having t grav reformulated, one can simply use Equation 15 to compute the infiltration characteristic length by:

An Approximate Explicit Expression of the Characteristic Time t grav
Although formally it is possible to evaluate the functional relationship F(β) numerically, the equation might be too impractical for many applications. Therefore, we use the 3T approximation (i.e., 3T TSE of the AAP model) alternatively to obtain an explicit formulation for F(β) and consequently for t grav . Calculating I grav at time t grav using the 3T approximation (Equation 11), and substituting it in Equation 15, yields: In this expression, the initial value of the hydraulic conductivity, K i , is neglected, meaning that the expression should be considered only for initially dry soils. Simplifying and rearranging Equation 29 results in a quadratic equation, where √ represents an unknown, and K s , S, and β are knowns: Solving Equation 30 for √ in terms of its positive root results in the following expression: which shows that F(β) now explicitly depends on β.

Test Data for Validation
The expressions above were obtained by using the AAP expansion. However, when evaluating their AAP based model, Parlange et al. (1982) made several assumptions that may not apply to real-world cases, such as the choice of the θ(z, t) profile as well as interpolation of the relationship between diffusivity and hydraulic conductivity. Their proposed relation between β and the soil hydraulic curves (see Equation 6 in Haverkamp et al., 1994) seems problematic and not applicable at all times. This point was discussed in-depth by Lassabatere et al. (2009), who showed that the relation holds poorly for some types of soils (see Figure 6 of Lassabatere et al., 2009). Apart from the AAP limitations, using infiltration data simulated with HYDRUS to validate t grav is more scientifically sound. For this reasons, we validated the proposed expressions of F(β, δ) against synthetic infiltration data generated with the HYDRUS-1D software (Šimůnek et al., 2008, 2016). For the HYDRUS simulations we used the soil hydraulic properties of the 12 USDA soil textural classes (Table 1) provided by Carsel and Parrish (1988). Table 2 summarizes the settings of the HYDRUS simulations.
The numerically generated data are considered as the reference data and used in the following to assess the accuracy of the approximate expansions, including those related to Philip's and the AAP TSE formulations.

Time Domain Validity and Time to Steady-State Infiltration
For this section we considered the scenarios detailed in Table 1, which were modeled with HYDRUS to obtain reference cumulative infiltration data. We then computed the Philip and AAP TSE expansions from the previously computed input parameters S, K s , K i , and β. Note that β was computed using Fuentes' equation (Equation 7). The initial hydraulic conductivity was computed using Equation 20. Lastly, sorptivity (S) data were obtained from horizontal infiltration simulations by Rahmati et al. (2020). The Philip and AAP TSE predictions were compared to the numerically generated data described in the previous section by considering pointwise differences between predicted ̂( ) and numerically simulated I(t) values: where d is the distance/difference between the targeted approximate expansion and numerically simulated infiltration per each unit cumulative infiltration at a certain time step t. Theoretically, when the time domain validity of the approximate expansions is targeted, d(t) will have lower values at initial times, but then will tend toward higher values at later times. Practically, the time where d(t) exceeds a critical value, d(t) ≥ d TDV , is considered as the time beyond which the approximate expansion is not valid anymore. For this analysis we used a d TDV threshold of 5%.
Parameters Note. The sorptivity (S) data were obtained from horizontal infiltration simulations by Rahmati et al. (2020). The tortuosity parameter l for the hydraulic conductivity was fixed at 0.5 as used by van Genuchten (1980). Initial water contents θ i at an initial pressure head of −10,000 cm were taken from Rahmati et al. (2020Rahmati et al. ( , 2021. The infiltration constant β was calculated using Equation 7. θ s , θ r , and θ i are the saturated, residual, and initial water contents; α, n, and m are parameters of the van Genuchten (1980) soil hydraulic model; K s is the saturated hydraulic conductivity; S is the soil sorptivity; β is an infiltration constant defined by Parlange et al. (1982) and formulated by Fuentes et al. (1992).

Table 1
Soil Hydraulic Parameters of the Mualem-van Genuchten (MvG) Model (van Genuchten (1980) for the Soil Water Retention and Hydraulic Conductivity Functions for the 12 USDA Textural Classes According to Carsel and Parrish (1988) (1980)).

