# A Percolation‐Based Approach to Scaling Infiltration and Evapotranspiration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}from the atmosphere, the former through photosynthesis, the latter through chemical weathering. In each case, water is critical for the chemistry: in transpiration, for transporting the CO

_{2}from the atmosphere, plant roots and decaying organic material; in infiltration, as a reagent and as an agent to bring nutrients to plant roots, and thus facilitate nutrient uptake.

## 2. Theory

#### Percolation Scaling Theory for Solute Transport

_{min}and D

_{b}are the chemical paths exponent and the fractal dimensionality of the percolation backbone, respectively. However, when the medium is highly heterogeneous, e.g., with a wide range of pore sizes, the length L of the optimal paths that provide the most rapid transport across the system is proportional to the system length to a different exponent, D

_{opt}, rather than to D

_{min}[28,29]. D

_{opt}is the so-called optimal paths exponent, and is determined for the heterogeneous limit [30]. The values of the exponents determined by [30] are mostly accurate to five significant figures, but the optimal paths exponent only to three. The most important distinctions in porous media that affect the exponents D

_{b}and D

_{min}relate to the connectivity of the paths and the specific percolation conditions: (i) imbibition (wetting) conditions correspond to site invasion percolation with trapping; (ii) drying conditions to invasion percolation with trapping; and (iii) saturated flow to random percolation [20]. Following [6] the minimum path length L is (using D

_{min}from [30]):

_{0}is the bond length, and 2D and 3D refer to two and three dimensional systems. In the case of porous media, x

_{0}corresponds to the inter-pore distance or a typical grain size. Numerical prefactors of order unity are less well constrained, hence omitted from Equations (1) and (2). The value of the exponent D

_{opt}in 2D (for all conditions of saturation), 1.21, is similar to the value of D

_{min}for 2D imbibition or drainage, 1.21, and the optimal path lengths L scale the same as the shortest paths for imbibition or drainage, L = x

_{0}(x/x

_{0})

^{1.21}. In 3D D

_{opt}= 1.46. Similarly, the most likely travel time $t$ is [30]

_{0}is the time for fluid to traverse a distance x

_{0}, such that v

_{0}= x

_{0}/t

_{0}is the fluid flow rate at the pore scale. The contrast between the exponent 1.87 for the most likely time and 1.37 for the shortest distance (for 3D saturated conditions) helps explain why paths that do not appear to be extraordinarily tortuous (20 m in a 1 m column) can still retard the arrival of solutes by an enormous factor [11] (over 1000). Since we are interested in (1) vegetation growth and transpiration in 2D (optimal paths exponent); and (2) unsteady flow in 3D (for the wetting conditions in Philip’s infiltration theory), we select here the following exponent values (from Equations (1)–(4)): (i) D

_{opt}(2D) =1.21 for vegetation growth, and (ii) D

_{b}(3D imbibition) = 1.861 for vertical infiltration. These two exponents are essentially the same as the two values assumed relevant [9,12] for soil formation (1.87) and vegetation growth (1.21), although as a result, our analogy to soil formation and chemical weathering is not quite precise. In [12], for example, soil was assumed to form under either saturated or saturating conditions, but only the exponent 1.87 was used. Given all the uncertainties in soil formation processes operating at time scales of up to tens of millions of years, ignoring the possibility that the appropriate exponent might, for some fraction of the time, be 1.861, seemed justified. However, in unsteady infiltration experiments, of much shorter duration and with much greater control, the exponent is constrained conceptually to be 1.861.

## 3. Experimental Data

## 4. Relevance of Percolation Scaling to Vegetation Growth and Soil Formation

_{b}= 1.87 (for 3D flow connectivity) and x

_{0}a typical grain diameter predicts soil depths over an extensive range of time scales (a decade to 100 Ma), where v

_{0}= x

_{0}/t

_{0}was taken to be the pore-scale infiltration rate. Equation (5) provides a soil formation rate of R

_{s}= dx/dt = 0.53v

_{0}(x/x

_{0})

^{−0.87}or 0.53v

_{0}(t/t

_{0})

^{−0.46}. Furthermore, ref. [12] showed that the related scaling of solute transport rates (from the velocity of the centroid of the solute spatial distribution) could predict soil formation rates at time scales of up to 60 Ma, and that that the predicted proportionality of R

_{s}to v

_{0}is in accordance with the observed four orders of magnitude variability in R

_{s}when precipitation also varied over four orders of magnitude (from the Atacama desert to the New Zealand Alps). Thus, rate of flow of water through the surface medium is responsible for much of the variability in soil formation rates.

