Oral presentation - Universität Kassel sand pack from (Pre-Experiment 2) Calculate the tank...

Tank Experiments and Numerical Modeling of Macrodispersion of Density-Dependent Transport in

Stochastically Heterogeneous Porous Media

Mehran Iranpour Mobarakeh

and

Manfred Koch

Oral presentationInterPore 2017

Overview

Introduction and Background

Research Objectives

Literature Review

Experimental Set-up

SUTRA (deterministic) Analysis

Results and Discussion

Conclusions and Future works

Introduction

kjhb

Seawater intrusion

Saltwater upconing

Groundwater Contaminant

plume

What is porous media?Solute transport in the absence of

sorption and decay:

Diffusion

Advection

Mechanical Dispersion

Hydrodynamic dispersion

Equations of Groundwater Flow:

Darcy’s Law: � = −�

�� ⋅(�� − ��)

The law of mass conservation or continuity principle:�

��(��)= −� ⋅ �� + ��

Flow equation: ��

��= � ⋅

�

��. �� − �� + ��

Equations of Solute Transport:

Solute transport: �(��)

�� = � ⋅(�� )− � ⋅ ��

Equation of state: � = ��∗

Dependency of transverse dispersion:

Dependency on fluid velocity by Peclet number

�� =��

�� = 0.28(��)�.��

�

Dependency on variable density

Dependency on heterogeneity

Dependency on scale

Literature Review

Investigator Year Research Work

Henry 1964

resolving the difficulties of modeling the

narrow transition zone between outflowing

fresh water and inflowing salt water

Kobus and Spitz 1985

Studying the quantitative evaluation of the

effects of permeability heterogeneity

interacting with density coupling

Schincariol and Schwartz 1990

illustrating the combined effect of

heterogeneity and solute concentration on

the vertical transvers e dispersion of a

continuously injected plume sinking in a

model aquifer


Welty and Gelhar 1991, 1992

Finding that downward solute movement

due to gravity can enhance effective

vertical mixing or dispersion of a solute

plume

Tchelepi 1994

Finding that for fluids which also contrast

in density as well as viscosity, the tendency

of the overriding of the lighter fluid above

the less dense displaced one is damped by

permeability heterogeneity

Gleick et al. 2000

Investigating that global warming can be

expected to exacerbate seawater intrusion

into freshwater aquifers

Starke 2005

Studying the effect of heterogeneity on

density dependent transport in tank

experiment


Schotting and

Gutierrez2008

Their investigation deals with the interplay of flow and transport

processes that affect dispersion. They found the fact that transverse

dispersivity was to be dependent on fluid velocity and density

difference contradicts the premise that dispersivity is a porous

medium constant.

Raoof and

Hassanizadeh2013

They did research about dispersivity dependence on saturation. In

this study, using a pore network model, they studied how the

transverse dispersivity varies nonlinearly with saturation. They

schematize the porous medium as a network of pore bodies and pore

throats with finite volumes to mimic the microstructure of real

porous media.

Karadimitriou et

al.2014

They did simultaneous thermal and optical imaging of two-phase

flow in a micro-model of the aquifer. They found that thermal

effects increase the dispersivity.

Experimental Set-up

Tank dimensions� = �.� � length, � = �.� � width and � = �.�� hieght

Outflow Chamber ( Boundary Condition)

Schematic view of the Tank

Grain size distribution curveStochastic packing of the tank: Grain size distribution curve for grain class 8.

