Department of Mathematics

IDEA: Internet Differential Equations Activities

Irrigation and Conservation

This exercise uses a single linear differential equation to investigate the question of when conservation of water by agriculture is useful and genuine.


Scarce water supplies in the western United States support fragile ecosystems and billions of dollars worth of agricultural, industrial and municipal use. Often, these water uses are competitive. For example, water consumed in irrigating crops may decrease the supply for industrial, municipal and ecosystem uses. In the past, the West expanded water supplies by constructing federally-subsidized water development projects designed to capture, store and transport the Spring freshet. These projects include the Columbia Basin Project in central Washington state, the Newlands Project in western Nevada, and the Central Valley Project spanning California. Expanding water supplies via expensive water projects is no longer feasible due to a variety of factors, including a lack of productive new sites for dams, political opposition to increased expenditures, and environmental opposition to adverse ecological impacts. Consequently, many western states are busy designing water conservation policies to utilize existing supplies more effectively. Since agriculture accounts for around 90% of the water consumed in the West, one popular water conservation policy is to encourage farmers to adopt more technically-efficient on-farm irrigation technologies.

This module investigates agricultural water conservation within the context of a farmer diverting water from a river to irrigate crops. Crops generally do not consume all of the water diverted into irrigation. The portion of diverted water that is not consumed by crops may re-enter the river as an "irrigation return flow," thereby augmenting stream flow for downstream users. Alternatively, unconsumed diverted water may escape the hydrologic system as an "irretrievable loss." Return flow may occur directly from the field, or in the form of groundwater that flows into a river or is pumped from a well. Irretrievable losses occur through evaporation or a variety of other means. The success of crops in consuming large portions of diverted water depends on the farm's irrigation efficiency, h in (0,1), defined as the ratio of the consumptive water use by crops, e, to the total water diverted into irrigation, q:

(1) h = e/q,

where e and q are both in units of water per square kilometer (w/km). Highly efficient irrigation systems apply water more uniformly over fields in a more timely manner, thereby enabling crops to increase their consumptive water use from smaller irrigation diversions. Farmers can increase on-farm irrigation efficiency by switching to water-application technologies better suited to field characteristics (e.g., replacing flood with sprinkler irrigation on sloped fields).

Exercise 1: Use equation (1) to demonstrate how improving on-farm irrigation efficiency from 25% to 80% reduces the irrigation diversion needed to meet a prior demand for 2 units of consumptive water use by 5.5 units.

The million dollar question for agricultural water conservation policy is whether the 5.5 units of reduced irrigation diversion due to an increase in irrigation efficiency in Exercise 1 constitutes conserved water increasing instream flows. Western states are divided over the answer. In Oregon, the answer is "yes". Oregon water law equates water conservation with reduced diversion, and grants the efficiency-improving irrigator a portion of the "conserved" water to irrigate additional acreage or sell. In California, the answer is "not necessarily". California water law measures water savings as a reduction in consumptive water use or irretrievable water losses--not reduced diversion. Which state's measure of water conservation is correct? Is Oregon's relatively liberal measure prone to view water as conserved when it truly is not? Is Calfornia's relatively strict measure liable to recognize too little water as conserved?

The answers to these questions depend on the hydrology of the river from which irrigation water is diverted. This module analyzes the relationship between increased on-farm irrigation efficiency and water conservation for two opposite hydrologic cases. In the first case we assume that applied water unconsumed by crops is irretrievably lost to the river system. In the second case we assume that unconsumed applied water returns to the river system to be used by downstream irrigators. Each case is based on a single differential equation measuring the spatial rate of change in instream flow. We analyze the solution to each differential equation to determine the extent to which increased irrigation efficiency leads to increased instream flows, i.e., water conservation, and to determine the correct measure of conservation.

Case I: Unconsumed Irrigation Diversions Are Irretrievably Lost to the River Basin

Figure 1: Irretrievable Water Losses

Figure 1 is a stream-flow diagram illustrating water use in a hypothetical river basin over a single cropping season. The vertical line represents the stretch of river within the basin. There are three farms in the river basin, each of which has water rights to divert 10 units (downward-pointing arrows). Each has an identical irrigation efficiency of 50%, which means that half of the irrigation diversion is consumed by crops (the encircled 5 units). The other half of the diversion is assumed to be irretrievably lost to the river basin (upward-pointing arrows). The quantity of water entering the basin is assumed to be 35 units. The numbers below the 35 units keep a running account of the water remaining in the river after irrigation diversions.

Consider now a more general version of the river basin depicted in Figure 1. Let x represent the distance of a farm from the "top" of the basin (i.e., the entry point of the river at x = 0). The quantity of basin inflow in a single cropping season is z(0), and the level of instream flows at a given location x is z(x). Denote the identical irrigation diversion of each farm as qx = q liters of water per square kilometer, and the uniform dimensionless irrigation efficiency level as hx = h. The level of consumptive water use by crops at each farm is ex = hq, and the portion of diverted water irretrievably lost to the basin is (1-h)q [by equation (1) above].

