Sensitivity Analysis for Stream Water Quality Management

Abstract

The sensitivity equations of stream water quality parameters are presented, and their practical applications to stream pollution control scientifically illustrated. Non-tidal streams are classified into: (a) clean or slightly polluted swift non-tidal streams, (b) moderately polluted swift non-tidal streams, (c) heavily polluted swift non-tidal streams, (d) clean or slightly polluted, intermediate non-tidal streams, (e) moderately polluted intermediate non-tidal streams, (f) heavily polluted intermediate non-tidal streams, (g) clean or slightly polluted slow non-tidal streams, (h) moderately polluted slow non-tidal streams, and (i) heavily polluted slow non-tidal streams.

Tidal streams are classified into: (a) clean or slightly polluted tidal streams, (b) moderately polluted tidal streams, and (c) heavily polluted tidal streams. The characteristics and water quality parameter ranges of different types of receiving streams are presented. The significance of water quality sensitivities and dissolved oxygen deficits for water quality management are systematically identified by the author's mathematical models.

Appendices

Appendix 1: Water Quality Models

For Non-tidal Streams

• 1. Biochemical Oxygen Demand (L) Model

$$ \mathrm{L}={\mathrm{L}}_{\mathrm{O}}\left[ \exp \left(-{\mathrm{K}}_1\mathrm{t}\right)\right] $$

(8.14)

• 2. Dissolved Oxygen Deficit (D) Model

$$ \begin{array}{l}\mathrm{D}={\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-1}\left[ \exp \left(-{\mathrm{K}}_1\mathrm{t}\right)\ \hbox{-}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\right]\\ {}+{\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_{\mathrm{n}}\right)}^{-1}\left[ \exp \left(-{\mathrm{K}}_{\mathrm{n}}\mathrm{t}\right)\ \hbox{-}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\right]\\ {}+\left(\mathrm{B}-\alpha \right){\left({\mathrm{K}}_2\right)}^{-1}\left[1\ \hbox{-}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\right]+{\mathrm{D}}_{\mathrm{O}} \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\end{array} $$

(8.15)

For Tidal Streams

• 1. Biochemical Oxygen Demand (L) Model

$$ \mathrm{L} = {\mathrm{L}}_{\mathrm{O}}\left[ \exp\ \left({\mathrm{J}}_1\mathrm{X}\right)\right] $$

(8.16)

• 2. Dissolved Oxygen Deficit (D) Model

$$ \begin{array}{l}\mathrm{D}={\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-1}\left[ \exp\ \left({\mathrm{J}}_1\mathrm{X}\right)- \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}+{\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_{\mathrm{n}}\right)}^{-1}\left[ \exp\ \left({\mathrm{J}}_{\mathrm{n}}\mathrm{X}\right)- \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}+\left(\mathrm{B}-\alpha \right)\ {{\mathrm{K}}_2}^{-1}\left[1- \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\ \right]+{\mathrm{D}}_{\mathrm{o}} \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\end{array} $$

(8.17)

Appendix 2: Sensitivity Formulas for Non-tidal Streams

• 1. The sensitivity of L to K₁:

$$ {\mathrm{S}}_{\mathrm{L},\ \mathrm{K}1}=-{\mathrm{L}}_{\mathrm{O}}\mathrm{t}\left[ \exp \left(-{\mathrm{K}}_1\mathrm{t}\right)\right] $$

(8.18)

• 2. The sensitivity of D to K₁:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}1}={\mathrm{K}}_2{\mathrm{L}}_{\mathrm{o}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-2}\left[ \exp \left(-{\mathrm{K}}_1\mathrm{t}\right)- \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\ \right]\\ {}\kern4em -{\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}\mathrm{t}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-1} \exp \left(-{\mathrm{K}}_1\mathrm{t}\right)\end{array} $$

(8.19)

• 3. The sensitivity of D to K_n:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}\mathrm{n}}={\mathrm{K}}_2{\mathrm{N}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_{\mathrm{n}}\right)}^{-2}\left[ \exp \left(-{\mathrm{K}}_{\mathrm{n}}\mathrm{t}\right)\ \hbox{-}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\right]\\ {}\kern4em -{\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}\mathrm{t}{\left({\mathrm{K}}_2-{\mathrm{K}}_{\mathrm{n}}\right)}^{-1} \exp \left(-{\mathrm{K}}_{\mathrm{n}}\mathrm{t}\right)\end{array} $$

(8.20)

• 4. The sensitivities of D to α and B:

$$ {\mathrm{S}}_{\mathrm{D},\ \mathrm{B}}=-{\mathrm{S}}_{\mathrm{D},\alpha }={\left({\mathrm{K}}_2\right)}^{-\mathrm{l}}\left[1\ \hbox{-}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\right] $$

