WSP11764

• File Number: 8273.100
• Description: Colorado River Basin Salinity Control - Federal Agencies - Bureau of Reclamation
• Basin: Colorado Mainstem
• Water Division: 5
• Date: 1/1/1984
• Author: BOR
• Title: Water Quality of the Colorado River System - Historical Trends in Concentration/ Load/ and Mass Fraction of Inorganic Solutes
• Water Supply Pro - Doc Type: Report/Study

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CONCENTRATION-STREAMFLOW
RELATIONSHIPS

In unregulated streams salinity generally varies
inversely with streamfloW. Possible causes of this
inverse relationship and a review of several mathe-
matical models proposed to describe this relation-
ship are listed by Lane [9]. In evaluating the
beneficial effects of salinity control projects,
concentration-streamflow models are useful to
correct the salinity for the effect of the natural
variability of streamflow before and after construc-
tion of the project. In this report, these models are
used in multiple regression analyses to remove
predictable, short-term fluctuations in concen-
tration, thus allowing more precise estimates of
long-term changes in concentration and load. This
chapter develops a theoretical model with two
parameters that have physical meaning and
compares the results of this model with those
obtained from the power equation.

The power equation is an empirical model com-
monly used to describe the concentration-stream-
flow relationship [9]. This equation (with solute
subscripts omitted) is:

C=b,Qb2

where: C = Concentration
Q = Streamflow
b" b2 = Empirical parameters estimated
by regression

The parameter b2 ranges from zero, where con-
centration is independent of streamflow, to minus
one, where load is independent of streamflow.
Equation 6 yields unrealistically high estimates of
concentrations at low flows, because C approaches
infinity as Q approaches zero.

Because the power model fails at low flows and
because the parameter b, and b2 have no physical
meaning, a theoretical residence time model is
derived based on the assumptions listed below.
Assumptions 2 and 4 may represent serious
oversimplifications of the physical system.

1. The solute concentration in rain or snowmelt
is ,1egligible.

2. The contact area of dissolvable solids is
independent of the streamflow.

3. The mass transfer coefficient for the dis-
solution of solids is constant.

4. The solubility of any solute is constant; i.e.,
the solubility is independent of the chemical
composition of both the dissolving solids and
the aqueous solution.

The differential equation (with solute subscripts
omitted) for dissolution is:

QdC' = -k(C'-Cs)dA'

(7)

where: Q = Streamflow (m3/s)
C' = Solute concentration after Q has
contacted A', the area of solid sur-
face (g/m3)
Cs = Solute solubility (g/m3)
k = Mass transfer coefficient (m/s)
A' = Solid-liquid contact area (m')

By assumption 1, the boundary condition is:

C(A'=O) = 0

(8)

By assumptions 3 and4, the solution to equation 7
is:

C = Cs [l-exp (-T)]

(9)

where T, the dimensionless residence time, is
(6) defined by:

_leA
T-a-

(10)

andA is the total solid-liquid contactarlla upstream
of the gaging station. Equation 9 is of the form:

C = b,[l-exp (-b2/Q)] (11)

where b, and b2 are parameters to be estimated by
regression, based on observed values of concen-
tration and streamflow. The parameter b, repre-
sents the solute solubility, and b2 represents the
product of the mass transfer coefficient and the
total solid-liquid contact area. The residence time
equation yields realistic values of concentrations
at zero streamflow (infinite residence time), where
the solute concentration approaches solubility,
and at infinite streamflow (zero residence time),
where the concentration approaches zero (see
fig. 5).

To test the effectiveness of the power and resi-
dence time equations in describing the relationship
between monthly values of concentration and
streamflow, regressions were conducted for 6
solutes and their sum. Regressions were run for
each of the three seasons at 12 different gaging

9