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MODELINGFLOW AND SOLUTE TRANSPORT

IN THE SUBSURFACEby

JACOB BEAR

SHORT COURSE ON

Copyright © 2002 by Jacob Bear, Haifa Israel. All Rights Reserved. To use, copy, modify, and distribute these documents for any purpose is prohibited, except by written permission from Jacob Bear.

Lectures presented at the Instituto de Geologia, UNAM,Mexico City, Mexico, December 6--8, 2003

Professor Emeritus, Technion—Israel Institute of Technology,Haifa, Israel

Part 2:slides 41-80

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PIEZOMETRIC HEADp

h z

= Elevation of point = Specific weight of water = Pressure in the waterp

z

Piezometric head is measured

with respect to a

DATUM LEVEL.

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CONFINED AQUIFER: bounded from above and belowby impervious formations. Piezometric surface aboveceiling of aquifer.

CLASSIFICATION OF AQUIFERS ACCORDINGTO THE PIEZOMETRIC HEAD SYSTEM

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ARTESIAN AQUIFERPortion of a confined aquifer in which thepiezometric head is above ground surface.

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PHREATIC AQUIFER: bounded from above by a phreatic surface.

PERCHED AQUIFER: local phreatic aquifer abovethe phreatic surface of a major one.

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LEAKY (PHREATIC OR CONFINED) AQUIFER: bounded from above and/or below by an aquitard.

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MULTIPLE AQUIFERS Distorted scale!!

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AQUIFERS ARE VERY THIN DOMAINS, RELATIVE TO HORIZONTAL DISTANCES OF INTEREST

CONFINED AQUIFER

Very small vertical flow component.

CONFINED AQUIFER WITH PARTIALLY PENETRATING WELLS

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Leaky-confined aquifer

Phreatic aquifer

River

Note the streamlines

Recall: distorted scale!

LOCAL vs. REGIONAL PROBLEMS.

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Flow in a vertical cross-section, with horizontal water table

Flow near inflowAnd outflow boundaries.

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MOTION EQUATIONS FO SINGLE PHASE FLOW: DARCY's LAW

Two ways for presenting Darcy’s law

The MOTION EQUATION is an APPROXIMATE FORM of the AVERAGED MOMENTUM BALANCE EQUATION FOR A FLUID ---the NAVIER STOKES EQUATIONS

(neglecting inertial effects and terms that express energy dissipation as a result of momentum transfer within the fluid).

Or….the French engineer in charge of the water system in the city of Dijon, Henry Darcy, 1856, ..…conducted experiments on sand packed filter columns, reaching EMPIRICAL CONCLUSION

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1 2h hQ KA

L

K = coefficient of proportionality called hydraulic conductivity.Q = volume of fluid per unit time passing through a column of constant cross-sectional area, A and length L.h1, h2 = elevations of inflow and exit reservoirs of the column. z = elevation of the point at which the piezometric head ismeasured, above a datum level.p, = fluid's pressure and mass density.

ph z

g

z = elevation of the point at which the piezometric head is measured.p, = fluid's pressure and mass density

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In a compressible fluid, = (p), we define HUBBERT's POTENTIAL (Hubbert, 1940):

* *( , )( )o

p

p

dph h t z

g p x

In Darcy’s law,

h1 - h2 = h = Energy loss across the column due to friction at the microscopic solid-fluid interface.

HYDRAULIC GRADIENT:

1 2 ( ) ( )h h h x h x x

L L

J =

=Flow

x x+xx

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SPECIFIC DISCHARGE:

discharge per unit time through a unit

area normal to the flow.

Qq

A

(…NOT “apparent velocity”, “Darcy velocity”, etc).

q K JSo far... flow through a finite length, L. What happens AT A POINT?

s

s+ s

s- s

s

s1

2

1

2

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s

s+ s1

2

s- s1

2

s

qs

Consider flow in a column aligned in the direction of the unit vector s. Along the column, h = h(s). Use Darcy's law for the segment:

1 1

2 2( )s s s s

s

h hq s K

s

In the limit, as s 0, we obtain:

1 1

2 2

0lim .

s s s s

s

h hdh

s ds

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Thus, AT A POINT:

; ( , ); ;s s s

dh dhq K K h h s t

ds ds s sJ J

with qs considered positive when it coincides with the positive direction of the s-axis.