Table 2
Parameters and Conditions Used for the HYDRUS-1D Simulations to Generate the Synthetic Infiltration Data (Adopted From Rahmati et al., 2020Rahmati et al., , 2021 To determine the attainment of the steady-state infiltration, we simply computed a linear regression of the last points and verified that the slope is equal to K s . Then, having predicted the steady-state infiltration, the time where the numerically simulated HYDRUS-1D cumulative infiltration curve starts to converge to the steadystate line determines the attainment of steady-state infiltration. The same criterion as introduced in Equation 32 has been used to find the time for the attainment of the steady-state infiltration. Theoretically, if the steady-state attainment is reached, d(t) will be higher at short times and tend toward zero with progressing in time. Practically, the time when d(t) drops below a critical value, d(t) ≤ d SSI , is considered as the time for attaining steady-state infiltration. A d SSI threshold of 5% was used for all soils in this analysis. Figure 1 illustrates variations in the functional F(β, δ) given by Equation 25, t grav (i.e., the reformulated t grav obtained for the AAP solution when δ is not set to zero) and t grav,Philip for the 12 USDA soil classes examined. Note that the values of F(β, δ) correspond to the roots of Equation 21, with the values of β tabulated in Table 1   The results indicate that t grav varies between 15 min for sand and 996 hr for silty clay, with an average of 138 ± 80 hr and a median of 64 ± 60 hr over all soil classes. Excluding silty clay from the analysis, the range of t grav gets narrower by varying between 15 min for sand and 183 hr for silty clay loam, with an average of 60 ± 19 hr and a median of 42 ± 42 hr. The reason why the silty clay soil is an outlier is still unresolved. One reason may be that the MvG parameters were taken from the class pedo-transfer function of Carsel and Parrish (1988). A clear trend can be detected in that the coarser textures have lower t grav values. This agrees with the physics of the flow processes involved, in that lower soil capillary forces and higher flow rates are to be expected for the more coarse-textured soils with their larger pores.

Functional Relationships for F(β, δ) and t grav
Similarly to t grav , Figure 1 shows that F(β, δ) varies between 2.59 and 3.25, with the finer soil textural classes exhibiting higher F(β, δ) values. These results indicate that the reformulated t grav is approximately 2.59-3.25 times (with a mean value of 3.1) higher than t grav,Philip , since grav = ( ) × grav Philip . This probably also impacts the interpretation of infiltration data presented later in Section 3.5. If we consider the relationship between t grav,P&F and t grav,Philip (i.e., , & = (1∕ ) 2 × , ) and use the possible range for c parameter (1/3, 2/3) as reported by Philip (1969), t grav,P&F will be almost 2.3-9 times the value of t grav,Philip . Therefore, we believe that our reformulated t grav nicely defines this characteristic time by falling between those defined by Philip (1969) and Philip and Farrell (1964).
Next, the relationships between F(β, δ) versus β and δ were analyzed. As can be seen from the contour plot in Figure 3, F(β, δ) is mainly controlled by, and positivity correlated to, β throughout most δ values. Additionally, we found that at increasing values of δ, F(β, δ) is dominated by δ. The threshold, where δ dominates the function ranges from around δ = 0.1 in coarse-textured soils with lower β values, to around 0.005 in fine-textured soils with higher β values. However, closer inspection reveals that we rarely find situations in nature where δ exceeds these threshold values where a δ value of 1 corresponds to K i /K s of 0.5. An in-depth discussion considering the dependency of δ on m and S e,i is provided in the following section.

