Quick-return subsurface flow

Journal of Hydrology 8 (1969) 122-136; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

Q U I C K - R E T U R N SUBSURFACE F L O W D. G. JAMIESON and C. R. AMERMAN*

Abstract: One of the least understood of all hydrologic problems is that of lateral flow in the soil's upper horizons. The nature and quantity involved in this quick-return flow are unknown and have to be inferred from the shape of the observed hydrograph. An attempt has been made to mathematically simulate the flow phenomenon using rainfall as input and a system of interconnected, nonlinear reservoirs as the operator. 1. Introduction The parametric approach to engineering hydrology is to regard the landphase of the hydrologic cycle as an open-ended system with rainfall forming the input and streamflow, the output. However, there is not just a single route for the conveyance of precipitation excess from the point of entry to the point of exit, but several intrinsically interconnected paths. The tendency has been to treat each of these routes as a separate identity; for example, surface runoff and base flow are usually predicted independently and then merged to produce the total outflow hydrograph. The probable reason for this is the nebulous connection, linking the infiltrated rainfall with the recharge to the saturated zone. In the soil profile above the saturated zone, thresholds and lateral outflows deplete the potential recharge to an unknown extent. It is the lateral outflow from the soil's upper horizons that concerns this investigation.

2. Background In systems hydrology, it is commonly assumed that the operator, 4, acting on input x to produce output y, is linear. Use can then be made of the principles of superposition and proportionality. If

• (xl + x2) =

it follows

(cxl) = c49(xl),

(Xl) +

(x2)

where c is a constant

(l)

(2)

and the system is said to be linear. • Formerly Research Hydraulic Engineer, USDA Hydrograph Laboratory, Beltsville, Maryland [currently with Water Resources Board, Reading, England] and Research Hydraulic Engineer, U.S. Dept. of Agr., Agricultural Research Service, Soil & Water Conservation Research Div., Madison, Wisconsin. 122

QUICK-RETURN SUBSURFACE FLOW

123

This is known to be an oversimplification but it is a useful approximation for surface water response; the field application of unit hydrograph theory bears witness to this. However, nonlinearity is unavoidable in the upper portion of the subsurface flow system. Since quick-return flow is dependent upon infiltration as input, the complexity of the feedback between rate of infiltration and soil moisture status generally imparts nonlinearity. In addition, as lateral flow is not observed during all periods of infiltration, a threshold restraint is evident. Synthesis techniques usually begin with the postulation of a general conceptual model (Fig. 1). Two approaches towards simulation are feasible:

lP

p

PRECIPITATION

OF

OVERLANDFLOW

Fig. 1. Land phase of the hydrologic system, conceptual model. (1) For a particular model and watershed, parameters can be optimized to find the best relationship between precipitation and runoff. Such was the approach of Dawdy and O'Donnell (1965) who used an assumed model and synthetic, error-free data. Had real data been used, the optimized parameters may have had no physical meaning but merely represent a means of converting inflow into outflow. However, simulation of a watershed flow pattern is only the first step towards prediction, which requires the parameters to have physical dimensions if the model is to be applied to ungaged watersheds. (2) The alternative approach is to use a knowledge of physical hydrology, derived from studying subsystems, to fix certain parameters as measured quantities. The inevitable compromise as used in the Stanford Mark 1V model (1966), as well as this model, is to subjectively fix some parameters and search for the remaining.

