Effective unsaturated hydraulic conductivity for one-dimensional structured heterogeneity

Transcription

1 WATER RESOURCES RESEARCH, VOL. 41, W09406, doi: /2005wr003988, 2005 Effective unsaturated hydraulic conductivity for one-dimensional structured heterogeneity A. W. Warrick Department of Soil, Water and Environmental Science, University of Arizona, Tucson, Arizona, USA Received 25 January 2005; revised 20 May 2005; accepted 14 June 2005; published 10 September [1] Effective hydraulic conductivities of unsaturated soils are defined for onedimensional structured heterogeneity. The heterogeneity is defined using homogeneous sublayers forming repeated unit cells of length L. The effective conductivity is defined as the steady downward Darcian velocity in an infinitely deep profile. The dependence of the hydraulic conductivity upon the pressure head is found by computing the average value of pressure head within the repeating unit cells for each effective conductivity. For a cell of length L approaching zero, the effective conductivity becomes the harmonic average of the individual sublayer hydraulic conductivities weighted according to the cell fraction each occupies. In that case, as the profile approaches saturation, the effective conductivity is the same as the well-known result for flow through an array of saturated layers. For a large cell length L, the effective hydraulic conductivity approaches an arithmetic average of the pressure heads which develop in each sublayer. Examples are computed for finite cell lengths L for both binary and tertiary systems. The effective hydraulic conductivity functions for the finite cell lengths fall within the envelope formed by the two limiting cases for small and large cell lengths. For the binary system, all of the hydraulic conductivity functions fall between the envelopes formed by the two hydraulic conductivity functions for the individual sublayer materials. Citation: Warrick, A. W. (2005), Effective unsaturated hydraulic conductivity for one-dimensional structured heterogeneity, Water Resour. Res., 41, W09406, doi: /2005wr Copyright 2005 by the American Geophysical Union /05/2005WR W Introduction [2] Typically, gravity dominates movement of water in the vadose zone, and consequently flow is predominantly vertical. Another common occurrence is that soil profiles tend to be stratified in horizontal layers due to depositional processes and development. Particularly above deep water tables, flow is dominated by effective unit gradient conditions, although small-scale effects occur within sublayers contained in the overall profile. Clearly, for saturated flow, effective hydraulic conductivity values exist for either normal flow or tangential flow through strata. However, this is not the case for unsaturated conditions. In that case, the hydraulic conductivity is a function of not only space but also pressure head. Defining scale-appropriate soil hydraulic properties to describe subsurface properties is a difficult challenge for a variable-saturated system. This is exacerbated by a scarcity of meaningful definitions for effective or average hydraulic properties. [3] For a one-dimensional profile, Warrick [2003] and Jury and Horton [2004] present integral expressions and a number of original references which address steady, vertical flow. A key early study, which is closely related, is by Gardner [1958] describing evaporation from a shallow water table for a homogeneous profile. His analysis was completed for several different forms of hydraulic conductivity functions for unsaturated soils. Subsequently, Gardner s work was extended to include evaporation through a layered profile by Willis [1960]. Warrick and Yeh [1990] presented detailed results for steady flow either upward or downward and discussed the pressure head profiles that generally develop in such systems. Generally, all of the above results use an integrated form derived directly from Darcy s law [e.g., Warrick, 2003, equation 5 141]: Z h dh z z 0 ¼ 1 q=k ; ð1þ h 0 where z is elevation, h is the pressure head, and K is the hydraulic conductivity, which is dependent on h. The Darcian volumetric flux q is taken positive for downward flow and negative for upward flow. At the elevation z 0,the pressure head is h 0. This expression is valid for all steady flow conditions, including upward flow, downward flow, and finite soil profiles. The hydraulic conductivity K is a function of h. Indirectly, K can be a function of z since equation (1) can be applied to individual layers, each of which has its own K(h) relationships. [4] Recently, Preuss [2004] examined flow of water and solutes through unsaturated flow in layered sediments. Borrowing from results useful for saturated soil, he presents effective vertical unsaturated hydraulic conductivities consisting of weighted harmonic averages of the individual conductivity functions for the layers and at the specified pressure head. Also following the same form as for satu- 1of6

