CRiSP1.6 Theory & Calibration Manual: II.5 - Total Dissolved Gas INDEXTOCPREVNEXT

II.5 - Total Dissolved Gas

II.5.1 - Introduction

In a riverine environment, total dissolved gas at equilibrium should be in relative balance with the atmospheric pressure. Natural sources, such as waterfalls or organic inputs, can cause the level of gas to rise above the equilibrium level. The primary source of dissolved gas supersaturation in the Columbia and Snake rivers is spill from hydroelectric dams. As water flows over the spillway, air becomes entrained by the spill flow. As a result, the river becomes supersaturated in total dissolved gas. Sinks of dissolved gas are relatively insignificant for the Snake and Columbia rivers; therefore, in CRiSP.1, the river never falls below the equilibrium level.

In the model, dissolved gas can enter the system in two ways: 1) at a headwater, representing the amount of gas coming from upstream sources or 2) at a dam, resulting from spill. Headwater input is read into the model from data files. Dissolved gas production at a dam is calculated by the model based on the level of spill. Then dissolved gas is propagated downstream with the water according to a system of reach dynamics (see Section II.5.4 Reservoir Dissolved Gas Distributions).

II.5.2 - Gas Production Equations

Theory

For CRiSP.1 version 6, new equations have been implemented for gas production from spill. As a part of the Dissolved Gas Abatement Study conducted by the U.S. Army Corps of Engineers, Waterways Experiment Station (WES) developed these new equations as an improvement over GASSPILL, the previously predominant model for gas production.

The new equations are an empirical fit of spill data and monitoring data collected by the U.S. Army Corps of Engineers. The percent of total dissolved gas (TDG) exiting the tailrace of a dam is predicted as a function of the amount of discharge in kcfs. This level of TDG is not necessarily the highest level of gas reached, but rather the level of gas in the spill water after some of the more turbulent processes have stabilized. The calibration for each dam was fit to the nearest downstream gas monitoring station, which is typically about a mile downstream of the dam.

For the eight lower Snake and lower Columbia dams that were studied by WES, the gas production equations take one of three forms: linear function of total spill, a bounded exponential function of total spill, or a bounded exponential function of the spill on a per spillbay basis. These equations were adopted for all dams in CRiSP.1. See Section III.2 Total Dissolved Gas Calibration for more details.

Equations for TDG supersaturation are of two types. One type constitutes empirical equations with no underlying theory, but the equations provide a general fit to observed supersaturation data as a function of spill. The other type constitutes mechanistic equations which define TDG levels in terms of physical processes producing spill. CRiSP.1 contains four empirical models and two mechanistic models. In general, we recommend using the calibrated values for TDG.

TDG Empirical Models

WES Linear Equation

The gas production equation as a linear function of total spill is:

(86)

where

WES Exponential Equations

The gas production equation as a bounded exponential function of total spill is:

(87)

or as a bounded exponential function of the spill on a per spillbay basis is:

(88)

where

Different day and night spill patterns for adult and juvenile fish passage at the Snake River dams require different production equations. CRiSP.1 is currently configured so that a separate spill pattern, and thus a separate gas production function, for night and for day can be set for each dam. A spill pattern specifies which spill bays are used to discharge flow both in number and position. Once the number of spill gates n for a particular pattern is set, eq (88) is then converted into eq (87) by the relation qs = Qs/n. This conversion formula assumes that the amount of spill is uniformly distributed among the open spill gates. The model parameters for the day and night gas production can be different for a given dam, reflecting a change in the position or number of gates and hence in the dynamics of gas production.

Empirical Exponential Equation

An empirical TDG supersaturation equation based on an exponential relationship between spill flow and supersaturation in the spilled water can be expressed:

(89)

where

This alternative exponential equation was first developed and used in CRiSP.1 version 3, and it was retained in version 4 for backward compatibility of models. It is currently used as the backup model when spill exceeds a certain value for certain dams in certain years.

Empirical Hyperbolic Equation

The TDG supersaturation equation data can also be fit with a hyperbolic relationship between spill flow and supersaturation. The relationship is:

(90)

where

Although this submodel can produce a degree of supersaturation at zero spill flow (when h = 0), this does not contribute to supersaturation in the tailrace water since the contribution of spill water to the tailrace is zero with zero spill as is defined in eq (109).