124

D. G. JAMIESON A N D C. R. A M E R M A N

3. Hydrograph analysis Quick-return flow is one of the most difficult flow regimes to study, because input, output and the operator are generally unknown and have to be inferred or assumed. Its contribution to the total hydrograph is evident but not measurable, since it is obscured by surface-water runoff and base flow. Since the hydrograph shape is the basis for inferring quick-return flow, it is appropriate to begin with hydrograph analysis. Throughout this investigation, reference is made to data from the Agricultural Research Service's North Appalachian Experimental Watershed near Coshocton, Ohio; quickreturn flow is known to contribute to streamflow in this region. In particular, an attempt has been made to match the observed hydrographs corresponding to some of the major rainfall events on watershed-94, (2.37 square miles). One of the classical methods of deriving a storage/flow relationship makes use of a simple recession curve for the watershed in question. Starting at a known or assumed point of zero flow, q0, and zero storage, So, for any prior discharge on the recession curve, qf, the associated storage, Sf, is the amount of precipitation excess still contained in the system (Fig. 2). For various q and associated S, a storage/flow relationship can be plotted (Fig. 3). Since the lower subsurface flow regimes are not considered here, some systematic form of base-flow separation was necessary. A semilogarithmic plot of discharge against time sometimes yielded a distinct change of slope at the junction of the two flow regimes. For those storm events exhibiting the phenomenon, this point was taken as the end of precipitation excess, qo, So. Base flow was abstracted systematically as shown in Fig. 2, assuming zero base flow at peak discharge. precipitation excess

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Time

Fig. 2. Derivation of storage/flow relationship.

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Fig. 3. Storage/flow relationship for an unsustained recessioncurve. A plot of discharge against storage for each of these simple hydrographs gave a unique relationship initially before easing into a series of approximately parallel lines (Fig. 4). The gradients of these lines were designated K2 and Kc, respectively. For the remainder of the storm events, this K2 relationship has been used to select the position of qoSo; by successive approximation, the storage/flow relationship for each of the simple recession curves could be made to have an initial slope of K 2 (Fig. 4). Recession curves from high-intensity, short-duration storm events tended to deviate from K 2 into Kc very quickly, whereas those from long-duration, low-intensity events retained the K2 relationship either entirely or for considerably higher flows. K 2 is the storage coefficient associated with quickreturn flow and Kc is the channel storage coefficient. The overland-flow storage coefficient, K~, cannot be obtained from the recession curve directly but is evident in the upper portion of the storage/flow relationship. Similar storage coefficients can be derived for the various regimes of base flow (Onstad and Jamieson, 1968). Therefore, it is proposed that the land phase of the hydrologic system can be approximated by a combination of reservoirs termed overland flow, quick-return flow, delayed-return flow, etc. Each flow regime is identified by a corresponding storage coefficient Ka~KN, the larger values inferring a more devious path to the point of exit. Subsequently, each of these flow regimes is cascaded into a common, surface channel whose storage coefficient is designated K~. The resulting combination of reservoirs is similar to the concept of Sugawara and Maruyama (1956). One or more regimes may not be evident, depending on the relative size

126

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of the watershed, the nature of the input, and antecedent conditions. To demonstrate the effect of size, two watersheds at Coshocton have been selected; the smaller one of 1.27 acres forms an integral part of the larger 303-acre watershed. While quick-return flow is evident in the larger, the hydrograph shape of the smaller indicates none (Fig. 5). The presumption is made for the smaller, that quick-return flow has not yet reappeared as surface runoff and, therefore, has flowed under the gaging station. The nature of the rainfall input has a profound effect on the route taken by the excess water. For example, a short-duration, high-intensity summer storm event may yield almost entirely overland flow with subsequent channel flow (Storm 46, Fig. 4) while a low-intensity, long-duration storm is more likely to produce subsurface flow in addition to channel flow (Storm 23, Fig. 4). However, a major storm event generally produces a combination of all flow regimes (Storm 29, Fig. 4).

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4. Mathematical simulation The importance of infiltration in the role of determining surface runoff from rainfall is such that improvisations and empiricisms are used as an approximation prior to the availability of more deterministic methods. One of the more sophisticated empiricisms was presented by Holtan (1961). Although any infiltration model could have been applied, it was convenient to use Holtan's as its concept already envisioned the upper horizons of the soil to act as a nonlinear reservoir. Infiltration forms the input to this reservoir, which can be depleted by one or more outflows. Thus the rate of infiltration is expressed as a function of the current storage available above a restricting horizon; the seepage through this confining layer constitutes one of the exhaustion rates. Time, therefore, is an implicit variable. A threshold criterion divides the moisture status above the restricting horizon into plant-available water capacity, Ca, and gravitational water. Moisture below the threshold value, C2, is exhausted only by evaporation and transpiration, whereas water in excess of the threshold value can also be depleted by downward seepage through the restricting horizon at a constant rate,/2. The vegetative cover of the soil in question exerts its influence on the rate at which the available porosity, or storage, is exhausted. With all other factors equal, the rate of infiltration for fallow soil is less than for a soil with