FLOW TAKES PLACE FROM HIGH PIEZOMETRIC HEAD (ENERGY) TO LOW PIEZOMETRIC HEAD. NOT NECESSARILY FROM HIGH TO LOW PRESSURE.

Do not use the piezometric head when the densityvaries (by temperature and/or concentration changes).

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VELOCITY, V, is the distance traveled per unit time. f

V V

Q qV

A

(Omit averaging symbol)

EFFECTIVE POROSITY. Part of the void space is not available to fluid flow, due to dead-end pores (immobile Fluid), zone with fine grained material.

eff

qV

HYDRAULIC CONDUCTIVITY, K (dims. L/T). Can be defined as: Specific discharge per unit gradient(in 1-d flow in an isotropic porous medium). The hydraulicconductivity depends of fluid properties and void space configuration (width of passages and tortuosity).

PERMEABILITY, k (dims. L2), depends only on void space Configuration.

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g gK k k

= dynamic viscosiy

= kinematic viscosity

UNITS for HYDRAULIC CONDUCTIVITY:m/d, cm/s, ft/d, gal/d-ft2 , (SI: m/s)

UNITS for PERMEABILITY:m2, cm2, ft2 , (SI: m2)

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In petroleum Engineering:

3 21cm /s/cm 1centipoise1darcy

1atmosphere

k g hQ

L

FORMULAE FOR PERMEABILITY:Empirical and semi-empirical formulae:

2k CdC = dimensionless coefficient.d = effective grain diameter, say d10.

(and many other formulae)

Changes of permeability with time, due toCompaction by external load.Dissolution of solid.Precipitation.Clogging by fines, Biological activities...Shrinkage of clay soil.

RANGE OF VALIDITY OF DARCY’s LAW

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Experiments:

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In pressurized conduits, and channels, the (dimensionless) REYNOLDS NUMBER, Re, expresses the ratio of inertial to viscous forces acting on a fluid.Helps to distinguish between LAMINAR FLOW, at low Re and turbulent flow at higher Re.

By analogy, for flow through porous media:

Re ,qd

d = some representative (microscopic) length characterizingvoid space, e.g. d10. = kinematic viscosity of fluid (e.g.,

Darcy's law is valid as long as the Re, that indicates the magnitude of the inertial forces relative to the viscous drag ones, does not exceed a value of about 1 (but sometimes as high as 10):

Re 1.

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EXTENSIONS OF DARCY'S LAW:

To three dimensions.To compressible fluids.To inhomogeneous porous medium.To inhomogeneous porous medium.To anisotropic porous medium.

Remember: Darcy's law is a simplified form of the averaged momentum balance equation.

THREE-DIMENSIONAL FLOWFor a homogeneous isotropic porous medium,K = a constant SCALAR

( )K grad h K h qSince the specific discharge is a VECTOR:

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( )K grad h K h q

x

y

z

h

J

J J

J

axes of Cartesian coordinate system

, , ,

, ,

, , ,

x x y y z z

x y z

q K q K q K

x y z

h h h

x y z

J J J

J J J

and, , , , ,x y z x y zq q q J J J

Are components of the VECTORS q,

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Recall: The hydraulic gradient is a vector equal to the negative of the gradient vector.

h J

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COMPRESSIBLE FLUID:Here, = (p), and we use the motion equation (Darcy's law) written in terms of h* (= Hubbert’spotential):

*K h q

INHOMOGENEOUS POROUS MEDIUM: We use K = K(x,y,z) in Darcy’s Law.

ANISOTROPIC POROUS MEDIUM

If the permeability at a given point depends on direction, the porous medium at that point is said to be ANISOTROPIC.

Reasons for anisotropy: layering, shape of grains, verticalstress.

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Darcy's law for an anisotropic porous medium:

x xx x xy y xz z

y yx x yy y yz z

z zx x zy y zz z

q K K K

q K K K

q K K K

J J J

J J J

J J J

This is a linear relationship between the components of q and the components

of .

, ,x y zq q q , , ,x y zJ J J

J

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The permeability is represented by NINECOEFFICIENTS: Kxx, Kxy, Kxz,…., etc.

Kij may be interpreted as the contribution to qi by a unit of .xJComponents of the SECOND RANK TENSOR OF.

