Published online 12 March 2007
Published in Soil Sci Soc Am J 71:289-297 (2007)
DOI: 10.2136/sssaj2006.0123
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SOIL PHYSICS
Analytical Solutions to Evaluate the Stream Tube Approach for Field-Scale Modeling of Evaporation
F. J. Leija,
A. Sciortinoa and
J. H. Daneb,*
a Dep. of Civil Engineering, California State Univ., Long Beach, CA 90840
b Dep. of Agronomy & Soils, Auburn Univ., Auburn, AL 36849
* Corresponding author (danejac{at}auburn.edu).
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ABSTRACT
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Quantifying upward flow across larger areas of the landscape is important for water resources management and research involving global exchange processes. The stream tube or parallel column model may be used for this purpose. Columns should be narrow enough to account for variability of local parameters and wide enough to minimize the effect of lateral flow. To provide guidance with the selection of minimum column width, an approximate analytical solution was obtained for steady flow in a medium composed of two vertical soil columns. The normalized matric flux potential, 
, was used as a dependent variable. The two-dimensional solution consisted of a one-dimensional term and a perturbation term to ensure continuity in water flux and soil pressure head, h. Comparison of two-dimensional and one-dimensional stream tube results showed little difference in h and , except near the interface. The greatest effect was found for finer textured soils. An analytical expression for the lateral flux was obtained from the solution for . For "wet" conditions involving a shallow water table and high surface water contents, the lateral flux was found to be no more than 5% of the evaporative flux. For "dry" conditions with miniscule fluxes, the evaporative and lateral fluxes may be of the same order of magnitude; application of the stream tube model could involve considerable error. The analytical solutions are convenient to evaluate the impact of stream tube width for different evaporation scenarios and also to verify more elaborate numerical schemes.
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INTRODUCTION
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Simulating the upward flux and the soil water pressure head during evaporation across large areas of the landscape is important for the management of water resources and applications involving remote sensing and exchange processes. It is well known that a lack of knowledge of the vadose zone properties, i.e., the water retention and conductivity functions, and the nonlinearity of these functions complicate large-scale modeling near the soil surface (e.g., Dooge and Bruen, 1997; Stewart et al., 1998). Vadose zone properties exhibit considerable spatial variability (Nielsen et al., 1973; Vauclin et al., 1994). There are two strategies to obtain soil hydrological information at larger application scales: (i) effective vadose zone properties are defined from local-scale parameters and vadose zone modeling is subsequently done at the desired application scale (a priori aggregation), or (ii) vadose modeling occurs at the local scale and output results for the larger scale are obtained from local model results (a posteriori aggregation). An "effective" medium approach has been used in several large-scale evaporation studies (Feddes et al., 1993; Mohamed et al., 2004), but a posteriori aggregation is generally more accurate (Soria et al., 2003) This type of aggregation is used when a heterogeneous field or watershed is viewed as an assembly of stream tubes or parallel columns in which water flow (or solute transport) is quantified with simple one-dimensional models (e.g., Dagan and Bresler, 1979; Rubin and Or, 1993; Toride et Leij, 1996).
Because the stream tube concept does not allow for lateral flow, a posteriori aggregation involves errors that will depend on stream tube size, vadose zone parameters, and physical conditions. It is clear that the error will increase if columns are narrower and longer or exhibit a greater contrast in model parameters. Especially the direction parallel to the surface is of interest for the discretization of a field or watershed into stream tubes. The area of a tube should not exceed the pixel size for which the soil properties and model parameters are effectively uniform. To assess the conflicting demands on stream tube width, it is desirable to compare modeling results that do account for lateral flow (e.g., a two-dimensional model) with the one-dimensional results. We used analytical modeling of water flow in the vadose zone of two vertical soil columns. Although analytical solutions require considerable simplifications of the mathematical problem and the physical system, they are convenient and instructive to evaluate the sensitivity of modeling results to column dimensions, the evaporation condition at the soil surface, the depth of water table, and the magnitude of vadose zone parameters.
