the mississippi delta is sinking into the gulf of mexico

National Center for Earth-surface DynamicsContribution from the National Center for Earth-surface

Dynamicsfor the Short Course

Environmental Fluid Mechanics: Theory, Experiments and Applications

Karlsruhe, Germany, June 12-23, 2006DELTA-SCALE NUMERICAL MODELING: A TOOL FOR DECISIONMAKING IN THE REHABILITATION OF THE MISSISSIPPI DELTAAdapted from a lecture given by Gary Parker, University of Illinois

on behalf of the National Center for Earth-surface Dynamics (NCED) at the ASCE World Environmental and Water Resources Congress, May 25, 2006.

The Mississippi Delta is sinking into the Gulf of Mexico.




Karlsruhe, Germany, June 12-23, 2006THE PROBLEMS OF THE MISSISSIPPI DELTA AND THE RISK TO NEW

ORLEANS WERE WELL-KNOWN LONG BEFORE HURRICANE KATRINA

Scientific American, October, 2001




Karlsruhe, Germany, June 12-23, 2006THE MISSISSIPPI DELTA FALLS WITHIN THE CLASS OF FANS AND FAN-DELTAS

The Okavango Inland Fan, Botswana. From NASA

The Nile Delta, Egypt. From NASA




Karlsruhe, Germany, June 12-23, 2006THE FAN-DELTA AT THE MOUTH OF THE YELLOW RIVER IS A GOOD ILLUSTRATION

Mouth of the Yellow River. From NASA.




Karlsruhe, Germany, June 12-23, 2006THE PROCESSES

Channel aggradation and progradation

• overbank sediment deposition as channel aggrades

• delta extension as channel head progrades

• channel migration or avulsion to shorter path to sea

The processes that build fans and fan-deltas are:




Karlsruhe, Germany, June 12-23, 2006THE ENTIRE NORTH CHINA PLAIN IS AN ALLUVIAL FAN

Over thousands of years of historical time the Yellow River has repeatedly shifted between courses north and south of the Shandong peninsula.

Present delta

Shandong peninsula




Karlsruhe, Germany, June 12-23, 2006WHAT HAPPENS WHEN HUMANS INTERFERE WITH THIS PROCESS BY BUILDING DIKES AND PREVENTING AVULSIONS?

Adapted from Hu Yisan and Xu Fuling (1989)

10

20

30

20 25 30 35 10 45 50 55155 100

S N

Distance (km)

Yellow River 20

m

One reach of the Yellow River is perched 10 m above the surrounding North China Plain




Karlsruhe, Germany, June 12-23, 2006THE KUSATSU RIVER FORMS A FAN-DELTA ON LAKE BIWA, JAPAN WHICH IS DENSELY POPULATED

It has not been allowed to avulse since the 10th Century.




Karlsruhe, Germany, June 12-23, 2006OVER GEOMORPHIC TIME THE MISSISSIPPI RIVER HAS REPEATEDLY AVULSED TO FORM NEW LOBES




Karlsruhe, Germany, June 12-23, 2006

Under natural conditions, this subsidence is balanced by overbank deposition of sediment and avulsion into low areas.

Mississippi River and levees downstream of New Orleans.

Subsiding fan-delta surfacebehind levees south of New

Orleans.

LIKE MANY LARGE DELTAS, THE MISSISSIPPI DELTA SUBSIDES BY COMPACTION UNDER ITS OWN WEIGHT




Karlsruhe, Germany, June 12-23, 2006THE RIVER IS DIKED ALONG ITS ENTIRE LENGTH AND IS NOT ALLOWED TO AVULSE

Sediment is either stored in-channel or funneled out to sea.




Karlsruhe, Germany, June 12-23, 2006SO THE RIVER BED GETS HIGHER AND HIGHER E.G. AT THE OLD RIVER CONTROL STRUCTURE,




Karlsruhe, Germany, June 12-23, 2006THE RIVER MOUTH EXTENDS FARTHER AND FARTHER INTO THE

GULF OF MEXICO,




Karlsruhe, Germany, June 12-23, 2006AND THE REST OF THE DELTA SUBSIDES UNDER COMPACTION, WITH NO REPLACEMENT SEDIMENT, CAUSING THE SHORELINE

TO ADVANCE

“At this rate, New Orleans will be exposed to the open sea by 2090.”

