modules_Quinton (03/02/16)

Quinton

This module defined in Classquinton  handles the subsurface drainage from hummock-covered hillslopes in the Arctic tundra.  An HRU is a one-metre wide strip of land having defined hydrological properties that is aligned perpendicular to a selected channel.  All measurements of depth are referenced using the ground surface as the datum unless otherwise stated and taken positive.

Observations

• p (mm Dt) – depth of precipitation (rainfall) received at the ground surface in time step. In basic module only.

Variables

• theta (m3/m3) - volumetric soil moisture of layer.

• layerwater (m) - depth of  liquid water in layer.  It consists of melt, precipitation and inflow received from other HRUs.

• dmelt (m) - depth to the frost table.

•  wdrained (m) – depth to the frozen saturated layer when dmelt is in the frozen unsaturated layer above.  Otherwise 0.0.

• Depth (m) – height of liquid water above the frost table.

• watertable (m) - depth from the surface to the surface of the water table. Is equal to (dmelt - Depth).

• d_surface (m) - depth  from the surface to the middle of the liquid water layer.  This depth is used to determine the average horizontal hydraulic conductivity.  Is equal to (dmelt - Depth/2).

• k (m/day) - HRU horizontal hydraulic conductivity.

• flow (m Dt) – volume of subsurface flow per unit plan area of hummock HRU in a time step – expressed as an average depth of water (m3/m2)

• flowm3 (m3) – volume of subsurface flow from hummock HRU in time step.

• cumflow (m) - cumulative volume of subsurface flow per unit plan area of hummock HRU – expressed as an average depth of water (m3/m2).

• runoff (m Dt) - volume of surface runoff per unit plan area of hummock HRU in a time step – expressed as an average depth of water (m3/m2).

• runoffm3 (m3/int) - volume of surface runoff from hummock HRU in time step.

• cumrunoff(m/m2) – cumulative volume of surface runoff per unit plan area of hummock HRU – expressed as an average depth of water (m3/m2).

• flowin (m/int) - volume of inflow from external sources (melt, drainage, other) entering an unit plan area of a hummock HRU in a time step – expressed as an average depth of water (m3/m2).

• flowinm3 (m3/int) - volume of inflow from external sources (melt, drainage, other) entering a hummock HRU in a time step.

• cumflowin (m/m2) - cumulative volume of inflow from external sources (melt, drainage, other) entering an unit plan area of a hummock HRU – expressed as an average depth of water (m3/m2).

• loss (m Dt) – loss of water from all layers of hummock HRU per unit plan area in a time step– expressed as an average depth of water (m3/m2) over the time step.

• capillary (m) –depth of the capillary water in the layer.

•  tension (m) – head of water in the layer.

• transit (day) - time for a particle of water to traverse an HRU.  It is equal to the length of the HRU.

• Cvis (J/m3/K) - heat capacity of layer saturated, frozen condition.

• Cvisa (J/m3/K) - heat capacity of layer in unsaturated, frozen condition.

• Cvws (J/m3/K) - heat capacity of layer in saturated, unfrozen condition.

• Cvwsa (J/m3/K) - heat capacity of layer in unsaturated, unfrozen condition.

• lamis (W/m/K) - thermal conductivity of layer in saturated, frozen condition.

• lamws (W/m/K) - thermal conductivity of layer in saturated, unfrozen condition.

• lamwsa (W/m/K) - thermal conductivity of layer in saturated, unfrozen condition

Parameters

• Type () - HRU land type, 0=NOTUSED, 1=DRIFT, 2=HUMMOCK.
• length (m) - length of HRU.  HRU has unit width.
• K_btm (m/d) - horizontal bottom hydraulic conductivity.
• K_top (m/d) - horizontal top hydraulic conductivity.
• ztrn (m) - transition depth.
• DrainTo () - If drift HRU,  the Hummock HRU drained to, 0=NOWHERE or HRU# (1 to MAXHRU).
• soil_type() - 0=organic1, 1=organic2, 2=organic3, 3=sand, 4=clay.
• tinit (°C) - initial layer temperatures.
• slope (m/m) - HRU average slope.
• Residual () - organic non-drainable porosity.  Also referred to as specific yield.
• d (m) - soil layer thickness.
• Drained (m) - depth of water table, i.e. saturated organic material (ice or water).
• FrozenTo (m) - initial depth of frost table.
• flowLag (hours) - lag inserted into flow out.
• flowstorage (days) - storage inserted into flow out.
• runoffLag (hours) - lag inserted into runoff.
• runoffstorage (days) - storage inserted into runoff.
• Pors (m3/m3) - used to redefine the porosity a soil type if the value is greater than zero. Affects all HRU's using this soil type.
• n () - an empirical constant used in the Van Genuchten calculation.  If zero Van Genuchten is not used to determine the soil tension.
• a () - an empirical constant used in the Van Genuchten calculation.