2 W09406 WARRICK: EFFECTIVE UNSATURATED HYDRAULIC CONDUCTIVITY W09406 individual sublayers within the cell. An inverse relationship can also be taken in the form q ¼ F h avg ; p; g : ð4þ [7] The last expression is of the same form as is commonly used for a homogeneous soil when the unit hydraulic gradient relationship applies: q ¼ Kh ð Þ: ð5þ Figure 1. Binary (two sublayers) and tertiary (three sublayers) structured profiles. rated flow, for horizontal flow through the layers, an arithmetic average of the individual layers was taken. Finally, an effective conductivity tensor was represented by a diagonal matrix with entries corresponding to the arithmetic average for the horizontal principal axes and the harmonic average for the vertical principal axis. These are essentially the same forms used earlier by Mualem [1984]. [5] The objective of this study is to define effective hydraulic conductivity functions consistent with onedimensional steady vertical flow in a heterogeneous profile which is characterized by a repeating pattern of sublayers. For each effective hydraulic conductivity value, the corresponding pressure head is found by averaging over the profile. 2. Theory [6] We define a heterogeneous soil profile as a sequence of layers which form a characteristic unit cell of length L. Each cell consists of n sublayers with sublayer thicknesses b 1 L, b 2 L,...b n L and with b 1 + b b n = 1. Figure 1 illustrates a binary and a tertiary structure with n = 2 and 3. We assume that the soil profile is variably saturated with an average downward Darcian flow velocity of q. The depth of the profile is sufficient that the pressure heads are repeating over each cell length L. An average pressure head h avg for the cell is defined by h avg ¼ 1 L Z L hdl 0 ; ð2þ Thus we write an effective unsaturated hydraulic conductivity K eff related to pressure head h and its inverse form h in terms of K eff : K eff ðhþ ¼ Fh; ð p; gþ ð6þ h ¼ fðk eff ; p; gþ: ð7þ For a homogeneous case, the effective conductivity simplifies to a function of h and a single set of hydraulic parameters p; similarly, for a homogeneous case an inverse exists giving h in terms of only a single K function which includes p. For the general heterogeneous case, the hydraulic properties and geometries of all of the individual sublayers must be included for both (6) and (7). [8] Figure 2 illustrates the general pattern of the pressure profile. For simplicity, a binary structure is assumed and the flow velocity is sufficiently small in order that the pressure head h is negative throughout, although the overall analysis remains valid when h becomes positive at localized regions within the unit cell. The values of the pressure head fall between h 1 and h 2 defined from the relationships q = K 1 (h 1 )=K 2 (h 2 ) with K 1 (h) and K 2 (h) the hydraulic conductivity functions corresponding to the two sublayers. In this illustration, the lower sublayer is designated by numeral 1 and exists for 0 < z < b 1 L. Within this region the pressure head decreases with increasing elevation and moves toward the limiting value of h 1. Within the other sublayer 2 defined by b 1 L < z < L, the pressure head increases with increasing elevation and moves toward the with L the length of the cell and the integral taken over one cell length (alternative averages may be possible, but use of the arithmetic average can easily accommodate both positive and negative pressure heads). The average pressure head will depend on the volumetric flux q, the soil properties, and the thicknesses of the sublayers within the unit cell. In functional form the dependence of pressure head on these factors can be expressed by h avg ¼ fðq; p; gþ: ð3þ The arguments p and g are arrays of hydraulic and geometric parameters, respectively, which define the 2of6 Figure 2. Schematic pressure head profile for a binary structured profile. The values of h are confined within the vertical envelope defined by q = K 1 (h 1 )=K 2 (h 2 ) with K 1 (h) the hydraulic conductivity function corresponding to the region 0 < z < b 1 L and K 2 (h) corresponding to b 1 L < z < L.

