Soil Science Society of America Journal 67:1122-1126 (2003)
© 2003 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Effect of Travel Time on Management of a Sequential Reuse Drainage Operation
William A. Jury*,
Atac Tuli and
John Letey
Dep. of Environmental Sci., Univ. of California, Riverside, CA 92521
* Corresponding author (wajury{at}mail.ucr.edu)
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ABSTRACT
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Sequential reuse of agricultural drainage water recovered in tile lines has been proposed as a strategy for reducing drainage volume in the western San Joaquin Valley of California. In this system, high-quality water is used to grow a salt-sensitive crop, and the drainage from this operation is collected by tile lines and subsequently used on a more salt-tolerant crop. This process is continued on progressively more salt-tolerant species until the final residual is collected and sent to evaporation ponds. In this paper we develop a transfer function model for simulating the drainage concentrations of each stage of a sequential reuse project. Transient salt concentrations are calculated for typical drain spacings and water management strategies, under the assumption that downward movement of water is eventually restricted by a subsurface barrier. Results of the calculations show that response times of fields managed in this way are extremely long, so that the drain lines primarily capture resident ground water for decades or more after the operation is started, especially if the barrier is assumed to be at a substantial depth below the surface. As a result, the system will never reach steady state in any practical period of time, and system design strategies based on steady state behavior will be flawed.
Abbreviations: PDF, probability density function • CDF, cumulative probability function
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INTRODUCTION
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CALIFORNIA'S WESTERN San Joaquin Valley is experiencing a variety of irrigation-induced problems such as water scarcity, deteriorating water quality, and salinization of agricultural soils (Letey et al., 1986; San Joaquin Valley Drainage Program, 1990; Letey, 2000). A largely impermeable subsurface layer has caused drainage water to accumulate across many years, resulting in rising water tables and saline seepage into low-lying flood plains (San Joaquin Valley Drainage Program, 1990). As a consequence, agricultural fields in this region are commonly tile drained to keep the root zone aerated and free of salts. Earlier plans for surface detention and export of excess drainage water to the ocean through the San Francisco bay area were canceled because of high selenium levels in evaporation ponds, and concerns that export through the bay would have adverse environmental consequences for the local habitat (Letey, 2000). Since that plan was disrupted, efforts to control salinization have focused more on water reuse or various forms of land management (Letey and Oster, 1993; Belitz and Philips, 1995).
One strategy for salinity control consists of periodic leaching of the crop root zone to reduce the dissolved salt content to the tolerance limits of the crop to be grown (Biggar et al., 1984). Rhoades (1984) has proposed alternating saline and nonsaline irrigation waters together with crop rotation, including both salt-sensitive and salt-tolerant crop species. Rhoades et al. (1988) tested this cyclic strategy in a field experiment with crops having different salt tolerance levels, and showed that high crop yield and quality could be achieved. Moreover, they concluded that the soil salinity level could be maintained within acceptable limits across time. Bradford and Letey (1992) used computer modeling to confirm the results of the Rhoades et al. (1988) experiments. They also showed that the soil salinity at the beginning of the cropping season was a significant factor influencing crop yield.
Another idea for drainage and salt control that is attracting considerable attention is a water-management program involving reuse of drainage water on successively more salt-tolerant crops, sometimes referred to as sequential reuse (Drainage Reuse Technical Committee, 1999). In this system, high-quality water is used to grow a salt-sensitive crop, and the drainage from this operation is collected by tile lines and subsequently used on a more salt-tolerant crop. This process is continued on progressively more salt-tolerant species until the final residual is collected and sent to evaporation ponds. Since substantial water is evaporated at each stage, the drainage water volume available for irrigation is progressively reduced, while the salt concentration is correspondingly magnified.