Variable Inputs

• Qg (MJ/m2/Dt) - surface ground flux from module Qmelt.
• driftmelt (m3/Dt) - drift melt and precipitation from module Qdrift.
• hru_p (mm/int) - precipitation in Variation#1

Heat capacity of layer soil/organic matter:

   frozen/saturated

    Cvis = Cv_i*por_s[soil_type_lay] + Cv_s[soil_type_lay]*(1.0-por_s[soil_type_lay]), (J/kg/K).

   frozen/unsaturated - drained to the irreducible water content

    Cvisa = Cv_i*por_s[soil_type_lay]*Residual + Cv_s[soil_type_lay]*(1.0-por_s[soil_type_lay])  + Cv_a*(por_s[soil_type_lay] - Residual), (J/kg/K).

   unfrozen/saturated

    Cvws = Cv_w*por_s[soil_type_lay]) + Cv_s[soil_type_lay]*(1.0-por_s[soil_type_lay]), (J/kg/K).

   unfrozen/unsaturated - drained to the irreducible water content

    Cvwsa = Cv_w*Residual *por_s[soil_type_lay] + Cv_s[soil_type_lay]*(1.0-por_s[soil_type_lay]) + Cv_a*(por_s[soil_type_lay] - Residual), (J/kg/K).

Thermal capacity of layer soil/organic matter:

Xs = 1.0 - por_s[soil_type_lay]

Xw = por_s[soil_type_lay]-Residual

Xa = 1.0 - Xs - Xw

n = por_s[soil_type_lay]

if(Xw >= 0.09)

    ga = 0.333-Xa/n*(0.333-0.035)

else

    ga = 0.013 + 0.944*Xw

gc = 1.0 - 2.0*ga

Fs = 1.0/3.0*(2.0/(1 + (ks_s[soil_type_lay]/lam_w-1.0)*0.125)+ (1.0/((1 + (ks_s[soil_type_lay]/lam_w-1.0)*0.75))))

Fa = 1.0/3.0*(2.0/(1 + (lam_a/lam_w-1.0)*ga) + (1.0/((1 + (lam_a/lam_w-1.0)*gc))))

a = Farouki_a(por_s[soil_type_lay])            solution of the relationship:   fractional porosity = 3a² - 2a³

frozen/saturated

lamis_lay = lam_i*a*a + lam_s[soil_type_lay]*sqr(1.0-a) + lam_s[soil_type_lay]*lam_i*(2*a-2*sqr(a))/(lam_s[soil_type_lay]*a + lam_i*(1.0-a))

unfrozen/saturated

lamws_lay = lam_w*a*a + lam_s[soil_type_lay]*sqr(1.0-a) + lam_s[soil_type_lay]*lam_w*(2*a-2*sqr(a))/(lam_s[soil_type_lay]*a+lam_w*(1.0-a))

unfrozen/unsaturated

lamwsa_lay = (Xw*lam_w + Fa*Xa*lam_a +Fs*Xs*lam_s[soil_type_lay])/(Xw + Fa*Xa + Fs*Xs)

Calculation of melt

Latent heat of fusion,  Hf = 334.4E+3 (J/kg).

frozen/saturated

melt = Qm/(-tlayer_lay *Cvis + por_s[soil_type_lay]*Hf)/1000 (m),

frozen/unsaturated - drained to the irreducible water content

melt = Qm/(-tlayer_lay *Cvisa + Residual*Hf)/1000 (m)

Assuming drainage is controlled by the Van Genuchten estimation.

It is assumed that when the soil tension is less than the bubbling pressure that the soil drains by gravity. Otherwise is controlled by:

          q _f = (f - q _r )*(2ⁿ )^-m + q _r   where y _b = 1/a.  Refer to capillary for additional information.