Symmetric tensor, i.e., Kij = Kji

Six distinct components.

The coefficients Kij are non-negative.

HYDRAULIC CONDUCTIVITY TENSOR

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In matrix forms, in 3-d domain:

K =

xx xy xz

yx yy yz

zx zy zz

K K K

K K K

K K KIn 2-d domain:

K =

xx xy

yx yy

K K

K K

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Other (compact) forms of Darcy"s law:Vector form:

h q K KJIndicial form: h

- i ij j ijj

q K Kx

J

subscripts i,j indicate xi, xj.

Einstein's summation convention:subscript (or superscript) repeated twice and only twicein any product or quotient of terms is summed over the entire range of values of that subscript (or superscript).

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PRINCIPAL DIRECTIONS OF A SECOND RANK TENSOR

for all and for . 0 0ij ijK i j K i j

IN AN ANISOTROPIC POROUS MEDIUM, FLOW AND GRADIENT ARE NOT CO-LINEAR!

When principal directions are used as a coordinate system: 0 0

K = 0 0

0 0

xx

yy

zz

K

K

K

0K =

0 xx

yy

K

K

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GENERALIZED DARCY LAW

By averaging the momentum balance equation for a Newtonian fluid, and introducing the simplifying assumptions:

Inertial effects are negligible relative to viscous ones.Shear stress WITHIN the fluid is negligible incomparison with the drag at the fluid-solid interface.

we obtain for saturated flow:

- s p g z

k

V - V

V, p, , = (average) velocity, pressure, density, and viscosity of the fluid, respectively.Vs = (average) velocity of the solid.z = elevation. k(x,y,z) = permeability tensor.

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r s

r

( - )

k = -

r

p g z

q V V V

q

For Stationary, non-deformable porous medium, Vs = 0:

= - p g z

k

q

When x,y,z are PRINCIPLE DIRECTIONS:

xxx

yyy

zzz

k pq = -

x

k pq = -

k pq = -

y

gz

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What about 2-d flow in an aquifer?

MODELING FLOW IN AN AQUIFER----ESSENTIALLY HORIZONTAL FLOW

USE DARCY'S LAW

- ,

- .

x x

y y

hQ KB KB

xh

Q KB KBy

J

J

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….and in vector form when K and T are tensors: :

' -

- grad - g .

Q B grad h

h rad h h

K

T T T T J

Q’= Discharge per unit width through entire aquifer thickness.h=h(x,y,t) = Average head along aquifer's thickness, B.

The product KB is called TRANSMISSIVITY.

When K=K(x,y,z),

T=T(x,y) = K(x,y,z) dz,( , )B x y

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FLOW IN A CONFINED AQUIFER.

Recall: on phreatic surface, p = 0, h = z.

h = h(x,y,z,t) = Piezometric head.H=H(x,y,t) = Elevation of water table above datum level.

- - - sin s

h Hq K K K

s s

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Introduce DUPUIT (1863) ASSUMPTION(s)

In vector notation:

' - grad - K h h Kh h QWhen bottom is at elevation = (x,y):

ESSENTIALLY HORIZONTAL FLOW IN THE AQUIFER . Equivalently: Equipotentials are vertical, Pressure distribution is hydrostatic along theoretical, Velocities are uniform along the vertical.

= - (h- ) grad h - (h- ) h.K K Q'K(h - ) plays the role of transmissivity of the phreatic aquifer. It is a tensor in an anisotropic medium.

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' ( ) const. ' ( )dH

Q KH x Q dx KH x dHdx

Boundary conditions. What about the seepage face?

When employing the Dupuit assumption, we neglect theseepage face.

By integration, we get the DUPUIT-FORCHHEIMER DISCHARGE FORMULA:

2 2-' .

2o Lh h

Q KL

Use Dupuit assumption:

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HOWEVER….non-horizontal flow:

End of part 2

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MODELINGFLOW AND SOLUTE TRANSPORT

IN THE SUBSURFACE

by

JACOB BEAR

WORKSHOP I

Part 2:slides 41-80

Copyright © 2000 by Jacob Bear, Haifa Israel. All Rights Reserved.

The Second International conference on Salt Water Intrusion and Coastal Aquifers--Monitoring, Modeling and Management