128

D. G. JAMIESON A N D C. R. AMERMAN

a substantial cover where more pore-spaces are surface connected by the root system. Consequently, a vegetation factor is required as an index of surface penetration. The form of Holtan's infiltration equation is f = a.S~ "4

(3)

if moisture content is below the threshold value, and f = a'Sa1''~ + f2

(4)

if moisture content is above the threshold value, where f = capacity rate of infiltration f2 = exhaustion rate through restricting horizon a =surface-penetration index S a = storage currently available above restricting horizon. Allowance is also made for depression storage, C1, which can infiltrate after the termination of rainfall or during periods when infiltration capacity rate exceeds the rainfall rate. The effect of having quick-return flow, QRF, is that of a lateral outflow from the soil reservoir in addition to a vertical exhaustion rate, f2 (Fig. 1). The assumption is made that QRF is a hydraulic flow and as a first approximation, has been made a linear function of moisture in excess of field capacity, that is, gravitational water. Ignoring evaporation and transpiration during periods of storm rainfall, the storage available at any time t + 1 ( S a ) t + 1 --- ( S a ) t -- f , "At

(5)

(Sa)t+ 1 = (Sa)t "[- (f2 -- f l ) "At q- 42"At

(6)

below the threshold, and

above the threshold, where subscript t denotes time and At, the time increment determined by rainfall input or by a fixed value after the cessation of r a i n f a l l ; f 1( ~ f ) , is actual infiltration and q2, the lateral outflow, QRF. Most simulation models tend to treat the watershed as a lumped system, whereas the prototype is distributed. It is known that the areal rainfall pattern and the variability in soil type generate spatially varying amounts of precipitation excess. However, for simulation, some degree of lumping is necessary. England and Holtan (1967) postulate that there is a correlation between soil types and their elevation sequence. They suggest that for hydrologic purposes, the complex distribution of soils can be simplified into three groupings or response zones, namely, upland, hillside and bottomland. Upland soils are characterized by fairly deep, residual soils while hillside soils tend to be shallower and steeply inclined and, therefore, more liable to generate precipitation excess. Bottomland soils are essentially deep alluvial-

QUICK-RETURN SUBSURFACE FLOW

129

colluvial complexes, capable of absorbing large quantities of rainfall. Each of the response zones for Coshocton W-94 was determined by examination of soil and topographic maps and treated as a separate identity for the purpose of calculating infiltration. (Fig. 6). Parametric hydrology is directed towards time-invariant systems. However, the watershed does not usually conform since, for one thing, the vegetative cover changes with seasons. For this reason, the surface penetration index a is not held constant but allowed to vary with month of the year. In addition, during the winter months, the ground may be partly frozen, which accentuates runoff. While the surface soil layers may be frost-free, the lower horizons could still be frozen (Razumova, 1965); this phenomenon can be simulated by reducing the vertical seepage rate, f2.

5. Routing Zoch (1934) introduced the simplest form of linear reservoir routing when he found the relationship between the input function, p ( t ) , and the output function, q(t), assuming the operator to be a single linear reservoir; the discharge, q, was related to storage, S, by the linear relationship (7)

S = Kq.

Combining this assumption with the continuity equation, (8)

p = q + dS/dt

gives a first-order, linear differential equation (9)

p = q + K(dq/dt).

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.

