88  

5. HILLSLOPE HYDROLOGY 5.1 Overviews As we saw on page 1, the essence of physical hydrology is in: (1) Generation of runoff in headwaters (2) Routing of runoff water through streams

Since much of the headwaters are hillslopes of varying size and gradient, it is obviously important to understand the hydrologic function of hillslopes. → Brutsaert (p. 442)

Hillslope runoff processes can be conceptually classified into overland flow and subsurface storm flow. In reality, these two processes occur simultaneously in different parts of the hillslope and interact with each other. 5.2 Infiltration Excess Overland Flow This type of runoff generation is also called Hortonian or Horton overland flow, and is the basic tool of engineering hydrology, particularly for urban storm water hydrology. The concept can be best demonstrated using the infiltration theory from pages 36-39. For example, the Green-Ampt theory (Eq. 2-39) predicts very high infiltration capacity (ic) at early time. If the “ponding” condition is maintained at the surface, ic decreases with continued infiltration and approaches the saturated conductivity (K0) at steady state. In real storm events, however, the ponding does not occur until the soil is completely saturated. To apply the Green-Ampt theory to non-ponding situations, we consider two cases.

(1) Rainfall intensity (qr) is smaller than K0. In this case, infiltration capacity is never reached and no runoff is generated.

(2) Average rainfall intensity is considerably greater than K0. The soil surface may become saturated at some point, causing the ponding condition. Infiltration rate (i) is equal to rainfall intensity until ponding. After ponding, i is equal to ic. The “excess” rainfall will become overland flow.

For storm runoff modeling, it is important to determine the length of “time to ponding” (tp). Let I(tp) and ic(tp) be cumulative infiltration and infiltration capacity at time tp. By definition, we must have: qrtp = I(tp) (5-1)

qr = ic(tp) (5-2)

89  

From the derivation of the Green-Ampt equation on pages 37-38,

10

0 L

hhKi f

c (5-3)

I = L(0 – i) (5-4)

where L is the depth to the infiltration front. From Eqs. (5-1) and (5-4):

i

prp

tqtL

0

)(

From Eqs. (5-2) and (5-3):

11

)(00

00

0pr

if

p

fr tq

hhK

tL

hhKq

prifpr tqKhhKtq 0000

2

ifrrp hhKqKqt 0000

2

0

2

0

000

2 Kqq

S

Kqq

hhKt

rrrr

ifp

(5-5)

where S is the sorptivity defined in Eq. (2-37), which is measurable in the field.

Note that tp → ∞ as qr → K0 and Eq. (5-5) is valid only for qr > K0. While this approach appears to be reasonable, qr is almost always smaller than K0 of the top soil in vegetated areas.

For example, in croplands and grasslands of the Canadian prairies, K0 is typically greater than 10-5 m s-1 = 36 mm hr-1. In comparison, average rainfall intensity rarely exceeds 25 mm hr-1.

→ In 2004-2015, the maximum 1-hr average rainfall intensity measured in Spy Hill Farm was 23 mm hr-1 (June 5, 2007).

Therefore, Hortonian overland flow almost never occurs in “natural” hillslopes, even though it may be the main runoff mechanisms in urban areas and disturbed areas with compacted soil. → Brutsaert (p. 443)

A notable exception is frozen soil where K0 is significantly lowered due to the ice crystals blocking soil pores. 5.3 Saturation Overland Flow In Hortonian overland flow, the soil is saturated from the top. In most natural hillslopes, overland flow occurs as the soil becomes saturated from the bottom.

90  

Infiltration causes the water table to rise during a storm. In the riparian zone, where the water table is close to the surface, it takes a relatively small amount of infiltration to saturate the entire soil. → Capillary fringe, see page 24. The upward hydraulic gradient under saturated areas causes: (1) all rain water to flow overland. (2) some subsurface water is forced out to the surface.

The area of saturation expands during a storm. → variable source area concept, see Brutsaert (p. 445). Hillslopes commonly consist of high-K0 soil underlain by relatively impermeable material (e.g. bedrock, frozen soil). In such case, thickness of the high-K0 zone has a strong influence on the occurrence of saturated areas. On a given hillslope, antecedent moisture condition has a major effect on saturated areas. 5.4 Subsurface Storm Flow Overland flow (Hortonian or saturation) explains the rapid response of stream hydrograph to storm events. However, studies using chemical and isotopic tracers have convincingly shown that the storm runoff water from forested hillslopes consists mostly of “old” or “pre-event” water stored in the soil.

