JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION DECEMBER
AMERICAN WATER RESOURCES ASSOCIATION
2003
CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES1
Amanda B. White, Praveen Kumar, Patricia M. Saco, Bruce L. Rhoads, and Ben C. Yen2
ABSTRACT: The objective of this work is to determine the effects of extension of a stream network through land drainage activities during the late 1800s on the hydrologic response of a watershed. The Mackinaw River Basin in Central Illinois was chosen as the focus and the pre-land and post-land drainage activity hydrologic responses were obtained through convolution of the hillslope and channel responses and compared. The hillslope response was computed using the kinematic wave model and the channel response was determined using the geomorphologic instantaneous unit hydrograph method. Our hypothesis was that the hydrologic response of the basin would exhibit the characteristic effects of settlement (i.e., increases in peak discharges and decreases in times to peak). This, indeed, is what occurred; however, the increase in peak discharges diminishes as scale increases, leaving only the decrease in times to peak. At larger scales, the dispersive effects of the longer hillslope lengths in the pre-settlement scenario seem to balance the dispersive effects of the longer path lengths in the postsettlement scenario, thus the pre-settlement and post-settlement peak discharges are approximately equivalent. At small scales, the dispersion caused by the hillslope is larger in the pre-settlement case; thus, the post-settlement peak discharges are greater than the pre-settlement. (KEY TERMS: surface water hydrology; watershed management; drainage; hydrologic response; hillslope; network; Mackinaw.) White, Amanda B., Praveen Kumar, Patricia M. Saco, Bruce L. Rhoads, and Ben C. Yen, 2003. Changes in Hydrologic Response Due to Stream Network Extension Via Land Drainage Activities. Journal of the American Water Resources Association (JAWRA) 39(6):1547-1560.
INTRODUCTION Humans can have a pronounced effect on the hydrologic characteristics of watersheds. Modification of rainfall/runoff processes by urbanization and
agricultural development often leads to fundamental changes in basin hydrologic response (Jones, 1997). The extent of these changes is directly related to the intensity of human alteration of watersheds. Throughout much of the Midwestern United States, including Illinois, widespread alteration of vegetation cover and channel networks occurred in the late 1800s and early 1900s as flat, wetland prairies were cleared and drained so the land could be farmed. In east central Illinois, virtually all of the 22,000 square kilometers of tall grass prairie that existed prior to European settlement has been eradicated (IDENR, 1994). Moreover, historical information, especially General Land Office survey records, indicates that the need for improved drainage resulted in the modification of entire drainage networks, as numerous artificial channels, or ditches, were dug to replace poorly defined prairie sloughs (Rhoads and Herricks, 1996). Although human induced change in stream networks undoubtedly has greatly modified the hydrologic response of watersheds in drained landscapes of Illinois, the magnitude of this effect is unknown. Past work in the coupling between watershed geomorphology and hydrology suggests that variability in the arrangement of network pathways has a substantial impact on basin hydrologic response (RodríguezIturbe and Valdés, 1979), an effect known as geomorphologic dispersion. The concept of geomorphologic dispersion in turn suggests that the alteration of headwater streams via land drainage activities should have important implications for downstream flow regimes.
1Paper
No. 03027 of the Journal of the American Water Resources Association. Discussions are open until June 1, 2004. Graduate Student, Associate Professor, and Graduate Student, Environmental Hydrology and Hydraulics Engineering Program, Department of Civil and Environmental Engineering, University of Illinois, 205 North Mathews Avenue, Urbana, Illinois 61801; Professor, Department of Geography, University of Illinois, 205 North Mathews Avenue, Urbana, Illinois 6180; and formerly, Environmental Hydrology and Hydraulics Engineering Program, Department of Civil and Environmental Engineering, University of Illinois, 205 North Mathews Avenue, Urbana, Illinois 61801 (deceased) (E-Mail/Kumar:
[email protected]). 2Respectively,
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WHITE, KUMAR, SACO, RHOADS, AND YEN The purpose of this study is to explore how modification of drainage network structure through the addition of headwater channels affects the hydrologic response of the Mackinaw River Basin, a typical low relief agricultural watershed in central Illinois. Analytical models of overland flow and geomorphologic instantaneous unit hydrographs (GIUH) are used to compute and compare pre-settlement and postsettlement hydrologic conditions, thereby isolating the influence of changes in network structure on basin response. The analysis also provides important information for assessing the influence of future stream management scenarios on the hydrology of the Mackinaw River.
example can easily be extended to a basin of arbitrary order Ω.
BACKGROUND The concept of the GIUH provides the theoretical framework to determine the stream network response, such as in human modified drainage systems. The kinematic wave model for overland flow is used to determine the hillslope response, and the convolution of the network and hillslope responses defines the total basin hydrologic response. Network Response The GIUH theory postulates that the distribution of arrival times of water particles at the outlet of a basin depends on the topological structure of the river network. Consider a Horton-Strahler third-order watershed exemplified in Figure 1. Utilizing Strahler’s ordering system (Strahler, 1957), a set Γ of several pathways γ can be defined that represents the various transitions from the injection point of an effective raindrop to the subsequent streams it flows through to the outlet of the basin. For the third-order watershed in Figure 1, the set of pathways Γ = {γ1, γ2, γ3, γ4} is defined as
Figure 1. A Typical Third-Order Watershed.