For the unfrozen depth of a soil layer and the liquid water in the thawed layer the actual soil moisture is calculated.  If  q >=  q _f, all of the melt is transferred to the variable,   layer_water_lay otherwise, the amount transferred is determined by the soil tension and the Van Genuchten estimation.

    excess = capillary_lay -  q _f * d_lay /por_s

Assuming all melt drains by gravity

layerwater_lay = layerwater_lay + melt distributed over applicable layers.

Depth of free liquid water in organic layers;

    Depth = Sum((layerwater_lay)) (m).

Depth from surface of the water table;

    d_surface = dmelt - Depth/2.0 (m).

Calculation of Horizontal Hydraulic Conductivity.

     Three power series were fitted to the horizontal hydraulic conductivity versus depth data collected at Siksik, Granger Basin and Scotty Creek sites.   One series used all the data and the other two power series were created using only the data greater than the first power series trendline and the second series used only the values less than the first series trendline.  These two power series were created to simulate the two extremes, i.e. minimum and maximum possible horizontal hydraulic conductivities.

The continuous function for hydraulic conductivity k are:

   log(K(z)) = log(Kbtm) - (log(Ktop) - log(Kbtm))/(1 + (z/ztrn)ⁿ)

Calculation of Horizontal Flow.  

   lossD = k*Depth*slope (m/day/m2).

    loss = k*Depth*fall/length/Freq (m/m2/int). Where FREQ is the number of intervals/day.

Water balance.

    The horizontal drainage is taken from the uppermost layerwater first and any left is taken from the next lower layer.

    layerwater_lay = layerwater_lay - Loss/( por_s[soil_type_lay]*(1-Residual)) (m)

The out flow is calculated from the losses in the layerwater layers;

    flow = Sum(Loss) (m).

N.B. units of (m/m2) are used for convenience.

Replenishment from snow drift melt;

flowin = driftmelt*length_drift_hru/length_hummock_hru (m) or from a hummock HRU is

flowin = flow*length_contributing_hummock_hru/length_hummock_hru.

layer maximum replenish depth:

maxdepth = dmelt - top_of_layer,

maximum replenish water (m),

maxfree = maxdepth*por[soil_type_lay]*(1.0 - replenish),

surplus after satisfying maxfree for melted layers is runoff (m).

Calculation of theta for each layer.

layerwater_ht = sum(layerwater_lay/por[soil_type_lay] (m),

if(dmelt - layerwater_ht) <= top_of_layer then

    theta = 1.0

else if(dmelt - layerwater_ht) >  (top_of_layer + layer_depth) then

     theta = residual

else

    theta = (depth_w +(layer_depth - depth_w)*residual)/layer_depth,

        where depth_w = top_of_layer + layer_depth - (dmelt - free_water_ht/por[soil_type_lay] (m).

Constants

g = 9.81 (m/s2)

visc_w = 0.0018 at 0°C (N.s/m2) (kg/(m.s) water

rho_a = 1.2  (kg/m3) air

rho_i = 920.0  (kg/m3) ice

rho_w = 1000.0  (kg/m3) water

c_a = 1010.0  (J/kg/K) air

c_i = 2120.0  (J/kg/K) ice

c_w = 4185.0  (J/kg/K) water

Cv_a = 1212.0  (J/m3/K) air

Cv_i = 1950400.0  (J/m3/K) ice

Cv_w = 4185000.0  (J/m3/K) water

lam_a = 0.025  (W/m/K) air

lam_i = 2.24  (W/m/K) ice

lam_w = 0.57  (W/m/K) water

order[ LOAM1, LOAM2, LOAM3, SAND, CLAY]

rho_s[] = { 41.1, 75.2, 91.4, 1300.0, 1300.0}  (kg/m3) density

c_s[] = {1920.0, 1920.0, 1920.0, 890.0, 890.0}  (J/m3/K) specific heat

Cv_s[] = {78912.0, 144384.0, 175392.0, 1157000.0, 1157000.0}  (J/m3/K) heat capacity

lam_s[] = {0.21, 0.21, 0.21, 2.50, 2.50}  (W/m/K) thermal conductivity

ks_s[] = {450.0, 154.0, 13.0, 5.0, 3.0}  (m/day) hydraulic conductivity

por_s[] = {0.96, 0.9, 0.87, 0.43, 0.43, }  () porosity

Notes on HRUs.