Analytical solutions have been derived for evaporation by Gardner (1958), Warrick (1988), and Yuan and Lu (2005), among others. The solutions, which often account for water uptake by plants, have typically been obtained for a uniform soil with idealized vadose zone properties. Very few analytical solutions exist for water flow in layered media. Srivastava and Yeh (1991) considered infiltration in stratified soils. The interface conditions were simplified by considering the medium to be composed of semi-infinite soils, i.e., the properties of downstream layers do not affect the flow in overlying soil layers. No such simplification can be made if the flow is predominantly parallel to the interface. Warrick and Knight (2002) used an equal capillary length for two flow domains. This greatly simplifies the interface condition but does not provide a realistic contrast between soil layers.
The objective of this study was to derive an approximate analytical solution for the soil water pressure head and the lateral flux during steady evaporation from the groundwater in a composite medium consisting of two vertical soil columns. The solutions are applied to investigate the discretization of stream tubes to represent flow in nonuniform porous media for large-scale hydrological modeling. If the (two-dimensional) solutions yield similar results as one-dimensional solutions, there are only slight errors involved in adopting the stream tube concept for modeling water flow in large-scale systems.
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SOIL WATER PRESSURE HEAD
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The governing equation for two-dimensional steady flow in an anisotropic medium is given according to Richards (1931):
 | [1] |
where h is soil water pressure head [L], which is negative for unsaturated conditions, K is the hydraulic conductivity [L T–1], and z and y are the vertical and horizontal coordinates [L]. We will use z as depth from the soil surface. We consider evaporation from a composite medium made up of two uniform parallel soil columns. The mathematical conditions for which Eq. [1] is to be solved are sketched in Fig. 1
. The upward flux is negative. Within the scope of this study, an upward or evaporative flux is simply established by imposing a fixed pressure head, he, at the soil surface at z = 0. Of course, actual evaporation will also depend on atmospheric conditions and there will be vapor movement in the soil. Brutsaert (1982), among others, discussed evaporation in greater detail. Furthermore, no attempt is made to account for water uptake by plants by including a sink term. The water table is located at depth z = L for both media. For the horizontal boundary condition away from the interface, we assume a zero gradient at infinity to circumvent the problem of trying to formulate a condition at the finite distances that denote column widths (i.e., y = –W1 or y = W2). At the interface, we impose continuity of water flux and pressure head.
We follow the common practice of using the matric flux potential as a dependent variable and to assume an exponential conductivity function (e.g., Pullan, 1990; Warrick, 2003):
 | [2a,b] |
where 
is the matric flux potential [L2 T–1] and 
is a coefficient [L–1], which is inversely related to the capillary length. Conversions between matric flux potential and pressure head or conductivity are done according to
 | [3a,b] |
Equation [1] now becomes a linear differential equation:
 | [4] |
For the formulation of boundary conditions it is convenient to use the following normalized dependent variable:
 | [5] |
where e and w are the matric flux potential at the evaporating surface and at the water table.
One-Dimensional Problem
It is instructive to first consider the one-dimensional problem:
 | [6] |
subject to
 | [7a,b] |
The solution is given by
 | [8] |
This solution will later be used to solve the two-dimensional problem. The normalized conductivity function, K/Ks, is illustrated for = 0.5, 1, 2.5, and 10 m–1 in the range –10m < h < 0 (Fig. 2
). Values of for actual soils can be found in Leij et al. (1997) and Warrick (2003), among others. Table 1 shows representative values of and Ks for a fine-, intermediate-, and coarse-textured soil (Soils A, B, and C, respectively). For our calculations, we will assume that he is no less than –10 m; in the field, appreciable evaporation may occur at much lower pressure heads. Even for fine-textured soils, the hydraulic conductivity would decrease by at least three orders of magnitude from h = 0 to –10 m (cf. Leij et al., 1996). If a lower value were selected for he, the upper part of the soil would be so dry that upward flow in the liquid phase would become negligible (Jury and Horton, 2004). The magnitude of the upward flux, qz, will be discussed below.