Let the specific path γ be defined as a collection of states γ = {x1,x2,...,xj,...,xk} where x1 = oω, x1 = cω with ω as one of {1,...,Ω}, xj with j = {3,...,k-1} as one of {cω+1,...,cΩ-1} and xk = cΩ. The probability p(γ) that a droplet will follow any path to the outlet is simply p(γ) = πx1px1,x2px2,x3 ... pxk-1,xk
γ1 = o1 → c1 → c2 → c3 → outlet
where πx1 is defined as the initial probability that a droplet will begin in state x1, and pxi,xj is defined as the probability that a droplet will transition from state xi to state xj. The travel time through a particular path is the sum of the travel times spent in each individual state.
γ2 = o1 → c1 → c3 → outlet γ3 = o2 → c2 → c3 → outlet γ4 = o3 → c3 → outlet
(1)
where oω denotes the overland flow state that directly contributes to a stream of order ω and cω represents the channel state of order ω. It is noted that the above
JAWRA
(2)
Tγ = Tx1 + Tx2 + ... + Tx j + ... + Txk =
1548
k
∑ Tx .
i=1
i
(3)
JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION
CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES Hence, the travel time distribution through each individual path γ is fγ(t) = fx1(t) * fx2(t) * . . . * fxj(t) * . . . * fxk(t)
hydrodynamic dispersion coefficient for each particular state xi, respectively. The kinematic wave celerity can be computed as:
(4)
uxi = λux*i
where fxi(t) is the travel time distribution through each individual state xi of path γ and the * denotes the convolution operator. The travel time distribution fb(t) at the outlet of the basin, when the rainfall is uniformly distributed over the entire basin, is determined by randomizing over all possible paths fb (t) =
∑ p(γ ) fγ (t) = ∑ p(γ ){ fx
γ ∈Γ
γ ∈Γ
1
where λ is an empirical constant dependent upon the channel geometry (typically 3/2 for a triangular channel and 5/3 for a rectangular channel (Chow et al., 1988)) and ux*i is the steady state flow velocity under uniform flow conditions in state xi. For our calculations, we chose λ to be 3/2. The hydrodynamic dispersion coefficient can be calculated as
}γ
(t) * L * f xk (t) . (5)
DLx =
It follows that the three elements needed to fully characterize the basin GIUH fb(t) are:
∂T
+ ux i
∂x
= DLx
i
∂ 2 hxi ∂x 2
(
− Lxi − uxi exp 4 DLx t 4 πDLx t 3 i i Lxi
f xi (t) =
)
2
.
(9)
Through this formulation of the travel time distribution for the individual channels, they obtained the network travel time distribution fn(t) as
fn (t) =
1 4 πDL t
3
∑
γ ∈Γ
(
− L − ut γ p(γ ) Lγ exp 4 DL t
)
2
(10)
– where the mean path length Lγ is obtained as Lγ =
∑
xi ∈γ
Lxi
(11)
and the hydrodynamic dispersion coefficient DL is reduced to DL =
uh* 3S
(12)
where h* and S are the steady state flow depth under uniform flow conditions for all states xi and the mean channel bed slope for all states xi, respectively. This form of the network travel time distribution (Equation 10) will be used to compute the network response.
(6)
where h x i , u x i , and D L x are the flow depth, the i kinematic celerity of a traveling wave, and the JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION
(8)
3Sxi
where h *xi and S xi are the steady state flow depth under uniform flow conditions in state x i and the mean channel bed slope in state xi, respectively. Using the above equations, Rinaldo et al. (1991) obtained the residence time distribution in each state as
All of these elements are determined directly from the geometry of the network organization; however, the last term also involves some knowledge of the channel geometry. The initial probabilities πxi are obtained by multiplying the number of channels of order i by the fraction of the basin contributing directly into channels of order i, thus obtaining the probability that a raindrop will begin in a channel of order i. The transition probabilities pxi,xj are computed by multiplying the mean number of channels of order i draining into channels of order j by the ratio of number of channels of order j to number of channels of order i (RodríguezIturbe and Valdés, 1979). Several methods have been used to determine the residence time distributions in each individual state, such as assuming an exponential distribution (Rodríguez-Iturbe and Valdés, 1979), a uniform distribution (Gupta et al., 1980), and a gamma distribution (van der Tak and Bras, 1990). Rinaldo et al. (1991) used an advection/dispersion equation derived from the Saint-Venant equations of momentum balance (Lighthill and Whitham, 1955) to describe the flow through individual streams as ∂hxi
uxi hx*i
i
1. The initial probabilities πxi of beginning in a particular state xi. 2. The transition probabilities pxi,xj of being transported from state xi to state xj. 3. The residence time distribution in each individual state fxi(t).
∂hxi
(7)
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WHITE, KUMAR, SACO, RHOADS, AND YEN Hillslope Response
1 mσh
The overland flow is modeled using the kinematic wave model, following the work of Eagleson (1970), Chow et al. (1988), and Akan (1993). The momentum equation for a kinematic wave is So = Sf
m−1
∂q ∂q + = ie . ∂t ∂x
Based on the definition of flow velocity in Equation (14), one arrives at mσh m − 1 = mu =
(13)
where So is the topographic slope and Sf is the friction slope under steady state flow conditions. For a wide, open channel in which the hydraulic radius R is approximately equal to the flow depth h, Equation (13) can be expressed in terms of flow velocity u or discharge per unit width of the channel q, respectively, as
dh = ie dt
(14)
and similarly,
q = σhm
(15)
dq = ie . dx
where σ=
teq
(18)
l = m−1 σie
1/ m
.
(26)
Integrating Equation (23), assuming an initially dry surface under constant rainfall excess and an upstream discharge per unit width of zero, gives q = iex .