Fischetti (2001), Scientific American





•The Mississippi Delta rapidly subsides by compaction under its own weight.

• Under natural conditions this subsidence is balanced by overbank deposition of sediment abetted by channel avulsion.

• The mud that would construct the floodplain is held behind levees and delivered out to sea.

• Meanwhile the sand deposits on the channel between the levees as it elongates.

• As a result, the levees and the prevention of avulsion is causing the shoreline to advance, not in geomorphic time, but in engineering time.

THE PROBLEM IN A NUTSHELL




Karlsruhe, Germany, June 12-23, 2006THE MISSISSIPPI DELTA PROJECT OF THENATIONAL CENTER FOR EARTH-SURFACE DYNAMICS (NCED)

NCED has been developing large-scale morphodynamic models to evaluate the rehabilitation of the Mississippi Delta.




Karlsruhe, Germany, June 12-23, 2006HURRICANE KATRINA HIGHLIGHTED THE NEED FOR THE MISSISSIPPI DELTA PROJECT

which is being conducted as as a joint effort between NCED’s Stream Restoration and Subsurface Architecture programs




Karlsruhe, Germany, June 12-23, 2006THE MISSISSIPPI DELTA/NEW ORLEANS PROBLEM IS OFTEN THOUGHT OF AS A HURRICANE/STORM-SURGE PROBLEM

There is no hope of alleviating the storm surge problem without building land.

This house is not in standing water

because of storm surge!The root of the problem, however is the disappearance of delta land as sediment that would replenish the sinking delta is instead channeled by dikes straight out to the Gulf of Mexico.




Karlsruhe, Germany, June 12-23, 2006CAN WE BUILD LAND

BY MEANS OF A PARTIAL DIVERSION (CONTROLLED

AVULSION) OF THE MISSISSIPPI RIVER?

HERE?

OR HERE?




Karlsruhe, Germany, June 12-23, 2006PROOF OF CONCEPT: THE WAX LAKE DELTA

During the Great Flood of 1973, the Mississippi River nearly avulsed into the Atchafalaya River

and part of the Atchafalaya River avulsed into a drainage channel (Wax Lake Outlet) to form the Wax Lake Delta




Karlsruhe, Germany, June 12-23, 2006THE WAX LAKE DELTA HAS BEEN BUILDING NEW LAND SINCE 1973

The Atchafalaya River has been receiving 30 ~ 60% of the sediment of the Mississippi River,

and the Wax Lake Delta has been receiving about half of the sediment of the Atchafalaya River.




Karlsruhe, Germany, June 12-23, 2006CAN WE CAPTURE THIS LAND BUILDING IN A NUMERICAL MODEL?

• Cooperating researchers at Louisiana State Univ. (Roberts, Twilley), Univ. of Louisiana Lafayette (Meselhe), Univ. of New Orleans (McCorquodale) and Tulane Univ. (Allison).

• NCED researchers at the Univ. of Minnesota (Paola) and the Massachusetts Institute of Technology (Mohrig), and

A first model has been developed by NCED at the Univ. of Illinois, in cooperation with




Karlsruhe, Germany, June 12-23, 2006ELEMENTS INCLUDED IN THE MODEL

• Deposition of mud

• Channel-floodplain co-evolution

• Self-channelization

• Flood hydrology

• Deposition of sand

• Sea level rise

• Delta progradation

• Subsidence




Karlsruhe, Germany, June 12-23, 2006THE GEOMETRYFluvial reachxv = downvalley coordinate on fluvial reachLf = reach lengthBfc = channel bankfull width on fluvial reachBf = floodplain width on fluvial reach

Fan-delta reachrf = downfan radial coordinateru = value of rf at upstream end of fan reachrd = rd(t) = value of rf at downstream end of fan-delta reach = fan-delta angleBd = r = fan-delta widthBdc = width of channel (or amalgamated channels) on fan-delta reach

Bf

Bfc

Bdc

Bd = r

Lf

rd

xv

ru

rf




Karlsruhe, Germany, June 12-23, 2006SOME ASSUMPTIONS

• Sand deposits in both the channel and the floodplain, but mud deposits only in the floodplain, For each volume unit of sand deposited in the channel-floodplain complex units of mud are deposited in the channel-floodplain complex.

