EFFECTIVE HYDRAULIC CONDUCTIVITY FOR A SOIL OF VARIABLE PORE SIZE WITH DEPTH

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1 Nigerian Journal of echnology (NIJOECH) Vol. 34 No. 3, July 05, pp Copyright Faculty of Engineering, University of Nigeria, Nsukka, ISSN: EFFECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF WIH DEPH * DEPARMEN OF CIVIL ENGINEERING, UNIVERSIY OF NIGERIA NSUKKA, NSUKKA, ENUGU SAE. NIGERIA address: chidozie.nnaji@unn.edu.ng ABSRAC wo models were derived for the estimation of effective hydraulic conductivity (K e) ) of a soil layer ased on exponential and inverse square variation of hydraulic conductivity with soil depth. Darcy s law was applied to a vertical soil stratum sudivided into a finite numer of layers. he relationship etween K e and layer thickness is of quadratic form with R.As the layer thickness increases, the values of K e for the exponential model increases drastically, exceeding the K e estimate of the power model. he percentage difference etween the two models assumes an asymptotic form to the y-axis at a percentage difference of 5%, as the size of layer approaches zero. power model gives lower estimates of K e than the exponential model within soil depth range of.5m D 3.m for n = ; 0.8m D.9m for n = 50; and 0.7m D.9m for n = 00 and aove. For very shallow soil stratum (D ), the exponential model gives etter and more realistic estimates of K e than the power model; while for medium to deep soil stratum (D ), the power model gives etter estimates of K e. Keywords: hydraulic conductivity, soil, infiltration, permeaility, water. INRODUCION Accurate determination of hydraulic conductivity is very crucial for infiltration and runoff estimation. Factors which affect water infiltration in the soil include hydraulic conductivity, wetting front and soil sorptivity [], ponding depth [], depth to water tale [3] and impact of rain drops [4]. Amongst these factors, the importance of hydraulic conductivity cannot e overemphasized. Hydraulic conductivity is a very important soil parameter used y the Green-Ampt model for estimation of infiltration and runoff. It is asically an intrinsic property of the soil and [5] summarized the hydraulic conductivities of various textural classes of soil. Selker [6] oserved that though soil hydraulic conductivity decreases with soil depth from the surface, yet many infiltration models ignore this fact. Beven [7] was the first to apply this fact to a small catchment y expressing oth hydraulic conductivity and soil moisture content as a function of soil depth as: N O (P) = N (R S P) and U O (P) = U (R S P) V. Here, K s is the saturated hydraulic conductivity, θ s is the saturated moisture content, Z is increasing depth of soil for a soil of total soil depth (D) while K *, θ * n, and m are site specific soil parameters to e fitted to soil data. he paper [8] recognized the fact that heterogeneity of hydraulic conductivity is a significant driver of net asin infiltration. One would ordinarily assume that decrease in soil permeaility with depth is as a result of decreasing porosity resulting from greater packing density of soil particles. However, the experimental results of [9] refute this supposition. Selkar [6] also found that hydraulic conductivity is affected y characteristic pore size rather than porosity. He oserved that as pore size decreases, hydraulic conductivity decreases with square of pore size. Askari, et al [0] noted that hydraulic conductivity is influenced y the presence of organic matter and montmorillonitic mineral materials. While there exists a gradual variation of hydraulic conductivity through a homogeneous soil profile, [] noted that soil hydraulic properties can vary in a nested fashion as a result of surface disturances due to tillage, pore size distriution due to structural cracks and root development and decay, textural layering and geology. his is usually the case in agricultural soils. Variaility in soil hydraulic conductivity is more pronounced and sporadic in agricultural soils as a result of alterations of soil * Author el