(27)
Assuming xo = 0 and incorporating Equation (25) into Equation (27) produces
(19)
q = σ(iet)m
Substituting Equation (19) into Equation (17) yields
JAWRA
(25)
where xo is the initial along channel displacement. Assuming xo = 0 and x = l, where l is the total overland flow length, the time to equilibrium t eq is obtained from Equation (25) as
Taking the derivative of Equation (15) with respect to time t, and assuming that σ is independent of x, gives ∂q ∂h = mσh m −1 . ∂t ∂t
(24)
x = xo + σiem − 1t m .
(17)
)
(23)
Substituting Equation (24) into Equation (21) and integrating yields
where ie = i - f ´ is the rainfall excess after infiltration losses are accounted for, i is the rainfall intensity, f ´ is the rate of infiltration, t is the time, and x is the displacement distance along the channel. Combining Equations (15) and (17), assuming that σ is independent of x, yields
(
(22)
h = iet.
m = 3/2, g = 9.81 m/s2 is the acceleration due to gravity, and f is the Darcy-Weisbach friction coefficient. The kinematic wave continuity equation, or the conservative form of the continuity equation, is
∂h ∂ ∂h ∂h + + mσh m −1 = ie . σh m = ∂t ∂x ∂t ∂x
(21)
Integrating Equation (22), assuming an initially dry surface under constant rainfall excess and an upstream depth of zero, gives
(16)
∂h ∂q + = ie ∂t ∂x
dx . dt
Thus, from Equations (18) and (20), taking Equation (21) into account and assuming that σ and m are constants, it can be surmised that
u = σhm-1
8 gSo , f
(20)
1550
(28)
JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION
CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES where q is the discharge per unit width for time 0 < t < teq. At t = teq, substituting Equation (26) into Equation (28), the peak equilibrium discharge per unit width qeq is
fconv (t) = g(t) * f (t) =
(33)
STUDY BASIN AND METHODOLOGY
(29)
qeq = iel.
Overview of Mackinaw River Basin
Hence, from Equations (28) and (29), the hillslope Shydrograph can be computed and the hillslope IUH determined through numerical differentiation of the S-hydrograph.
The Mackinaw River Basin (MRB) was chosen for this study because extensive hydraulic geometry relations are available through the work of Stall and Fok (1968). In their research, they developed descriptions of flow parameters such as volume, velocity, depth, top width, etc., using nine stream gaging stations within the MRB and characterized these properties in a Horton-Strahler framework. The MRB was also chosen because modifications of the channel network in this watershed are representative of those that occurred throughout many watersheds in east central Illinois. In addition, a portion of the MRB has been designated as an Illinois Resource Rich Area, meaning that this area has significant natural community and species diversity. Within the MRB are three nature preserves: Mehl’s Bluff Nature Preserve, ParkLands Nature Preserve, and Ridgetop Hill Prairie Preserve. In the 1950s, the Mackinaw Valley Improvement Association was formed and, together with the Nature Conservancy and the Illinois Environmental Protection Agency, became the Mackinaw River Partnership in 1996. Their goal is to “preserve and enhance the natural resources of the Mackinaw River” (Mackinaw River Partnership, 1996). The Mackinaw River is approximately 200 kilometers in length (Figure 2) and its headwaters are located near the city of Sibley in Ford County, Illinois (Figure 3). From Sibley, the river flows west to meet the Illinois River south of Pekin, Illinois. The drainage area of the MRB is approximately 2,950 km2 with a majority of the watershed in Tazewell, McLean, and Woodford Counties, and small portions in Mason, Livingston, and Ford Counties (Figure 3). Physiographically, most of the MRB lies within the Bloomington Ridged Plain, yet the western portion falls within the Springfield Plain (Leighton et al., 1948). These regions are subdivisions of the Till Plains Section of the Central Lowland Province and are characterized by a series of end moraines. The Bloomington and Shelbyville moraines are the largest of these moraine systems. The sediments that overlie the bedrock consist of glacial till, glacial outwash, and windblown silt (loess), and are generally 30 to 120 meters thick (IDNR, 1997). The landscape is characterized by uplands on the end moraines, lowlands along the valleys or floodplains, and areas of alluvial
Basin Response Once the network and hillslope responses are computed, they are convolved to determine the hydrologic response of the entire Mackinaw River Basin. This is accomplished as follows. In general, the form of the convolution of the hillslope and network IUHs, or the basin IUH fb(t) is (recall from Equation 5)
∑ p(γ ){ fx
fb (t) =
t
∫0 g(τ) f (t − τ)dτ .
γ ∈Γ
1
}γ
(t) * L * f xk (t)
∑ p(γ ){ fh (t) * fc
1
γ ∈Γ
}γ
(t) * L fck (t)
(30)
where fh is the hillslope IUH, or the travel time distribution in the initial state, fci is the individual channel IUH of a stream of order i, and * signifies the convolution operator. For example, an expansion of the previous equation for a third order basin, making use of Equation (1) will be
{
}
{
fb (t) = p(γ 1) fh1 * fc1 * fc2 * fc3 + p(γ 2 ) fh1 * fc1 * fc3
{
}
{
+ p(γ 3 ) fh2 * fc2 * fc3 + p(γ 4 ) fh3 * fc3
}
}
(31)
or simply
{
}
{
fb (t) = p(γ 1) fh1 * f pγ 1 + p(γ 2 ) fh1 * f pγ 2
{
}
{
}
+ p(γ 3 ) fh2 * f pγ 3 + p(γ 4 ) fh3 * f pγ 4
}
(32)
where fpγi is the travel time distribution for each path γi. The convolution operation is performed using the convolution integral
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WHITE, KUMAR, SACO, RHOADS, AND YEN
Figure 2. Location of the Mackinaw River Basin and the Ten Major Basins Located Within the MRB.