•The channel(s) migrates or avulses across the floodplain (width Bf) or fan-delta (width Bc = r) to fill all the available space.

• The channel is meandering and has sinuosity f on the fluvial reach and d on the fan-delta reach. Averaging over many bends, the relation between down-channel coordinate x and down-valley coordinate xv in the fluvial reach is given as

• Where r is down-channel coordinate on the fan-delta, the corresponding relation for the fan-delta reach is

fvx

x

dfr

r




Karlsruhe, Germany, June 12-23, 2006SOME ASSUMPTIONS (contd.)

• The bulk porosity of the deposit in the channel-floodplain complex of the fluvial reach is pf. The corresponding value for the fan-delta is pd.

• The mean elevation of the bed of the channel-floodplain complex of the fluvial reach is f; the corresponding value for the fan-delta reach is d.

• Well below the surface channel-floodplain complex is a basement with elevation bf in the fluvial reach and elevation bd in the fan-delta reach. These basements are subsiding under compaction at the respective fluvial and fan-delta rates f and d, where

dbd

fbf

t;

t

xv

f

bf f




Karlsruhe, Germany, June 12-23, 2006EXNER EQUATION OF SEDIMENT CONTINUITY: FLUVIAL REACH

The volume flood transport rates sand and mud during floods load in m3/s are denoted as Qs and Qm, respectively. Flood intermittency is If so that e.g. IfQs = mean annual volume sand load. Sand and mud are transported down the channel(s). Only sand deposits in the channel; sand and mud deposit across the entire channel-floodplain complex.

vvv xxmsfxmsf

bffvfpf

QQIQQI

)(xB)1(t

Qs, Qm

f

bf

f

Bf

xv xv+xv

Qs, Qm

Reduce with ffb

t

xx,

x

Q)1(

x

)QQ(

v

s

v

ms

to get x

QI)1(

tB)1( s

fff

fp




Karlsruhe, Germany, June 12-23, 2006EXNER EQUATION OF SEDIMENT CONSERVATION: FLUVIAL REACH

AND FAN-DELTA REACHES

Fluvial reach: sediment is transported in the channel but deposited over the entire floodplain width to model long-term floodplain deposition, channel migration and avulsion. Where the subscript “f” denotes “fluvial reach”:

x

QI)1(

tB)1( s

ffffff

fpf

Fan-delta reach: sediment is transported in the channel, or an amalgamation of more than one active channels, but deposited over the entire width Bd = rf of an axially symmetric fan-delta to model long-term floodplain deposition, channel migration and avulsion. The analogous form of sediment conservation, for which the subscript “d” denotes “fan-delta reach”, is:

r

QI)1(

tr)1( s

dfdddd

fpd




Karlsruhe, Germany, June 12-23, 2006CLOSURE FOR SELF-FORMED CHANNEL GEOMETRY

Channel geometry is described using bankfull parameters. A Chezy resistance relation with constant Cz = Cf

-1/2) is applied to gradually varied flow. Where Ubf denotes flow velocity at bankfull flow, the bed shear stress is given as (slide 34 of lecture on hydraulics)

The sand in the streambed is characterized with a single grain size D. A constant channel-forming Shields number is assumed; recalling Qbf = UbfBbfHbf,

2bffb UC

form2

bf2bf

2bff

2bffb

bf HRgDB

QC

RgD

UC

RgDwhere from slide 25 of the lecture on hydraulic geometry form

* ~ 2.16. The sand transport relation is that of Engelund and Hansen (1967); reducing the relation of slide 11 of the lecture on hydraulics with Qs = Bbfqt = sand load at bankfull flow,

2/5form

fbfs )(

C

05.0DRgDBQ




Karlsruhe, Germany, June 12-23, 2006IS THE ASSUMPTION OF CONSTANT CHANNEL-FORMING SHIELDS

NUMBER REASONABLE?