This is only possible if the storm water has gone through the soil.

In most of the previous studies conducted in headwater catchments, subsurface stormflow was the dominant flow mechanism for hillslope runoff generation. → Brutsaert (p. 446)

It took a long time for engineering hydrologists to accept this paradigm, but they now think that subsurface stormflow is important. Why has it taken such a long time?

Compared to the simplicity of Hortonian overland flow, the mechanism of subsurface stormflow is very complex. There appears to be a range of flow mechanisms depending on the environment and season.

There is no simple and universally applicable numerical algorithm of subsurface stormflow for practical modeling purposes.

The science is still “young”. New discoveries still occur in hillslope hydrology. (1) Macropore or “pipe” flow

Even though the lateral flow of subsurface water may be slow (≅K0 × slope) in the soil matrix, the network of large diameter (mm-cm) conduits can transmit storm water at a velocity

91  

comparable to overland flow. This type of flow is called “macropore flow”, “preferential flow”, “pipe flow”, “bypass flow”, etc.

The conduits may be provided by: - desiccation cracks - decaying roots - animal burrows

It has been reported that initially small, disconnected macropores may develop into a larger, connected network of pipes over the years. → Sidel et al. (2001, Hydrological Processes, 15:1675-1692) (2) Rapid delivery of infiltration by macropore flow may be coupled with a fast lateral flow in the weathered zone above relatively impermeable bedrock (3) Soil water capacity function (Cw, see page 31) is very small within the capillary fringe. A small addition of water can cause a dramatic change in matric potential head (hm). → Resulting in the rapid saturation, allowing water to flow through the top soil having many macropores. The soil underlying hillslopes commonly have a certain degree of stratification with a relatively impermeable layer.

It has become clear in the past decade that the topography of the impermeable surface (e.g. bedrock) has a decisive role in runoff processes. In particular:

- Subsurface flow does not happen until the “depression storage” on the bedrock is filled.

- Subsurface flow paths are channel-like, flowing in the bedrock valleys.

The conceptual model is referred to as “fill and spill”. → Spence & Woo (2003, Journal of Hydrology, 279:151-166) Tromp-van Meerveld & McDonnell (2006, Water Resources Research, 42, W02411) Read the following paper for a review of important concepts and recent advances.

McGlynn et al. (2002, Journal of Hydrology, 257:1-26) Weiler & McDonnell (2004, Journal of Hydrology, 285:3-18) 5.5 Conceptual Model of Subsurface Flow The lateral flow of soil water and groundwater plays a major role in storm runoff generation and also provides baseflow during dry periods.

92  

Actual flow mechanisms are complex due to geological heterogeneity, macropore flow, and other unknown factors. However, a relatively simple model serves as a tool linking a small-scale physical process to large-scale behavior of watersheds. Subsurface flow in a hillslope underlain by impermeable bedrock is described by the Richards equation (Eq. 2-27 in page 33). Due to the non-linear dependence of K on hm (or ) it is usually impossible to obtain an analytical solution of Eq. (2-27). Although it is possible to solve the equation numerically (e.g. HYDRUS3D) for a given set of physical parameters and boundary conditions, the utility of such approach is limited, due to the large degree of uncertainty in parameter values. In the following, we will develop simplified analytical solutions that approximate Eq. (2-27). Dimensional analysis provides a useful guide to the simplification process. We start by assuming that the flow occurs predominantly parallel to the slope, where the cross slope flow is negligible. This reduces the dimension of Eq. (2-27) to two:

tz

hK

zx

hK

xTT

Noting that hT = hm + z and K(hm) = Ks Kr(hm),

tK

z

hK

zK

x

hK

xK r

mrs

mrs

(5-6)

To convert Eq. (5-6) to a dimensionless form, we define dimensionless variables:

D

xx *

D

zz *

rs

s

B

tKt

*

rs

r

*

mhh * (5-7)

→ see van Genuchten equation (Eq. 2-8 in page 25) for the definition of r, s, and . Using the variables defined above, Eq. (5-6) can be written:

*

*

*

*

**

*

* tB

KK

z

h

D

K

zD

K

x

h

D

K

xD

K

rs

rssr

rsrs

*

*

*

*

**

*

* tB

DDDK

z

hK

zx

hK

x rrr

(5-8)

Since Kr and * are the function of h* (= hm), Eq. (5-8) indicates that the solution of the 2-D Richards equation for the hillslope setting is invariant of scale and soil parameters for a given set of dimensionless variables D and D/B, and n in Eq. (2-8).