deposition. Elevations range from 263 meters above mean sea level in the uplands to 136 meters above mean sea level in the lowlands. Associated with bluffs along the main river valley, the greatest local relief is approximately 46 meters (IEPA, 1994). In its present state, a majority (90.3 percent) of the MRB is used for agricultural purposes. However, before this area was settled during the 1800s, it was mostly prairie and timberlands. Virgin prairies can scarcely be found now, although the upland plains of the MRB were dominated by wet prairie, prairie “pothole” ponds (named as such because of their circular nature), and floodplain forest prior to settlement and farming activities (Reber, 1997). The total area of JAWRA
presettlement prairie in McLean, Tazewell, and Woodford Counties is estimated at 89.5, 67.6, and 69.9 percent, respectively, and the total area of presettlement wetlands is estimated at 26, 24, and 20 percent, respectively, for the same counties (IDENR, 1994). Initial efforts to improve the drainage of the upper MRB are described in a history of Anchor County and vicinity (Bi-Centennial Committee, 1976:53): “Due to the straightening and deepening [of the river], it was expected that the Mackinaw would empty itself 18 times in the same length of time it formerly took to drain itself.” In addition to the “ditching,” as the channelization and dredging process is often called, drainage was also increased by using 1552
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CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES
Figure 3. The DEM Extracted Drainage Network for the Mackinaw River Basin. The DEM and the drainage basin boundary extracted from the DEM are shown in addition to the county boundaries.
“day laborers [who] dug lateral ditches by hand. This man-made drainage made it possible to plow the prairie fauna for the first time ... [In addition], the advent of tile drainage early in the [18]80s completed the transformation of the prairie into ordinary farmland... .” (Herre, 1940:43). The climate of the MRB is classified as humid continental, hence the summers are generally hot and humid and the winters are cold and dry. The mean annual precipitation is 95.89 centimeters. The mean January maximum and minimum temperatures are 1˚C and -9˚C, respectively; the mean July maximum and minimum temperatures are 31˚C and 18˚C, respectively (IDNR, 1997). In the MRB water is stored in man made reservoirs impounded by dams. It should be noted that these dams could cause flow retardation and, thus, could lead to deviations of flow velocities from those calculated using the hydraulic geometry in Stall and Fok (1968).
Illinois, 1807 to 1822 and 1830 to 1862) were prepared from field notes of surveys conducted between 1804 and 1855. The field notes are a conglomeration of federal land surveyors’ field notes and field notes of private surveys. Each plat contains natural and manmade features, such as bodies of water, wagon trails, town sites, and vegetation type (i.e., prairie, timber, fields, etc.). Although there may be errors associated with these surveys due to the fact that they were performed by several different entities and that standards were not always adhered to uniformly (Rhoads and Herricks, 1996), they are the only records available of the pre-settlement landscape of the MRB. Thus, the network from the surveying plats will henceforth be referred to as the pre-settlement network and the DEM extracted (RMC, 1995) network obtained by pruning the space filling network such that it is consistent with Version 3 of the River Reach File (USEPA, 1995) will hereafter be referred to as the post-settlement network. In comparing the presettlement and post-settlement networks, it was found that two orders of tributaries were added upstream of the pre-settlement network. Thus, the first order streams of the pre-settlement network correspond to the third order streams of the post-settlement network (Figure 4). The goal of this study is to isolate the influence of the network structure on possible changes in hydrologic response, thus channelization of the streams was not taken into account, although,
Pre-Settlement and Post-Settlement Networks To determine the approximate number of HortonStrahler orders that were added to the MRB’s river network, the streams shown on plats from the 1800s were compared with the current network structure. The current network structure was obtained from the DEM extracted river network (Figure 3). The Federal Township Plats (Surveyor General of the State of JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION
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WHITE, KUMAR, SACO, RHOADS, AND YEN
Figure 4. Comparison of the Pre-Settlement and Post-Settlement River Networks. The idealized watersheds to the right illustrate the change in network configuration, and thus hillslope configuration, before and after the Mackinaw River basin was settled.
channelization of a watershed has been shown to increase the peak discharge and decrease the time to peak of the hydrologic response to a rainfall event (Simpson and Newman, 1982). To compare the hydrologic response of the entire MRB in its pre-settlement and post-settlement states, it is necessary to consider the hillslope response, as well as the network response. The hillslope response is developed using the kinematic wave model outlined in the Background Section and the network response is developed using the GIUH method also outlined in the Background Section. The total basin response is then obtained through convolution of the hillslope and network responses (also discussed in the Background Section).
