development and application of groundwater flow … and application of groundwater flow and solute...

Technische Universität Darmstadt - 30. Mai 2006

Development and Application of Groundwater Flow and Solute Transport Models

Randolf Rausch


Overview

• Groundwater Flow Modeling

• Solute Transport Modeling

• Inverse Problem in Groundwater Modeling


Groundwater Flow Modeling


Flow modeling in fractured and karstified media

Model approaches for fractured / karstified sytems

Double porosity flow models

Parameter identification


Problem: Flow in fractures and pipes


Large dynamics in head and discharge


Matrix and pipe network


Matrix and pipe network + observation wells


Interpolated head distribution


Interpolated head distribution and reality


Classification of fractured media

Mesh of fracturesand impermeablematrix

Fractures withpermeablematrix

Fractured systemwith minorkarstification

Conduit systemwith a karstifiedrock matrix

Granite Sandstone Limestone Karstified Limestone

Waste Disposal Water Supply


Possible model approaches

Single PorosityModel

Double PorosityModel

Discretemodel

Equivalentcontinuum

Two coupledfracturesystems

Twoequivalentcontinua

Fracturenetwork and continuum


Double continuum flow model

Linear exchange depends on:

• head difference• exchange coefficient

Different aquifer parameters:

• Hydraulic conductivity

• low in fissured system• high in conduit system

• Storage coefficient:

• high in fissured system• low in conduit system

Karst system Double continuum system


System of coupled flow equations:

( )

( )

)t,z,y,x(fhand)t,z,y,x(fh:Solution

hhWt

hSzhk

zyhk

yxhk

x

hhWt

hSzhk

zyhk

yxhk

x

ba

babb

bb

bzz

bbyy

bbxx

baaa

aa

azz

aayy

aaxx

==

−α−+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−α++∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

000

000


Parameter identification – interpretation of heads

Classification by standard deviation

heads from fracture system:

large standard deviationsame range as continuum b (conduit system)

heads from matrix system:

only one point with same standard deviation as continuum a (fissured system)


Double continuum model: “Stubersheimer Alb“

• steady state model calibrationusing measured head and discharge data

⇒ distribution of hydraulicconductivity

• sensitivity study for transientflow: simulate the measuredcharacteristics to investigate theexchange behavior


Transient exchange betweenfracture and matrix system

Simulation results

Continuum a: smooth curveContinuum b: large dynamics

Double continuum model showssame characteristics as measureddata

Heads

Discharge


Solute Transport Modeling


Application of solute transport models

• interpretation of concentration data

• mass balance of contaminants

• predictions of pollutant plumes

• design of pump and treat management

• planning of monitoring strategy

• risk assessment in case of waste disposals


Simulation of a contamination plume


Relation between groundwater flow and transport models


Representation of transport processes

Advection

Advection + Dispersion (+ Diffusion)

Advection + Dispersion + Adsorption

Advection + Dispersion + Adsorption + Decay

Transport in 1-D-aquifer


( ) )( inf

ccnquccD

tc

−−+−∇⋅∇=∂∂ σ

Transport equation

The temporal change of pollutant mass for every volumeelement is given by the in- / outflows from

• Dispersion

• Advection

• Reaction

• Sources / Sinks

( )cD∇⋅∇

( )uc⋅∇−

)( inf

ccnq

−−

σ


1-D transport equation

Measure for the relation advective / dispersive transport is given by the PECLET-number:

xcu

xcD

tc

∂∂

−∂∂

=∂∂

2

2

uuL

DuLP

LL

eα

==

L: typical length scale of transport problem

Pe = 0 pure dispersive transport

Pe = pure advective transport∞


Solution methods for transport equation

- Analytical solutionsSimple flow conditions / simple initial and boundary conditions, homogeneity

- Neglecting Diffusion / DispersionPath lines, travel times, concentration along path lines

- Numerical SolutionsGrid methods: FD, FV, FE

Particle-Tracking Methods: MOC, Random-Walk


Numerical solution methods

Grid Methods:Finite Differences Finite Elements

Finite Volumes

Particle-Tracking Methods:Method of Characteristics

Random-Walk-Method


Problems with grid methods

- Numerical dispersion

- Oscillations

Possible solution: grid refinement


Possible solution: adaptive grid

Adaptive gridding methods consists in dynamically refining the grid size to eliminate numerical dispersion

Example of grid refinement


Adaptive refinement: nested iteration


Typical error indicators are

• The gradient of the solution ,

• Concentration variation between neighbor elements.