� = �� Hazen Method

Measured characteristics of the sands used to pack the tank:Class �0(�� ) �100(�� ) �10 (��) �60(��) �50(��) �� (� �⁄ ) Hazen

2G 2 14 3.6049 6.2381 5.7224 1.7 1.0 1.5 × 10−1

2F 1 9 1.9662 4.0210 3.5787 2.0 1.0 4.5 × 10−2

3Old 0.5 5 0.9237 1.8203 1.6112 2.0 1.0 9.9 × 10−2

3New 0.6 6 1.7331 3.0530 2.7905 1.8 1.0 3.5 × 10−2

5G 0.6 6 1.2455 2.7392 2.4435 2.2 1.0 1.8 × 10−2

5F 0.5 4 0.8909 1.5638 1.4217 1.8 1.0 9.2 × 10−3

7 0.3 3 0.4678 0.8980 0.7979 1.9 1.0 2.5 × 10−3

8 0.16 2 0.2891 0.6539 0.5764 2.3 1.0 9.7 × 10−4

9H 0.06 09 0.1402 0.3444 0.3080 2.5 1.2 2.3 × 10−4

GEBA 0.04 0.0865 0.087 0.1797 0.1615 2.1 1.0 8.7 × 10−5

Hydraulic conductivityDetermine hydraulic conductivity of the

sands and TBM analysis to fill the tankfrom Pre-Experiment1

The modified permeameter forDarcy experiment-constant head.

�� =��

�Δh�

Measured Hydraulic Conductivity of different Sand classes:

Sand classColor coding

��Hydraulic

conductivity (�)�� (�)

2G gray 5.7 0.111 -2.19823

2F gold 3.58 0.075 -2.59027

3-New white 2.8 0.050 -2.99573

5G yellow 2.44 0.0300 -3.50656

3-Old orange 1.6 0.017 -4.07454

5F magenta 1.42 0.0120 -4.42285

7 red 0.8 0.0048 -5.33914

8 green 0.58 0.00143 -6.55269

9H blue 0.31 0.00046 -7.68428

GEBA black 0.16 0.000130 -8.94798

Hydraulic conductivity

Coarse

Fine

Turning Band Method (TBM)

Histogram of TBM blocks 392*98

Histogram of downscaled version of 392 × 98 blocks into 49 × 49 blocks

Calibrating the concentration and the electrical conductivity of two differentkinds of salt used in the experiment

NaCl concentration calibration

Breakthrough curve

�� = 1000 ppm

�ℎ = 80 ��

� =��

��% =

9.8�

129600 �= 6.48 ⋅

�

��

Breakthrough curve�� =

�/�

�=

2.63 ⋅10��

7.5 ⋅10��= 0.35

� = 3.22 ⋅10��

Piston flow:without molecular diffusionand mechanical dispersion

A = 0.1225

Calculate the tank effective porosity

From the breakthrough curve Pre-Experiment2

Experimental Methodology

Determine hydraulic conductivity of the sand classes (Pre-Experiment1) and TBM analysis to fill the tank and calculating hydraulic conductivity of

the sand packfrom (Pre-Experiment 2)

Calculate the tank effective porosity

From the breakthrough curve Pre-Experiment3

Running 12 main Experiments to

measure the microdispersion

Plot outlet breakthrough

curveFrom Experiment

Estimation of steady-state

conditionFrom SUTRA

Plot the Concentration profile by sampling from different observation ports From

Experiment after steady-state

Calculate the dispersion width From Experimentally

Plotted Concentration Profile of 9 columns

Plot the variance (proportional to dispersion

width power two) versus tank length and getting a linear regression From

Experiment

Calculate transverse macrodispersivity From

solute plume spatial moments (Garabedian et

al. 1991) from Experimental Results

�� (��) �� Start date End date

250 4 14 18.01.2016 03.02.2016

250 8 40 31.1 04.05.2015 13.05.2015

250 Quality Control 12 51.87 11.12.2016 18.12.2016

250 12 43.36 08.02.2016 17.02.2016

5000 4 15.57 05.11.2015 19.11.2015

5000 8 31.81 24.11.2015 03.12.2015

5000 Quality Control 8 30.46 20.10.2016 31.10.2016

5000 12 74.48 26.02.2016 04.03.2016

20000 4 7.8 30.05.2016 12.06.2016

20000 8 29.46 09.05.2016 21.05.2016

20000 Quality Control 8 30.8 14.11.2016 25.11.2016

20000 12 45.26 24.04.2016 01.05.2016

35000 4 18.56 19.06.2016 06.07.2016

35000 8 34.13 21.07.2016 01.08.2016

35000 12 52.38 30.08.2016 06.09.2016


Varing parameters are:

Injected saltwater concentration

Seepage velocity of the fresh and saltwater

Start macrodispersionExperiment by

running twelve cases

12 E

xperim

ents +

3 E

xperim

ents are Q

uality

Co

ntrol


Plot outlet breakthrough

curveFrom Experiment

y = 3E-08x3 - 4E-05x2 + 0.0112xR² = 0.9158

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280

C/C

_exp

ecte

d [

]

Elapsed Time [hr]

Outlet Breakthrough Curve

�� = � × �� × ��

�� = 4�

�→ �� = 0.061

�

��

�� = 8�

�→ �� = 0.122

�

��

�� = 12�

�→ �� = 0182

�

��

�� = � × �� × ��

�� = 4�

�→ �� = 0.056

�

��

�� = 8�

�→ �� = 0.112

�

��

�� = 12�

�→ �� = 0168

�

��

�� = 5000 ppm

u = 8 �/��Quality control experiment


�� = 5000 ppm


Estimation of steady-state

conditionFrom SUTRA


�� = 5000 ppm


Plot the Concentration profile by sampling from different

observation ports From Experiment


�� = 5000 ppm


Calculate the dispersion width From Experimentally

Plotted Concentration Profile

�� is the variance of the mixing width

� is the tank length at the observationpoint

� is the flow velocity

�� =�

�⋅

��

�⋅�

�� =�

�⋅

��

�

Transverse dispersion can be determined as:


�� = 5000 ppm


Plot the variance (proportional to dispersion

width power two) versus tank length and getting a linear regression From

Experiment

Calculate transverse macrodispersivity

From Linear Regression of

Experimental Results

�� =�

�⋅

��

��= �� = 0.182389 (mm)

Linear Regression

Approved


Effects of inflow velocities on the plume plateaus for the various experiments with different source-concentration values at observation ports based on experimentally measured data.

Experimental ResultsWidth of the dispersion zone (variance of the normalized concentration distribution):


Sudden drop in the middle of the tank

Sudden drop in the middle of the tank

Conclusions The transverse macrodispersivity was found to be proportional to the variance,

correlation length and anisotropy of the permeability distribution of the aquifer.

In the case of large molecular Peclet numbers, the diffusion is negligible, withlarge heterogeneity of the porous medium, as this work, the diffusion isnegligible. However, if the heterogeneity and the number of Peclet are small, thediffusion must be taken into account for the calculation.

The pressure distribution at the tank boundaries is a very sensitive parameter inthis model. Even slight deviations of the experimental conditions against thetheoretically determined pressure boundary conditions cause great changes in theflow and transport behavior of the system. The correct consideration andmaintenance of correct pressure boundary conditions is therefore indispensablefor an exact simulation of real processes.

Future works Experiments with variable granular structure dependent dispersivities �� and ��,

mean granular structure �� as well as attempts to study the homogeneous mediawhere influence of density is independent of the heterogeneity are necessary.

Because of the high pressure sensitivity at the boundaries of the model, anecessity of a continuous recording of the pressure levels using pressure probesis seen, which should be logged automatically.

In order to get the transport behavior data consistently everywhere in the tankand to prevent systematic errors, it’s reasonable to record the spatial andtemporal concentration process of water constituent (solute) with photometricand digital methods.

Thank you Vielen Dank

سپاسگزارم

Dependency of transverse dispersion:

Dependency on scaleHorizontal dispersivity Transverse dispersivity

Horizontal

Vertical

Longitudinal

Models of dispersion:

First order analysis

Perturbation analysis

Monte-Carlo analysis

This method was used in this work with Stochastic description ofhydraulic conductivity variation

comprehensive approach touncertainty propagation

deterministic solutions forthe numerically generatedrealizations

Piezometer boards

Two different Piezometerboards employed in differentstages of the experiment inorder to keep track of changesin the pressure inside andoutside of the tank.

