Notice: Wiley Online Library will be unavailable on Saturday 30th July 2016 from 08:00-11:00 BST / 03:00-06:00 EST / 15:00-18:00 SGT for essential maintenance. Apologies for the inconvenience.
 This study addresses infiltration from strip sources of water on the soil surface. The problem arises as the limiting case of furrow irrigation as well as for closely spaced surface drip emitters. Previously, the most common approach used for describing flow from strip sources was numerical modeling based on Richards' equation. This work investigates a third approach borrowed from previous applications for infiltration from disc sources. The assumption is that when the cumulative infiltration is expressed per unit area of the wetted strip, the difference of that value and one-dimensional infiltration is linear with time. We test the assumption directly by doing numerical experiments with six representative soils and using various strip widths, boundary, and initial water contents. The general conclusions are that the differences between the cumulative infiltration per unit source area for a strip and a planar source are linearly related to time but that a single value of the proportionality factor γ is generally inadequate. Values of γ are found between 0.64 and 1.16. An effort was made to relate γ to soil hydraulic parameters and strip width as well as to the boundary and initial conditions. Once γ is defined, the strip infiltration follows immediately if the corresponding one-dimensional solution is known without the necessity of performing a two-dimensional simulation.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 When water is applied over a long strip or furrow, we have essentially a two-dimensional infiltration process. The same scenario occurs with the application of water from closely spaced surface, drip emitters which results in a wetted strip. Early theoretical studies, for the most part, assumed a Gardner soil [Gardner, 1958] whose unsaturated hydraulic conductivity, K [L T−1] is
with Ks [L T−1] the saturated hydraulic conductivity, α [L−1] the sorptive number and h [L] the pressure head which is assumed nonpositive. Warrick and Lomen  and Batu  simplified the mathematical difficulty by specifying the Darcian flow across the soil surface rather than as a mixed boundary condition with a specified water content or pressure along the strip and negligible flow to the outside. Martinez and McTigue  used a numerical (boundary element) scheme to attack the same problem; Basha  used Green's functions to write down relevant solutions. One of Basha's innovations was to predefine a velocity profile across the soil surface and use an inverse technique to evaluate parameters describing the velocity profile in order to match a desired constant pressure head. The emphasis has been on steady flow, although Warrick and Lomen  and Basha  did consider transient cases by use of a “linear” soil water characteristic [Warrick, 2003, p. 65].
 A closely related problem is infiltration from a disc. Interest in flow from disc sources has been heightened with the popularity of disc tension infiltrometers. Haverkamp et al.  expressed the three-dimensional cumulative infiltration per unit source area I3D [L] as a simple sum:
The I1D [L] is the one-dimensional cumulative infiltration, t is time [T], r0 [L] the radius of the disc, γ a dimensionless “constant”, S0 [L T−0.5] the sorptivity, θ0 (dimensionless) the volumetric water content at the disc source and θn (dimensionless) the initial water content in the profile. Haverkamp et al. rationalized that a reasonable bound on γ was 0.6 to 0.8. Note that once one-dimensional infiltration is described, results for the disc follow as a simple linear term with respect to time.
 The primary objective of the present study is to pursue a solution for the strip for which the cumulative infiltration is a sum of the one-dimensional term and an “edge effect”. The edge term will be taken proportional to time giving a solution analogous to (2). A second objective is to determine a range of values of the proportionality constant γ and determine how it is affected by texture, strip width, boundary conditions and initial conditions.
 The last term in (2) is based on linear diffusion from an edge of a one-dimensional semi-infinite strip [Turner and Parlange, 1974; Smettem et al., 1994]. The derivation is such that the cumulative infiltration per unit length of the edge of the wetted soil is multiplied by the circumference 2π r0 giving the results for a disc. If cumulative flow due to a unit length of the edge is defined to be Qedge [L3 L−1], inspection of the last term in (2) reveals
To verify consistency, note that when Qedge is multiplied by the circumference and divided by the area of the circle the result is the last term in (2). With x0 [L] the semiwidth, results for a strip are found by multiplying Qedge by 2 to account for two sides of the strip and dividing by the area per unit length (2x0). The result in a form analogous to (2) is
with I2D [L] the two-dimensional cumulative infiltration per unit area of strip. The corresponding infiltration rate per unit area of the strip i2D [L T−1] follows from the derivative of (4):
For large times the derivative of the one-dimensional infiltration rate dI1D/dt is known to be Ks. Thus the steady state rate is
The results apply for all soil hydraulic properties. For the special case of the Gardner soil, the same arguments used to find (3) lead a to a special form of (6):
We can refer to iW,2D as a “two-dimensional Wooding equation” as it is analogous to the well-known result of Wooding  for a disc or shallow pond.
 The objectives will be achieved using (4) and simulations based on the two-dimensional Richards' equation:
where z [L] is the depth. A constant initial water content of θ = θn is assumed. The strip surface is assumed to be at a constant water content θ = θ0 for 0 < x < x0. Beyond the strip, surface flow is assumed zero and an infinite domain is considered. The flow domain and the boundary conditions are illustrated in Figure 1. The solution can be written in the form
where p, b and i are vectors representing hydraulic parameters boundary and initial conditions, respectively. The first objective reduces to comparing f (t, p, x0, b, i) with the last term in (4).
where Θ is the effective saturation (dimensionless), θr and θs denote the residual and saturated water contents (dimensionless), respectively; and αvg [L−1] and n (dimensionless) are empirical shape parameters.