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Fig. 2. Normalized hydraulic conductivity, K/Ks, as a function of pressure head, h, for exponential conductivity model with = 0.5, 1, 2.5, and 10 m–1.
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Table 1. Hydraulic properties of soils and upward flux, qz, for three combinations of surface pressure head, he, and depth of water table, L.
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Figure 3
shows profiles of the normalized matric flux potential and pressure head for these values for a 2-m-deep profile using he = –10 m. For fine-textured media, with smaller , decreases gradually with distance from the water table (z = 2 m). On the other hand, decreases more rapidly for coarse-textured media with as extreme a step change from = 1 to 0 just above the water table when 
. The corresponding pressure head profile is bounded by the hypothetical cases: (i) = 0 with maximum upward flow in a saturated medium [i.e., K(z) = Ks] and (ii) with no upward flow and a unit gradient opposite to the gravitational head [i.e., K(z) = 0]. A substantial part of the drop in pressure head occurs near the surface, especially for coarser textured media. The head profiles will obviously change with depth of water table or he.
Two-Dimensional Problem
The two-dimensional problem is given by
 | [9a,b] |
where k equals 1 for y < 0 and 2 for y > 0 (i.e., the left and right column, respectively). This equation is solved subject to the following conditions:
 | [10a,b] |
 | [11a,b] |
 | [12a,b] |
 | [13] |
 | [14] |
These conditions follow directly from those sketched in Fig. 1 except for interface condition Eq. [13]. Appendix A contains a further discussion of this condition and the formulation for the auxiliary function 
(z). Solution of this problem, obtained as outlined in Appendix B, yields
 | [15] |
where
 | [16] |
 | [17] |
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After obtaining from Eq. [15], the pressure head follows from Eq. [5] and [3b]. Figure 4
shows the distribution of and h for the three possible soil combinations; the finer textured soil is to the left of the interface (i.e., y < 0). The vertical distributions away from the interface (e.g., y = ±1 m) follow directly from the one-dimensional solution (Fig. 3). In the coarser soil, decreases rapidly near the water table, whereas the greatest decrease for h occurs near the soil surface. At the interface (y = 0), there is a clear discontinuity of . Except for the AB medium, the horizontal gradient in each soil (tube) is quite small and dividing the soil into stream tubes with discrete properties seems to have a negligible effect on vertical soil water transport. The pressure head profiles show a more pronounced gradient in the horizontal direction, especially near the interface in the coarser soil. At the interface, h is approximately continuous. It appears that the effect of layering diminishes rapidly away from the interface. The magnitude of the horizontal flux is a more precise attribute for the effect of layering. An expression for both the vertical (upward) and horizontal (lateral) flux is given below.
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Fig. 4. Distribution of normalized matric flux potential, , and pressure head, h, for the three soil combinations assuming surface pressure head he = –10 m and water table depth L = 2 m.
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We further examine the (y) and h(y) distributions by taking slices out of the contour maps in Fig. 4 at depths z = 1 and 1.8 m for the AB and BC media. Especially for the coarser media, will be very small closer to the surface. Figure 5
shows the lateral profiles and also the results for the one-dimensional solutions of the single stream tubes (dashed lines). Notice the difference in scale for the axes. For the finer textured AB medium, y extends from –10 to 10 m while for the coarser BC combination, y is between –1 and 1 m. A different scale is also used for both dependent variables and h. The normalized matric flux potential (and also ) is discontinuous across the interface, whereas the pressure head h is approximately continuous. At z = 1.8 m, where the medium is close to saturation (h > –20 cm), there is a discernible difference between the one- and two-dimensional solutions for h. This difference remains for the drier conditions at z = 1 m for the finer textured AB medium. The one- and two-dimensional solutions do not converge for up to several meters from the interface. For the AB medium there is also a difference between the one- and two-dimensional profiles due to lateral flow. For the BC medium, where values are smaller because of the coarse Soil C, there are only very slight differences between one- and two-dimensional solutions.