(8.21)

• 5. The sensitivity of D to K₂:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}2} = -{\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}{\left({\mathrm{K}}_2 - {\mathrm{K}}_{\mathrm{1}}\right)}^{-2}\left[ \exp \Big(-{\mathrm{K}}_1\mathrm{t}\right)\ \hbox{--}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\Big]\\ {}\kern4.25em + {\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}\mathrm{t}{\left({\mathrm{K}}_2 - {\mathrm{K}}_{\mathrm{1}}\right)}^{-1} \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\\ {}\kern4.2em -{\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}{\left({\mathrm{K}}_2 - {\mathrm{K}}_{\mathrm{n}}\right)}^{-2}\left[ \exp \Big(-{\mathrm{K}}_{\mathrm{n}}\mathrm{t}\right)\ \hbox{--}\ \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\Big]\\ {}\kern4.3em + {\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}\mathrm{t}{\left({\mathrm{K}}_2 - {\mathrm{K}}_{\mathrm{n}}\right)}^{-1} \exp \left(-{\mathrm{K}}_2\mathrm{t}\right)\\ {}\kern4.3em + \left(\alpha - \mathrm{B}\right){\left({\mathrm{K}}_2\right)}^{-2}\left[1 - {\mathrm{e}}_{\mathrm{X}}\mathrm{p}\Big(-{\mathrm{K}}_2\mathrm{t}\right)\Big]\\ {}\kern4.3em - \left(\alpha /{\mathrm{K}}_2 - \mathrm{B}/{\mathrm{K}}_2 + {\mathrm{D}}_{\mathrm{O}}\right)\ \mathrm{t}\ \left[ \exp \Big(-{\mathrm{K}}_2\mathrm{t}\right)\Big]\end{array} $$

(8.22)

• 6. The sensitivity of DO to K:

$$ {\mathrm{S}}_{\mathrm{C},\ \mathrm{K}}=-{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}} $$

(8.23)

Appendix 3: Sensitivity Formulas for Tidal Streams

• 1. The sensitivity of L to K₁:

$$ {\mathrm{S}}_{\mathrm{L},\ \mathrm{K}\mathrm{l},\ \mathrm{t}} = -{\mathrm{L}}_{\mathrm{O}}{\mathrm{m}}_1\mathrm{X}\left[ \exp\ \left({\mathrm{J}}_1\mathrm{X}\right)\right] $$

(8.24)

• 2. The sensitivity of D to K₁:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}\mathrm{l},\ \mathrm{t}}={\mathrm{K}}_2\mathrm{L}\mathrm{o}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-2}\left[ \exp\ \left({\mathrm{J}}_1\mathrm{X}\right)\ \hbox{-}\ \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.5em -{\mathrm{K}}_1{\mathrm{m}}_1{\mathrm{XL}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-1} \exp\ \left({\mathrm{J}}_1\mathrm{X}\right)\end{array} $$

(8.25)

• 3. The sensitivity of D to K_n:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}\mathrm{n},\ \mathrm{t}}={\mathrm{K}}_2{\mathrm{N}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-2}\left[ \exp\ \left({\mathrm{J}}_{\mathrm{n}}\mathrm{X}\right)\ \hbox{-}\ \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.5em -{\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}\mathrm{t}{\left({\mathrm{K}}_2-{\mathrm{K}}_{\mathrm{n}}\right)}^{-1} \exp \left(-{\mathrm{K}}_{\mathrm{n}}\mathrm{t}\right)\end{array} $$

(8.26)

• 4. The sensitivities of D to α and B:

$$ {\mathrm{S}}_{\mathrm{D},\ \mathrm{B},\ \mathrm{t}}=-{\mathrm{S}}_{\mathrm{D},\alpha,\ \mathrm{t}}={\left({\mathrm{K}}_2\right)}^{-1}\left[\mathrm{l}\ \hbox{-}\ \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right] $$

(8.27)

• 5. The sensitivity of D to E:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{E},\ \mathrm{t}} = {\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}\mathrm{X}{\left({\mathrm{K}}_2 - {\mathrm{K}}_1\right)}^{-1}\left[{\mathrm{n}}_1 \exp\ \left({\mathrm{J}}_1\mathrm{X}\right) - {\mathrm{n}}_2 \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.25em + {\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}\mathrm{X}{\left({\mathrm{K}}_2 - {\mathrm{K}}_{\mathrm{n}}\right)}^{-1}\left[{\mathrm{n}}_{\mathrm{n}} \exp\ \left({\mathrm{J}}_{\mathrm{n}}\mathrm{X}\right) - {\mathrm{n}}_2 \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.25em + \left(\alpha /{\mathrm{K}}_2 - \mathrm{B}/{\mathrm{K}}_2 + {\mathrm{D}}_{\mathrm{O}}\right){\mathrm{n}}_2\mathrm{X}\left[ \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\end{array} $$