by the discharge per unit channel width at equilibrium qeq, and (3) taking the numerical derivative of the normalized hillslope S-hydrograph to compute the hillslope IUH. The parameters required for Equations (28) and (29) are: (1) the topographic slope of the hillslope So, (2) the Darcy-Weisbach friction coefficient f, (3) the constant rainfall excess ie, and (4) the total overland flow length l. Recalling from Figure 4, the pre-settlement hillslope corresponds to the first through fourth orders of the space filling network and the post-settlement hillslope corresponds to the first and second orders of the space filling network. Thus, the average topographic slope of the pre-settlement and post-settlement hill– slopes S o was computed as
Pre-Settlement and Post-Settlement Hillslopes
So =
It should be noted that the hillslope of the MRB was quite different in its pre-settlement state than in its post-settlement state. The thick, pre-settlement prairie grasses and numerous storage ponds would have generated a much slower response time to a rainfall event, as compared to the tile drained, postsettlement cropland. However, differences in surface roughness and tile drainage were not taken into account to focus on the effect of the network structure on the basin response. Computation of the hillslope IUH for the MRB in both pre-settlement and postsettlement states was accomplished by: (1) determining the hillslope S-hydrograph using Equations (28) and (29); (2) normalizing the hillslope S-hydrograph
JAWRA
κ
1 Si κ i=1
∑
(34)
– where Si is the average slope of channels of order i, κ = 4 (fourth order) for the pre-settlement scenario, and κ = 2 (second order) for the post-settlement – scenario. In the pre-settlement scenario, So = 6.255 x – -3 10 and in the post-settlement scenario, So = 4.541 x -3 10 (see Table 1). The Darcy-Weisbach friction coefficient f for shallow overland flow with Reynolds number ℜ < 900 was computed using the following nondimensional equation developed by Li and Shen (1973).
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CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES TABLE 1. Parameters Describing the Mackinaw River Basin’s River Network. Space Filling Network Order
Pre-Settlement Network Order
Post-Settlement Network Order
Along Channel Slope (S)
Along Channel Length (L) (km)
Number of Streams (N)
Basin Area (A) (km2)
1 2 3 4 5 6 7 8
Hillslope Hillslope Hillslope Hillslope 1 2 3 4
Hillslope Hillslope 1 2 3 4 5 6
9.145E-03 6.794E-03 5.181E-03 3.901E-03 2.574E-03 1.873E-03 3.401E-03 7.830E-04
0.444 0.736 1.094 2.095 5.873 10.313 34.865 81.740
24,039 5,745 1,262 253 51 11 2 1
8.243E-02 3.476E-01 1.349E+00 6.775E+00 3.869E+01 1.391E+02 8.808E+02 3.009E+03
0.4 i 24 + 660 3 gυ f = ℜ
To generate a hillslope IUH compatible with the network IUH, a rainfall intensity i that is dependent upon the return period was required. This is because the network IUHs discussed in the Background Section are a function of the frequency of occurrence of a particular discharge F, as opposed to being a function of the rainfall intensity i. The following relationship was used to determine the effective rainfall rate ie, which is assumed to be equal to the actual rainfall rate i (thus, the rate of infiltration f ´ is zero).
(35)
where υ = 1.115 x 10-6 m2/s is the viscosity of water and the Reynolds number ℜ can be calculated as ℜ=
4uh 4 q = . υ υ
(36)
ie =
According to the computations performed herein, the Reynolds numbers did not exceed 900 for any order subbasin within the MRB. The velocity u and flow depth h were computed using the downstream hydraulic geometry equations presented in Stall and Fok (1968) ln u( F , ω) = a −
bp b − cF + ln Aω* q q
sp s ln h( F , ω) = r − − tF + ln Aω* q q
(39)
where Q is the frequency dependent discharge, k is the saturated fraction of the drainage basin (i.e., the fraction of the basin contributing to overland flow), and A is the total area of the basin. The discharge Q was determined using the downstream hydraulic geometry equation presented in Stall and Fok (1968) (similar to Equations 37 and 38).
(37)
ln Q( F , ω) = x −
(38)
yp y − zF + ln Aω* q q
(40)
where Q has units of cubic feet per second, x, y, z, p, and q are empirical constants, and A*ω is the area of the subbasin and has units of square miles. Therefore, Q is dependent upon F, as well as ω. The values for x (0.00), y (1.34), and z (7.52) were obtained from the published results of Stall and Fok (1968). Incorporating Equation (40) into Equation (39) gives
where u has units of feet per second, h has units of feet, a, b, c, r, s, t, p, and q are empirical constants, and A*ω is the area of the subbasin (dependent upon the Horton-Strahler order ω of the subbasin) and has units of square miles. The empirical constants p (-2.13) and q (1.55) were obtained through the linear regression of the basin area with respect to order where p is the y-intercept and q is the slope. The values for a (0.25), b (0.12), c (2.26), r (-1.03), s (0.47), and t (3.13) were obtained from the published results of Stall and Fok (1968). The area of the various subbasins are in Table 1.
JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION
Q kA
i( F , ω) =
1555
[
exp a − bF + c ln Aω* kAω
]× 1
ξ3
(41)
JAWRA
WHITE, KUMAR, SACO, RHOADS, AND YEN where a = x −
yp y , b = z, c = , ξ is a conversion factor q q
(Table 1). It should also be noted that the average hillslope length will be greater in the pre-settlement scenario due to the smaller total channel length than in the post-settlement scenario (per Figure 4 and Table 1).
between English and S.I. units, Aω is the area of the subbasin and has units of square meters, and i(F, ω) has units of meters per second. Recall that we assume that the rainfall excess i e is equal to the rainfall intensity i(F, ω), or that the rate of infiltration f'´ is zero. Thus, the rainfall excess is a function of frequency and order and is denoted as ie(F, ω). Our interests lie in the rainfall excess corresponding to a certain flow in the channel, therefore the abstractions are excluded. Also, this assumption is plausible because, for a kinematic wave, the flow is steady and uniform and when the hillslope is saturated (a majority of overland flow does not occur until the soil becomes saturated), steady, uniform flow is more feasible. The last parameter required is the total hillslope length l. Horton (1932, 1945) suggested that the average hillslope length can be approximated as half the distance between channels, and hence l=
1 A = 2 D 2 Ltot
TABLE 2. Pre-Settlement Values for ni,w, Which is the Average Number of Channels of Order i Draining into Channels of Order w. The horizontal headings represent i and the vertical headings represent w.