Table 1. Hydraulic Properties for the Six Contrasting Soilsa
 The HYDRUS-2D software package [Šimůnek et al., 1999] was used for the numerical solution of the Richards equation for the two-dimensional case and HYDRUS-1D [Šimůnek et al., 1998] for the one-dimensional. In the HYDRUS-2D a vertical domain was selected such that the outer boundaries did not affect the flow field inside of the domain. The computational domain (50 × 70 cm) was discretized into an unstructured finite element mesh with 1223 nodes (with 2325 triangular finite elements) with significantly finer detail immediately surrounding the source (Figure 1). In HYDRUS-1D we selected a domain for every soil such that the wetting front will be far from the bottom boundary and the infiltrations will not be affected from this boundary. The domain was discretized into 201 nodes with finer detail near the surface. The initial and boundary water contents were the same as the two-dimensional simulation (i.e., θ (0, z) = θn and θ (t, 0) = θ0).
 The sorptivity S0 needed in (4) was evaluated with Philip's iterative solution for one-dimensional infiltration. Details for this solution can be found in Warrick [2003, equations 4-51 to 4-72]. This procedure calculates the dimensionless sorptivity, S* while only m, Θ0 (boundary effective fluid saturation) and Θn (initial effective fluid saturation) are needed. The result of S0 is obtained with the following relation [Warrick, 2003, equation 4-72]:
 The “edge effect” I2D − I1D is shown in Figure 2 as a function of t for the 6 representative soils with x0 = 10 cm, Θ0 = 1 and Θn = 0.1. Linearity, as projected in (4), is clearly evident and the “edge effect” remains linear as t increases. The r2 values for the 6 cases were from 0.9992 to 0.9999.
 The value γ as a function of m for 6 representative soils is presented in Figure 3a with x0 = 10 cm, Θ0 = 1, Θn = 0.1. The values for γ tend to be higher than the 0.6–0.8 and the range is wider than found by Haverkamp et al. . While the middle class texture (loam) with middle m resulted in a value of γ ≈ 0.8, γ increases to approximately 1.1 for both small and large values of m (sand, clay). For completeness, values of γ were found for the analogous three-dimensional infiltration case and presented in Figure 3a. Simulations made for r0 = 10 cm Θ0 = 1, and Θn = 0.1. The disc results followed the same trend as the strip results. For smaller m values, the strip results were slightly smaller while for the largest m values (for the 2 coarsest textures) the disc results were about the same to smaller.
The proposed advantage is that p is more basically related to the shape of the retention curve than m. Figure 3b presents γ as function of p for the same soils. The trend of γ with respect to p is very similar to that of γ with respect to m.
 The effect of the source semiwidth is shown by plotting x0 versus γ for the loamy sand and silt loam soils in Figure 4. (Overall the edge effect becomes a less significant part of the total flow as the strip width becomes larger, but the absolute amount contributed at the edge remains nearly the same). Values of the reduced saturation for the boundary and initial conditions are Θ0 = 1, Θn = 0.1. Increasing x0 results in an increase of γ up to a constant value of γ = 0.997 for the loamy sand soil and γ = 0.865 for the silt loam soil.
 The effect of the source boundary pressure head h0 on γ is presented in Figure 5. Again the loamy sand and silt loam soils were used. The strip semiwidth was x0 = 10 cm and the initial value of the effective saturation was Θn = 0.1. For the silt loam the values of γ decreased slightly for the wetter boundary conditions (left to right); for the loamy sand the trend was reversed with a larger change in γ. The effect of the initial reduced water content Θn is given for loamy sand and silt loam soil in Figure 6 with x0 = 10 cm, Θ0 = 1. The loamy sand has increasing γ values for wetter initial conditions; the silt loam shows a small decrease as Θn increases.
 A final example compares iW,2D the “2-D Wooding equation” (7), with the results of Basha [1994, Table 1] for steady state infiltration for a Gardner soil. The results (Figure 7) agree quite well for αx0 greater than about 0.4. For the smaller values of αx0, qW,2D is larger than Basha's solution which is assumed “correct.” This is somewhat expected, based on Weir's  findings showing that Wooding's  results became less reliable as the radius of a disc source, in that case, became small.
5. Discussion and Conclusions
 A two-dimensional infiltration equation for strip sources has been investigated. The main conclusion is that the differences between the cumulative infiltration per unit source area for the two-dimensional strip and the one-dimensional sources are linearly related to time. This was shown for all six contrasting soils, all widths considered and all initial and boundary conditions.
 Once γ and I1D are known, I2D can be calculated easily by adding the edge effect. Not only are the computations much less tedious than for a two-dimensional numerical solution, but infiltration amounts can easily be computed independently for any time considered without running a full simulation. Of course, the subsurface water distribution is not found which could be a disadvantage. However, there is a strong advantage when massive amounts of calculations are required such as for many sensitivity analyses, inverse estimations of soil hydraulic parameters or in surface-subsurface flow during furrow irrigation [e.g., Abbasi et al., 2003; Wöhling et al., 2004]. Thus the new method provides a simple way to solve two-dimensional infiltrations similar to the simplicity of analytical solutions. Furthermore, this model can be implemented in spreadsheet software potentially providing a simple framework for many applications [e.g., Wraith and Or, 1998]. Better methods are needed in order to refine estimates of γ. This is a subject for future research. One possible approach is to use artificial neural networks, which is beyond the scope of this study.
 This work was supported by BARD (project grant agreement US-3662-05R) and Western Research Project W-1188.