3 W09406 WARRICK: EFFECTIVE UNSATURATED HYDRAULIC CONDUCTIVITY W09406 Table 1. Coefficients Describing the Hydraulic Relationships for Soils a limiting value h 2.Atz = L the pressure head returns to the same value as for z = 0 and the cycle is repeated. Similarly, the pattern is also repeated below for z < 0. If q had corresponded to the conductivity at the crossover point of the hydraulic conductivity functions (where h 1 and h 2 become equal), then h would simply be a constant throughout the profile. The relative values of h 1 and h 2 will reverse when the limiting unit gradient corresponds to opposite sides of the crossover point Limiting Effective Conductivity Relationships [9] The effective conductivity functions (6) and (7) simplify for extreme values of the block length. We examine those relationships for L! 0 and L!1. [10] For very fine sublayers, L is small. Darcy s law gives within any sublayer the approximation q K i ðhþ 1 þ Dh i ; ð8þ b i L where h is the pressure head and Dh i is the difference in pressure between the top and bottom of the sublayer i. For small values of L, there is only a small variation in pressure head. Furthermore, the ratio is a valid approximation of the pressure head gradient. Rearranging (8) results in q b i L K i ðhþ 1 Dh i : Adding the results for i =1,2...n gives X n a, cm 1 m q s q r K s,cms 1 Loamy sand (10) 3 Sand (10) 3 Sandy clay loam (10) 4 a After Carsel and Parrish [1988]. q b i L K i ðhþ 1 Xn Dh i : ð9þ ð10þ The sum of the changes in pressure head must be zero as h is repetitive at opposite ends of the cell. Setting the right side of the last equation equal to 0 and solving for q gives the effective conductivity relation when the block length L approaches 0: " # 1 q ¼ K eff ðhþ ¼ Xn b i : ð11þ K i ðhþ The last expression gives an implicit result for q in terms of h; it also defines the effective conductivity as a function of h. Each of the K i functions increases monotonically with increasing values of the pressure head h; therefore it follows that the effective conductivity will also increase with increasing h. Interestingly, the effective conductivity is analogous to the effective conductivity for a saturated system. The difference between (11) and the well-known harmonic average for a saturated system is that the value of h must be consistent with q, which defines the effective conductivity, whereas for the saturated system the conductivities are simply constants. Equation (11) is also consistent with that used by Mualem [1984] and Preuss [2004], who used the same form for all vertical flow cases. [11] For a large block length L, the length of each sublayer will become sufficiently large that the average pressure within each sublayer will approach the limiting value defined by q = K i (h i ). These limiting values are illustrated for a binary system by h 1 and h 2 in Figure 2. Therefore the cell average for pressure head is equal to the weighted average over the cell length. To emphasize that h i is dependent on q, which in turn is equal to K eff (h 1 ), we write an inverse relationship consistent with (7): h ¼ Xn b i h i ðk eff Þ: ð12þ The maximum value of the effective conductivity that can be accommodated with L! 1 will be the minimum saturated conductivity for any of the layers. (For L finite, localized intervals of positive pressure can develop.) Equation (12) is different than from the method of Figure 3. Effective conductivity relationships for a binary system (sand and sandy clay loam). Limiting values for small and large cell lengths L are given for (a) b 1 = 0.1 (b 2 =1 b 1 ), (b) b 1 = 0.5, and (c) b 1 = of6

4 W09406 WARRICK: EFFECTIVE UNSATURATED HYDRAULIC CONDUCTIVITY W09406 are also crossover points at 57.4 cm (between K 1 and K 3 ) and 12.1 cm (between K 2 and K 3 ). Figure 4. Effective conductivity relationships for a tertiary system (sand, sandy clay loam, and loamy sand) for b 1 = 0.33, b 2 = 0.33, and b 3 = Limiting values of the functions for small and large cell lengths L are shown in Figure 4a along with the curves for the individual soils. Results for finite cell lengths of L = 10, 100, and 1000 cm are in Figure 4b along with the limiting relationships for large and small cells Example 1: Pressure Head Profiles [15] Pressure head profiles were calculated for q/k 1 = for a binary structure. This value of q = 3.64(10) 8 cm s 1 (10.1 cm yr 1 ) is appropriate for longterm, deep drainage in a semiarid region. The first computations are with L = 10, 100, and 1000 cm in order to examine the effect of cell length. The sublayer intervals are defined by b 1 = b 2 = 0.5. [16] Computations were performed in Mathcad 11 [Mathsoft, 2002]. The algorithm consists of computation of (1) over a single cell (z =0toz = L). Initially, the boundary condition at z = 0 was set to the pressure head defined by K 2 (h) =q. The profile of h was found for the cell Mualem [1984] and Preuss [2004], who used equation (11) for all vertical flow cases. [12] In addition, to average values of h correspond average values of water content q. These are evaluated using the water retention relationships between h and q and appropriately averaging the q values for the individual layers. Note that the relationship between h and the effective K is dependent only on the layering and the conductivity relationships of the individual layers and independent of the water retention. 3. Examples and Calculations [13] For example calculations, it is assumed that van Genuchten s [1980] functional forms are applicable giving the water content q and the relative hydraulic conductivity K/K s as a function of pressure head h: q ¼ q r þ ðq s q r Þð1 þ uþ m ð13þ K ¼ ð1 þ uþ 0:5m ½1 u m ð1 þ uþ m Š 2 ð14þ K s 1= 1 m u ¼ jahj ð Þ : ð15þ The parameters m, a, q r, q s and K s must be defined for each sublayer. [14] Hydraulic characteristics chosen from the group averages of Carsel and Parrish [1988] for a loamy sand, sand, and a sandy clay loam are presented in Table 1. Plots of the hydraulic conductivity functions of two of the soils are included in Figure 3 and labeled as K 1 for the sandy clay loam and K 2 for the sand (repeated in each panel). At the crossover point, h is 15.9 cm and K 1 = K 2 = 1.49(10) 5 cm s 1. In Figure 4a the functions are plotted for all three soils with the loamy sand given by K 3. With three soils, in addition to the crossover point at 15.9 cm, there 4of6 Figure 5. Pressure profiles resulting from q = 3.64(10) 8 cm s 1 (10.1 cm yr 1 ) with contrasting cell lengths of L = 10, 100, and 1000 cm. The sublayer intervals are defined by b 1 = b 2 = 0.5. The dashed lines indicate the limiting pressure level values for the two sublayers.