93  

Since -1 is related to the thickness of capillary fringe (Assignment #2, Question 3e), 1

D

D

represents the relative thickness of the groundwater zone (hm > 0) to the capillary fringe. The D/B indicates the shape of flow domain, with smaller value indicating that vertical flow is less important. From the above, when D/-1 and B/D are sufficiently large, we can approximate the model by:

1) neglect the flow in the vadose zone 2) neglect the flow in vertical direction

These are called the Dupuit’s assumptions, and the theory based on them is called Dupuit-Forchheimer theory. → Brutsaert (p. 382)

Suppose a slope of gradient β. The thickness of groundwater, measured perpendicular to the bedrock surface is (x). Flow is strictly along the x-direction. There are two ways of deriving the Dupuit-Forchheimer equation for a sloping bed (Childs, 1971. Water Resources Research, 7: 1256-1263). We follow the approach described by Brutsaert (p. 382-384). Note that Fig 10.15 in Brutsaert has the direction of the x-axis reversed. We note that = zcos in the diagram. Also, recall the definition of hydraulic head:

g

ph

w

wT

where pw is pressure

Since the flow is negligible in z-direction,

0

1

z

p

gzz

h w

w

T

cosg

zg

z

pww

w

Noting that pw(x, z) = 0 when z = (i.e. the water table),

pw = wg( z)cos

Differentiating the above with respect to x,

cosgxx

pw

w

Also note that sin

x.

cossin1

xx

p

gxx

h w

w

T

(5-9)

Applying Darcy’s law to express specific discharge qx,

94  

sincos00 xK

x

hKq T

x (5-10)

where K0 (m s-1) is the average saturated hydraulic conductivity of the slope material. The water balance equation for a control volume is given by (see page 33):

t

niqx ex

(5-11)

where i (m s-1) is the infiltration flux, taken to be normal to the x-axis. The storage coefficient ne is called drainable porosity or specific yield, representing the “amount of water (m3) drained from unit area (m2) per unit drop (m) of the water table”. → Discussed later. Substituting Eq. (5-10) into (5-11),

ee n

i

xxxn

K

t

sincos0 (5-12)

Note that the same concept can be extended to a three-dimensional slope,

eyxyx

e n

i

yxyyxxn

K

t

sinsincoscos0 (5-13)

where x and y are slope angles in x- and y-direction. When is sufficiently small, i.e. < 5o, Eq. (5-12) can be written:

ee n

i

xxn

K

t

0 (5-14)

The transient flow equations, (5-12), (5-13), and (5-14) are called the Boussinesq equation. → Brutsaert (p. 384) 5.6 Application of the Boussinesq Equation Consider a simple case of steady state, where Eq. (5-14) becomes

0K

i

dx

d

dx

d

(5-15)

with the boundary conditions: =0 at x = 0

0

dx

d

at x = B (no flow)

Integrating Eq. (5-15) with respect to x yields:

95  

1

0

CxK

i

dx

d

(5-16)

Separating variables and integrating,

21

0

CdxCxdxK

id

21

2

0

2 22 CxCxK

i (5-17)

Using the boundary condition at x = B for Eq. (5-16),

C1 = iB / K0

Using the other boundary condition for Eq. (5-17),

C2 = 02 / 2

2

0

20

2 2 xBxK

i

(5-18)

The solution predicts the curved surface, with the degree of curvature depending on i/K0. The nonlinear Boussinesq equation (Eq. 5-14) is difficult to solve. Brutsaert (pp. 390-393) describes the application of the Boltzmann transform technique to solve a special case of

xxn

K

t e

0

(5-19)

with the initial condition

= D (complete saturation) at t = 0 for all x > 0

and the boundary conditions:

= Dc at x = 0

0

x

at x = B (drainage divide)

This set of partial differential equation and the initial and boundary conditions simulates the drainage of a hillslope after a major storm event. Brutsaert presents a solution for the above problem for a special case of Dc → 0 and B → ∞, which gives the time dependent drainage flux Q (m2 s-1) per unit width of a slope as

tDnK

aQ e

1

23

0 (5-20)

where a is a dimensionless constant. Brutsaert (p. 393) suggests a ≅ 0.66. An alternative approach to obtain the approximate solution of Eq. (5-19) is linearization (Brutsaert, pp. 398-404). In this approach, we expand the right hand side of Eq. (5-19) to

96  

2

20

xxxn

K

t e

(5-19b)

and make two assumptions:

1) Magnitude of 2

x

is much smaller than

2

2

x .