Cervinka et al. (1999) presented results from a sequential reuse demonstration project conducted on irrigated farmland in the western San Joaquin Valley. The crop areas, drainage volumes, anticipated dilution, and expected yields were all designed based on the steady state assumption that the concentration of water collected in the drainage system of each stage was the same as the concentration of water leaving the root zone. However, drain water concentrations during the years of operation of this project have differed significantly from their anticipated steady state values, suggesting that the transition times to adjust to the new management may be considerable, and that correct design of a sequential reuse system will require knowledge of the response time of the soil to a change in surface management (Letey et al., 2001). Jury (1975a) calculated the response time of tile-drained fields as a function of their drainage rates, drain spacing, and depth to barriers reducing or preventing downward flow, and found that it may take years to leach existing salt out of fields of the type found in the San Joaquin Valley. This delay time might substantially affect the performance of a system designed for water reuse or remediation of saline soil (Jury, 1975b).
This paper develops a conceptual-mathematical model of water and chemical movement through the soil and to the tile drain that is used to represent the sequential reuse system and calculate the buildup of salinity in the soil and drainage water over time.
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METHODOLOGY
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The purpose of this analysis is to construct a simple model of the tile-drain concentrations in a system where agricultural drainage water is sequentially pumped to fields containing increasingly more salt-tolerant crops. Tile drain concentrations are easily modeled as transfer functions, where the input concentration to the field Cin is converted to an output concentration Cout at the drain using the drainage probability density function (PDF) f(Q), where Q is the amount of drainage required to move a mobile chemical from the surface to the drain (Jury and Roth, 1990). This PDF is assumed to be a unique function of the geometry and soil properties, so that a single PDF is all that is required to model the system. If portions of the system are draining at different rates, the cumulative drainage-time function Q(t) may be used to convert the modeled concentration outflow to a time record. The following assumptions are used to construct the model: each stage of the system has the same drain depth H, and drain spacing 2S; the soil is homogeneous and bounded by a vertical flow barrier at a depth D below the drain; the initial salinity level Cs in the soil and throughout the ground water is uniform; the first stage of the system using high quality irrigation water of concentration Cirr is leached at a low drainage rate R1 and leaching fraction 
1; subsequent stages are leached at a higher drainage rate R2 and leaching fraction 2; the curvature of the water table is neglected and the lateral ground water velocity is zero.
The travel time to the drain from any point of entry on the surface is the sum of two terms: (i) the travel time tu through the unsaturated zone (assumed to be constant); and (ii) the variable travel time ts from the point of entry at the water table to the tile drain. The saturated zone travel time ts (and hence the PDF) is calculated by a model.
The two-dimensional calculation of the travel time ts from the water table to the tile drain is described in detail in Jury (1975a). It begins with the Kirkham and Powers (1972) solution for the stream function 
(x,z) through the saturated zone of a tile-drained system with drain spacing 2S and depth to barrier D. Lines of constant stream function represent water flow paths. First, the geometry of a given field is scaled by defining new variables X = x/S, Z = z/S. Then the streamlines of the scaled system are calculated as a Fourier series. This solution is given by (Jury, 1975a):
 | [1] |
Streamlines are constructed for equally spaced points of entry into the saturated zone along X separated by 
X. An illustrative set of streamlines for the particular geometry 
= S/D = 1 is given in Fig. 1.
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Fig. 1. Water flow streamlines through the saturated zone of a tile line with = S/D = 1, where S is half the distance between drains and D is the height of the tile above an impermeable barrier. The symbols x and z refer to horizontal and vertical directions, respectively.
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The travel time from a point of entry at the water table to the drain is calculated from this information as follows. Since any water entering between two adjacent streamlines must remain between the lines all the way to the drain, the travel time of any mobile solute in the water is approximately equal to the amount of time required to replace the amount of water in storage between the lines. The water in storage is the area a(x) between the lines times the saturated volumetric water content 
s. If the unsaturated zone is draining at a steady flux rate R, then the amount of water added in time t between two adjacent lines separated by x is R x x x t. Thus, the travel time to the drain from a point of entry that is a distance x lateral from the drain line is equal to
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where A(X) = a(x)/S2 is the dimensionless area between the two streamlines originating at the water table at X - X and X. A(X) is estimated by numerical integration.