5 W09406 WARRICK: EFFECTIVE UNSATURATED HYDRAULIC CONDUCTIVITY W09406 Table 2. Average Values of h and Corresponding Average Values of q for Varying L Values and Steady Vertical Flow Rate K eff (h)/k s,1 = q/k s, h, cm q h, cm q h, cm q h, cm q h, cm q 1 = 0.1, 2 = 0.9 a L = = 0.5, 2 = 0.5 a L = = 0.9, 2 = 0.1 a L = a Units of L are centimeters. using the built-in adaptive Runga-Kutta function (Rkadapt). The value of h found at the upper limit of the cell (at z = L) was then substituted for the lower boundary condition (at z = 0) and the values of the profile were recalculated. This was continued until the pressure head at each level within the cell no longer changed appreciably. Without loss of generality, the lower sublayer was always taken as that with the lowest value of saturated K. [17] Results for L = 10, 100, and 1000 cm are given as Figure 5. Note that the ordinate is for a normalized length z/l. In each case, the pressure profile remains between an envelope defined by the limiting pressure head values h = 146 and 43 cm given by (5). For the smallest cell length of L = 10 cm, the value remains close to the wetter limiting value h = 43 cm. By contrast, for the largest cell length of L = 1000 cm, the profile approaches the drier limit (h = 146 cm) within the first sublayer and then returns to the wetter value in the second sublayer. The profile for the intermediate case (L = 100 cm) is less extreme and decreases toward the drier limit in the first sublayer, but not to the extent of that for the largest cell length Example 2: Conductivity Relationships for the Limiting Cases [18] The effective hydraulic conductivity was found as a function of pressure head for the limiting cases of L! 0 and L!1and for binary systems with b 1 = 0.1, 0.5, and 0.9 (b 2 =1 b 1 ). The results are plotted in Figure 3. Note that for all cases, the effective conductivity falls within the envelope formed by the two conductivity functions K 1 (h) and K 2 (h) and all of the relationships pass through the crossover point where K 1 (h) =K 2 (h). As is expected intuitively, for b 1 small the curves are closer to K 2 than for the larger b 1 cases and vice versa. Also, the resulting curves are closer together for the wetter end. Note that we are using a log scale and the conductivities are much larger on the left side of each plot Example 3: Conductivity Relationships for the Finite and Nonzero Cell Lengths [19] Values of the conductivity were calculated for a binary system with L = 10, 100, and 1000 cm and b 1 = 0.1, 0.5, and 0.9 as before. The resulting average pressure heads are given in Table 2 for effective conductivities defined by K eff (h)/k s,1 = q/k s,1 = 0.9, 0.1, 0.01, , and Also, given are the limiting values for L! 0 and L!1. Note that in each case the smallest L value (10 cm) approaches the limiting value for L! 0. The largest difference in h corresponding to each effective conductivity occurs on the drier end, as would be expected. For completeness, corresponding water content values are also given Example 4: Conductivity Relationships for a Tertiary Structure [20] A tertiary structure was defined by using the same sandy clay loam for sublayer 1 with b 1 = 0.33; the sand for sublayer 2 with b 2 = 0.33; and a loamy sand for sublayer 3 with b 3 = Again, the three individual conductivity functions as defined from Table 1 are plotted in Figure 4a. As mentioned previously, with three hydraulic conductivity functions, there are three crossover points (at h = 12.1, 15.9, and 57.4 cm). Also shown in Figure 4a are the limiting values of the effective conductivity function for small (L! 0) and large (L!1) cell lengths. For the limiting cases, the largest effective value of hydraulic conductivity corresponds to the lowest saturated hydraulic conductivity which is for the sandy clay loam (K 1 ). The drier range of values for the effective conductivity function for small (L! 0) cells tends toward the finer-textured materials (sandy clay loam and loamy sand); for the large 5of6

6 W09406 WARRICK: EFFECTIVE UNSATURATED HYDRAULIC CONDUCTIVITY W09406 (L!1) cells the relationship tends toward the coarsest material (sand). The effective conductivities were also evaluated for three finite values of cell length (L = 10, 100, and 1000 cm) for three contrasting values of effective K. The calculated values are shown in Figure 4b and fall between the limiting cases and in the expected order (L = 10 the closest to the results for L! 0, etc.). Computations were repeated after interchanging sublayers 2 and 3 for the finite cell lengths. The results (not shown) were changed only slightly (typically no more than in the third significant figure). 