2) There exists a value 0 that represents the average of in 0 < x < B. Then, Eq. (5-19b) can be approximated by

2

2

2

200

xD

xn

K

t he

(5-21)

where e

h n

KD 00 is hydraulic diffusivity (m2 s-1).

We can solve Eq. (5-21) with the following initial and boundary conditions:

= D at t = 0 for 0 ≤ x ≤ B

= Dc at x = 0 for t > 0

0

x

at x = B for t > 0

To make the boundary condition homogeneous, we define:

' = Dc (5-22)

Then, the problem is replaced by:

2

2

xD

t h

(5-23)

' = D Dc at t = 0 for 0 ≤ x ≤ B (5-24)

' = 0 at x = 0 for t > 0 (5-25)

0

x

at x = B for t > 0 (5-26)

Equations (5-22)-(5-26) can be solved by the Fourier series method (see pages 47-53). We assume that ' (x, t) = G(t)F(x).

2

2

dx

FdGD

dt

dGF h

2

211

dx

Fd

Fdt

dG

GDh where is a constant

97  

0 GD

dt

dGh and 0

2

2

Fdx

Fd

The solution for G is given by:

dtD

G

dGh

lnG = Dht + C

G = eC exp(Dht) (5-27)

For F(x), it can be shown that (see page 48) = 0 and < 0 do not yield a non-trivial solution that satisfies the boundary conditions:

F = 0 at x = 0 from Eq. (5-25)

0

dx

dF

at x = B from Eq. (5-26)

The solution for > 0 is

xbxaxF sincos)(

From Eq. (5-25), a = 0

xbxF sin)( and

xbdx

dF cos

From Eq. (5-26), 0cos Bb

B

nn 2

12 n = 1, 2, 3 …

B

xnbxF n 2

12sin)(

Since the differential equation is linear and the boundary conditions are homogeneous, the superposition principle can be used to write the solution as:

tDB

n

B

xnbtx h

nn 2

22

1 4

12exp

2

12sin,

(5-28)

This is a Fourier series solution. To satisfy the initial condition, Eq. (5-24),

1 2

12sin

nnc B

xnbDD

The Fourier coefficients are given by (see page 53):

B

cn dxB

xnDD

Bb

0 2

12sin

2

98  

12

4

2

12cos

12

22

0

n

DD

B

xn

n

B

B

DDb c

B

cn

txDtx c ,',

tDB

n

B

xn

n

DDD h

n

cc 2

22

1 4

12exp

2

12sin

12

14

Noting that 1/(2n 1) = 1, 1/3, 1/5 … for n = 1, 2, 3…

(2n 1)2 = 1, 9, 25 … for n = 1, 2, 3…

2

2

4exp

2sin

4,

B

tD

B

xDDDtx hc

c

(5-29)

2

2

4exp

2cos

2

B

tD

B

x

B

DD

xhc

tnB

K

B

DDK

xKQ

e

c

x2

002

00

000

4exp

2 (5-30)

where Q is the drainage flux (m2 s-1) per unit width of the slope (Brutsaert, p. 403, Eq. 10.115). So far, we assumed that 0 exists, but did not examine its value. At an early stage when = Dc at x = 0 and ≅ D at x = B, it is reasonable to assume that 0 is an average between Dc and D.

i.e. 0 = (Dc + D) / 2

As t increases and the slope is drained, → Dc at x = B.

0 ≅ Dc at late stage

If Dc and D are not dramatically different, it is reasonable to use a constant value of 0 = (Dc + D) / 2. → Brutsaert (p. 400)

t

nB

DDK

B

DDKQ

e

cc2

0222

0

8exp

(5-31)

5.7 Drainable Porosity So far, we have assumed a unique constant ne, which relates the change in water storage to the magnitude of water table change (page 94). Operationally, ne is defined using the drainage of water from a soil column having a cross-sectional area A (m2). Suppose that a water table drop of ∆ (m) results in a drainage of ∆V (m3).