2 EFFECIVE FECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF O WIH DEPH structure and properties during agricultural activities. It was in recognition of this that [4] employed empirical approach to estimate effective hydraulic conductivity under meadows conditions. Leconte and Briseette [] also oserved that the impact of rain drops on are soil surfaces give rise to stale crust layer, having a saturated hydraulic conductivity significantly lower than soil elow it. Having adequately estalished that hydraulic conductivity of soils is as variale as many other soil properties, the ojective of this paper is to otain simple and easy to apply expressions for effective hydraulic conductivity in a soil of uniform profile. he traditional infiltration models use a constant hydraulic conductivity to evaluate infiltration and this possily accounts for disparity etween calculated and field infiltration results.. MEHODOLOGY Figure elow shows the vertical profile of a homogeneous soil stratum of depth D whose hydraulic conductivity gradually decreases in the vertical direction. his stratum has een sudivided into n minute layers of thickness Z, each of which is assumed to have a constant hydraulic conductivity. he hydraulic conductivity of each minute layer is represented y the average of the hydraulic conductivities of the immediate upper and lower orders. First we derive the expression for effective K for n finite layers of clearly distinct K. Figure : Soil Stratum Sudivided Into Layers According to Darcy s law, X = NY Zh () Z\ herefore, Zh = Z\ = P. he total headloss ]^_ ]^_ through the layers is given y dh = dh + dh + dh 3 +..dh n. Hence, for equal layers of soil: X(P) N c Y = XP N d Y + XP N e Y + + XP N Y () P N c Y = P N d + P N e + + P N (3) N c = h N i (4) Since a representative value of K is required for each layer, it can safely e assumed that for the infinitesimal layer etween any two oundaries, say 0 and, N k d = N k + N d ;.N d e = N d + N e ;. N l e = N e + N l i ;.N = N + N i (5) Sustituting Equation (5) into Equation (4), we have: = h (6) N c N + N i An expression is needed for the variation of K with depth. wo cases were considered: (i) a power law - N(R) = N k (mr) je and (ii) an exponential function - N(R) = N k nop (SmR)as given y [6].. Power Law Beginning from the first layer and using n layers such that D = nz, N d = N k mp je, N e = N k (mp) je, N l = N k (m3p) je Hence N = N k (mp) je (7) Also i N = N k m je P je (q S ) je + N k m je (Pq) je ( 8) i N = N k m je P je r q e + ( S q) es (9) Sustituting Equation (9) in Equation (6): = h N c N k m je P je r (0) q e + ( S q) es = me P e h N c N k r () q e + ( S q) es Recalling that D = nz: he term d i t + = me P e N c d v u (vwx)t e N k h r q e + ( S q) es () y can e simplified as follows: q e + ( S q) e = qe S q + q e (q e S q + ) Hence, Equation () ecomes: (3) Nigeria Journal of echnology Vol. 34 No. 3, July

3 EFFECIVE FECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF O WIH DEPH = me P e N c e h qe (q e S q + ) N k q e S q + h qe (q e S q + ) q e S q + h qe (q e S q + ) q e S q = h(qe S q) As n ecomes very large, the solution converges, (5) (4) he expression d e (qe S q) can e split into two as follows: h(qe S q) = hq e + hq} (6) he two sum of series can further e simplified as follows: hq = hq e = ( + ) ( + )( + ) 6 (7) (8) Hence Equation (4) ecomes: = me R e ( + ) ( + )( + ) N c l ~ + (9) N k 6 Let the layer thickness =α N c = 3N k m e R e ~ e e S (0) N c = 3N k m e α e e S (). Exponential Function Following the steps already introduced for the power function aove, the exponential variation of hydraulic conductivity (K) through the soil profile for the n th layer can e expressed as: N = N k exp ƒsm(p) () i N = N k exp ƒsmp(q S ) + exp(smpq) (3) Sustituting Equation (4) in Equation (4), we otain: = h N c exp(smpq)[ + exp(smp)] [ + exp( βd / n) ] (4) nk0 Ke = n (5) βdi exp i= n he numerator can further e simplified as follows: hexp r mrq s = exprmr s + expr4mr s + expr 6mR s + expr8mr s + + exp (βd) (6) Equation (6) can easily e recognized as the sum of a geometric progression with the following properties. α = ˆ = expr mr s (7) Hence hexp r mrq s = exprmr S exp(mr) s S expr mrq Š (8) s Sustituting Equation (7) in Equation (5) yields N k [ S exp] N c = expr mr (9) s S exprmr s N k N c = [ S exp(mr)] exprsmr s S expr mr s (30) In terms of layer thickness, we have N k N c [exp(sβα) S exp(βα)] (3) [ S exp(mr)] Equations () and (3) are the expressions for effective hydraulic conductivity for a uniformly varying saturated soil, assuming power and exponential functions respectively. 3. RESULS AND DISCUSSION he models were illustrated using the experimental data otained y [9]. Of course this was for a specific soil. he use of Equations () and (3) will require knowing the values of the soil parameters K 0 and β. For the power function (Equation 30), these parameters can e otained y rearranging the expression N(R) = N k (mr) je as follows: \(N) = \ r P es + \(N km je ) (3) he plot of Ln (K) versus \ u d ty is a fairly straight line with intercept \(N k m je ) as shown in Figure. Hence, the value of N k m je can e determined. From the plot shown in Figure, the \(N k m je ) = 0.98 so that N k m je =.67m 3 /hr. For the exponential model, the soil parameters were determined y rearranging the expression N(R) = N k exp(smr) as follows: \(N) = \N k S mp (33) Nigeria Journal of echnology Vol. 34 No. 3, July