kAω 2 Ltotω
(42)
(44)
i=1
κ = 4 (fourth order) for the pre-settlement scenario, κ = 2 (second order) for the post-settlement scenario, ηi,ω is the average number of channels of order i draining into channels of order omega (Tables 2 and – 3), and A i is the average area of a subbasin draining into a channel of order i (Table 1). Note that Equation (44) computes the area draining directly into channels of order ω through overland flow by computing the area of the subbasins that were pruned from the space filling network. Therefore, the i in this equation refers to orders of the space filling network. Ltotω is simply the mean total channel length for each order ω of the pre-settlement and post-settlement networks JAWRA
2
0.91
0.00
0.00
3
6.50
1.00
0.00
4
6.00
5.00
0.00
1
2
3
4
5
2
1.28
0.00
0.00
0.00
0.00
3
4.08
1.49
0.00
0.00
0.00
4
8.00
2.09
0.91
0.00
0.00
5
31.00
11.50
6.50
1.00
0.00
6
73.00
29.00
6.00
5.00
0.00
– Once the parameters S o, f, ie(F, ω), and lω were determined, the S-hydrograph was computed. To develop the S-hydrograph, a very small initial discharge per unit width qo = 1 x 10-10 m2/s was chosen to calculate an initial Reynolds number (Equation 36), an initial Darcy Weisbach friction coefficient (Equation 35), and an initial value for σ (Equation 16). From there, the discharge per unit width was computed using Equation (28). The parameters ℜ, f, and σ were then recalculated with the new discharge per unit width and the process was repeated using Equation (28) until t = te, at which point Equation (29) was utilized. Finally, the S-hydrograph was normalized by the discharge per unit width at equilibrium qeq to determine the unit step response function, which produces the impulse response function, or the hillslope IUH, as discussed below (Dooge, 1959). The response to a unit input I(τ), where τ is the time at which I(τ) is applied, can be determined using the convolution integral (after Equation 33)
κ
∑ ηi,ω Ai ,
3
(43)
where Aω is Aω =
2
TABLE 3. Post-Settlement Values for ni,w, Which is the Average Number of Channels of Order i Draining into Channels of Order w. The horizontal headings represent i and the vertical headings represent w.
L where D is the drainage density tot , Ltot is the A total length of all of the channels in the basin, and A is the total area of the basin. Adapting this relationship for our purposes such that the contributing hillslope area is the saturated fraction of the total area of the subbasin produces the following lω =
1
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CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES g(t) =
t
∫0 I (τ) f (t − τ)dτ
the time to peak discharge decreases and the peak discharge increases from the pre-settlement to postsettlement state for equivalent orders. This is a result of the decrease in overland flow length from the presettlement to post-settlement configuration.
(45)
where g(t) is the unit step response function, or the normalized S-hydrograph, and f(t - τ) is the impulse response function, or the hillslope IUH if I(τ) is unit and instantaneously applied. Thus g(t) =
t
∫0 f (t − τ)dτ .
(46)
Taking the derivative of both sides, assuming that the unit input is applied at τ = 0, gives f (t) =
dg(t) . dt
(47)
Therefore, the hillslope IUH f(t) equals the derivative, or slope, of the normalized S-hydrograph g(t) with respect to time.
RESULTS Figure 5 portrays the normalized hillslope S-hydrographs g(t) for the pre-settlement and post-settlement MRB for orders ωpost = {3,...,6}, or ωpre = {1,...,4}, for a frequency of occurrence of a particular discharge of F = 0.1, and a saturated fraction of the basin of k = 1. Recall that the post-settlement network is ordered differently than the pre-settlement network (Figure 4), hence ωpost = ωpre + 2. A saturated fraction of k = 1 was used throughout this analysis to obtain the hydrologic response when the entire basin is contributing to the overland flow. The time to reach equilibrium increases as the order increases because basin area and hillslope length increase as order increases, thereby increasing the time required to reach steady state. Also, the time to reach equilibrium decreases from the pre-settlement to post-settlement scenario for equivalent orders (i.e., ωpost = ωpre + 2). This is a result of the two orders of tributaries added to the pre-settlement network, creating less overland flow length for the post-settlement basin, thus decreasing the time required for the water to reach the channel and the time to reach steady state. The hillslope IUHs fhω for the pre-settlement and post-settlement MRB corresponding to the normalized S-hydrographs (Figure 5) are presented in Figure 6. As can be seen, the time to peak discharge increases and the peak discharge decreases as the order increases. This was expected due to the increase in hillslope length as basin order increases. In addition,
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Figure 5. Hillslope S-Hydrographs g(t) to the Pre-Settlement (top) and Post-Settlement (bottom) Mackinaw River Basin for Orders wpost = {3,...,6}, Corresponding to wpre = {1,...,4} (Figure 4), for a Frequency of F = 0.1, and for a Saturated Fraction of k = 1.
Figure 7 presents the network IUHs fnω for the presettlement and post-settlement MRB for orders ωpost = {3,...,6}, or ωpre = {1,...,4}, for a frequency of F = 0.1, and for a saturated fraction of k = 1. Recall that the network IUHs were obtained using the GIUH method described in the Background Section. For orders ωpost = 3 and 4, or ωpre = 1 and 2, the pre-settlement peak discharge is greater than the post-settlement and the pre-settlement time to peak is slightly less than the post-settlement. However, for orders ωpost = 5 and 6, or ωpre = 3 and 4, the pre-settlement and post-settlement network IUHs are approximately equivalent. Also, as order increases, the differences between the pre-settlement and post-settlement network IUHs
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Figure 6. Hillslope Instantaneous Unit Hydrographs fhw for the Pre-Settlement (top) and Post-Settlement (bottom) Mackinaw River Basin for Orders wpost = {3,...,6}, Corresponding to wpre = {1,...,4} (Figure 4), for a Frequency of F = 0.1, and for a Saturated Fraction of k = 1.