TDG Mechanistic Models

The TDG mechanistic models based on the physical process on spilling water and dissolving excess TDG in the tailrace water was developed by Water Resource Engineers, Inc. (Roesner and Norton 1971) for the U.S Army Corps of Engineers. Relevant parameters in the mechanistic submodels are illustrated in Fig. 38.

Fig. 38 Representation of spillway and stilling basin.



The mechanistic model begins with an equation for TDG concentration as:

(91)

where

(92)

(93)

(94)

(95)

(96)

The coefficients are estimated using different relationships depending upon the dam. These are known as GasSpill 1 and GasSpill 2 and are detailed as follows.

GasSpill 1

GasSpill 1 is a three-parameter multiplicative model previously used by the U.S. Army Corps of Engineers at Bonneville Dam only. The equation is:

. (97)

With c = 0, this model is identical to the two-parameter multiplicative model developed by Water Resources Engineers (WRE), Inc. (Roesner and Norton 1971).

GasSpill 2

GasSpill 2 is a three-parameter additive model previously used by the U.S. Army Corps of Engineers at all other dams. It is defined:

(98)

where

(99)

II.5.3 - Tailrace Dynamics

Introduction

Extensive field studies led by the U.S. Army Corps of Engineers have provided insights into how dissolved gas exits the dams and is transported downstream. CRiSP.1 now allows for different scenarios on how the spill and powerhouse flows exit a dam.

Flow enters a dam containing a certain amount of dissolved gas (forebay gas level). This flow is routed in part through the powerhouse and the rest through the spillway. Spill produces gas in the tailrace flow that generally exceeds incoming levels, whereas the flow exiting through the powerhouse retains the forebay gas level. The interaction between these two flows in the tailrace is dynamic. Currents can dilute the supersaturated spill by inducing mixing with the less-gassed powerhouse flow or the powerhouse flow can be entrained into the spill flow and also become gassed as a result. Varying flow and spill conditions can change the level of entrainment and mixing, as well as the amount of dissolved gas being produced.

In CRiSP.1, both tailrace mixing and entrainment can be specified at a dam. It is likely that some dilution is represented by these coefficients because most of the data used to calibrate the gas production equations came from gas monitoring stations downstream of the spillway. In addition, there is very little data from the powerhouse flow after it exits the dam, so it is also difficult to measure entrainment directly. To avoid over-determination due to too many parameters and too little data, this calibration was kept simple by using an all or nothing approach to mixing in the tailrace based on observations from field studies rather than a statistical fit of the tailrace mixing parameter.

The final measure of CRiSP.1's calibration is the accuracy of the modeled forebay levels. If the amount of gas in the downstream forebay was underestimated then the entrainment function was used to adequately adjust the total amount of gas being added to the system. This was done using the procedure described in Entrainment.

Separate Flows

For the majority of dams on the Columbia and Snake rivers, the flows exit the dams as separate flows. The spill flow will exit the dam with a dissolved gas value produced from spill. The powerhouse flow will often contain a lower gas level, typically closer to the level of gas in the forebay. This motivated a two-flow model for the river. The two flows are denoted (looking downstream) as "left flow" and "right flow." Currently, only the amount of flow and the dissolved gas level vary between the left and right flows in a reach or at a dam.

For each dam a spill_side token and value is designated in the columbia.desc file. For example, looking downstream at Ice Harbor Dam, the spillway is on the right side of the dam, so the spill_side value is right, and consequently the spill flow is the right flow and the powerhouse flow is the left flow. For some projects, this is a simplified view. In these cases, if a bias in the spill flow exists as it exits the dam then that side was chosen as the spill_side. CRiSP.1 assigns spill_side to right if the spill_side is not designated in the river description file. Table 30 contains the spill_side values used by the model.

It should be noted that for some of these dams, there is essentially complete mixing in the tailrace of the two flows and hence both flows will exit the dam with the same dissolved gas level. The spill_side in this case will have no real impact. The next section discusses mixing in more detail.