F(β, δ) Versus F(β)
The effect of δ on t grav was explored by comparing F(β, δ) and F(β), where F(β) is equal to F(β, δ) when δ is set to zero, that is, F(β) = F(β, δ = 0). Our analysis in the previous section showed that δ has a minimal influence on F(β, δ) and that setting δ = 0 did not lead to considerable changes in the F(β, δ) values, and consequently also not in t grav . Our numerical calculations also showed that by setting δ = 0, one will obtain the same results for both cases as shown in Figure 1, which may be a consequence of the δ values being very close to zero for all examined soil classes, except for clay and silty clay, and to a lesser extent also for silty clay loam (Figure 2).  Table 1. For a more detailed examination of the effect of δ on F(β, δ), and consequently on t grav , the surface response function between F(β, δ), m, and S e,i was explored (Figure 4), whereby all parameters were allowed to vary within a physically meaningful range. To do this, β values must be determined from the m parameter since there is logical relationship between both parameters. In this sense, an empirical linear function of ( ) = 2.141 − 2.421 × with R 2 value of 0.999 was used to determine β for different m values based on the β and m values tabulated in Table 1 for the different soils. The m values were allowed to vary between 0 and 0.7, the latter corresponding to the highest n value (here m = 1−1/n) among the studied soil classes. Note that smaller m values generally correspond to more fine-textured soils, while higher m values correspond to coarser-textured ones. We varied S e,i between 0 and 0.9 since S e,i values above 0.9, which correspond to near saturation, may lead to lower infiltration rates. As can be seen from Figure 4, the contour lines are horizontally aligned for the case m < 0.4 (medium-and fine-textured soils) and S e,i values less than 0.6, indicating that S e,i starts to affect F(β, δ) only when the soils are close to saturation. The horizontal contour lines can be found for all soils with any possible m value as long as S e,i is lower than 0.5. Figure 4 therefore clearly indicates that S e,i , and consequently δ, have considerable effect on F(β, δ) and t grav only when infiltration occurs in coarse-textured soils at a relatively high degree of saturation (e.g., S e,i > 0.6) at the start of the infiltration process. The latter condition seldom occurs in nature since coarse-textured soils usually are drained fast after saturation. On the other hand, the possible effects of the m parameter on F(β, δ) and t grav is well explained by the β parameter due to their interlinked relationship. Hence, we are confident that ignoring δ in our formulation will not impact the overall predictions. Consequently, we use F(β, δ = 0) whenever the AAP solution for t grav is used, while for simplicity we use only the term F(β) throughout the text.

Implicit Versus Explicit Solutions of t grav
Still needed is a verification that the much simpler explicit solutions given by Equation 31 provide accurate approximations of the implicit solutions of t grav as given by Equation 21. Figure 5 shows variations in F(β) versus β for the 12 soil textural classes when using AAP and the related 3, 4, and 5T expansions. In case of the implicit solution of t grav , F(β) was always higher than 2.6 for all textural classes, with F(β) linearly increasing with increasing β (Figure 5, AAP). In contrast, a second-order polynomial trend between F(β) and β is present when the 3T explicit solution is used, with a maximum of about 3.6 in F(β) when β values were about 1.3, and lower F(β) elsewhere. A similar trend as for the implicit solution of t grav , albeit with a smaller slope, existed between F(β) and β when the 4T expansion was used. The 5 T approximate expansion also showed a second-order polynomial trend between F(β) and β, but less nonlinear and with mostly a negative slope. Figure 5 indicates that the approximate expansions are relatively far away from the implicit solution of t grav , especially at long infiltration times, with the F(β) values showing that the reformulated equation for t grav is 2.69-3.25 times larger than the classical t grav model given by Equation 3. These results indicate that precise calculations of t grav requires Parlange's AAP solution instead of the AAP approximate expansions. We explored this more directly by comparing the obtained t grav values from the implicit and explicit solutions, Equation 31. Figure 6 shows the expected differences in t grav between the implicit and explicit solutions. We excluded the 4 and 5T expansions since they are complicated to solve and still inaccurate compared to the AAP implicit solution (especially for large β values). To check the null hypothesis that the pairwise difference between the t grav vectors has a mean equal to zero, a paired-sample t-test at a probability of p < 0.05 was performed. The t grav values obtained from the  implicit solution was considered to be the reference value (̂ ) for comparisons. The results (Table 3) indicate that the differences in the t grav values obtained from the implicit and explicit solutions are significant. As shown in Figure 7, for some soils, the relative difference, ( grav −̂grav ) ∕̂grav × 100 , is relatively high. For example, for the loam soil, the difference reached almost 18% (Figure 7). Overall, the relative difference was soil type-dependent, with the fine-and coarse-textured soils showing lower (±5%) differences, while for the medium-textured soils the t grav explicit solution was overestimated by 10%-18%. Therefore, we recommend using the implicit solution for future use. Still, using this solution is very time-consuming and complicated since it requires a numerical solution to estimate t grav using the AAP formulation. Therefore, we suggest applying the following empirical expression between F(β) and β, which is obtained for the implicit solution from the data shown in Figure 5: Using this empirical expression, F(β) can be obtained for β values between 0.6 and 2.