Figure 7. Network Instantaneous Unit Hydrographs fnw for the Pre-Settlement (top) and Post-Settlement (bottom) Mackinaw River Basin for Orders wpost = {3,...,6}, Corresponding to wpre = {1,...,4} (Figure 4), for a Frequency of F = 0.1, and for a Saturated Fraction of k = 1.
diminish. This occurs because the pre-settlement network has less upstream tributaries than the post-settlement network, thus at order ωpre = 1, there are no upstream tributaries contributing to the channel, yet for the equivalent order ωpost = 3, there are two orders of upstream tributaries contributing to the channel. The upstream tributaries disperse the flow, causing larger variances in the IUH, hence the lower peaks and longer durations of the low order, post-settlement network IUHs. This effect is reduced as upstream tributaries are added to the pre-settlement network; therefore, the high order subbasins of the pre-settlement and post-settlement networks have almost identical network IUHs. The basin IUHs fbω for the Mackinaw River Basin for orders ωpost = {3,...,6}, or ωpre = {1,...,4}, for a frequency of F = 0.1, and for a saturated fraction of k = 1, are shown in Figure 8. For all orders, the time to peak is shorter in the post-settlement scenario than in the pre-settlement. The peak discharge is greater in the post-settlement case for orders ωpost = 3 and 4,
or ω pre = 1 and 2, yet is approximately equal for orders ωpost = 5 and 6, or ωpre = 3 and 4. Thus, as order increases, the differences in magnitude of the peaks decreases, however, the differences in time to peak persist. Overall, in the post-settlement scenario, the time to peak is shorter than in the pre-settlement scenario. This time lag is a result of the shorter hillslope lengths in the post-settlement case, causing shorter travel times to the channel than in the pre-settlement case. At large scales, or high orders, the dispersive effects of the longer hillslope lengths in the pre-settlement case seem to balance the dispersive effects of the longer path lengths in the post-settlement case. Hence, the pre-settlement and post-settlement peak discharges are approximately equivalent. At small scales, the dispersion caused by the hillslope is larger in the pre-settlement scenario, thus the post-settlement peak discharges are greater than the pre-settlement.
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CHANGES IN HYDROLOGIC RESPONSE DUE TO STREAM NETWORK EXTENSION VIA LAND DRAINAGE ACTIVITIES shorter hillslope lengths. The basin hydrologic responses of both the pre-settlement and post-settlement scenarios were computed and compared. The kinematic wave model was utilized to calculate the hillslope response and the network response was computed using the inverse-Gaussian formulation of the geomorphologic instantaneous unit hydrograph. The total basin response was determined by convolving the hillslope and network responses. The time to reach the peak discharge for the postsettlement case was shorter than for the pre-settlement, regardless of order. This is due to the shorter hillslope lengths in the post-settlement scenario, resulting in shorter travel times. At large scales, the dispersive effects of the longer hillslope lengths in the pre-settlement scenario seem to balance the dispersive effects of the longer path lengths in the postsettlement scenario, thus the pre-settlement and postsettlement peak discharges are approximately equivalent. At small scales, the dispersion caused by the hillslope is larger in the pre-settlement case, thus the post-settlement peak discharges are greater than the pre-settlement. Also, at all scales, the distribution of the hillslope IUHs appears to have a large influence on the results. At large scales, the network IUHs are equivalent in both the pre-settlement and post-settlement scenarios, yet the post-settlement basin IUHs have shorter times to peak than the pre-settlement. Thus, the origin of this deviation must be attributable to the dispersion caused by the hillslope. At small scales, the pre-settlement network response has shorter times to peak than the post-settlement; however, the presettlement basin response has longer times to peak than the post-settlement. Again, the basis of this transition must be a result of the dispersion caused by the hillslope. These two examples illustrate the considerable influence of the hillslope on the basin’s hydrologic response. These results indicate that the addition of tributaries for land drainage purposes produces the characteristic effects of settlement on the hydrologic response of a basin (i.e., increases in peak discharges and decreases in times to peak). In general, these traits are more distinguishable at small scales, yet at large scales, the decrease in times to peak continues to persist. This study is important to watershed management agencies because it illustrates the effects of increasing the channel length, and thus decreasing the hillslope length, on hydrologic conditions in a watershed. It emphasizes the importance of hillslopes in determining the hydrologic response of watersheds in which the stream network has not been extended headward artificially. This information is particularly valuable for planning future watershed management activities and practices. The approach developed in
Figure 8. Network Instantaneous Unit Hydrographs fbw for the Pre-Settlement (top) and Post-Settlement (bottom) Mackinaw River Basin for Orders wpost = {3,...,6}, Corresponding to wpre = {1,...,4} (Figure 4), for a Frequency of F = 0.1, and for a Saturated Fraction of k = 1.