(10)

When a linear reservoir discharges into a second linear reservoir which have storage coefficients K and Ko, respectively, an instantaneous input of unit volume into the first reservoir causes outflow into the second: p(2) = 1 / K ' e -t/K. The outflow from the second reservoir can be derived by convoluting the IUH of the second reservoir with this inflow: e-t/K _ e-t/Ko

u (0, t) -

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This model of two, unequal reservoirs has been suggested by Sugawara and Maruyama (1956) and by Singh (1962). The concept of cascading through two linear reservoirs has been used to route both surface and subsurface flows (Fig. 6). The reservoir storage coefficient, K, pertains to the imaginary reservoir simulating the particular flow regime in question (K= K~, K2, K 3 etc.) and K c is the storage coefficient of the common channel into which they discharge. If, however, Kc is equal to one of the other K-values, Eq. (11) is not applicable. Nash (1957) represented the runoff phenomena by n cascading linear reservoirs, each having the same storage coefficient and obtained the expression

u(0, t) = K"

"(n - 1)!.e

(12)

Therefore, in the specific case of two equal reservoirs, this simplifies to: u(O, t) = t / K 2 . e -'/K .

(13)

Equation (13) has been used for routing overland flow for Coshocton W-94, since K 1 for overland flow was found to equal Kc of channel flow. The channel storage coefficient (Kc) and the quick-return flow storage coefficient (K2) can readily be determined from hydrograph analysis (Fig. 4); these coefficients have been taken as constants for Coshocton W-94. As the overland flow storage coefficient cannot be determined from examination of the simple recession curve, the method of moments as suggested by Nash (1958) has been used. For a short, intense storm: 1st moment of hydrograph about the origin = K 1 + K~.

(14)

The first moment is also equal to the lag between the mid-volume of precipitation excess and the mid-volume of runoff. The routing model is essentially a storage model with no separate allowances for translation effects. However, as Laurenson (1962) emphasizes, implicit in any storage model are the combined effects of both attenuation and translation.

6. Application The overland flow and quick-return flow accounting and routing procedures have been programmed for an IBM 1620 computer. Fifteen of the major rainfall events occurring on Coshocton W-94 formed the input to the model. These events consist of the original ten simple storms used in the hydrograph analysis in addition to five complex storms. On each occasion, the antecedent soil moisture status and all parameters but one were inserted

132

o. G. JAMIESON AND C. R. AMERMAN

as measured variables. The surface-penetration index, a, for any one m o n t h was established by iteration. The portion o f overland flow, OF, and quick-return flow, Q R F , has been estimated f r o m the total observed h y d r o g r a p h for subsequent comparison with the accumulated predicted amounts, (Table 1). To illustrate the reTABLE 1

Observed and predicted totals of overland and quick-return flows Storm No.

2 4 14 19 23 29 30 35 37 46 3 10 25 28 40

Date

8-21-60 6-28-57 7-11-46 6-12-57 8-31-65 6-04-41 2-10-59 7-20-46 4-25-61 8-02-64 3-09-64 8-27-40 3-06-45 6-24-57 8-30-40

Rainfall Total Duration (h) (in) 3.90 3.40 2.77 2.16 2.30 2.10 1.97 1.71 1.82 2.25 3.70 2.63 2.46 2.22 1.87

18.0 17.2 2.4 2.8 29.0 12.4 27.2 32.4 11.9 1.5 36.4 21.1 25.8 12.7 25.3

Observed Z'OF+SQRF (in)

SOF

0.92 2.17 0.74 0.73 0.07 0.83 1.33 0.80 1.16 0.15 2.32 0.14 1.43 0.23 0.20

0.85 1.25 0.64 0.77 0.00 0.60 0.49 0.57 0.57 0.16 0.15 0.23 0.15 0.15 0.00

(in)

Predicted SQRF Z'OF-/-XQRF (in) (in) 0.30 0.66 0.01 0.10 0.03 0.14 0.77 0.12 0.31 0.00 1.93 0.11 1.17 0.19 0.37

1.15 1.91 0.65 0.87 0.03 0.74 1.26 0.69 0.88 0.16 2.08 0.34 1.32 0.34 0.37

sponse of the model to rainfall inputs, three radically different storm events are presented. Storm 46, (Fig. 7) is indicative o f short-duration, high-intensity rainfall which results in negligible subsurface flow. Storm 29, (Fig. 8) is representative of longer duration, intense rainfall. Storm 25 (Fig. 9) is a complex, low-intensity rainfall event on partly frozen g r o u n d resulting in a large a m o u n t o f quick-return flow. Each o f these three hydrographs have been added to the base flow, if existent. Base flow was c o m p u t e d by routing the d o w n w a r d seepage, (f2 o f Eq. (4)), through two lower reservoirs by the techniques o f Onstad and Jamieson (1968).