99  

Then,

AVne (5-32)

This definition, however, is over simplistic. The characteristic of real soils has a gradual decrease of water content () with matric potential head, not an abrupt drop from s to s – ne implied in Eq. (5-32). In real soil, drainage of water occurs from the entire vadose zone. Therefore,

AV

ne is dependent on the depth to

the water table, even for a uniform soil. In addition, the drainage is not an instantaneous process. Therefore ne is dependent on time as well. For these reasons, ne of real soil is not a unique property (Brutsaert, pp. 378-379). The concept of drainable porosity (= specific yield) needs to be applied with caution, especially for the transient processes occurring over time scales of hours and days. 5.8 Baseflow Recession Analysis A watershed can be seen as a collection of hillslope “strips”, with each strip contributing baseflow. Suppose that a hillslope strip contributes baseflow q(t) per length of the bank (m2s-1). Conceptually, the total baseflow Q (m3s-1) is given by:

L

rightleft dsstqstqtQ0

),(),()( (5-33)

where s denotes the coordinate along stream channel L is the total channel length → Brutsaert (pp. 415-418) In general, Q(t) decreases with time after a storm. If the aquifer underlying the watershed has a linear reservoir like behavior (see pages 7-9), we expect:

100  

*0 exp)(

T

tQtQ (5-34)

where T* is the reservoir response time Q0 is the flow rate at the beginning of the baseflow period If the aquifer behaves like a linearized Boussinesq aquifer, the baseflow is represented by Eqs. (5-30):

t

nB

K

B

DDKLtQ

e

c2

002

00

4exp

22)(

(5-35)

Where the leading 2 indicates the contributions from the left and right banks, and next L indicates the integration over the channel length. All parameters are understood to be some average values representing the watershed. If the aquifer behaves like an initially saturated Boussinesq aquifer, i.e. Eq. (5-20):

2

13

02

66.02)(

tDnKLtQ e (5-36)

Brutsaert (pp. 390-391) suggests that Eq. (5-36) describes the short-term response of fully saturated slopes, whereas Eq. (5-35) is more suitable for long-term response. One can gain insights into the hydrological characteristics of watersheds by comparing field data with theoretical equations. Two methods are commonly used: (1) Log-log plot (Brutsaert, pp. 420-425) We write theoretical equations in a form:

baQdt

dQ (5-37)

where a (m3(b-1) sb-2) and b (dimensionless) are constants to be determined. For example, by differentiating Eq. (5-35) with respect to t,

t

nB

K

nB

K

B

DDLK

dt

dQ

ee

c2

002

200

2

004

exp4

4

Q

nB

K

e2

002

4

b = 1 and enB

Ka

200

2

4

(5-38)

Differentiating Eq. (5-36) with respect to t,

101  

2

33

02

66.0 tDnK

L

dt

dQe

Noting that 2

33

03

033 )()66.0(

tDnKDnKLQ ee

33

02)66.0(2

1Q

DnKLdt

dQ

e

3

02)66.0(2

1

DnKLa

e

and b = 3 (5-39)

Equations (5-38) and (5-39) give theoretical values of a and b, to be compared against empirical values determined by curve fitting the field data. For plotting field data of daily discharge, we define:

21

iiav

QQQ

t

QQ

dt

dQ ii

1

(5-40)

where Qi is the daily discharge of i-th day Qav is the average Q between i-th and i+1-th day dQ/dt is the daily recession rate of Q.

By plotting

dt

dQ versus Qav for baseflow periods ( 0

dt

dQ)

on log-log axes, one can draw “envelope” lines (Brutsaert, pp. 424-425). These envelopes indicate the lowest rate of recession for a given Qav, which is taken to be baseflow recession rate. The envelope with b = 3 indicates the short-term response corresponding to rapid drainage of fully saturated slopes. The envelope with b = 1 indicates the long-term, slow drainage of slopes. Noting that Eq. (5-37) can be written as:

Qba

dt

dQlogloglog

The value of can be determined from the intercept given by Q = 1 m3 s-1 → log(Q) = 0 (2) Master recession curve (MRC) method

102  

Continuous records of Q(t) are broken up into individual baseflow segments, and then “pasted” together to form a single recession curve representing the “integrated” response of the watershed (Posavec et al. 2006, Ground Water, 44: 764-767).