Influence of β on t grav
Given the fact that β is very difficult to estimate and its value has little effect on 1D cumulative infiltration rates (e.g., Latorre et al., 2018;Rahmati et al., 2020), we analyzed the effect of β on t grav using a constant values of 0.6 for β of 0.6 as suggested by Haverkamp et al. (1994), rather than a soil-dependent β according to Equation 7, for both the implicit and explicit solutions. As can be seen from Figure 8, a constant β value of 0.6 does not represent t grav very well for none of the USDA soil classes (except for sand Figure 6. Variations in the soil characteristic time, t grav , determined using the implicit and explicit solutions (based on three terms) of t grav for the examined soil textural classes. The inset shows differences between the two approaches for the coarser soils. Note that the ordinate is split for better visualization. Logarithms (log 10 ) of t grav data were used in the analysis to ensure a normal distribution of the data.

Results of the Paired-Sample t-Test Between t grav Values Obtained From Implicit and Explicit Solutions
which originally had a β value of 0.6), neither when the implicit nor explicit solution was used. In both cases, using a constant β value resulted in up to 25% underestimations in the t grav predictions. The course the soil texture, the less the underestimation was caused by a constant β value. In other words, the finer the soil texture, the higher the deviation of t grav values obtained when constant and soil-dependent β values were used. The reason is that fine-textured soils usually have β values almost double or even larger than the constant value of 0.6.
To compare the soil-dependent and constant β cases when the implicit or explicit solution is used, a paired-sample t-test at p < 0.05 showed ( Table 4) that our null hypothesis that the pairwise difference between the t grav vectors   (Haverkamp et al., 1994) are applied. In both solutions, t grav values obtained from soil-dependent β values are considered as reference (̂ ) for comparisons. has a mean equal to zero is rejected, meaning that the differences between t grav values are significant. We hence strongly recommend using soil-dependent β values in future applications, whether using implicit or explicit solutions.

Application of t grav
This section briefly discusses selected top-of-mind examples where the used of t grav is suggested. As mentioned in the Introduction, several previous studies used t grav from different perspectives. Philip (1969) himself used the parameter (denoted here as t grav,Philip ) as a priory indicator to determine the time domain validity of his TSE solution and its truncated forms. We therefore briefly explored if t grav and t grav,Philip accurately represented the time domain validity of AAP TSE approximate expansions (which are identical to Philip's TSE approximate expansions) by analyzing 1D simulated infiltration curves obtained with HYDRUS-1D. Results are shown in Figure 9 for four soil textural classes.
As can be seen from the results in Figure 9, t grav,Philip was not a good time domain validity indicator for the TSE approximations. It gave a far too high value its safe use. Interestingly, Figure 9 shows that the AAP 2T equation diverged from the simulated data already at small infiltration times in nearly all cases, thus suggesting that this approximation should be used with caution, especially for late infiltration times. The 3T equation represented infiltration well, even at later times when its curve diverges from the simulated HYDRUS-1D curves. Furthermore, the reformulated t grav seemed to be a good indicator for the time domain validity of the 3T equation.
As mentioned before, Ross et al. (1996) used t grav,Philip and I grav,Philip to scale times ( * = 2 ∕ grav,Philip ) and the cumulative infiltration curves ( * = 2 ∕ grav,Philip ) to obtain a dimensionless implicit analytical equation for infiltration. Their equation could be used to determine dimensionless parameters of the normalized soil water retention and hydraulic conductivity functions. Using reformulated t grav and I grav values instead of t grav,Philip and I grav,Philip to scale the infiltration data would produce different scaling factors and indirectly lead to varying predictions of the hydraulic parameters. While beyond the scope of our current study, this topic should be analyzed in future research.
We also briefly evaluated if the reformulated t grav or t grav,Philip parameters were able to accurately represent the time to reach steady-state infiltration, as done by Reynolds et al. (2000). For this we fitted linear steady-state lines to the HYDRUS data to determine steady-state infiltration (I ssi ) attainment as explained in Section 2.6. The results in Figure 10 show that although t grav with a value of 0.24 hr is still slightly conservative for coarse-textured soils, it served as a useful indicator for attainment of steady-state infiltration (with t ssi of 0.2 hr) not captured by Philip's t grav,Philip value of 0.1 hr. By comparison, t grav of the fine-textured soils was too conservative (much to the right of the tangent departure point), where t grav (∼80 hr) was almost three times the value of t ssi (30 hr). In contrast, t grav,Philip was located to the left, indicating early times for steady-state infiltration with a t grav,Philip value of 26 hr. Still, it seems that times slightly beyond t grav,Philip would capture the time to steady-state infiltration more accurately. In case of intermediate soil textures (loam, silt, silty clay, and silty clay loam), both criteria seemed to fall beyond the time when steady state is reached (with t grav and t grav,Philip being 5-10 times larger than t ssi ). Therefore, neither t grav,Philip nor t grav would be suitable estimates for the time when steady-state infiltration will be reached. We note that t grav , which defines the time during which the truncated approximate expansions of AAP TSE are valid, can serve as an indicator for the attainment of steady-state infiltration if the approximation is accurate for all times until steady state is reached (i.e., remaining valid beyond the time when steady state is reached). Figure 10 shows that t grav,Philip violated this condition by for fine-textured ( Figure 10a) and coarse-textured (Figure 10d) soils. Lassabatere et al. (2006)  Logarithms (log 10 ) of t grav data were used in the analysis to ensure a normal distribution of the data.  Haverkamp et al. (1994) method may result in different results for the S and K s predictions. Using similar concept as those presented here, Rahmati et al. (2020) also showed that inferring soil hydraulic parameters (S and K s ) from infiltration data works better when the infiltration data are measured until t grav is reached. They furthermore showed that infiltration measurements for durations shorter than t grav could lead to significant errors in the S and K s predictions, particularly when estimating K s . Their results also indicated that the infiltration measurements should be long enough for robust interpretation of real field measurements.