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Fig. 5. Lateral normalized matric flux potential (y) and pressure head h(y) profiles at depth z = 1 and 1.8 m assuming surface pressure head he = –10 m and water table depth L = 2 m.
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SOIL WATER FLUX
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 | [18] |
where qz [L T–1] is the volumetric flux per cross-sectional horizontal distance (and per unit width) with qz < 0 for upward flow. For the one-dimensional problem, the following expression can be readily obtained with Eq. [8]:
 | [19] |
This expression may also be used as an approximation for the two-dimensional problem, for which there is no convenient solution of the upward flux. Values for qz are given for Soils A, B, and C for three combinations of he and L (Table 1). For relatively wet conditions (i.e., he = –3 m or "field capacity" and L = 2 m), the gravitational gradient mitigates the upward flux. If he is reduced to –10 m, the (absolute) evaporative flux increases for the finer textured soils despite the somewhat drier conditions. If the soil becomes even drier by lowering the water table to L = 5, the upward flux becomes rather small. In particular for Soil C, the hydraulic conductivity near the surface is very low and, consequently, the hydraulic gradient will be small throughout the profile, except in the upper part (cf. Fig. 3b).
The horizontal flux for intra- and intercolumn water movement is defined as
 | [20] |
where qk [L T–1] is the volumetric flux per cross-sectional vertical distance (and per unit width) at an arbitrary time. Differentiation of Eq. [15] and use of Eq. [9b] yields
 | [21] |
where uk(z) and Ck(z) were defined by Eq. [16] and [17].
Figure 6
displays the distribution of qy for the three soil pairs that we have been considering. The horizontal flux is from the coarser soil on the right to the finer soil on the left. The largest flux occurs at the interface. For the AB medium, the maximum occurs approximately halfway between the water table at z = 0 and the soil surface at z = L = 2 m. Because the finer Soil A has a relatively high conductivity, the flux contours extend somewhat farther to the left of the interface than to the right. Near the interface, the lateral flux is sizeable and may even exceed the upward flux for Soil B (cf. Table 1). For the other two pairs that include Soil C, the horizontal flux is considerably smaller (notice the 10-fold change in qy for every unit of the legend). Horizontal flow occurs mostly near the water table between z = 1.5 and 2 m. The upward flux for Soil C is minuscule; even the small values for the lateral flux may still be several orders of magnitude larger than the vertical flux in the one-dimensional case of Soil C. Clearly, the stream tube concept would introduce considerable error in the upward flux when applied for this scenario (he = –10 m and L = 2 m with a relatively coarse medium). Note that the error in evaporation at the stream tube interface may be obtained from the integral qy at the interface.
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Fig. 6. Distribution of horizontal flux, qy, for the three soil combinations assuming surface pressure head he = –10 m and water table depth L = 2 m.
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The stream tube concept is based on the assumption that horizontal flow is negligible compared with vertical flow. This assumption was examined with contours of the ratio of horizontal to vertical flux, |qy/qz|, for three combinations of he and L for the AB medium (Fig. 7
). The values for qz are for one-dimensional uniform media, and are given in Table 1. For relatively "wet" conditions (i.e., he = –3 m and L = 2 m), the upward flux is constrained because the relatively small gradient in soil pressure is mostly cancelled by the gravitational head. Horizontal flow is not directly affected by gravity, but is relatively small because both soils are relatively wet. The ratio |qy/qz| exhibits the same distribution for both media. This ratio remains similar for somewhat drier conditions (he = –10 m and L = 2 m), the slight increase in qz is accompanied by an increase in lateral flux qy—especially for the coarser Soil B. In both soils, the error may be >40% near the interface. At 2 m away from the interface, the error will be, at most, 2.5% for Soil A and 5% for Soil B. For the driest conditions (he = –10 m and L = 5 m), there is very little upward flow in Soil B and any lateral flow represents a large error in the coarser Soil B, whereas the stream tube concept involves little error in the finer Soil A. The concept may not be appropriate for these conditions, but one should also consider the small value for qz and the uncertainty of describing K with Eq. [2b] for dry conditions.