The diagram below is from the lecture on bankfull hydraulic geometry

y = 3.2613x-0.0198

y = 0.2801x0.1527

y = 2.1332x-0.3567

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

Qhat

Bti

l, ta

us,

S

WidthSlopetaustarav taustarPower (taustar)Power (Width)Power (Slope)

Q

S,,B~

bf

B~

bf

S

y = 3.2613x-0.0198

y = 0.2801x0.1527

y = 2.1332x-0.3567

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

Qhat

Bti

l, ta

us,

S

WidthSlopetaustarav taustarPower (taustar)Power (Width)Power (Slope)

Q

S,,B~

bf

B~

bf

S

18.2bf




Karlsruhe, Germany, June 12-23, 2006BACKWATER AT BANKFULL FLOW IN A SELF-FORMED CHANNEL

The full backwater equation is (slides 34 and 25 of the lecture on hydraulics)

A constant channel-forming Shields number form* implies a bankfull velocity

Ubf that is constant in the downstream direction:

The backwater equation applied to bankfull flow thus reduces to

2fb

b22

UC;x

gHx

Hg

2

1

x

HU

formbf RgDU

bf

formfbf2bf H

RD

xx

H)1(

Fr

Where Frbf = bankfull Froude number. In low-slope sand-bed streams, Frbf2

is usually < 0.1, and can be neglected, yielding

bf

formfbf

H

RD

xx

H




Karlsruhe, Germany, June 12-23, 2006ARE THE ASSUMPTIONS OF CONSTANT BANKFULL FLOW VELOCITY

AND Frbf2 << 1 REASONABLE?

RgD

UU bf

1

10

100

1.E+10 1.E+11 1.E+12 1.E+13

Qhat

Uh

at

UhatAverage

U

Q

0.01

0.1

1

0.00001 0.0001 0.001 0.01

S

Fr b

f2

U

We use the data of the lecture on bankfull hydraulic geometry, and define

Constant not too bad! Frbf2 << 1 generally good!




Karlsruhe, Germany, June 12-23, 2006BACKWATER CALCULATION FOR BANKFULL DEPTH Hbf

It is useful to change the backwater calculation from down-channel coordinates r and x, respectively on the fan-delta and fluvial reach to down-fan-delta and down-valley coordinates rf and xv, where

The backwater equations for the respective reaches become

dd

fv r

r,

x

x

The calculation is performed upstream from the location of the topset-foreset break, located at r = rd(t). The elevation of standing water d(t), which may change in time to include sea level rise, must be specified at r = rd. In addition, the water surface elevation must be continuous at the junction between the fan-delta and fluvial reachs. This leads to the conditions

bf

formf

dbf

bf

formd

dbf

H

RD

xx

H,

H

RD

rr

H

fud Lxbffrrbfddrrbfd HH,)t(H





Once Hbf is calculated everywhere, the relations below

can be reduced to the following forms for the sand load Qs at bankfull flow and bankfull width:

CALCULATION OF SAND LOAD, BANKFULL WIDTH

2/5

form

1

bf

form

fbfs D

HCQ05.0Q

2/5

form

sbf

DRgDD05.0

QB

form2bf

2bf

2bff

HRgDB

QC 2/5form

fbfs )(

C

05.0DRgDBQ





Upstream boundary condition: specified feed rate of sand:

TRANSLATION OF EXNER EQUATIONS FROM DOWN-CHANNEL COORDINATES TO DOWN-VALLEY COORDINATES

dd

fv r

r,

x

x

v

sffff

ffpf x

QI)1(

tB)1(

f

sfddd

dfpd r

QI)1(

tr)1(

Continuity condition at junction between fluvial and fan-delta reachs:

)t(QQ sfeed0xsv

uffv rrsLxs QQ

Bf

Bfc

Bdc

Bd = r

Lf

rd

xv

ru

rf




Karlsruhe, Germany, June 12-23, 2006The topset-foreset break, which serves as a surrogate for shoreline, is located at rf = rd(t).

