Encyclopedia of Environmental Science and Engineering, Volume I and II

718 MODELING OF ESTUARINE WATER QUALITY

in the water or other chemical forms. Such models may be

three dimensional to represent transport of material down

the estuary as well as laterally and vertically, or they may

be two dimensional to represent transport of material down

the estuary and laterally or vertically, or they may be one

dimensional to represent transport of material down the

estuary. Because the complexity of developing and solving

mathematical models decreases as the number of dimen-

sions included are decreased, the one dimensional model

has received the widest attention in terms of development

and use. This type of model is most advantageously applied

to linear type estuaries, that is, estuaries which have little

or limited variation in cross sectional area and depth with

distance down the estuary. Examples of such models include

the model of the Thames River in England, the Delaware

River in New Jersey, the Potomac River in Maryland, and the

Hudson in New York (see Thomann and Mueller 1987 for an

excellent introduction to such models).

Water quality models are usually derived from the fol-

lowing basic three dimensional continuity equation:









C

t x

E

C

xy

E

C

yz

E

C

z

x

u

xyz

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

(CC

y

vC

z

)()() wc S



∂

∂ ∑

where

E  dispersion coefficient along each of the three axes

x , y , and z

u , v , w  velocity in x , y , or z direction respectively

S  source or sink for material

C  concentration of material.

This equation expresses a relationship between the flux

of mass caused by circulation and mixing in the estuary and

the sources and sinks of mass. In the one-dimensional form

in which the assumptions have been made that concentra-

tions of some material are of homogeneous concentration

laterally and vertically (the y and z directions, respec-

tively) and that the net transport of the material through

the estuary is of concern, then the following equation has

been used (O’Connor and Thomann, 1971; Thomann and

Mueller 1987):







C

tAx

EA

xAx

QC S

11 ⎛

⎝⎜

⎞

⎠⎟

where

A  cross sectional area of estuary

Q  freshwater inflow

E  dispersion coefficient in x direction

and other terms are the same as above. Such models may

be used to determine changes in material concentration with

time for materials whose rate of entry to the estuary and/or

loss from the estuary in a sink are steady or only slightly vari-

able. A further assumption is to select the steady state situa-

tion, the condition in which the concentration of the material

does not change with time. For this condition 
C / 
t in the

above equation is to set to zero and the equation solved.

Recently two dimensional models have been developed.

These models often assume that vertical stratification does

not occur in the water column and that lateral stratification

does occur. Such models are most appropriately applied to

estuaries with large surface areas and shallow waters. Such

models have been developed for many estuarine systems.

Feigner and Harris (1970) describe a link-node model devel-

oped specifically for the Francisco Bay-Delta Estuary, but

applicable elsewhere. It models the two-dimensional flow

and dispersion characteristics of any estuary where strati-

fication is absent or negligible. Hydrological parameters of

tidal flow and stage are computed at time intervals ranging

from 0.5 to 5.0 mins and at distance intervals ranging from

several hundred to several thousand feet. Predictions of qual-

ity levels are computed on the same space scale, but on an

expanded time scale, ranging from 15 to 60 mins. The model

is thus truly dynamic in character. It predicts fluctuating

tidal flows and computes tidally varying concentrations of

constituents, in contrast to a non-tidal model based on the

net flow through the estuary such as that developed for the

Delaware estuary. It can also accommodate both conserva-

tive and non-conservative constituents.

First, the hydraulic behavior of the estuary is modeled.

Having established channel directions both in the actual

prototype channels and (artificially) in the bay areas, the

authors use one dimensional equations based on the follow-

ing assumptions :

a) Acceleration normal to the x -axis is negligible.

b) Coriolis and wind forces are negligible.

c) The channel is straight.

d) The channel cross-section is uniform throughout

its length.

e) The wave length of the propagated tidal wave is

at least twice the channel depth.

f) The bottom of the channel is level.

Equations of motion and continuity are, respectively





u

t

u

u

x

Ku u g

H

x

and



H

tbx

uA

1

()

where

u  velocity along the x -axis

x  distance along the x -axis

H  water surface elevation

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