7. Conclusions A n understanding of the rainfall-runoff relationship has been the focal point o f m a n y investigations, both recently and historically. Unfortunately,

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the methodology of the historic still influences that of the recent period. The majority of investigations deal entirely with surface runoff because of its relatively greater importance. Invariably, each begins with an arbitrary separation of base flow. Dickinson (1963) summarized most of these methods and concluded that an adequate method of baseflow separation has not yet been determined. The lack of discernment usual to methods of baseflow separation has been such that the evidence of quick-return flow could be completely obliterated from the total hydrograph. Indeed, its very existence has been questioned by some (Roche, 1963). Despite this, lateral flow from the soil's upper horizons has been observed and measured at certain locations. These locations have been either forested (Whipkey, 1962), steeply inclined (Hewlett, 1961) or with soils displaying anisotropic hydraulic conductivity (Amerman, 1965). In the specific case of Coshocton W-94, a combination of all three conditions probably contributed to the flow phenomenon. The storage coefficient pertaining to quickreturn flow, K 2, was similar for watersheds of various sizes within the same drainage basin, inferring that K 2 is a function of soil type and perhaps land slope.

References AMERMAN,C. R., The use of unit-source watershed data for runoff prediction. Water Resources Res., 1 (1965) 499-507 CRAWEORD,N. H. and R. K. LINSLEY,Digital simulation in hydrology. Stanford Watershed Model IV, Tech. Rept. No. 39, Dept. Civil Eng., Stanford Univ. 0966) DICKINSON,W. T., Unit hydrograph characteristics of selected Ontario watersheds. M.S.A. Thesis (Univ. of Toronto, 1963) ENGLAND,C. B. and HOLTAN,H. N., Geomorphic grouping of soils in watershed engineering. Presented at Amer. Soc. Agron., Washington, D.C. (Nov. 1967) HEWLETT, J. O., Soil moisture as a source of base flow from steep mountain watersheds. U.S. Forest Service, Southeast Forest Expt. Sta., Paper No. 132 (1961) HOLTAN,n . N., A concept for infiltration estimates in watershed engineering. U.S. Dept. Agr., ARS 41-51 (1961) LAURENSON,E. M., Hydrograph synthesis by runoff routing. Univ. New South Wales, Water Res. Lab Rept. No. 66 0962) NASH, J. E., Determining runoff from rainfall. Proc. Inst. Civil Engrs., 10 0958) 163-184 NASH,J. E., Systematic determination of unit hydrograph parameters. J. Geophys. Res. 64(1) (1959) 111-115 ONSTAD,C. A. and D. G. JAMIESON, Subsurface flow regimes of a hydrologic watershed model. Presented at Second Seepage Syrup., Phoenix, Arizona, (March 1968) RAZUMOVA,L. A., Basic principles governing the organization of soil-moisture observations. Design of Hydromet. Networks Syrup. (Quebec, 1965) ROCHE,M., Hydrologie de surface (Gauthiers Villars Editeur, Paris, 1963) SmGn, K. P., Nonlinear instantaneous unit-hydrograph theory. Amer. Soc. Civil Engrs., J. Hydr. Div., 90(HY2) (1964) 313-347 SUGAWARA,M. and F. MARUYAMA,A method of prevision of the river discharge by means

136

D. G. JAMIESON AND C. R. AMERMAN

of a rainfall model. Publ. No. 42, de L'Ass. Int. d'Hydrologie Symp. Darcy, Tome III, 71-76 (1956) WHIPKEY, R. Z., Subsurface stormflow in forest soil. Presented at 43d Annual Meeting, Amer. Geophys. U., Was hington, D.C. (April 1962) ZOCH, R. T., On the relation between rainfall and streamflow, I, II, III. Monthly Weather Review, 62(9) (1934) 315-322; 64(4) (1936) 105-121 ; 65(4) (1937) 135-147