(8.28)

• 6. The sensitivity of D to K₂:

$$ \begin{array}{l}{\mathrm{S}}_{\mathrm{D},\ \mathrm{K}2,\ \mathrm{t}}=-{\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}{\left({\mathrm{K}}_2-{\mathrm{K}}_1\right)}^{-2}\left[ \exp\ \left({\mathrm{J}}_1\mathrm{X}\right)\hbox{--} \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.25em - {\mathrm{K}}_1{\mathrm{L}}_{\mathrm{O}}{\mathrm{Xm}}_2{\left({\mathrm{K}}_2\hbox{--}\ {\mathrm{K}}_1\right)}^{-1} \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\\ {}\kern4.25em - {\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}{\left({\mathrm{K}}_2\hbox{--} {\mathrm{K}}_1\right)}^{-2}\left[ \exp\ \left({\mathrm{J}}_{\mathrm{n}}\mathrm{X}\right)\hbox{--} \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.25em - {\mathrm{K}}_{\mathrm{n}}{\mathrm{N}}_{\mathrm{O}}{\mathrm{Xm}}_2{\left({\mathrm{K}}_2 - {\mathrm{K}}_{\mathrm{n}}\right)}^{-1} \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\\ {}\kern4.25em + \left(\alpha - \mathrm{B}\right){\left({\mathrm{K}}_2\right)}^{-2}\left[1- \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\right]\\ {}\kern4.25em - \left[\left(\alpha - \mathrm{B}\right)\ {\left({\mathrm{K}}_2\right)}^{-1} + {\mathrm{D}}_{\mathrm{O}}\left]{\mathrm{m}}_2\mathrm{X}\right[\ \exp\ \left({\mathrm{J}}_2\mathrm{X}\right)\ \right]\end{array} $$

(8.29)

• 7. The sensitivity of DO to K:

$$ {\mathrm{S}}_{\mathrm{C},\ \mathrm{K},\ \mathrm{t}}={\mathrm{S}}_{\mathrm{D},\ \mathrm{K},\ \mathrm{t}} $$

(8.30)

Glossary [31–35]

Ammonia nitrogen

A common way to report ammonia concentration (expressed as ammonia-nitrogen).

Ammonification

A process of formation of ammonia nitrogen from reduced organic nitrogen compounds.

Biological oxidation

A process by which living organisms in the presence of oxygen convert organic matter into a more stable or a mineral form.

Carbonaceous

Containing carbon and derived from organic substances such as coal, coconut shells, and organic waste.

Denitrification

A biochemical process of conversion of nitrite nitrogen and nitrate nitrogen to molecular nitrogen, nitrogen dioxide, or a mixture of these two gases, under reducing conditions in the absence of free dissolved oxygen.

Deoxygenation

It is a process for depletion of the dissolved oxygen in a liquid either under natural conditions associated with the biochemical oxidation of organic matter present or by addition of chemical reducing agents.

Deposit

Material left in a new position by a transporting agent such as earth quake, gravity, human activity, ice, water current, or wind.

Dissolved gases

The sum of gaseous components, such as oxygen, nitrogen, carbon dioxide, methane, hydrogen sulfide, etc. that are dissolved in water.

Dissolved oxygen (DO)

The concentration of oxygen dissolved in water, which is often expressed in units of mg/L.

Dissolved oxygen deficit (D)

The difference between the dissolved oxygen saturation concentration (C_s) and actual dissolved oxygen concentration at time t (Q) in a receiving water (such as river) at some downstream distance away from the point of waste discharge (D = C_s − C). See dissolved oxygen deficit and dissolved sag curve.

Dissolved oxygen sag curve (DO sage curve)

A stream water quality curve that represents the profile of dissolved oxygen concentration along the course of a stream resulting from deoxygenation associated with biochemical oxidation of organic matter and reoxygenation through the absorption of atmospheric oxygen and biological photosynthesis. Also called oxygen sag curve.

Dissolved oxygen saturation concentration (C_s)

The maximum concentration (mg/L) of dissolved oxygen in water under specific water temperature , pressure and salinity .

Dissolved solids

The constituents in water that can pass through a 0.45-μm pore-diameter filter.

