fachgebiet geohydraulik und ingenieurhydrologie fachbereich bauingenieurwesen prof. dr. rer. nat....

LABORATORY EXPERIMENTAL INVESTIGATIONS AND NUMERICAL (MONTE

CARLO)SIMULATIONS TO VALIDATE A THEORY OF HYDRODYNAMIC MACRODISPERSION IN

DENSITY-DEPENDENT TRANSPORT PROCESSES OF GROUNDWATER FLOW IN

STOCHASTICALLY STRONGLY HETEROGENEOUS POROUS MEDIA

(MODEL AQUIFER)

FachgebietGeohydraulik und IngenieurhydrologieFachbereich BauingenieurwesenProf. Dr. rer. nat. Manfred Koch

Department of Geo-hydraulics and Engineering HydrologyUniversity of Kassel

PhD Student and Research Assistant: Mehran Iranpourand

Supervisor: Prof. Dr. Manfred Koch

Transition Zone:

• Relative Densities of sea water•Tides•Pumping wells•The rate of ground water recharge•Hydraulic characteristics of the aquifer

Sea Water intrusion

Seawater intrusion is a process that occurs in virtually all coastal aquifers, where they are in

hydraulic continuity with seawaterSalt water intrusion happens when salt water is drawn in to fresh water aquifers. The behavior is caused because sea water has a higher density

than fresh water. This difference in density causes the pressure under a column of the same

height of fresh water . If these 2 columns are connected at the bottom then the pressure difference would cause a flow of saltwater column to the freshwater until the pressure

equalize.

http://en.wikipedia.org/wiki/Coast

Experimental InvestigationModel Configuration

Liinear regressiion::AT = 0,5·d/dx

𝛼∁

Model design

Mean, Variance, Correlations

Q

QInitial ConcentrationSpecified Pressure( Boundry Conditions)p ( z) = rh (c = 0 ) * g * zMesh Structure(392*98)Time stepsEach element: 2.5 *1.25 cm

Permeability K of the present tank packing with the positions of six sampling ports indicated

Sieve Analysis: Grain Curve of Sand GEBA

Sieve Analysis: Grain Curve of Sand 5G

Sieve Analysis: Grain Curve of Sand 2G


Klightest Density

q = - K grad h

Generalization of Darcy’s column

h/L = hydraulic gradient

q = Q/A

Q is proportionalto h/L

-In 1856, Henry Darcy studied the movement of water through porous material. He determined an equation that described groundwater flow. The following description tell how Darcy determined his equation:-It may be noted that this velocity is not quite the same as the velocity of water -flowing through an open pipe. In an open pipe, the entire cross section of the pipe conveys water. On the other hand, if the pipe is filed with a porous material, say sand, then the water can only flow through the pores of the sand particles. -Hence, the velocity obtained by the above expression is only an apparent velocity, with the actual velocity of the fluid particles through the voids of the porous material is many time more.

Din Standard

How to generate numbers with TBM (Turning Band Method)Batchfile for running all programs to generate a sandpackUser’s Guide for Turning Band CodePack generation to illustration in SURFER and pack generation for Excel-data with PACKBILD-VALUESa.turn2d.exeCopy turn2d.exe & turn2d1.dat & turn2d2.dat in a new folder.The file turn2d.exe needs two input files named turn2d1.dat & turn2d2.dat which have been written by Editor/WordPad/Notepad, so we need also put these two in the folder.In the file turn2d1.dat we can change mean conductivity and standard deviation. For example, mean of conductivity is between 0, 0016 and 0, 0036, and standard deviation is between 1 to 3. For example, we can assign 49 to “nx”, etc.