_{opt}= 1.21 (2D) and x

_{0}taken as a typical pore diameter (~10, the geometric mean plant xylem diameter, Watt [33] hold for time scales ranging from minutes to 100 kyr. In this case, x is either the plant height, or the root radial extent, which have been shown to be closely equivalent over time scales from a growing season to about 40 years [10,34,35,36,37]. Once again, v

_{0}= x

_{0}/t

_{0}is the pore-scale flow rate, ranging here from about 24 cm/year to 20 m/year. This is a similar, though narrower range of flow rates than those relevant for soils under typical conditions i.e., about 2.0 cm/year to 20 m/year. The soil depths and the linear extent of natural vegetation computed from Equations (5) and (6) are given in Figure 1. The vegetation data incorporate a wide range of data sets: at time scales from minutes to days, these are root tip extension rates, for weeks to hundreds of years, they are plant heights (which are a proxy for root radial extent over most of this range), and for thousands of years up, the measurements are subsurface extents of clonal organisms, such as Posidonia oceanica and various tree species, or fungi. We note that the leaf area index could also be used to characterize plant growth [38], instead of plant height. At the longest time scales, the soil data are mostly paleosols and laterites. At the shortest time scales, the soil data include primarily soil formation after disturbances, such as landslides, glacial retreat, or tree-throw. For typical pore and particle size of 10 μm and 30 μm, respectively, the presented pore scale velocities bound the data. We point out that since larger pores have a smaller retention capacity as they are easily drained, their contribution to plant growth is smaller [33]. We will apply essentially the same Equations (5) and (6) to vertical infiltration and horizontal flow associated with transpiration, respectively.

## 5. Potential Relationship with Unsteady Flow

_{0}= x

_{0}/t

_{0}. Thus, x

_{0}/t

_{0}→ x

_{g}/t

_{g}, where x

_{g}is the transpiration during a growing season (measured typically as a depth) and t

_{g}is the length of the growing season. Using this hypothesis, we rewrite the scaling for vegetation growth in Equation (6) as

_{g}(instead of transpiration data, which is much less abundant) that range from 0.02 m (the Namibian desert at the dry end of the scale [39]) to 1.65 m (tropical rainforests and savannahs [40]) for a growing season of t

_{g}= 0.5 year, provides x = 0.02 m (t/180 days)

^{0.83}and x = 1.65 m (t/180 days)

^{0.83}, for a minimum and a maximum transpiration, or, equivalently, by Equation (7), for a minimum and maximum growing season plant height, respectively. As shown in Figure 2, these values of x (t

_{g}) bound the world’s plant heights [31] at a time corresponding to the length of the growing season, meaning that Equation (7) generates essentially identical scaling predictions of plant growth rates as depicted in Figure 1.

_{0}values consistent with a gradation of flow rates across their study sites. Such a close correspondence between predicted upper and lower vegetation growth limits and the observed range of actual vegetation growth rates in Figure 2 (the independent meta-data set of [31]) implies the possibility of applying an analogous upscaling to the horizontal soil moisture transport associated with the plant transpiration. Further, it also suggests a close relationship between transpiration “depths” and plant heights, which is also indicated by examining record breaking crop heights [9]. For amaranth, sunflowers, hemp, and corn, these record heights all are approximately 10 m in a growing season, and for tomatoes, 20 m in a year. Such rates correspond fairly closely to some classic literature source values for the saturated hydraulic conductivity, K

_{S}. In particular, a geometric mean K

_{S}value for soils is roughly 1 μm/s [43,44] and typical saturated subsurface flow rates are of the same order of magnitude [45]. An amount of 1 μm/s corresponds to a flow distance of about 32 m in 1 year. Thus, such high transpiration rates required for growth rates on the order of 20 m/year would require the existence of very nearly saturated conditions much of the time in a typical soil. This approximate correspondence suggests that the soil saturated hydraulic conductivity may generate a rough maximum for plant growth rates, if the water (and nutrient) supply is adequate. We note that in the above we consider saturation as optimal conditions in terms of conducting water and nutrients; unsaturated conditions are optimal in terms of aerobic root metabolism.