    The Quinton modules are not areal models but written to handle a one-metre wide strip of land that is aligned perpendicular to a selected channel The strip is made up of , each having a distinct personality that is determined by the parameters assigned to it.  At present there are two types of HRUs’, drift and hummock.  Any number of drift and  hummock HRUs can be included in a model.  Drift HRUs must drain into hummock HRUs,  not into another drift HRU. Multiple drift HRUs can drain into one hummock HRU. Drift HRUs must be defined before the hummock HRUs they drain into.  Hummock HRUs may be cascaded,  draining into one another before finally draining into the creek.  At present it is assumed that if an areal water balance is required for a basin,  the ouput of the CRHM model will be further processed in a spreadsheet.

Water Balance.

    To confirm proper operation of the modules a water balance was added to the Quinton and Qdrift modules.  When the model is initialised at the beginning of a model run the water content defined in the initial conditions is tabulated in the Log/Debug Output window.  Similarly at the completion of the model run the final water content is tabulated.  An example of the results of a model run follow.

    The outputs from the HRUs are given twice.  Once, as the volume (m3) from the HRU given that the area is equal to the length of the HRU times one metre width and secondly as the equivalent water depth over the area of the HRU.

02 05 19 00 30 Initial

HRU 1: (Drift ) - water content (m3) (m/m2): 5.40 0.200

HRU 2: (Hummock) - water content (m3) (m/m2): 4.16 0.297

HRU 3: (Hummock) - water content (m3) (m/m2): 4.16 0.297

HRU 4: (Hummock) - water content (m3) (m/m2): 4.16 0.297

HRU 5: (Hummock) - water content (m3) (m/m2): 4.16 0.297

HRU 6: (Hummock) - water content (m3) (m/m2): 4.16 0.297

HRU 7: (Hummock) - water content (m3) (m/m2): 4.16 0.297

HRU 8: (Hummock) - water content (m3) (m/m2): 4.16 0.297

02 08 01 00 00 Final

HRU 1: (Drift ) - water content (m3) (m/m2): 0.00 0.000

HRU 1: (Drift ) - total precip (m3) (mm/m2): 0.00 0.000

HRU 1: (Drift ) - water storage (m3) (m/m2): 0.00 0.000

HRU 2: (Hummock) - water content (m3) (m/m2): 3.11 0.222

HRU 2: (Hummock) - cumulative flowin (m3) (m/m2): 5.40 0.386

HRU 2: (Hummock) - cumulative precip (m3) (m/m2): 0.00 0.000

HRU 2: (Hummock) - cumulative flowout (m3) (m/m2): 3.90 0.278

HRU 2: (Hummock) - flowout in storage (m3) (m/m2): 0.00 0.000

HRU 2: (Hummock) - cumulative runoff (m3) (m/m2): 2.55 0.182

HRU 2: (Hummock) - runoff in storage (m3) (m/m2): 0.00 0.000

HRU 3: (Hummock) - water content (m3) (m/m2): 3.11 0.222

HRU 3: (Hummock) - cumulative flowin (m3) (m/m2): 6.45 0.460

HRU 3: (Hummock) - cumulative precip (m3) (m/m2): 0.00 0.000

HRU 3: (Hummock) - cumulative flowout (m3) (m/m2): 5.43 0.388

HRU 3: (Hummock) - flowout in storage (m3) (m/m2): 0.00 0.000

HRU 3: (Hummock) - cumulative runoff (m3) (m/m2): 2.06 0.147

HRU 3: (Hummock) - runoff in storage (m3) (m/m2): 0.00 0.000

Processing of parameters Drained and FrozenTo.

    When the frost table is below the water table the initialization consists of moving the water contained between the two tables into the variable layerwater_lay??? for the applicable layers.  When the (frozen) water table is below the frost table the quantity of heat required to melt the frozen soil is less as instead of saturated s. Don’t understand?? Do you mean? When the water table falls below the frost table the quantity of heat required to melt the drained soil is less because its’ moisture content is set to the residual value

Derived from Letts et al. (2000).