Theoretical equations (e.g. Eq. 5-35 and 5-36) are matched to the master recession curve (MRC) using the non-linear regression technique (e.g. solver tool in Microsoft excel).

The slope of best-fit curves is compared against theoretical slopes to determine K0, D Dc, etc. → homework

5.9 Lumped Watershed Response In the previous sections, we analyzed the hydrological responses of individual hillslopes to storm events using the Boussinesq equation. This is sometimes called “reductionist” approach, where the whole-basin response is given by the sum of all the pieces. A completely opposite view is offered by “holistic” approach, which tries to understand the watershed behavior without explicitly concerning individual slopes. Note that the holistic approach is philosophically different from the “black box” approach, such as the linear reservoir model (page 3). The holistic approach attempts to incorporate the understanding of hydrological process in the manner that is consistent with field observation. TOPMODEL (Beven and Kirkby, 1979, Hydrological Science Bulletin, 24: 43-69) is an elegant and successful example of the holistic approach. The most critical processes for hillslope runoff are: - Saturation overland flow (page 89) - Subsurface storm flow (page 90) TOPMODEL is designed to calculate the area of saturation and the magnitude of subsurface stormflow. Suppose a hillslope strip that is connected to a stream. The length of contour lines is c (m), where the strip meets the stream. The area covered by the strip is A (m2). The magnitude of subsurface stormflow entering the stream is given by:

Q = cT tan (5-41)

where T (m2 s-1) is transmissivity tan is the slope gradient Equation (5-41) assumes that the slope of the water table is similar to the topographic slope. Unlike the Dupuit-Forchheimer theory, which uses T = K0, TOPMODEL calculates T as:

103  

)()exp()( 00

bw

b

w

b

w

fzfzsz

z

z

z

ssat eef

KdzfzKdzzKT

(5-42)

where Ks0 (m s-1) is saturated hydraulic conductivity at the surface, f (m-1) is a decay coefficient, zw (m) is depth to the water table, and zb (m) is the depth to the bedrock. Equation (5-42) captures the essential characteristic of most hillslopes, where Ksat quickly

decreases with depth. Since bw fzfz ee , it is commonly assumed that:

wfzs ef

KT 0 (5-43)

Noting that it is difficult to estimate the distribution of zw over the watershed, zw is replaced by soil water deficit s (m).

s = zwne or zw = s / ne (5-44)

m

s

f

K

n

sf

f

KT s

e

s expexp 00 (5-45)

where m = ne / f

The T takes the maximum value Tmax when the soil is completely saturated (i.e. s = 0, zw = 0).

Tmax = Ks0 / f

Using Tmax and m, Eq. (5-41) can be written:

tanexpmax

m

scTQ (5-46)

The “great leap” of TOPMODEL is its use of “successive steady state”, where the subsurface drainage (Q) is equal to the total rainfall input (RA) at any given instant. → Hornberger et al. (2014, Elements of Physical Hydrology, p. 278)

tanexpmax

m

scTRA

m

s

T

Raexp

tan max where

c

Aa

m

s

T

Ra

max

lntan

ln

tanlnln

max

am

T

Rms (5-47)

The last term in Eq. (5-47),

tan

lna

, is called the topographic index.

104  

Note that m, R, and Tmax are assumed spatially uniform, while a and tan are variables. We define the watershed average of s by

m

T

Rms

max

ln (5-48)

where

Aw

dAa

A

tanln

1 and Aw (m2) is the watershed area.

Subtracting Eq. (5-48) from (5-47):

m

amss

tanln

tanln

amss (5-49)

Equation (5-49) indicates that s is smaller in areas with high

tan

lna

.

Note that tan is slope gradient and a = A/c is the contributing area per unit contour length.

Since

tan

lna

is greatest near the stream, s is smallest near the

stream. As s decreases during a storm event, s in some areas decreases to 0, meaning complete saturation. → Overland flow is generated. TOPMODEL uses water balance to compute s at each time step, and calculates the total area of s < 0 using Eq. (5-49).