## 6. Relevance of Percolation Scaling to Infiltration

_{S}, ($1/3\le A/{K}_{S}\le 2/3$), while S is proportional to the square root of K

_{S}, suggesting a dependence of S ~ A

^{0.5}, as noted (and also investigated) by [32]. However, as we show below, our treatment yields a different proportionality, S ~ A

^{0.77}.

_{b}= 1.861 for conditions of imbibition, rather than 1.87, appropriate for saturated conditions. Although the two resulting exponents (1/D

_{b}) of 0.5348 (saturated) and 0.537 (imbibition) are quite similar, since the medium is unsaturated by definition, one should use the latter, as in Equation (9).

_{0}= x

_{0}/v

_{0}, and v

_{0}is proportional to the saturated hydraulic conductivity, K

_{S}. Most formulations (see e.g., [51]) of K

_{S}lead to a proportionality to x

_{0}

^{2}(as does the critical path analysis of [52]), such that x

_{0}/v

_{0}is proportional to x

_{0}

^{−1}. This makes A proportional to x

_{0}

^{2}(or K

_{S}) but S proportional to x

_{0}

^{1.53}, or K

_{S}

^{0.77}, suggesting a different proportionality S ≈ A

^{0.77}, than that in Philip’s theory, S ≈ A

^{0.5.}

^{−0.55}than for xt

^{−0.5}, meaning that an exponent value of 0.537 is a better approximation than 0.5. Although the values of the exponents in [30] are exceptionally well constrained, it is known that this percolation scaling is precise only in the limit of large system sizes [15]. Systems may be considered large at upwards of 100 bonds on a side, which for typical soil particle sizes in the tens of microns means length scales of millimeters and up [25].

^{0.5}, independent of the specific values of A and S, as stated by [32], but generate a slightly different result if I is given by Equation (9). In particular, we have

_{0}

^{0.04}), generally in conformance with the results. However, in order to determine with certainty whether the residual scatter can be attributed to the additional time factor produced by our particular theoretical description, it would be necessary to know the individual values of t

_{0}. Unfortunately, these values are not obtainable from the published data. It is clear, however, that the deviation must be due to a discrepancy in the power, because the formulation of the scaling functions [32] eliminates any scatter due to variability in the coefficients. For a first estimate of how much uncertainty is introduced by the factor t

_{0}

^{0.04}, note that t

_{0}is proportional to τ

^{−1.06}. From this, we estimate the variability in t

_{0}using the largest and smallest τ values in the data set, 0.271 and 5.8 × 10

^{−4}, respectively, providing minimum and maximum multiplicative factors of 1.05 and 1.32. Multiplication of the second term by these values gives the lower and upper bounds, respectively, on the theoretical prediction in Figure 3. It should be noted finally that additional terms in the infiltration treatment due to Philip would also generate deviations in scaling. Interestingly, however, our lowest order theoretical treatment appears to generate just the right magnitude of scaling deviations to be compatible with experiment.

^{0.727}, which is in reasonably good agreement with our prediction in the discussion following Equation (8) (0.77, 6% difference), but deviates significantly (by 45%) from the value predicted by the classical Philip’s infiltration theory, 0.5.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Vegetation growth and soil formation as a function of time. Scaling predictions for vegetation growth, both natural (green) and intensively managed (blue), using maximum pore scale flow rates of 20 m/year, and minimum flow rates of 0.24 m/year. We assume pore and particle sizes of 10 μm and 30 μm, respectively, in accord with typical silt particle sizes and the usual assumption that pore diameters are about 30% that of particles [46]. The vegetation data are divided into crops [9], for which nitrogen and other nutrients were supplied; “fast plants” [9,10] and BAAD [31]. The sources for the soil database are referenced in [9] and [10]. The same maximum flow rate prediction (marked “Flow”), but using vertical solute transport scaling (“Physical Transport”), generates the upper bound on soil production as on plant growth (“Biological Transport”) using the horizontal optimal paths scaling. “Phys Min” and “Bio Min” are the corresponding predictions at the given lower flow rates. The lower bound for soil formation is consistent with a smaller flow velocity than for vegetation growth, in accord with the fact that soil formation, albeit at a very slow rate, can occur in regions where precipitation is insufficient to support plant growth (e.g., the Atacama desert).