SUMMARY AND CONCLUSIONS The objective of this research was to explore the effects of the modification of network structure via land drainage activities on contemporary hydrologic conditions. This objective was accomplished by comparing the drainage network on plats from surveys performed during the early 1800s (denoted as the presettlement scenario) to the DEM extracted drainage network (denoted as the post-settlement scenario) to determine how the network has changed due to European settlement. The Mackinaw River Basin in Illinois was chosen as the focus for this analysis due to the availability of data and its popularity among watershed management agencies and nature conservationists. The post-settlement scenario was shown to have had two orders of tributaries added upstream of the headwater channels, thus the pre-settlement network has shorter path lengths to the outlet, yet longer hillslope lengths, whereas the post-settlement network has longer path lengths to the outlet, yet JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION
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WHITE, KUMAR, SACO, RHOADS, AND YEN this study can be extended by incorporating effects of changes in land cover (i.e., prairie grass versus cropland), on the response time of the basin. Thick grass and numerous wetlands characteristic of prairies should decrease the peak discharge and increase the time to peak in the pre-settlement scenario. Also, channelization of the streams and the presence of tile drains must be taken into account. These attributes also contribute to the increase in peak discharge and decrease in time to peak for post-settlement conditions.
Rhoads, B. L. and E. E. Herricks, 1996. Naturalization of Headwater Streams in Illinois: Challenges and Possibilities. In: River Channel Restoration: Guiding Principles for Sustainable Projects, A. Brookes and F. D. Shields Jr. (Editors). Wiley, Chichester, United Kingdom, pp. 331-367. Rinaldo, A., A. Marani, and R. Rigon, 1991. Geomorphological Dispersion. Water Resources Research 27(4):513-525. RMC (Rocky Mountain Communications, Inc.), 1995. 3-Arc-Second Elevation Data on CD-ROM. Golden, Colorado. Rodríguez-Iturbe, I. and J. B. Valdés, 1979. The Geomorphologic Stucture of Hydrologic Response. Water Resources Research 15(6):1409-1420. Simpson, P. and J. Newman, 1982. Manual of Stream Channelization Impacts on Fish and Wildlife. U.S. Department of the Interior, Fish and Wildlife Services, Washington, D.C. Stall, J. B. and Y. Fok, 1968. Hydraulic Geometry of Illinois Streams. Technical Report, University of Illinois Water Resources Center for Research, Report No. 15, Illinois State Water Survey, Urbana, Illinois. Strahler, A. N., 1957. Quantitative Analysis of Watershed Geomorphology. EOS Transactions AGU 38(6):913-920. Surveyor General of the State of Illinois, 1807-1822; 1830-1862. Federal Township Plats, Illinois. USEPA (U.S. Environmental Protection Agency), 1995. Version 3 of the River Reach Files. Available at www.epa.gov/OST/BASINS/ STATES/IL. Accessed in July 2000. van der Tak, L. D. and R. L. Bras, 1990. Incorporating Hillslope Effects into the Geomorphologic Instantaneous Unit Hydrograph. Water Resources Research 26(10):2393-2400.
LITERATURE CITED Akan, O. A., 1993. Urban Stormwater Hydrology. Technomic Publishing Co., Inc., Lancaster, Pennsylvania. Bi-Centennial Committee, 1976. History of Lawndale, Martin, and Anchor Townships and the Villages of Colfax and Anchor, McLean County, Illinois. Cornbelt Press, Fairbury, Illinois, 164 pp. Chow, V. T., D. Maidment, and L. Mays, 1988. Applied Hydrology. McGraw-Hill, New York, New York. Dooge, J. D. I., 1959. A General Theory of the Unit Hydrograph. Journal of Geophysical Research 64(2):214-256. Eagleson, P. S., 1970. Dynamic Hydrology. McGraw-Hill, New York, New York. Gupta, V. K., E. Waymire, and C. T. Wang, 1980. A Representation of an IUH from Geomorphology. Water Resources Research 16(5):885-862. Herre, A. W., 1940. An Early Illinois Prairie. American Botanist 46:39-44. Horton, R. E., 1932. Drainage-Basin Characteristics. EOS Transactions AGU 13:350-361. Horton, R. E., 1945. Erosional Development of Streams and Their Drainage Basins: Hydrophysical Approach to Quantitative Morphology. Geological Society of America Bulletin 56:275-370. IDENR (Illinois Department of Energy and Natural Resources), 1994. The Changing Illinois Environment: Critical Trends. Technical Report, ILENR/RE-EA-94/05, Vols. 2 and 3, IDENR, Springfield, Illinois. IDNR (Illinois Department of Natural Resources), 1997. Mackinaw River Area Assessment. IDNR, Springfield, Illinois, Vols. 1 and 2. (IEPA (Illinois Environmental Protection Agency), 1994. Illinois Water Quality Report. IEPA Bureau of Water, Springfield, Illinois, Vol. II. Jones, J. A. A., 1997. Global Hydrology: Processes, Resources and Environmental Management. Addison Wesley Longman, Harlow, New York. Leighton, M. M., G. E. Ekblaw, and L. Horberg, 1948. Physiographic Divisions of Illinois. Journal of Geology 56:16-33. Li, R.-M. and H. M. Shen, 1973. Effect of Tall Vegetations on Flow and Sediment. Journal of the Hydraulics Division, ASCE 99(HY5):793-814. Lighthill, M. J. and G. B. Whitham, 1955. On Kinematic Waves: I. Flood Movement in Long Rivers. In: Proceedings of the Royal Society of London, Series A, Vol. 229, pp. 281-316. Mackinaw River Partnership, 1996. Available at http://dnr.state.il. us/orep/c2000/manage/MACKINAW/Mackhome.htm. Accessed in August 2002. Reber, R., 1997. The Mackinaw. The Illinois Steward 5:13-20.
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