The foreset-basement break is located at rf = rb(t). Both of these boundaries migrate outward as the delta progrades.

The basement over which the delta progrades is assumed to have constant slope slope Sb in the analysis described here.

The slope of avalanche Sa of the delta face is also assumed to be a prescribed constant here.

CONDITIONS AT THE DELTA FACE

Bd = r

rd

rfBdc

ru

rb

topset

foreset

rb

foreset

shelf floor

d

rd

d

Sa

fan-delta elevation profile

Sb

topset




Karlsruhe, Germany, June 12-23, 2006Let fore(r) denote elevation profile on the foreset. The slope profile over the foreset is given as

Now then

where

SHOCK CONDITION AT THE DELTA FACE

Bd = r

rd

rfBdc

ru

rb

topset

foreset

rb

foreset

shelf floor

d

rd

d

Sa


Sb

topset

)]t(rr[S]t),t(r[ dfaddfore

dddar

dfore r)SS(tdt

d

d

rateonprogradatidt

drr dd

df rrf

ddd r

S

fan-delta slope at topset-foreset break




Karlsruhe, Germany, June 12-23, 2006Now assuming spatially constant subsidence rate d and vanishing sand transport at the base of the foreset (Qs = 0 at r = rb) integrate Exner on the foreset from rd to rb,

and reduce with the relations of the previous page to get the following relation for shoreline progradation speed

SHOCK CONDITION AT THE DELTA FACE

Bd = r

rd

rfBdc

ru

rb

topset

foreset

rb

foreset

shelf floor

d

rd

d

Sa


Sb

topset

b

d

b

d

r

r ff

sfdd

f

r

r dfore

fpd

drr

QI)1(

drt

r)1(

d

d

rrspd

dd

dddadr

d2d

2b

Q)1(

)1(

r)SS(t

)rr(2

1




Karlsruhe, Germany, June 12-23, 2006Elevation must be continuous at the foreset-basement break. Here the basement is taken to be horizontal and also subsiding at rate b. The continuity condition is

Taking the derivative of both sides of the above relation with respect to t and noting that - b/rf = Sb and b/t = -d, it is found that

ELEVATION CONTINUITY AT THE FORESET-BASEMENT BREAK

Bd = r

rd

rfBdc

ru

rb

topset

foreset

rb

foreset

shelf floor

d

rd

d

Sa


Sb

topset

)]t(r)t(r[S]t),t(r[)t,x( dbaddb

dr

ddddabba

dt

r)SS(r)SS(




Karlsruhe, Germany, June 12-23, 2006The relation governing the progradation rate of the topset-foreset break (shoreline) is:

The relation governing the progradation rate of the foreset-basement break is

SUMMARY OF MOVING BOUNDARIES

Bd = r

rd

rfBdc

ru

rb

topset

foreset

rb

foreset

shelf floor

d

rd

d

Sa


Sb

topset

dr

ddddabba

dt

r)SS(r)SS(

d

d

rrspd

dd

dddadr

d2d

2b

Q)1(

)1(

r)SS(t

)rr(2

1

dr

br




Karlsruhe, Germany, June 12-23, 2006FIRST APPLICATION OF THE MODEL TO THE WAX LAKE DELTA

v

s

f

ffff

fpf x

Q

B

)1(I

t)1(

r

Q

r

)1(I

t)1( sdfd

ddd

pd

The model was solved using a fixed grid for the fluvial reach and a moving-boundary coordinate system for the fan-delta reach.