4 EFFECIVE FECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF O WIH DEPH Figure : Plots for Determination of Soil Parameters for the Power Model Figure 3: Plots for Determination of Soil Parameters for the Exponential Model From the plot of LnK versus Z shown in Figure 3 elow, the values of K 0 and β can e respectively determined as 6.4m/hr and.009/m. In order to ascertain the effect of layer size on the values of K e, plots of K e versus layer thickness were produced as shown in Figure 4. As the layer thickness ecomes inœinitesimal (α 0), the plot of K e versus layer thickness ecomes asymptotic to the layer thickness axis for the two models. However, as the layer thickness increases, the values of K e for the exponential model increases drastically, exceeding the K e estimate of the power model. Figure 4 also shows that the relationship etween K e and layer thickness is of quadratic form with R. As is to e reasonaly expected, etter estimates of K e can e otained y sudividing the soil stratum into infinitesimal layers. o further understand the effect of soil stratum sudivision on accurate estimation of K e, a plot of K e versus numer of layers (n) was produced a shown in Figure 5. Figure 5 shows that the power model consistently gives lower values of K e than the exponential model. However, the rate of convergence is identical in oth cases. Both models converged when the soil was sudivided into twenty (0) layers, with further sudivision (up to n = 00) resulting in no further improvement in K e values. Also as n approaches, the K e estimate of the power model tends to infinity. his reveals a structural deficiency in the underlying assumption that hydraulic conductivity varies with the inverse square of depth the relationship is not that simple. Figure 6 is a plot of numer of layers and size of layers versus percentage difference etween the K e values of the two models. As the size of layer increases, the percentage difference increases, approaching 5% at a layer size of m and n =. However, percentage difference etween the two models assumes an asymptotic form to the y-axis at a percentage difference of 5%, as the size of layer approaches zero. he percentage difference drastically reduces to approximately 5% at a layer size of 0.36m. his shows that the models ecome very close as layer size ecomes infinitesimal, ut can never completely agree. Likewise, as the numer of layers increases, the percentage difference approaches 5% asymptotically; and as the numer of layers increases, the percentage difference increases drastically. A close look at Equation (0) shows that it converges to a definite value as n. his eliminates the need for sudividing the soil stratum into layers and shorten the computation time. Hence as n, N c = 3N k m e R e (34) For the soil parameters used in this research, the value of K e otained y using Equation (33) is.477m/hr. If the K e were to e determined y sudividing the soil stratum, it would require hundred layers for the power law model to yield this value. he exponential model gave a value of.608m/hr using hundred layers. Figure 5 shows that oth the power and exponential models tend towards the K e estimate of Equation (33). Hence it can e surmised that Equation (33), though simple in nature, gives a reliale estimate of K e. Figure 7 shows the percentage difference etween the K e values of the two models using different numer of layers and layer sizes, and the K e estimate of Equation (33). Figure 7a shows that the percentage difference for the power model quickly approaches zero even at an n value of 30 and is proportional to the inverse of the square of the numer of layers. Douling the numer of layers, reduces the percentage difference y a quarter (5%). On the other hand, douling the numer of layers for exponential model reduces the percentage difference y aout 30%; ut the percentage difference does not exceed 5% even at an n value of 000. Generally, the exponential model gives higher values of percentage Nigeria Journal of echnology Vol. 34 No. 3, July