Conclusions
The seminal papers of Philip ( , 1969 describing cumulative infiltration by a TSE is still a widely used approach to analyze the infiltration process, where water enters the soil profile via the surface. Regardless of whether the gravity time, t grav , serves as a good indicator for the time domain validity of Philip's TSE expansion, it is of great interest to reformulate t grav using all possible terms of the TSE, or by means of another general infiltration model that is valid for all infiltration times. This because the higher-order terms in the TSE are known to exert a significant impact on infiltration, especially at longer times where the truncated two-term equation of Philip fails to describe infiltration appropriately. On the other hand, theoretically, t grav is a general concept based on equal contribution of capillarity and gravity on cumulative infiltration, that remains valid regardless of the infiltration model being used. For these reasons, we reformulated t grav to be grav = ( ) 2 ∕( − ) 2 using the AAP of Parlange et al. (1982), which has no time constraints. Using AAP, we were able to explore the effects of a fourth parameter (a soil specific shape parameter β) on t grav in addition to S, K s , and K i . The effects of β on t grav are implemented in the form of a functional relationship F(β). Finally, we provided an implicit and an explicit solution for F(β) and examined the effects of β (both constant and soil-dependent) and initial soil moisture conditions on F(β), and consequently on t grav . Finally, we analyzed the use of the classical t grav,Philip as well as a reformulated t grav on the plausibility of these two time-indicators to serve as time domain validity criteria for AAP TSE and related approximate expansions (which are assumed to be identical to Philip's TSE approximate expansions), and as a criterion for determining the time when steady state infiltration is reached. We also discussed the possible accuracy of the inferred soil hydraulic properties when classical and reformulated t grav definitions are considered. Based on the results, the following conclusions can be drawn: 1. The reformulated t grav is 2.59-3.25 times (3.1 on average) larger than the classical t grav . The differences between the classical and reformulated t grav parameters seem to be much higher in the case of fine-textured soils. 2. Practically, a linear relationship exists between F(β) and β, with higher β values leading to higher F(β) values. 3. The initial soil moisture content did not cause any changes in F(β) and consequently also not in t grav , thus showing that for nearly all soil saturation degrees and soil types, the term δ related to K i can be set to zero with little or no effects on the results. 4. Finding considerable differences between the implicit and explicit solutions of F(β), final calculations of t grav also showed significant differences between the implicit and explicit solutions, particularly for intermediate soil texture classes. We therefore recommend usage of the implicit solution. 5. The use of a constant value of 0.6 for β, as suggested by Haverkamp et al. (1994), resulted in erroneous t grav values compared to soil dependent β values when the explicit solution was used. This was not the case when the implicit solution was used. We therefore strongly suggest avoiding a constant β for practical implications, or to use a constant β only along with the implicit solution. 6. The reformulated t grav appeared to be a better indicator for the time domain validity of Philip's TSE and the AAP 3T approximation compared to the classical t grav,Philip . 7. The reformulated t grav expressions were found to be suitable for the time when steady state infiltration is reached in coarse-textured soils. However, both classical and reformulated t grav expressions performed poorly when applied to fine-textured soils, for which the reformulated t grav was too conservative and t grav,Philip too short.