If we regard the saturated zone as a solute transport volume, we may calculate the travel time probability distribution from the solute concentration arriving at the drain (Jury and Roth, 1990). At time t = 
, the total volume per unit width of drainage that has entered the drain line since t = 0 is Q() = R x x x . If we added a uniform concentration C0 to the top of an initially salt-free saturated zone at t = 0, then those streamlines originating at parts of the field nearer to the drain than x() will be discharging C = C0 to the drain and those farther from the drain will still be adding salt-free water. Thus, a fraction X[Q()] = x()/S of the chemical added to the field at t = 0 has arrived at the drain after cumulative drainage Q, and we may interpret Ps(Q) = X(Q) = x(Rt)/S as the cumulative probability density function (CDF) of the saturated zone travel time. Thus the PDF of the cumulative drainage through the saturated zone is given by fs(Q) = dPs(Q)/dQ.
Under these assumptions, the sequential outflow may be solved by the mathematics of transfer functions as follows. In general, a given tile drain outflow concentration Cout(Q) is expressed as a function of its input concentration Cin(Q) at the water table by Eq. (3) (Jury and Roth, 1990):
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Note that Eq. [3] contains two terms, one representing the arrival of the drainage water that has reached the tile and a second describing the resident water in the saturated zone (assumed to be spatially uniform at t = 0) being pushed out ahead of the incoming water. We may easily combine the saturated zone and unsaturated zone travel times (expressed as cumulative drainage), since the latter is assumed to be constant. The CDF of the total cumulative drainage is thus
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where Qu is the amount of drainage required to move solute from the soil surface to the water table at depth H.
We may now apply the model to the sequence of drains, noting that as the irrigation water passes through the unsaturated zone of a field with a leaching fraction i, it is concentrated by a factor Ni = 1/i. Thus, if the irrigation water added to the first stage has a constant concentration Cirr, a drainage rate R1, and a leaching fraction 1, then the outflow concentration in the first drain is equal to
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Similarly, since all other stages have drainage rates R2 and leaching fractions 2, the outflow in the Jth drain is equal to
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These expressions become very simple after Laplace transformation (Jury and Roth, 1990):
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where
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is the Laplace transform of f(Q). The recursive Eq. [7] and [8] may be combined into a general expression for the solute concentration entering the Mth drain as follows:
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Thus, if the Laplace transform of f(Q) is known and Eq. [10] can be inverted, the tile drain concentration from each stage of the sequential reuse system may be calculated as a function of cumulative drainage and converted to time.
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RESULTS AND DISCUSSION
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Figure 2 shows the dimensionless CDF P(T) as a function of the scaled travel time T = Rt/sS = Q/sS for drain spacing to barrier depth ratios = S/D of 1, 5, and 20. In all calculations the dimensionless unsaturated zone travel time was set equal to 0.005. Each of these curves were parameterized approximately by fitting to the exponential model
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The representation in Eq. [11] becomes exact for shallow barriers (large ), but underestimates long times and overestimates short times when the barrier is deep (small ). With this parameterization, the solution to Eq. [10] may be obtained analytically or by numerical inversion of the Laplace transform. The solution may also be converted to actual time when the parameters of a particular system are known. We illustrate the calculation using the data shown in Table 1 from the Red Rock Ranch sequential reuse demonstration facility (Cervinka et al., 1999). This facility overlies the Corcoran clay barrier at a substantial depth below the surface. However, it is possible that soil layering could distort the streamlines before they reach deep into the saturated zone. For this reason, we have calculated results for both a shallow barrier (D = 3 m) and a deep one (D = 60 m). We use the small drainage rate and leaching fraction in Table 1 for Stage 1 and the larger leaching fraction and drainage rate for all subsequent stage calculations.
Figure 3 shows the response time and equilibration time of the first stage alone for the shallow and deep barrier cases. Even when only the first stage is considered, it is obvious that the system is inherently transient and does not reach steady state in any practical period of time. Moreover, the response time is lengthened considerably as the depth to barrier is increased. During the early stages, the drainage water concentration consists primarily of resident ground water that was removed by the tile lines.
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Fig. 3. Response time and equilibration time of the first stage of a system overlying a shallow (3-m) and deep (60-m) barrier, using parameters from the Red Rock Ranch in Table 1.