**Figure 2.**Allometric data of plant height vs. time and age (5539 data points representing the unmanaged plants in the BAAD) [31] is compared with our predicted upper and lower limits on growth rates using the pore-scale flow rates discussed in the text. “Regnans maximum” is the maximum plant height of E. regnans computed from Equation (7) with 1 μm/s flow rate [9], a value which corresponds to the prediction from the analogous equation at larger scales using a corresponding ET of ~1650 mm in a growing season. “Regnans minimum” is generated from “regnans maximum” by multiplying by a factor 1/20, in general accord with the variability in precipitation to pan evaporation ratio determined in [41]. “Overall minimum” is obtained by multiplying “regnans maximum” by the ratio of smallest to largest ET values in the BAAD, 1/82.5. The hydraulic limit is postulated as a cavitation limit [47]. Note that the minimum and maximum scaling relationships could be equally expressed in terms of the limiting growing season transpiration values, 20 mm at the dry end of the spectrum [39] and 1650 mm at the wet end [40], which are plotted on the graph at the time of six months, a typical growing season. The correspondence between plant height and growing season transpiration suggests the viability of growth models in terms of growing season transpiration values.

**Figure 3.**Scaled parameter $\beta $, determined from Equation (10) vs. scaled time $\tau $, calculated from Equation (11), for 26 experiments from [32] (digitized from their Figure 2b). The infiltration data represent field measurements using double-ring infiltrometers and 26 infiltration tests carried out on a 9.6-ha grassland watershed (called R-5) near Chickasha, Oklahoma. Given that t

_{0}is proportional to τ

^{−1.06}, we estimated the variability in t

_{0}in Equation (12) using the largest (0.271) and smallest 5.8 × 10

^{−4}values of τ in the dataset. This provides minimum and maximum multiplicative factors of 1.05 and 1.32. Multiplication of the second term in Equation (12) by these values gives the lower and upper bounds, respectively for the theoretical predictions. The blue and red curves are the minimum and maximum expected from the theory in this study. Philip’s infiltration theory allows no scatter, and would, to within the resolution of the graph, coincide with the blue curve.

**Figure 4.**Replotting the data and predictions given in Figure 3 bi-logarithmically (same color coding) demonstrates that the data agree with the statistical nature of percolation. In particular, as times and corresponding length scales increase relative to pore-scale times, the statistical constraints from percolation theory become more restrictive, and large fluctuations related to small sample size become less important.

**Figure 5.**Sorptivity S as a function of Philip model parameter A. Comparison of the prediction of percolation scaling (red) with the 26 experiments collected by [32], digitized from their Figure 5, for the relationship between S and A, the coefficients of the transient and steady-state terms of the infiltration equation, respectively. Note that the exponent observed from the data, 0.727 (black line), is in much closer agreement with our prediction using percolation theory, 0.77 (in red), than with the value in Philip’s infiltration theory, 0.5 (in blue).

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**MDPI and ACS Style**

Hunt, A.G.; Holtzman, R.; Ghanbarian, B.
A Percolation‐Based Approach to Scaling Infiltration and Evapotranspiration. *Water* **2017**, *9*, 104.
https://doi.org/10.3390/w9020104

**AMA Style**

Hunt AG, Holtzman R, Ghanbarian B.
A Percolation‐Based Approach to Scaling Infiltration and Evapotranspiration. *Water*. 2017; 9(2):104.
https://doi.org/10.3390/w9020104

**Chicago/Turabian Style**

Hunt, Allen G., Ran Holtzman, and Behzad Ghanbarian.
2017. "A Percolation‐Based Approach to Scaling Infiltration and Evapotranspiration" *Water* 9, no. 2: 104.
https://doi.org/10.3390/w9020104