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Fig. 7. Distribution of lateral with respect to one-dimensional upward flux, |qy/qz|, for the AB medium assuming: (a) surface pressure head he = –3 m and water table depth L = 2 m, (b) he = –10 m and L = 2 m, and (c) he = –10 m and L = 5 m.
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It is also of interest to quantify how the upward flux—with evaporation at z = 0 being a special and important case—changes due to the presence of a different soil nearby. No closed-form solution exists for the upward flux in the two-dimensional problem. Instead, we estimate it by numerical approximation of the last term of Eq. [18] using the analytical results for the matric flux potential. Figure 8
shows the vertical distribution of qz for the three scenarios from Fig. 7 at 25 cm from the interface in Soils A and B. The results for the one-dimensional problem are included as dashed lines. The upward flux may deviate considerably from the one-dimensional value due to lateral water flow (cf. Fig. 7). Because of mass conservation, increases in the upward flux for soil are accompanied by decreases for Soil B and vice versa (cf. Fig. 8a and 8b). The fluctuations diminish at greater distance from the interface (i.e., wider stream tubes). There is little flow for the dry conditions when he = –10 m and L = 5 m, and the upward flux in Soil A remains virtually constant while there is considerable fluctuation—notice the logarithmic scale—for the coarser Soil B (Fig. 8c). Because of the mathematical description of the evaporation problem with a uniform boundary condition at z = 0, except for the discontinuity at y = 0, the evaporation flux is always very similar to that for the one-dimensional case.
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Fig. 8. Upward flux, |qz|, as a function of depth in the AB medium at y = –0.25 m (Soil A) and y = 0.25 m (Soil B) from numerical approximation of Eq. [18] (two-dimensional case) and Eq. [19] (one-dimensional case): (a) surface pressure head he = –3 m and water table depth L = 2 m, (b) he = –10 m and L = 2 m, and (c) he = –10 m and L = 5 m.
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SUMMARY AND CONCLUSIONS
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We obtained an approximate analytical solution for steady-state evaporation of two vertical soil columns to assess errors in modeling evaporation across larger areas of the landscape with the stream tube or parallel column concept. Other applications may also be envisaged such as the verification of numerical schemes. Figure 1 illustrates the mathematical problem. The Richards equation was linearized by assuming a Gardner-type hydraulic conductivity function (cf. Fig. 2) and by using a normalized matric flux potential, , as the dependent variable. As a starting point, we considered one-dimensional flow during evaporation in a uniform medium (Fig. 3). The one-dimensional solution constitutes the extremes for the two-dimensional solution of . At the (vertical) interface we stipulated that soil water flux and pressure head be continuous. Approximate continuity of pressure head, h, was imposed by using a geometric mean of the Gardner parameter in the one-dimensional solution (Appendix A). The two-dimensional solution consisted of the one-dimensional solution and a perturbation term. The latter was obtained by solving a simple ordinary differential equation subject to the lateral conditions (Appendix B).