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Figure 4 shows the drain concentrations from all four stages of a system overlying a shallow barrier at 3 m below the drain, using parameters from the Red Rock Ranch in Table 1. Several features are noteworthy. First, each successive field requires longer to reach steady state than the previous one. Second, steady state concentrations (shown as dashed lines in the figure) can be exceeded during the transient stage because the high concentration of the resident ground water can magnify as it passes through the system. Figure 5 is identical to Fig. 4, except that the barrier now resides at 60 m below the drain. The behavior of the field is qualitatively the same as the field with a shallow barrier, except the time scale for transition to the steady state is now expanded considerably because of the significantly greater volume of resident water that must be leached.
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Fig. 4. Drain concentrations from all four stages of a system overlying a shallow (3-m) barrier, using parameters from the Red Rock Ranch in Table 1. Dashed lines represent steady state concentrations.
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Fig. 5. Drain concentrations from all four stages of a system overlying a deep (60-m) barrier, using parameters from the Red Rock Ranch in Table 1. Dashed lines represent steady state concentrations.
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An approximate index of the response time of a field with a given geometry and flow rate may be estimated from Fig. 2. Table 2 gives the time in years for 25, 50, and 75% of the solute added at t = 0 to reach the drain for all of the cases studied. Median response times vary from a low of 2.1 yr for the shallow system leached at 0.50 m yr-1 to a high of 31.2 yr for the deep barrier system leached at 0.225 m yr-1. The systems are also very asymmetric, requiring much longer to leach the remaining percentages of the field than early ones. What this implies is that during the first few years after initiation of the sequential reuse strategy, drain concentrations for systems with a large spacing between drains will consist mainly of water extracted from the ground water at or near the ground water concentration (Fig. 5). Thus, the predominant early function of a sequential reuse operation would be ground water reclamation. This prediction is in qualitative agreement with the findings of Deverel and Fio (1991), who determined from oxygen 18 measurements that up to 60% of the water extracted by a drain tile in the San Joaquin Valley was from deep resident ground water. These authors also simulated water flow for the system they measured and calculated travel times of up to 34 yr for arrival at the drain (Fio and Deverel, 1991).
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Table 2. Time in years for P(t) = 0.25, 0.50, and 0.75 of the water added at the surface to arrive at the drain for a system with half spacing (S) = 60 m, saturated volumetric water content (s) = 0.45, and drainage rate (R) = 0.22 or 0.5 m yr-1.
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The extremely long transition times to steady state for the tile geometries studied indicates that any management design for sequential reuse based on steady state criteria will be flawed. For example, if crops are selected for the fields in succession based on their ability to tolerate the steady state salinity coming from the previous drain, the irrigation water may have to be diluted for a substantial period of time to avoid damaging the crop. This transient effect persists for many years and must be part of any management design. The problems associated with a steady state analysis are illustrated in Fig. 6, which shows the percentage dilution of the drainage water from Stage 1 required to maintain the irrigation water to Stage 2 at 5 dS m-1 or less (suitable for growing cotton at zero yield depression), using the Red Rock Ranch parameters in Table 1. Since the steady state concentration coming from Stage 1 is 4.2 dS m-1, the steady state design calls for zero dilution, whereas there is substantial dilution required for 20 yr in the shallow barrier case, and indefinitely in the deep barrier case.
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Fig. 6. Percentage dilution required to maintain the irrigation water of Stage 2 of a system overlying a shallow (3-m) and deep (60-m) barrier at 5 dS m-1 or less, using parameters from the Red Rock Ranch in Table 1. The predicted dilution using the steady state model is zero.
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CONCLUSIONS
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Our analysis demonstrates the dominant effect of transient solute movement on the performance of a sequential reuse operation. For drain tile geometries of the type found in the San Joaquin Valley, steady state will never be reached, and initially saline ground water will significantly affect the concentration of irrigation water for stages beyond the first for the entire practical lifetime of the project. Because this resident ground water becomes concentrated after passage through a root zone, substantial dilution may be required to reduce the drainage water to levels suitable for irrigation of all but the most salt-tolerant of species.
The conceptual model used in this analysis was intended primarily to demonstrate the dominant effects of travel time on performance of a sequential reuse operation. A number of potentially important factors such as soil layering, ground water movement, or deep seepage unimpeded by a barrier were neglected in this study and will be examined by numerical methods in a subsequent paper.