070802,1 idate,irun …currect date; run numbers49,49,1 nx,ny,nz …numbers of points in the x,y,z direction0.204, 0.0255, 0. dx,dy,dz …spacings in the x,y,z direction0.5,0.02,0. xl1,xl2,xl3 …correlation lengths (ξp is, intuitively, "some average distance of two sites belonging to the same cluster" (Stauffer and Aharony 1991, p.60), and it "is proportional to a typical cluster diameter" (ibid., p. 22). Stauffer and Aharony (1991, p.60) define it as ξ2 = Σr r2 g(r) / Σr g(r)). 1.6E-3,1.23 condg,sig …geometric mean Kg; standard deviation f---copy to above---

for 392x98070802,1 idate,irun392,98,1 nx,ny,nz0.025, 0.0125, 0. dx,dy,dz 0.5,0.02,0. xl1,xl2,xl3 1.6E-3,1.23 condg,sig

for 49x49070802,1 idate,irun49,49,1 nx,ny,nz0.204, 0.0255, 0. dx,dy,dz 0.5,0.02,0. xl1,xl2,xl3 1.6E-3,1.23 condg,sigGeometric mean:In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root (where n is the count of numbers) of the product of the numbers.For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is \sqrt{2\cdot 8}=4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is \sqrt[3]{4\cdot 1\cdot 1/32}=1/2

In the turn2d2.dat we can change value of seed between 48,000 to 52,000.(“iu” means “seed”)300,.025,500 nline, …numbers of lines to be used delzet, …physical spacing along each line nztest …equal to the maximum numbers of points simulated on any line50.,0.2 bigk,dk …maximum normalized frequency1,31345,12 nmont,iu,nl …numbers of Monte Caulo simulation to perbru …arbritary (willkürice) odd …iutagel; prime uniform ramdom numbers generator iu for fct. RAND≤220

-3.,3.,24 cmin,cmax, ...minimum& maximum values used iu computing Ndelt … the range of values cmin-cmax1,1,0 ks(1),ks(2),ks(3) ...sample euseuhe & single heplicate statistics for the N(0,1); Held are computed iu for j-ru direction1 ilog …=1; for N(0,1) held iu transforward according to log usualno file nfile …results of au Moute Caulo runs will be written “nfile=no file” nothing is writenlogcond.dat tfile …print out file When we run turn2d.exe, 1: "lnk.dat" for gam2 and 2: "perm.dat" for SUTRA, 3: logcon.dat (= real conduct.) 4.turn.log will be generated.Turn2d.exe calculated hydraulic conductivity for 392x98 Matrix for SUTRA.And recalculate hydraulic conductivity Kf to permeability K in SUTRA.

compute auto-correlation of TBM output with gam2parameter file for gam2

b.tbmtogeom.exe

If we run tbmtogeom.exe, 392x98 Matrix converts to 49x49 Matrix, and calculate mean of Kf.It generate file out_geom.grd.

How to correct TBM

1. Copy the original sources to another folder. 2. Check the routes of files “Correlation.out”,”Correlation.R”. Delete or put” #” before the

sentence to make it workless. 3. Check the file “Rscript_Correlation”. Make sure it’s the correct route to your software R.

"C:\Program Files\R\R-3.0.1\bin\R.exe“ CMD BATCH Correlation.R 4. Run “Packgeneration_New” 5. Copy three “lvl” files to this folder.

How to do with SURFER

1. Open SURFER 2. Map-Contour map-New contour map… 3. Open out_link_sand – Fill Contours- Levels –Load…- open -ok 4.

SUTRA model grid:* set up to mimic the x-z cross-sectional geometry of the tank.* each sand block was represented by 8 x 4 elements, resulting in a total of nx * ny = 392 x 98 = 38416 elements.

Numerical simulationsDescription and setup of numerical model

Boundary conditions:

Previously:outflow BC for p was set to be equal to the initially set outflow fresh water head (BC0), giving rise to some discrepancy in the experimental and modeled plume positions

Presently:use of Dirichlet BC for p that is more consistent with the pressure distribution of the homogeneously mixed fresh and saltwater in the outflow chamber at steady state conditions using the mixing concentration Cmix ~ 0.5* C0 at the saltwater inlet so that this pressure BC (denoted as BC1) at the outflow boundary varies with depth z asp_BC = (Cmix) *g *z

Numerical simulationsResults of deterministic models and

comparison with experiments

SUTRA-modeled concentration distributions using pressure boundary conditions BC1 (mixed saltwater head) at the outflow boundary for tracer case and C0=100000 ppm. Note the backwater effects at the outflow boundary for the high concentration case

• statistical deviation of the migration of a dissolved component in with respect to an average value• caused by microscopic velocity variations on the pore-scale• impossible to survey microscopic flow processes therefore dispersion is visualized by variations in the concentrations• macroscopic feature to describe microscopic flow processes

Basic Concepts of DispersionWhat is Dispersion?

