Balance de Masa- Fluido en Medio Poroso

7/28/2019 Balance de Masa- Fluido en Medio Poroso

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Modelacin Matemtica de Aguas Subterrneas

Eric Morales

Mass Continuity in Deformable Media (Saturated Flow)

Lagrangian approach: Consider a reference volume at time t0 including a fixed volumeVgof incompressible soil grains (somehow we are following the bulkvolume that

encompasses exactly the same volume of grains).

At t > t0 the same grains will generally occupy a different volume * due to soil

deformation.

Dibujos de t0 y t > t0 del volumen

Fluid mass conservation (the volume Vgis conserved by definition) requires

* *

* ( ) *gD

S d dDt

q q n

where

g

D

Dt t

q = Lagrangian derivative associated with Vg

S= fluid saturation, 0 S 1

= porosity (variable)

qqg= fluid flux relative to that of the grains

Apply Gauss divergence theorem

* *

* ( ) *gD

S d dDt

q q

In the limit as * d*

* ( ) *gD

S d dDt

q q

Note that we are leaving d* inside the parenthesis since it is a time varying

infinitesimal volume, d* = dVv + dVg, although dVgis time invariant.

Define V

g

Ve

V = void ratio = volume of voids / volume of grains.

From 111

g

V g

VV V e

, it follows that1

ee

* ( ) *1

g

D eS d d

Dt e

q q

Now* * *(1 )*

*1 1 1

v g g

g

dV dV dV eddV

e e e

time constant, which leads to


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Eric Morales

1

( )1

g

DSe

Dt e

q q

Using Darcys law g h q q K (assumes velocity of the grains is not too fast,

otherwise needs to couple a motion equation for the grains, as for example in clays and

peat where grain velocity may be important) leads to

1

1

DSe h

Dt e

K

Mass accumulation term

Now we will focus on

1 1

1 1 1 1

D e D e DS DeSe S S

Dt e e Dt e Dt e Dt

[fluid compressibility] + [saturation change] + [medium compressibility]

Saturation change (imbibiti on/drainage):DS

Dt

Define *dS

Cd

specific saturation capacity

Then*DS dS D DC

Dt d Dt Dt

F lui d compressibil ity:D

Dt

The compressibility of water is defined as1

w

dc

dP

Ifcw = constant, then can integrate

0 0

''

'

P

w

P

dc dP

to get the following state equation

00

wc P Pe

UsuallyP0 is the atmospheric pressure and it is thus set to 0,P0 = 0, so 0wc P

e

Now define the pressure head as

0 0

'

( ')

P

P

dP

P

, which implies (by Leibnitzs rule)

1

( )

d

dP P


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Eric Morales

Therefore wD d dP D D

cDt dP d Dt Dt

Medium compressibi li ty:De

Dt

The common approach is limited to vertical deformation, following the 1D consolidationtheory of Terzaghi (1925). It assumes for equilibrium (vertical stresses on horizontal

surface)

e P

where

e = effective (intergranular) stress

= total stress (overburden pressure) = total weight (solids + water) + applied stress at

surface (for example weight of water in reservoir)

If = constant = a, leads to a linear relation e a P

-----------------------------------------------Example: Hydrostatic case

We want to compute the effective stress on the horizontal plane at depth z= a + b

(dibujo)

1/(1 )

1/ 1

g g g g

d g

T V g g

W W V

V V V V e

= dry unit weight

(1 )1

g w g g w w g w

T g w

T V g

W W V V e

V V V e

= total unit weight

where

Vw = VV(saturation)

W= weight.

Subscripts refer to: g= grains, V= voids, w = water.

(1 ) (1 ) ( )(1 )d T g g w g wa b a b a b b

( )(1 )e g w wP a b b b

( )(1 ) ( 1)e g wa b b

[total solid weight] + [buoyant weight]

Buoyant weight = weight of water of the volume occupied by grains in the saturatedregion Archimedes principle

-----------------------------------------------

The Terzaghi relation implies that, for = constant, e P . This seems to work

well for saturated unconsolidated sediments. In consolidated rocks (Robinson and

Holland, 197?) and unsaturated media only a part of P converts into e (Bishop,


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Eric Morales

1960; McMurdie and Day, 1960). Then e P where 0 1 . is the

boundary porosity in the petroleum lingo or Bishops parameter in soils lingo . In

partially saturated media represents a measure of the fraction of pore surface in

contact with the fluid and is sometimes approximated as ( ) ( )P S P .

Figure. Variation of parameterwith degree of saturation Sr for four compacted partially

saturated soils. Values were determined from triaxial compression tests. (From Bishop

and Blight, 1963). AW Bishop and GE Blight, Some aspects of effective stress insaturated and partly saturated soils, Geotechnique,13, 177-197 (1963).

Uniaxial consolidation tests provide lab relations between e and e (or 10log e ). These

relations are obtained under equilibrium and disregard kinetics (assume e responds

instantaneously to P ). Soils engineers define

v

e

dea

d coefficient of compressibility (varies with applied stress e )


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Eric Morales

10(log )c

e

ec

compression index (best fit slope along virgin curve)

or10 10

(ln )

(log ) (ln ) (log )

e ec

e e e e

d dde dec

d d d d

2.303c v ec a

where 10 1010

logln 2.303log

log

ee e

e

and

1(ln )

e e

ee

e

d d

dd

cs= swelling index; best fit slope of rebound curve. Tipically cs


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*'1

1 1 1

v vw

a S a S D D DSe Sc C

Dt e e Dt e Dt

recall 1 1

1 1 1 1

D e D e DS DeSe S S

Dt e e Dt e Dt e Dt

For slight deformation qg 0, gD

Dt t t

q ; therefore

D

Dt t

and, since

h z , leads toh

t t

*'1

1 1 1

v vw

a S a S D hSe Sc C

Dt e e t e t

Defining thespecific storage * '

1

vs w

a SS Sc C e

leads to the final form of the flow equation

1

vs

a Shh S

t e t

K

For slightly compressible liquids 0 and thus

1

vs

a Shh S

t e t

K

In the saturated zone ( = 1, C*

= 0, S= 1) (1 )s w vS c a

In the unsaturated zone one commonly assumes ( ) and imbibition/drainage

dominates, so sd

S Cd

= specific moisture capacity

In hydrogeology one commonly treats Ss as a constant except in problems involving land

subsidence.

Note that rapid variations in surface loading appear as a source term in the flow equation.

If= constant, 0t

and the flow equation simplifies to

sh

h St

K

For incompressible fluid in rigid saturated media and/or steady state

0h K


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Eric Morales

Alternative expressions for Ss

The form ofSs depends on the definition of medium compressibility

(Gambolati, 1973)

Defineg= density of grains (constant)

1

g

gbe

= bulk density of grains

1 gbb

gb e

dc

d

= bulk medium compresibility

But21 (1 ) 1

gb g g gb

e e e e

d d de de

d d e e d e d

which leads to

1

1b

e

dec

e d

or

1

vb

ac

e

s w bS c c

Note: 11 1b b b b

b b

b b e e b e

d V d V dVc V

V d d V d

. If= constant,

1 bb

b

dVc

V dP

Thus, bc of Bear (1979, pp. 85-86)

Under yet another definition of medium compressibility

1 VVV

dVcV dP

; VV= volume of voids

Then 1 1V g V

V

V g

d V V adec

V V dP e dP e , but

1 1

vb V V

a ec c c

e e

, therefore

s w VS c c

which is common in the petroleum literature

Balance de Masa- Fluido en Medio Poroso

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