4. Discussion and Conclusions [21] Effective hydraulic conductivities of unsaturated soils are defined using steady flow through an infinitely deep profile consisting of a repeating pattern of sublayers. The effective conductivity relationship is found by relating the flow rate to the average pressure head which occurs. [22] The effective functions depend on the hydraulic properties of the sublayers and the geometric factors defining the unit cell over which the patterns are repeated and the relative depth of each sublayer given by b 1, b 2, etc. Limiting values for small and large cell lengths are based on appropriate harmonic and arithmetic means (11) and (12). An interesting observation is that the limiting value for small L is consistent with the well known effective average for a saturated layer. These are also consistent with the form used by Preuss [2004]. Results compare favorably with Figure 7 of Preuss using L =0.2m,b 1 = 0.95, and b 2 = 0.05 in equation (11). In this case, the repeating units characterized by L are sufficiently small that (11) is appropriate; for large L that would not be the case. [23] Computations for a binary system reveal that the effective hydraulic conductivity function is constrained within an envelope formed by the hydraulic conductivity functions for the individual sublayers. For contrasting relative depths of the sublayers, the effective conductivities approach the conductivity function of the sublayer of greatest relative thickness. When finite cell lengths are considered, the effective values are constrained between the two limiting cases calculated for L! 0 and 1. [24] Tertiary patterns are more complex than the binary case, but the limiting cases for small and large cell lengths L are easily found and generally are similar to the binary results. For finite cell lengths, computations again showed effective conductivity relationships constrained between the two limiting cases (for small and large cell lengths). [25] Several future applications are of interest and bring up questions beyond the scope of this study. We can speculate with partial answers to some of the more obvious questions which introduce varying degrees of complexity. The first is with respect to a profile of finite depth, rather than of infinite length. In this case, an effective conductivity could be defined in exactly the same way and likely would fall between the two limiting effective hydraulic conductivity functions defined for L. The flow velocity would be the same throughout, but the pressure head would obviously not be expected to repeat at the two ends of the domain. Differences in pressures at the two ends would be expected to have a negligible effect on the effective K on all but very shallow domains. A second question regards appropriate values for the hydraulic properties of time-dependent systems which are stratified. This question is more complex, in that it is necessary to further refine what is meant by equivalent hydraulic properties. One case of interest is for time-periodic conditions near the surface which will damp out over depth and be taken as steady state in much of the domain for which the present analysis is relevant. A third question is with respect to a profile of randomly defined strata. For any realization, the present analysis applies, but can the results be put in a context of unit cell lengths and sublayers which are defined statistically? [26] Acknowledgments. This work was supported by NSF (grant EAR ) and by Western Regional Research Project W-188. References Carsel, R. F., and R. S. Parrish (1988), Developing joint probability distributions of soil water retention characteristics, Water Resour. Res., 24, Gardner, W. R. (1958), Some steady state solutions of unsaturated moisture flow equations with application to evaporation from a water table, Soil Sci., 85, Jury,W.A.,andR.Horton(2004),Soil Physics, 6thed.,JohnWiley, Hoboken, N. J. Mathsoft (2002), Mathcad User s Guide: 2000 Professional, Cambridge, Mass. Mualem, Y. (1984), Anisotropy of unsaturated soils, Soil. Sci. Soc. Am. J., 48, Preuss, K. (2004), A composite medium approximation for unsaturated flow in layered sediments, J. Contam. Hydrol., 70, van Genuchten, M. T. (1980), A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, Warrick, A. W. (2003), Soil Water Dynamics, Oxford Univ. Press, New York. Warrick, A. W., and T. C. Yeh (1990), One-dimensional, steady vertical flow in a layered soil profile, Adv. Water Resour., 13(4), Willis, W. O. (1960), Evaporation from layered soils in the presence of a water table, Soil Sci. Soc. Am. Proc., 24, A. W. Warrick, Department of Soil, Water and Environmental Sciences, University of Arizona, 429 Shantz Building, Room 38, Tucson, AZ , USA. (aww@ag.arizona.edu) 6of6