5 EFFECIVE FECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF O WIH DEPH difference than the power model, for the same numer of layers. Figure 4: Effective Hydraulic Conductivity versus Layer hickness. Figure 5: Effective Hydraulic Conductivity versus Numer of Layers Figure 6: Percentage Difference etween the Power and Exponential Models Estimate of Ke Further investigations were made in order to understand the ehavior of the two models with respect to the depth of soil stratum and the numer of sudivisions. Generally, the K e estimates of the two models ecome increasingly smaller as the soil depth increases. Figure 8 clearly shows that the power model gives lower estimates of K e than the exponential model within soil depth range of.5m D 3.m for n = ; 0.8m D.9m for n = 50; and 0.7m D.9m for n = 00 and aove. For all other ranges of depth, the exponential model gives lower K eestimates. Besides for shallow soil stratum, the power model cannot e relied upon for good estimates of K e ecause the curve ecomes asymptotic to the K e axis as soil depth reduces. his is especially the case when very few sudivisions are used. Moreover, as soil depth increases, the K e estimates of the exponential model approach zero. his too is unreliale. Hence it can e clearly seen that for very shallow soil stratum (D ), the exponential model gives etter and more realistic estimates of K e than the power model; while for medium to deep soil stratum (D ), the power model gives etter estimates of K e. 4. CONCLUSIONS wo simple models for estimating hydraulic conductivity of uniform soils have een derived. he result of this research has shown that hydraulic conductivity does not exactly follow the inverse square law for shallow soils. Moreover, the two models are complementary to each other for varying ranges of soil depth. Besides, an infinite numer of soil layers is not necessarily required for the solution to converge. Overall, the power model is more advantageous than the exponential model in that it converges to a finite value for an infinite numer of sudivisions. his gave rise to Equation (33) which does not require layer sudivision and yet yields good results. 5. REFERENCES [] Regalado, C. M., Ritter, A., Alvarez-Benedi, J. and Munōz-Carpena, R. Simplified method to estimate the Green-Ampt wetting front suction and soil sorptivity with the Philip-Dune falling head permeameter, Vadoze Zone Journal, Vol. 4,005, pp9 99. [] Warrick, A. W., Derihun, D. Sanchez, C. A. and Furman, A. Infiltration under variale ponding depths of water, Journal of Irrigation and Drainage Engineering Vol. 3, Numer 4, 005, pp [3] Risse, L. M., Nearing, M. A. and Savai, M. R. Determining the Green-Ampt effective hydraulic conductivity from rainfall-runoff data for the WEPP mode, ransactions of the ASAE, Vol. 37, Numer, 994, [4] Zhang, X. C., Nearing, M. A. and Risse, L. M. Estimation of Green-Ampt conductivity parameters: Part II Perennial crops, ransactions of the ASAE, Vol. 38, Numer 4, 995, pp [5] Rawls, W. J., Brakensiek, D. L. and Miller, N. Green- Ampt infiltration parameters from soil data, Journal of Hydraulic Engineering, Vol. 09, Numer, 983, pp6 70. Nigeria Journal of echnology Vol. 34 No. 3, July

6 EFFECIVE FECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF O WIH DEPH Figure 7: Comparison of the Models Estimates of K e with that of Equation (33) Figure 8: Plots of K e versus Soil Depth Using Different Numer of Soil Layers. [6] Selker, S. J. Green and Ampt infiltration into soils of variale pore size with depth, Water Resources Research, Vol. 35, Numer 5,999, pp [7] Beven, K. Infiltration into A Class of Vertically Non- Uniform Soils, Hydrological Sciences Journal, Vol. 9, 984, pp [8] Craig, J. R., Liu, G. and Soulis, E. D. Runoff-infiltration partitioning using an upscaled Green-Ampt Nigeria Journal of echnology Vol. 34 No. 3, July

7 EFFECIVE FECIVE HYDRAULIC CONDUCIVIY FOR A SOIL OF O WIH DEPH solution, Hydrological Processes, doi:0.00/hyp.760, 00. [9] Childs, E. C. and Byordi, M. (969), he vertical movement of water in stratified porous material, I, Infiltration, Water Resources Research, Vol. 5, 969, pp [0] Askari, M. anaka,., Setiwan, B. and Saptomo, S. K. Infiltration characteristics of tropical soil ased on water retention data, Journal of Japan Society of Hydrology and Water Resources, Vol., Numer 3,008, pp5 7. [] Mohanty, B. P. and Zhu, J. Effective hydraulic parameters in horizontally and vertically heterogonous soils for steady-state landatmospheric interaction, Journal of Hydrometeorology, Vol. 8, 007, pp [] Leconte, R. and Brisseette, F. Soil moisture profile mdel for two-layered soil ased on sharp wetting front approach, Journal of Hydrologic Engineering, Vol. 6, Numer, 00, pp4 49. Nigeria Journal of echnology Vol. 34 No. 3, July