Initial dissolved oxygen deficit (D₀)

The difference between the dissolved oxygen saturation concentration (C_s) and actual dissolved oxygen concentration (C) in a receiving water (river or lake) at the point of waste discharge (D₀ = C_s −  C). See dissolved oxygen deficit.

NCKU (National Cheng Kung University) equations

They are reaeration coefficient equations developed by National Cheng Kung University, Taiwan, showing the effect of salinity on receiving water’s reaeration coefficient. The NCKU equations are modeled by K_2s = K_2f exp (0.0127 Chlorinity); K_2s = K_2f exp (0.0000127 Chloride); and K_2s = K_2f exp (0.007 Salinity); in which K_2s = reaeration coefficient of saline water, day⁻¹; K_2f = reaeration coefficient of fresh water, day⁻¹; Chlorinity = chlorinity of receiving water, g/L; Chloride = chloride concentration of receiving water, mg/L; and Salinity = salinity of receiving water, ‰, or ppt, or parts per thousand.

Nitrate nitrogen

A common way to report nitrate concentration (expressed as nitrogen).

Nitrification

A process of formation of nitrate nitrogen from reduced inorganic nitrogen compounds, such as ammonia nitrogen. Nitrification in the natural environment is carried out primarily by autotrophic bacteria.

Nitrite nitrogen

A common way to report nitrite concentration (expressed as nitrogen).

Non-tidal stream/river

A stream/river which water level and flow direction will not fluctuate and will not be affected by the action of lunar and solar forces upon the rotating earth.

Oxygen-sag curve

See dissolved oxygen sag curve.

Photosynthesis

The conversion of light energy to chemical energy. At night, this process reverses: plants and algae suck oxygen out of the water.

Reaeration

(a) The physical chemical reaction by which oxygen is absorbed back into water, (b) An aeration process by which oxygen in air is absorbed back into natural water, such as stream water and lake water, (c) A natural process of oxygen exchange between the atmosphere and a natural water body in contact with the atmosphere. Typically, the net transfer of oxygen is from the atmosphere and into the water, since dissolved oxygen levels in most natural waters are below saturation. When photosynthesis produces supersaturated dissolved oxygen levels, however, the net transfer is back into the atmosphere, (d) Reaeration process is modeled as the product of reaeration coefficient multiplied by the difference between dissolved oxygen saturation and the actual dissolved oxygen concentration, that is: F_c = K₂ (C_s − C) = (K_L/H) (C_s − C). Here F_c = rate or flux of dissolved oxygen across the water body. M/L³/T; C = dissolved oxygen concentration, M/L³, C_s = saturation dissolved oxygen concentration, M/L³, K₂ = reaeration coefficient, 1/T, H = water depth, L, K_L = surface transfer coefficient, L/T.

Reaeration coefficient

A mass transfer coefficient (K₂) in reaeration process. See reaeration and mass transfer coefficient.

Reaeration rate

(a) The rate at which oxygen is absorbed back into water. This is dependent, among other things, upon turbulence intensity, temperature, and the water depth, (b) The reaeration rate is defined as the rate of dissolved oxygen across the water body F_c = K₂ (C_s −  C). Here F_c = rate or flux of dissolved oxygen across the water body, M/L³/T; C = dissolved oxygen concentration, M/L³; C_s = saturation dissolved oxygen concentration, M/L³; K₂ = reaeration coefficient, 1/T.

Reaeration rate coefficient

See reaeration coefficient.

Receiving waters

(a) A river , lake, ocean, stream, or other bodies of water into which wastewater or treated effluent is discharged; (b) A distinct water body that receives run off, or wastewater discharges, such as streams, rivers, lakes, estuaries and oceans.

Saline water intrusion

The movement of saline groundwater into a formerly freshwater aquifer as a result of pumping in that aquifer usually near coastal areas where the source of saline water is the nearby ocean.

Sensitivity

(a) In analytical testing, the lowest practical detection level; (b) In microbiological testing, the likelihood that the test result will be positive when the target organism is present, (c) In water resources engineering, the smallest changes of certain physical parameters that will affect hydraulic or hydrological model’s solutions.

Sensitivity analysis

(a) A mathematical analysis of the sensitivity of the dependent variable in a mathematical expression as a function of variations in the value of any independent variables or coefficients associated with the independent variables, (b) A mathematical analysis which determines how much the value of Y is affected by changes in the values of a and b.

Tidal

Pertaining to periodic water level fluctuations due to the action of lunar (moon) and solar (sun) forces upon the rotating Earth.

Tidal current

A water current brought about or caused by tidal forces.

Tidal stream/river

A stream/river which is affected by tidal current and its water level and flow direction fluctuate due to the action of lunar and solar forces upon the rotating Earth.