Table 30 Spill side tokens for each dam
Dam spill side Dam spill side Dam spill side
Chief Joseph right Dworshak left McNary right
Wells left Hells Canyon right John Day right
Rocky Reach left Lower Granite right The Dalles right
Rock Island right Little Goose right Bonneville right
Wanapum right Lower Monumental left

Priest Rapids right Ice Harbor right



The spill fraction determines the amount of flow which is attributed to the spill_side flow of the river. The amount of dissolved gas in each of the flows depends on several factors: the amount of gas in the forebay of the dam, the amount of gas produced by the spill flow, and the amount of mixing and/or entrainment in the tailrace. Mixing and entrainment are both adjustable by dam and are explained in the following sections. Once mixing and entrainment are applied, a dissolved gas value is determined for each flow and passed as input gas values to the next reach.

Mixing

Theory

In CRiSP.1, for dams where there is a significant amount of mixing in the tailrace, the flows from spill and the powerhouse are averaged according to their flow fractions. The mixed TDG value is contained in both flows upon exiting the tailrace. This has the effect of diluting the spill flow and raising the level of dissolved gas in the powerhouse flow.

To allow for all possibilities between the extremes of separate flows and full mixing, CRiSP.1 includes a mixing coefficient for the dam which determines the amount of mixing to occur between the powerhouse and spill flows before exiting the tailrace of the dam.

Let and , where Sfr is the percent of the river in the spill side flow. Then mixing in the tailrace can be expressed by a decay process which decreases the difference between the two gas levels as a function of the mixing parameter set for each dam. At the dam, the spill flow gets the gas level Gspill and the powerhouse flow has the gas level of the forebay. Before exiting the tailrace, the difference in gas level between the two flows is decayed. This is represented by replacing Gdif with the expression .

After applying the mixing in the tailrace and solving for Gspill and Gphouse, the exiting gas levels are:

(100)

. (101)

where

Given these expressions for mixing, a value of leads to no mixing and the spill flow exits with the gas value generated by the gas production equations and the powerhouse flow retains the forebay gas value. For a value of , complete mixing is attained and both flows leave the dam with a gas level of Gmix, the flow weighted average of the two gas levels.

Parameter Determination

In the gas production field studies led by the Waterways Experiment Station (1996; 1997a), a significant amount of mixing was observed in the tailraces of The Dalles Dam and Bonneville Dam. For these dams, the gas production equations represent well-mixed powerhouse and spillway flows in the tailrace (U.S. Army Corps of Engineers 1996b). As a result, complete mixing was assumed in CRiSP.1 by setting . For the remaining dams on the mainstem Columbia River and lower Snake River, WES's work supported separate spill and powerhouse flows. This is represented by a zero mixing coefficient, . For these dams, their gas production equations represent the amount of gas in the spill flow.

On the upper Columbia River according to a field study for Chief Joseph Dam prepared by the Seattle District, U.S. Army Corps of Engineers, the spill and powerhouse flows exit Chief Joseph Dam as separate flows (U.S. Army Corps of Engineers 1998b). Separate flows were assumed for the remaining upper Columbia dams. This is represented by a zero mixing coefficient, .

Complete mixing at Dworshak Dam was also assumed based on the steep structure of the dam and its narrow tailrace. This is represented by setting .

Table 31 Tailrace Mixing coefficients
Dam Dam Dam
CHJ 0 HCY 0 MCN 0
WEL 0 DWR 10 JDA 0
RRH 0 LWG 0 TDA 10
RIS 0 LGS 0 BON 10
WAN 0 LMN 0

PRD 0 IHR 0



Entrainment

Theory

Entrainment refers to the phenomena that the powerhouse flow actually becomes entrained by the spill flow and is gassed as a result. In this scenario, the spill TDG levels are not diluted but rather more TDG is added to the system via the powerhouse flow. The entrainment function is an empirical relationship between the total amount of gas added to the powerhouse flow and the amount of flow going over the spillway. The higher the spill the more gas that is added to the powerhouse, with the level of TDG in the exiting powerhouse flow ranging from the forebay TDG level to the TDG level in the spill flow. This relationship was motivated by the heuristic that the greater the amount of spill, the greater the "plunging" force and hence the greater amount of energy in the spill flow. The relationship can be expressed:

(102)

where

Parameter Determination

The values for k_entrain are estimated annually and represent annual averages. They can be expected to vary from year to year as details of the annual spill patterns and other conditions vary.