Diffusion: process where a constituent moves from a higher concentration to a lower concentration, fick equation.Dispersion: mixing caused by physical processesThe hetrogeneity of porous media creats groundwater velocity fields that are highly complex at the pores . The hetogenity creat a varaince in the ground water velocity around the avarage linear velocity. In fact mechanical disperson is indirect transport process. Combined effects of mechanical dispersion and molecular is hydrodynamic dispersion.

time

Properties of transport equation:

2D, SATURATED POROUS MEDIA

MODELING PROCESS

Conceptual Model (Model Geometry, Boundaries,…) Mathematical Model Numerical Model Code Verification Model Validation Model Calibration Model Application Analysis of uncertainty and stochastic modeling Summery, conclusion and reporting

Bc: Drichlet , Neumann, cauchy Solutions of math models are analytical method, Numerical Method(finit eelement , finite difference)

SET OF ASSUMPTIONS model Calibration: changing values of model input parameters to match field condition.Sensitivity Analysis: a process of varying model input parameter over a reasonable range and observing the relative change in model response.

Thank you Vielen Dank سپاسگزارم



Tank Experiments and numerical Modeling of Macrodispersion of Density-dependent Transport in stochastically heterogeneous Media

PhD Candidate and Research Assistant: Mehran Iranpour

Supervisor: Prof. Dr. Manfred Koch

Aqueous phase liquid containments (APLCs) → Soluble in water → Transport and Propagation through


1. Saltwater attack to coastal aquifers (seawater intrusion) Bear et al (1999);

2. Saltwater upconing in formation aquifers Voss and Koch (2001);

3. Vertical seepage of brackish water from open ocean canals Koch and Zhang (1998);

4. Movement of brine solutions in salt domes that have been targeted as possible nuclear waste repositories Herbert et al (1988);

5. Infiltration of dense aqueous (miscible) or non-aqueous (immiscible) phase liquids Kimmel and Braids (1980).

Some typical examples:

, Hydrodynamically stable configuration: ↑ Dispersion ↓ Unstable layering ( denser fluid above): ↑ Dispersion ↑ Pc 1 Density affects on dispersion Tank of Kobus & Spiz(1985): ↑ ↓ , B ↓


According to the previous researchers:

Tank ExperimentTank design and experimental setup


Mean, Variance, Correlations

Q

QInitial ConcentrationSpecified Pressure( Boundry Conditions)p ( z) = rh (c = 0 ) * g * zMesh Structure(392*98)Time stepsEach element: 2.5 *1.25 cm



Digital conductivity meter:

Vial Holder

Valve

Vial

Needle

0 100 200 300 400 500 6000

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300f(x) = 0.539972802768155 xR² = 0.999870896341793

calibration curve: LF3000 ( 0 - 300 mg/L )

conductivity [µS/cm]

conc

entra

tion

[m

g/L]


Stochastic packing of the tank1. Order Sand, sieve analysis and measurement of hydraulic conductivity



Stochastic packing of the tank

Limits of validity of Darcy`s law were considered:

Re 10 : Laminar FlowRe 60 ~ 100 : Turbulent Flow 10 Re 60 : Transient Flow; flow is laminar but non-linear When the index m = 1 , the flow is linear. i (Darcy`s law)


Stochastic packing of the tank2. Generate stochastic numbers with Gaussian Distribution throw TBM



The optimum plot was obtained when we took the values of correlation lengths in x and y direction to be four times that of the block size.

49 x 49 = 2401 blocks for the whole tankThe latter is packed under water (to eliminate air bubbles)


Measurement of the hydraulic conductivity of the tank, experimentally:


0.0005 0.0007 0.0009 0.0011 0.0013 0.0015 0.0017 0.0019 0.00210

0.0000020.0000040.0000060.000008

0.000010.0000120.0000140.0000160.000018

0.00002A*dh/dl-Q

dh*A/dl

Q (m

3/da

y)

• Computation of the hydraulic conductivity of the tank (equivalent K), theoretically:


= 0.01

Thank you Vielen Dank سپاسگزارم


fachgebiet geohydraulik und ingenieurhydrologie fachbereich bauingenieurwesen prof. dr. rer. nat....

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