The solutions were used to simulate h and in a composite medium consisting of three combinations of soils with a surface head he = –10 m and a depth of the water table L = 2 m. The conductivity parameters of the soils are given in Table 1. Discretizing the nonuniform soil into stream tubes does not greatly affect h and , except in the immediate vicinity of the interface (Fig. 4 and 5). The greatest effect was simulated for the finer textured Soil A, which is more apt to transmit and retain water during the evaporative conditions. The relative magnitude of the lateral flux is a more pertinent estimate for the error associated with the stream tube concept. The contours in Fig. 6 showed that only for the finer textured Soils A and B there was substantial water movement across the interface (i.e., between stream tubes). Because the evaporative flux for the other soil combinations is tiny, even the minuscule lateral flux that we predicted is relatively large. The solutions are convenient to evaluate different scenarios. Figure 7 considered different values for he and L. During "wet" conditions, the ratio of lateral to upward flux was similar for Soils A and B. During "dry" conditions, the lateral flux may be more than 40% of the upward flux in the coarser Soil B. Consequently, considerably more care should be taken to specify sufficiently short and wide stream tubes for the coarser soil. The changing behavior of the upward flux due to lateral flow was illustrated in Fig. 8.
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APPENDIX A
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Interface Condition
The extremes of the matric flux potential away from the interface are determined by the one-dimensional solution, i.e., Eq. [8] with equal to 1 or 2. Closer to the interface, the effect of the other stream tube will be noticeable and as |y| 0, the pressure head distributions will coalesce, albeit the matric flux potential will be discontinuous at the interface. We postulate that the pressure head distribution at y = 0 may be inferred from Eq. [8] by using the geometric mean of 1 and 2. Other averaging schemes can readily be used.
It follows from Eq. [2b] and [5] that
 | [A1] |
We can then obtain
 | [A2] |
The expression for h that needs to be substituted follows from using the hypothetical distribution for , i.e., with the average , in Eq. [8]:
 | [A3] |
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APPENDIX B
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Solution of the Mathematical Problem
The lower boundary condition given by Eq. [11] becomes homogeneous through the following transformation of the dependent variable:
 | [B1] |
The governing equation is now
 | [B2] |
subject to the conditions:
 | [B3a,b] |
 | [B4a,b] |
 | [B5a,b] |
 | [B6] |
 | [B7] |
The solution of this problem is assumed to resemble the one-dimensional solution plus a perturbation term to satisfy the interface conditions:
 | [B8] |
where uk is the solution of the homogeneous one-dimensional problem, which follows readily from Eq. [8] and Eq. [B1] and is given by Eq. [16], and 
k is a perturbation function that will vanish for greater distances from the interface at y = 0.
The governing equation for k is obtained by substituting Eq. [B8] into [B2]:
 | [B9] |
Because uk is the solution of the one-dimensional homogeneous problem and because any derivative of k with respect to z is likely to be small, we may rewrite Eq. [B9] as
 | [B10] |
The general solution of this equation is
 | [B11] |
where we already used the finality constraint, i.e.,
 | [B12a,b] |
We still have the two interface conditions:
 | [B13] |
 | [B14] |
These are used to obtain the expressions for coefficients C1 and C2 given by Eq. [17]. The solution for k obtained in this manner is then used with Eq. [B1] and [B8] to determine the final solution for given by Eq. [15].
We still need to ascertain that derivatives k with respect to z are negligible. The following error term quantifies the ratio of terms in Eq. [B9] containing a longitudinal derivative of k—given in square brackets—to those with the second and first derivative of uk with respect to z:
 | [B15] |
Explicit evaluation of the derivatives will be cumbersome, but the terms can be readily evaluated with mathematical software. Figure 9
shows a plot of the absolute value of e as a function of depth (0 < z < 2 m) at y = 0.01 m (i.e., just to the right of the interface) for the five scenarios considered in this study. The results were obtained with the Mathematica software (Wolfram Research, Champaign, IL). The vertical gradient of will be larger when there is greater upward flow or a greater contrast between the two soils. The error is typically <1%; in many cases, it is far less. This suggests that the contribution by to vertical transport is minor compared to u and that we were justified in dropping the derivatives of with respect to z.
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Fig. 9. Absolute value of error e for neglecting terms with vertical gradients in given by Eq. [B15] as a function of depth at y = 0.01 m for five scenarios of soils and boundary conditions.
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Received for publication March 20, 2006.
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REFERENCES
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ABSTRACT
INTRODUCTION