Table 32 Estimations of k_entrain from CRiSP.1 runs using filtered Columbia River Data Access in Real Time (DART) data (observed and modeled TDG > 100%).
Location 1994 1995 1996 1997 1998 1999
CHJ




0.13
WEL
.143 0.00 .94 1 0.25
RRH
.001 .005 0.00 .002 0.00
RIS
.143 .004 .018 .014 0.00
WAN .052 .029 0.00 .054 .013 0.13
PRD
0.00 0.22 0.00 0.04 0.00
LWG
.009 .009 .012 .017 0.025
LGS
.143 .96 .555 .802 0.325
LMN
0.00 0.00 0.00 0.00 0.05
IHR
0.00 0.22 0.00 0.004 0.05
MCN
0.00 0.00 0.00 0.00 0.00
JDA
0.00 0.00 0.00 0.00 0.00
TDA 0.00 0.00 0.00 0.00 0.00 0.00


Modeled forebay levels at Little Goose, Lower Monumental, Wanapum and Priest Rapids dams with and without the entrainment coefficient at the previous dam are shown versus the observed forebay values in Fig. 40-Fig. 42.

Fig. 39 Lower Granite (LWG) production values with and without entrainment and observed data (points) at Little Goose Forebay.



Fig. 40 Little Goose (LGS) production values with and without entrainment and observed data (points) at Lower Monumental Forebay.



Fig. 41 Rock Island (RIS) production values with and without entrainment and observed data (points) at Wanapum Forebay.



Fig. 42 Wanapum (WAN) production values with and without entrainment and observed data (points) at Priest Rapids Forebay.



II.5.4 - Reservoir Dissolved Gas Distributions

Theory

The CRiSP.1 reservoir gas model has been reworked to model the movement and mixing of parcels of water distinguished by different levels of total dissolved gas. A quasi-2D river model describes the river as two flows, with each flow having its own TDG level. Looking downstream, there is the right-bank and the left-bank flow (see Fig. 43).

At a dam, the river is divided according to the proportion of spill from the nearest upstream dam. At a confluence, the river is divided according to the proportion of flow from the two converging rivers. At a reach where there has been no spill or upstream confluences, the gas levels on either side of the river are simply set to be equal and there is essentially one flow in the reservoir. Fig. 44 represents the case downstream of a dam. The right-bank flow in this case is just the spill flow, and the fraction of flow in the right-bank flow is simply the spill fraction.

Fig. 43 A Divided Reservoir



Fig. 44 Reservoir Gas Dynamics



TDG is mixed between the two flows and simultaneously dissipated as the water moves downstream, with the river velocity being estimated from the flow and reservoir geometry. In this manner, the model captures heterogeneous levels of gas. Fig. 44 also illustrates the gas dynamics modeled in the reservoir.

Each of the flows has an initial level of TDG which is then diffused through the boundary between them and also dissipated into the air. Simple exponential functions were used to achieve these processes in the model. These exponential functions were chosen for their simplicity; the sparseness of data and the added complexity discouraged the use of a full two-dimensional advection-diffusion model. Exponential functions were also used because the rate of change of an exponential variable is proportional to its value; this is representative of many decaying substances in nature.

The 2-flow model is shown in eq (103) and (104):

(103)

(104)

where

(105)

(106)

Using eq (103) and (104), we get:

. (107)

In other words, the difference between the two concentrations is decaying to zero due the diffusion factor and the dissipation factor . Similarly, with a little algebra, the total mass in the system can be shown to be:

. (108)

Thus the total mass (without the dissipation factor it remains at Gmix) is decaying to equilibrium level E. Hence the physical properties are captured with these two equations. Gright and Gleft are computationally inexpensive and their simplicity results in an easy fitting and integration.