Inlet Flowmeters

Flowmeters employed to track the flowrates of salt and freshwater at alltimes during the experiment to register an identical flowrate for both thefresh and salt water. The model of both flowmeters is Proline Promag 53,which operates based on electromagnetic fields.

Saltwater flowmeter

Freshwater flowmeter

NaCl concentration calibration

Digital conductivity meter(LF 3000) employed in orderto measure the conductivityof the fluid at each samplingport.

Constant head method

Modified

The modified permeameter forDarcy experiment-constant head.

Experimental Tank Hydraulic Conductivity

�ℎ (��) �ℎ ∗ �/�� (

��

�) �(

�

�) �(

�

�) �� (

�

�) �� (

�

��)

5 6.250E-04 6.015E-06 9.624E-03 4.910E-05 1.228E-04 10.6064

10 1.250E-03 1.171E-05 9.371E-03 9.562E-05 2.391E-04 20.6544

12 1.500E-03 1.412E-05 9.412E-03 1.153E-04 2.881E-04 24.8951

14 1.750E-03 1.658E-05 9.474E-03 1.353E-04 3.384E-04 29.2345

16 2.000E-03 1.899E-05 9.493E-03 1.550E-04 3.875E-04 33.4797

Determination of the hydraulic conductivity of sand-pack:

� =� ⋅��

� ∙ �ℎ0

0.000002

0.000004

0.000006

0.000008

0.00001

0.000012

0.000014

0.000016

0.000018

0.00002

0.0005 0.0007 0.0009 0.0011 0.0013 0.0015 0.0017 0.0019 0.0021

Q (

m3

/day

)

dh*A/dl

A*dh/dl-Q

Plot of Discharge vs. Hydraulic Gradient:

Theoretical method to compute k equivalent value:

�� (��) = 0.00948 (�

��

�� (��) = 0.01 (�

��

�� (��) =� �� (��)

Theoretical Tank Hydraulic Conductivity

�� [��]� [

�

��] �� = 0.5

��

��(experiment)

[mm]

ℎ� [�]

250 4 3.83416 0.80460739

250 8 1.428085 0.863336192

250 12 1.540404 0.8410545

5000 4 0.089688 0.895049979

5000 8 0.182389 0.836535124

5000 12 0.487421 0.738464787

20000 4 0.676848 0.878819497

20000 8 0.311913 0.83633463

20000 12 0.049672 0.859862721

35000 4 2.890137 0.793007219

35000 8 0.508847 0.832523

35000 12 0.51401 0.813182843


Numerical Analysis

Setting the right boundary condition (uncalibrated model):

Name a) b) c) d) e) f) g) h)

Time step 50 50 1500 1500 50 50 1500 1500

�[��] 250 250 250 250 35000 35000 35000 35000

BC_right��

= �

��

= ��

��

= �

��

= ��

��

= �

��

= ��

��

= �

��

= ��

�� = �� × 0.5 = ��

�� = 0 Freshwater

solute

SUTRA Analysis

Monte-Carlo simulation

Sand Pack�

= ��(�� )��

� ��[�] ��[�]

MC-Sandpack1 -5.6757 0.25 0.2 0.050

MC-Sandpack2 -5.6757 1.00 0.4 0.100

MC-Sandpack3 -5.6757 1.50 0.3 0.075

MC-Sandpack4

(real)-5.6757 3 0.75 0.100

The characteristics of the sandpacks aregoing to be much more heterogeneous bymoving from sandpack 1 to the sandpack4 (real case)

Monte-Carlo simulation

Oral presentation - Universität Kassel sand pack from (Pre-Experiment 2) Calculate the tank...

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