Karlsruhe, Germany, June 12-23, 2006DATA COLLECTION FOR THE MODEL WAS DONE BY 16 GRADUATE STUDENTS

in Parker’s class: River Morphodynamics, spring 2006




Karlsruhe, Germany, June 12-23, 2006WE ARE DEEPLY INDEBTED TO PROF. HARRY ROBERTS OF LOUISIANA STATE UNIVERSITY FOR HIS GENEROUS SHARING OF

CRUCIAL DATA




Karlsruhe, Germany, June 12-23, 2006SOME MODEL INPUT PARAMETERS

Sand size D = 0.10 mm Bankfull discharge Qbf = 4100 m3/sFlood intermittency If = 0.27Sand input rate Gtbfu = 6.6 Mt/year,Channel-forming Shields number bf* = 1.86Length of fluvial reach Lf = 25,000 mFloodplain width of fluvial reach Bf = 800 mFan-delta angle = 120Foreset slope Sa = 0.002Basement slope Sb = 0.00018Dimensionless Chezy resistance coef, Cz = 20Deposit porosity pf = pd= 0.6Channel sinuosity f = d = 1Fraction mud deposited per unit sand f

= d = 0.49Fan-delta subsidence rate d = 5.8 mm/yearFluvial subsidence rate f = 0Rate of sea level rise = 1.2 mm/year

d




Karlsruhe, Germany, June 12-23, 2006IT WORKS WITH A MINIMUM OF TUNING!

WE CAN MODEL LAND-BUILDING IN THE MISSISSIPPI DELTA!

Position of delta front:simulation and data

Delta area:simulation and data




Karlsruhe, Germany, June 12-23, 2006PROJECTION FOR WAX LAKE DELTA FRONT TO YEAR 2081




Karlsruhe, Germany, June 12-23, 2006BUT THE CODE IS MISSING KEY ELEMENTS

Peat’s coffee

humicmud

clay

silt

Pleistocene basememt

new sedimentbearing down on

old sediment

Peat’s coffee

humicmud

clay

silt



old sediment

Peat’s coffee

humicmud

clay

silt



old sediment

• Link to subsurface stratigraphy

• Link between sediment deposition and vegetation

• Link to coastal sediment processes




Karlsruhe, Germany, June 12-23, 2006AT PRESENT OUR PROJECT IS SMALL, BUT

WE PLAN TO EXPAND IT

• Non-NCED resources: 1. potential NSF subsurface effort2. Louisiana DNR Grant to E. Meselhe and others (with NCED cooperation)3. more possibilities with ExxonMobil4. eventual cooperation with more ecologists, coastal scientists, meteorologists, economists and (ultimately) politicians.

• NCED resources: time of 4 PI’s + one dedicated postdoctoral researcher + one or more graduate students.




Karlsruhe, Germany, June 12-23, 2006OUR ULTIMATE GOAL:

Provide a Predictive Scientific Basis for

Controlled Diversions to Rebuild The Delta

Should the diversion be upstream or downstream of New Orleans?

For example, can we rebuild Barataria Bay? How long will it take?

And how should the structure(s) be designed?




Karlsruhe, Germany, June 12-23, 2006REFERENCESA fairly complete set of references can be found in:Parker, G., Sequeiros, O. and River Morphodynamics Class of Spring, 2006, 2006, Large scale

river morphodynamics: application to the Mississippi Delta, Proceedings, River Flow 2006 Conference, Lisbon, Portugal, September 6-8, 2006, Balkema.

a pdf version of which has been supplied with the material for this short course as WaxLake.pdf.

See alsoParker, G., Muto, T., Akamatsu, Y. Dietrich, W. E. and Lauer, J. W. (submitted May, 2006),

Unraveling the conundrum of river response to rising sea level from laboratory to field. Part I. Laboratory experiments, Sedimentology.

Parker, G., Muto, T., Akamatsu, Y. Dietrich, W. E. and Lauer, J. W. (submitted May, 2006), Unraveling the conundrum of river response to rising sea level from laboratory to field. Part II. The Fly-Strickland River System, Papua New Guinea, Sedimentology.

Preprints of both of these papers can be downloaded ashttp://cee.uiuc.edu/people/parkerg/preprints.htm

the mississippi delta is sinking into the gulf of mexico

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