A given reservoir can have "slugs" of water which entered the reach under different initial conditions. Typically, these slugs are caused by varying spill conditions at an upstream dam. Conditions at a dam can vary on a dam time step (six hour) basis. Thus all water leaving the reach in a given dam time step is assumed to have the same initial conditions. At any given point in the reach, daily river velocities and the distance downstream in the reach are used to calculate the length of time the water has been in the reach. These travel times are used to capture the correct initial conditions and the amount of mixing and dissipation that have occurred in this slug of water. At any given point in the reach, the dissolved gas level is calculated by knowing the initial conditions for Gright and Gleft, and Sfr along with x (distance downstream).

Parameter Determination

In transect studies completed by the U.S. Army Corps of Engineers, gas data from lateral cross sections of the Snake and Columbia rivers were sampled to gather information on mixing characteristics in each of the reservoirs from Lower Granite to Bonneville. These pools were sampled under high and low flow conditions and showed that while the dam introduced a heterogeneous flow, the reservoirs were well-mixed by the next downstream forebay.

Because mixing rates vary according to dam operations, river velocity, and other conditions such as wind, a conservative estimate for mixing was fixed for all reaches. A value of 0.075 was used to fix the mixing rate so that the flows were 95% mixed in 40 miles. The transect data from the 1996 and 1997 studies showed that the difference between the left-bank and right-bank flows rarely differed by more than this in the downstream forebay (Waterways Experiment Station 1996, 1997a).

II.5.5 - Other Gas Inputs

In the last several years, more and more dissolved gas data has become available from the U.S. Army Corps of Engineers. Nearly every pool has at least 2 gas monitoring stations (Fig. 45), one in the forebay of the dam and one in the tailrace of the previous dam. For this reason, an input feature was added to CRiSP.1 to allow the direct input of dissolved gas data at any reach or dam in the model. This is achieved through the output_gas token in the yearly input data file. By default this feature is turned off, but if the line output_gas on appears in a reach or dam profile, then a vector of dissolved gas data of length num_days * num_dam_slices (currently 366*4) should be supplied.

The intention of this feature was to allow total dissolved gas to enter the system above the dams. In most data files, a vector of data is provided at two locations: Chief Joseph Pool for gas entering from the Columbia headwaters, and Lower Granite Pool for the gas entering from the upper Snake and Clearwater. For a more accurate description of dissolved gas, historic data could be used for all reaches where it is available, but generally this is turned off since gas production and distribution is well modeled.

The output_gas token has the effect of setting gas values that exit the reach on both sides of the river to the same value.

Fig. 45 Map of Columbia Basin showing dams, USACE Gas Monitor Stations, and USGS Streamflow Gaging Stations



Total Dissolved Gas in the Tailrace

Total dissolved gas supersaturation in the tailrace results from mixing spill water with water passing through turbines (Fig. 38). The equation is:

(109)

where

Total Dissolved Gas at a Confluence

The TDG at a confluence is determined by the addition of two flows with different TDG levels. The equation is:

(110)

where

Total Dissolved Gas Dissipation

Total dissolved gas levels above the saturation level are lost from the river as a first order process. WRE (Roesner and Norton 1971) defined this by a total flux equation for a segment as:

(111)

where

(112)

where

To express the loss in terms of concentration, we divided eq (111) by AD to give:

. (113)

Note that one mile = 16.0934 x 104 cm = 5280 ft, and one day = 8.64 x104 seconds. To put the calculation in units of miles and days, we express U in miles/day and D in feet and Dm in cm2/s. Thus the diffusion coefficient per unit square mile of river is:

(114)

where the coefficient k is expressed:

(115)

assuming:

TDG loss rate due to degassing can be expressed as a function of the residence time since the water entered the tailrace:

(116)

where

Noting that in the models G is in terms of percent above supersaturation, we then set Geq = 0.

Adjustments of k

The TDG dissipation coefficient depends on the average depth as defined in eq (115). The average depth is variable according to the geometry of the reservoir and the pool elevation. This depth is defined as:

(117)

where

CRiSP1.6 Theory & Calibration Manual: II.5 - Total Dissolved Gas INDEXTOCPREVNEXT
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