Note that Eq. (5-49) describes ss as a function of

tanln

a, but it does not explicitly

account for the spatial distribution of the saturated area. TOPMODEL is a lumped hydrological model. Using Eq. (5-49), Eq. (5-46) can be written as:

tantan

lnexpexpmax

a

m

scTQ

105  

tan

tan)exp(expmax

a

m

scTQ

Am

sT

exp)exp(max

Therefore, mean subsurface storm flow per unit area, q (m s-1) is:

m

sT

A

Qq exp)exp(max (5-50)

Equation (5-50) represents the watershed-scale storage-flow relationship. Such an equation offers the link between the physical processes and a large-scale hydrological model. → “Parameterization” of watershed hydrology. In TOPMODEL, Tmax and m are the essential model parameters that need to be “tuned” by calibration. In summary:

1) TOPMODEL keeps track of the watershed average saturation deficit s by:

s = precipitation – evapotranspiration – overland Flow – subsurface Flow 2) At each time step, the total saturated area (s ≤ 0) is computed from Eq. (5-49). 3) Saturation overland flow is given by:

QOF = precipitation × total saturation area 4) Subsurface flow is given by Eq. (5-50). 5) The fitting parameters m and Tmax are determined by calibration.

5.10 Distributed Approach Lumped models, such as TOPMODEL, provide useful tools for simulating the stream discharge hydrograph, but they offer little information on the explicit spatial distribution of “internal variables” such as: - water table, soil moisture - snow depth, snowmelt rate - evaporation Distributed hydrological models divide a watershed into numerous grid cells, and simulate the water balance of each grid cell, as well as cell-to-cell transfer of water. For example, the water balance of cell-i can be written as:

4

1,

kkiiii QETPS (5-51)

106  

where ∆Si is storage change Pi is precipitation ETi is evapotranspiration Qi,k is the flow into cell-i from k-direction

The Qi,k can be calculated using the Dupuit-Forchheimer equation applied to individual cells:

Qi,k = Ti,k tanw,k ∆l (5-52)

where Ti,k is transmissivity in k-direction tanw,k is the water table slope in k-direction ∆l is the cell size The Ti,k is calculated from equations similar to Eq. (5-42) using the depth to water table zw,i at cell-i. Equations (5-51) and (5-52) form the skeletons of many distributed hydrological models designed to simulate hillslope processes. e.g. Weiler and McDonnell (2004, Journal of Hydrology, 285: 3-18) Wigmosta et al. (1999, Water Resources Research, 35: 255-264) Distributed Hydrology-Soil-Vegetation Model (DHSVM) of Wigmosta et al. (1999) is widely used by the forest hydrology communities in Canada and the U.S.A. Hill Vi of Weiler and McDonnell (2004) provides an effective tool for conducting numerical experiments on hillslope drainage processes. 5.11 Distributed vs. Lumped Approach Lumped hydrological models have a very long history. They were developed as simple and practical engineering tools to predict storm flow hydrographs. The flood of new discoveries on hillslope processes in the 1960s and 1970s, coupled with the development of digital computers brought forth the idea of physically-based, distributed hydrological models. The “blue print” of a truly distributed model was proposed by Freeze and Harlan (1969, Journal of Hydrology, 9: 237-258), which is an astoundingly prescient view of hydrology. The hydrological research community took several decades to construct such a model, but finally the models started to emerge. → Vanderkwaak and Loague (2001, Water Resources Research, 37: 999-1013) Despite the appearance of physically-based model construction, distributed models, applied to data-sparse watersheds, often produce unrealistic results without site-specific calibration. So, we must accept the fact that distributed hydrological models do need calibration. → Bronstert (1999, Hydrological Processes, 13: 21-48)

107  

This poses a very serious challenge, because there are so many parameters to calibrate in distributed models. A model can produce almost identical results for numerous combinations of “calibrated” parameter values. This concept is called “equifinality”. → Beven (1996, In: Distributed hydrological modeling, Kluwer Academic Publising., pp. 255-278) In theory, only one set of parameter values is “correct”, and all other combinations produce “right” results for wrong reasons. Beven (2002, Hydrological Processes, 16: 189-206) goes to an extent of recommending that we should give up the blue print of distributed hydrological modeling. One significant issue is the scale- or model- dependence of hydrologic parameters such as Ksat. The model-calibrated values of Ksat are often orders-of-magnitude different from the mean value of Ksat based on field measurements (Bronstert, 1999). Therefore, proliferation of lumped models can potentially harm the research community by promoting a disconnect between experimentalists and modelers. These are major challenges, but also exciting opportunities for physical hydrologists.