The selection of elements for refinement and coarsening is basedon an error indicator.

c∇

In case of time step methods the solution of the proceeding time step is considered.

Error indicator for adaptive griding


Example of an adaptive refined grid from a simulation


The Inverse Problem in Groundwater Modeling


Direct problem

Given: parameters, boundary and initial conditions

Wanted: head / flow distribution (or concentration)

Usually parameters are not known completely!

Therefore:

Calibration (i.e. completion of parameters) using measurements of heads / flows (or concentration) is required.


Inverse problem

Given: heads / flows, (concentrations)

Wanted: parameter distribution

Problem: ill-posednessNo unique solution may existMeasurement errors make result unreliable

Ways out: Reduction of degrees of freedom and regressionIntroduction of “a priori” knowledgeJoint use of head, flow and / or concentration measurementsEstimate of uncertainty


Example for non uniqueness of the inverse problem

Q = B T (h1-h2)/L

Where:

Q: dischargeB: widthT: transmissivityh: headL: length

Identification problem of steady state calibration. Every T leads to the same head distribution, only Q varies!


Criterion for goodness of fit (maximum likelihood)

Without prior knowledge minimize

or with „a priori“ knowledge

Minimization can be done manually („trial and error“) or byautomatic methods: e.g. MARQUARDT-LEVENBERG algorithm

ikcomputed

ii

measuredim wpffppS 2

1 ))((),...,( −=∑

jj

priorjjik

computedi

i

measuredim wppwpffppS ′−+−= ∑∑ 22

1 )())((),...,(

(pj parameter, fi heads or flow, wi weights)


Concepts for parameterization of spatial structures

Reduction of degreesof freedom by:

Zonation (N zones)

Interpolation and pivotpoints


Spatial transmissivity distribution (Jurassic karst aquifer)

Frequency distributions:all data

valleys

plateau


Umm Er Radhuma aquifer system in Saudi Arabia


Geological and hydrogeological units of the Umm Er Radhuma aquifer system

I R A Q

I R A N

ARABIAN GULF

QATAR

K U W A I T

Euphrates

Tigris

BAHRAIN

U. A. E.

AS SULAYYIL

AZ ZULFI

RIYADH

BURAYDAH

HAFAR AL BATIN

AN NUYRIYAH

AL KHAFJI

AD DAMMAM

AL HUFUF

BUQAYQ

SALWAH

HARAD

LAYLA

YABRIN

AL KHARJ

AL KHUBAR

AL JUBAYL

Kilometers

0 100 20050

Legend

Umm Er Radhuma

Quaternary

Neogene

Dammam

Rus

Aruma

Tdm

Tsm

Tr

Ka

Q

Tu

Ka

Ka

Tu

Tu

Tr

Tdm

Tsm

Tsm

Q

Q

Tdm


Geological structure and stratification

Geological Cross Sections


Geological structure and tectonic features

Main Anticlinal Structures and Faults

Colored sub crop surface:

base of Umm Er Radhuma


Vertical hydraulic conductivity distribution

Rus aquitard Dammam / Neogene


Muschelkalk lithology


Time needed for the change of rocks within an aquiferdepends on rock type

Important features for the Muschelkalk are:

- the dissolution of evaporites, - and the karstification of carbonate rocks.

The processes depend on the exposition to theground surface.


Thickness reduction of the Mittlerer Muschelkalk by saltand sulphate dissolution


Consequence:

• 50 m thickness reduction within a relative short time

• Cracking of rocks• Intensification of fractures• Acceleration of karstification


What are the major processes for dissolution?

• The main factor is the exposition to the ground surface.

• The exposition to the ground surface depends on the cuesta development.


“Cuesta“ development in Baden-Württemberg

Location of cuestas and rivers during Oligocene and Late Miocene (34 – 20 Ma)



Location of cuestas and rivers during Upper Miocene(11 – 5 Ma)



Location of cuestas and rivers during Lower Pleistocene(0.8 – 1.8 Ma)


“Cuesta“ development in northern Baden-Württemberg

Cuesta development and location of Muschelkalk aquifers from Oligocene - Quarternary


Consequence

• Adjacent areas within the cuesta landscape developed successively over a long time and represent different states of karstification.