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ACKNOWLEDGMENTS
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The authors are grateful to the University of California Salinity Program for financial support of this research.
Received for publication May 13, 2002.
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REFERENCES
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- Belitz, K., and S.P. Philips. 1995. Alternative to agricultural drains in California's San Joaquin Valley: Results of a regional-scale hydrogeologic approach. Water Resour. Res. 31(8):1845–1862.
- Biggar, J.W., D.E. Rolston, and D.R. Nielsen. 1984. Transport of salts by water. Calif. Agric. 38(10):10–11.
- Bradford, S., and J. Letey. 1992. Cyclic and blending strategies for using nonsaline and saline waters for irrigation. Irrig. Sci. 13:123–128.
- Cervinka, V., J. Diener, J. Erickson, C. Finch, M. Martin, F. Menezes, D. Peters, and J. Shelton. 1999. Integrated system for agricultural drainage management on irrigated farmland. Rep. of Westside Resource Conservation District to U.S. Bureau of Reclamation, Grant 3:4-FG-20–11920. Dep. of the Interior, Washington, DC.
- Deverel, S.J., and J.L. Fio. 1991. Groundwater flow and solute movement to drain laterals, western San Joaquin Valley, California. 1. Geochemical assessment. Water Resour. Res. 27:2233–2246.
- Drainage Reuse Technical Committee. 1999. Task 1. Drainage reuse. Final report of the technical subcommittee of the University of California Salinity/Drainage Program to the the San Joaquin Valley Drainage Implementation Program. California Dep. of Water Resources, Sacramento, CA.
- Fio, J.L., and S.J. Deverel. 1991. Groundwater flow and solute movement to drain laterals, western San Joaquin Valley, California. 2. Quantitative hydrologic assessment. Water Resour. Res. 27:2247–2257.
- Jury, W.A. 1975a. Solute travel-time estimates for tile-drained fields: I. Theory. Soil Sci. Soc. Am. Proc. 39:1020–1024.
- Jury, W.A. 1975b. Solute travel-time estimates for tile-drained fields: II. Application to experimental studies. Soil Sci. Soc. Am. Proc. 39:1024–1028.
- Jury, W.A., and K. Roth. 1990. Transfer functions and solute transport through soil: Theory and applications. Birkhäuser Publ., Basel, Switzerland.
- Kirkham, D., and W. Powers. 1972. Advanced soil physics. John Wiley and Sons, New York.
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- Letey, J., S. Grattan, J.D. Oster, and D.E. Birkle. 2001. Findings and recommendations to develop the six-year activity plan for the department's drainage reduction and reuse program. Final Rep. Task Order No. 5. Contract No. 98–7200–B80933. California Dep. of Water Resources, Sacramento, CA.
- Letey, J., and J.D. Oster. 1993. Subterranean disposal of irrigation drainage waters in Western San Joaquin Valley. p. 691–697. In R.G. Allen (ed.) Proc. on Management of irrigation and drainage systems: Integrated perspectives. ASCE national conference on irrigation and drainage engineering, Park City, Utah. 21–23 July 1993. Am. Soc. of Civil Engineers, Park City, UT.
- Letey, J., C. Roberts, M. Penberth, and C. Vasek. 1986. An agricultural dilemma: Drainage water and toxics disposal in the San Joaquin Valley. Spec. Publ. 3319. Univ. of California, Div. of Agric. and Nat. Resources, Riverside, CA.
- Rhoades, J.D. 1984. Use of saline water for irrigation. Calif. Agric. 38(10):42–43.
- Rhoades, J.D., F.T. Bingham, J. Letey, A.R. Dedrick, M. Bean, G.J. Hoffman, W.J. Alves, R.V. Swain, P.G. Pacheco, and R.D. Lemert. 1988. Reuse of drainage water for irrigation: Results of Imperial Valley study. I. Hypothesis, experimental procedures, and cropping results. Hilgardia 56(5):1–16.
- San Joaquin Valley Drainage Program. 1990. A management plan for agricultural subsurface drainage and related problems in the Westside San Joaquin Valley. Final rep. San Joaquin Valley Drainage Program, Sacramento, Calif.
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ABSTRACT
INTRODUCTION