Exercise 2: Show that the differential equation measuring the spatial rate of change of instream flows at any location x is:

(2) dz/dx = - a q

where dz/dx is in units of water per kilometer of the river (w/km) and a represents the assumed constant width of the river basin in kilometers.

Exercise 3: Demonstrate that the solution to equation (2), together with an initial flow z(0), is:

(3) z(x) = z(0) - a qx

Explain why this solution makes hydrologic sense when unconsumed irrigation diversions are irretrievably lost to the river basin.

Exercise 4: What is the marginal impact on instream flow of incrementally increasing the irrigation efficiency, h, of a farm at location x while keeping its consumptive water use constant (i.e.the diversion qx=qx(hx) is reduced if efficiency is increased)? [Hint: Use equation (1) to write the right-hand side of equation (3) in terms of e and h, and then compute the sign of the partial derivative dz/dh.] Does increased irrigation efficiency lead to increased instream flow (water conservation) in the absence of irrigation return flow?

Case II: Unconsumed Irrigation Diversions Return to the River

In reality, irrigation return flows are an important component of stream flows in the West. Accounting for return flows in modelling is difficult because their location and timing is often not well known. For Case II we make the simplifying assumption that the irrigation return flow from a farm located at x re-enters the river at location x + t downstream. Another way of looking at this is that the instream flow enjoyed by the farm located at x is augmented by the irrigation return flow of a farm located upstream at x - t. In Case II we continue to assume that each farm along the river has the same irrigation efficiency (i.e., hx-t = hx = hx+t), and diverts an identical amount of water (i.e., qx-t = qx = qx+t). Figure 2 illustrates this hydrologic scenario by converting the irretrievable water losses in Figure 1 into irrigation return flows re-entering the river t km below the point of diversion (the leftward-pointing arrows).

Figure 2: Irrigation Return Flows

Exercise 5: Demonstrate that the spatial rate of change in instream flows is given in Case II by:

(4) dz/dx = -a qx + a qx-t(1 - hx-t)

The first term subtracts irrigation diversion, qx, from instream flow at location x. The second term adds irrigation return flow measured as the portion of the diversion unconsumed by crops at location x - t upstream.

Exercise 6: Demonstrate that the solution to equation (4) is:

(5) z(x) = z(0) - a qx x + a qx-t (1 - hx-t)x

Interpret the hydrologic meaning of the solution.

Exercise 7: What is the marginal impact on instream flow at location x of incrementally increasing the irrigation efficiency of a farm at location x - t upstream, while keeping its consumptive water use e constant? [Hint: Compute the sign of the partial derivative dz/dhx-t as in Exercise 4.] Does increased irrigation efficiency lead to increased water in the river (i.e., water conservation) in the presence of irrigation return flows?

The impact of increasing a farm's irrigation efficiency on the quantity of instream flow in a hypothetical river basin was analyzed under a range of hydrologic circumstances. Case I assumed that any diverted water unconsumed by crops was lost to the basin. Solving the relevant differential equation, and partially differentiating its solution with respect to irrigation efficiency, demonstrated that instream flow increases when a farm increases its irrigation efficiency in the absence of irrigation return flows. The underlying hydrologic reasoning is that increased irrigation efficiency allows a farm to meet the consumptive water needs of its crops with a smaller diversion of water from the river, and a smaller diversion means that less water is taken out of the river and irretrievably lost to the basin after irrigation. The water conserved by an efficiency-improving farm can be measured as the reduction in diversions (or identically irretrievable water losses) before and after an increase in irrigation efficiency. Under these hydrologic conditions, Oregon is correct in measuring conserved water as the reduction in diverted water before and after an increase in on-farm irrigation efficiency.

Alternatively, Case II assumed that any diverted water unconsumed by crops returned to the river at some downstream location as an irrigation return flow. This is a more realistic hydrologic assumption for the West than that made in Case I. Solving the relevant differential equation, and partially differentiating its solution with respect to an incremental increase in irrigation efficiency, demonstrated that instream flow remains unchanged when a farm increases its irrigation efficiency in the presence of irrigation return flows. This is not conservation. The underlying hydrologic reasoning is that increased irrigation efficiency decreases the portion of a farm's irrigation diversion returning to the river by increasing consumptive water use by crops, and in a return-flow system this is the only leakage to the system. Water conservation in a return-flow hydrologic system occurs principally as a reduction in consumptive water use by crops. Consumptive water use can be reduced by giving crops less water than they need for maximal growth (i.e., deficit irrigation), or by shifting to less water intensive crops. Oregon's measure of conservation as a reduction in diversion is invalid under these more realistic hydrologic circumstances. On the other hand, California's measure of conservation as a reduction in consumptive water use or irretrievable water losses is valid for both return-flow and non-return-flow hydrologic systems, and is thus the most accurate.


Copyright © 1997 Ray Huffaker
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