Present Muschelkalk aquifer systems

1 - 3: present areas of Muschelkalk aquifers


Development of karstification

mo: raremu: existing

mo: existing (wide areas)mm + mu: none

nonePerched aquifers

mo: very rare (sealing)mm: nonemu: existing

mo: manynoneCaves

mo + mm: rare (sealing)mu: existing

mo: manynoneAreas with no surface runoff

mo: numerous, but sealed

mm + mu: existing

mo: numerous

mm + mu: many

noneDolines, sinkholes

Salt completely dissoluted,Sulphates mostly dissoluted

Salt mostly dissoluted,Sulphates start of dissolution

nodissolution

Salinar in MittlerenMuschelkalk

OligoceneMioceneNot yetstarted

Start of karstification

321Area / Stage of Development


Palaeo-climatology

Recharge Estimation

0

10

20

30

40

50

60

10000 8000 6000 4000 2000 0

Years Before Present

Rec

harg

e [m

m/y

]

Precipitation Fluctuations

0

100

200

300

400

500

600

700

0200040006000800010000


Prec

ipita

tion

[mm

/yea

r]

BRICE 1978DIESTER-HAAS 1973

BRICE 1978

COLE 2004DIESTER-HAAS 1973VAN ZINDEREN BAKKER 1980

BRICE 1978

BRICE 1978

COLE 2004

COLE2004

VAN ZINDEREN BAKKER 1980

BRICE 1978

BRICE 1978

ISSAR2003

BRICE 1978BUTZER 1958

BARTH 1999BRICE 1978

Precipitation Fluctuations

0

100

200

300

400

500

600

700

0200040006000800010000


Prec

ipita

tion

[mm

/yea

r]

BRICE 1978DIESTER-HAAS 1973

BRICE 1978

COLE 2004DIESTER-HAAS 1973VAN ZINDEREN BAKKER 1980

BRICE 1978

BRICE 1978

COLE 2004

COLE2004

VAN ZINDEREN BAKKER 1980

BRICE 1978

BRICE 1978

ISSAR2003

BRICE 1978BUTZER 1958

BARTH 1999BRICE 1978

Temperature Fluctuations

23

24

25

26

27

28

0200040006000800010000


Tem

pera

ture

[°C

]


Isotopes information: estimation of river infiltration

Temporal distribution of d18O in river water and groundwater

Spatial distribution of Iller infiltration


Isotopes information: groundwater age

Simulated travel time to the Al Hassa oasis (mean residence time 12,000 a)

Groundwater Age Umm Er Radhuma Aquifer14C-Groundwater Age and 3H- Detection Line


Modeling under uncertainty

• worst case – best case analysis

• scenario techniques

• sensitivity analysis

• stochastic modeling


Stochastic modeling

Required: Parameter distribution, mean, variance, correlation length

Result: Mean, deviation, confidence limits

Methods: Monte Carlo Simulation

FOSM (First Order Second Moment)


Monte Carlo method

Principle:

- generation of a large number N of equally probable random realizations of the aquifer

- the ensemble of N calculated solutions (Zi) is analyzed statistically

1

)(1

2

1

−

−==

∑∑==

N

ZZ

N

ZZ

N

ii

N

ii

σ


Delineation of groundwater protection area

Sample problem:

Pumping rate:Q = -.005 m3/s

Groundwater recharge:I = 8 l/s/km2


Statistical analysis of field data


Simulation steps using Monte Carlo method



Repeated simulation of catchment areas

Unconstrained sampling

Sampling under calibration constraint



Probability distribution


Convergence of Monte Carlo method

A large number of realizations N may be necessary in order to get meaningful convergent statistics

Problem: this number is not known a priori !


Never forget:

• a good model includes important features of reality

• a model does not replace data acquisition

• a good modeler explores the uncertainty of her/his predictions

• what we really want are robust decisions

• do not overstretch a model

• a model is not reality

development and application of groundwater flow … and application of groundwater flow and solute...

Documents