Miscible and immiscible multiphase flow in deformable porous media

MATHEMATICAL COMPUTER MODELLING Mathematical

PERGAMON

and Computer

Modelling

37 (2003) 571-582 www.elsevier.nl/locate/mcm

Miscible and Immiscible Multiphase Flow in Deformable Porous Media G. KLUBERTANZ*, F. BOUCHELAGHEM, Soil Mechanics

Laboratory,

Swiss Federal

Institute

L. LALOUI AND L. VULLIET

of Technology,

CH-1015

Lausanne,

Switzerland

[email protected]

Abstract-In this paper, two coupled models are proposed for porous media containing pore fluids: an immiscible and a miscible one. Emphasis is placed on the comparison between them at different steps during development and application. Even if their microscopic basis may be rather similar, the macroscopic formulation is different, as must be expected due to the different physical processes modelled. Each model has its particular domain of applicability and validity. @ 2003 Elsevier Science Ltd. All rights reserved. Keywords-Porous

media, Miscible

and immiscible

fluids, Coupled processes

1. INTRODUCTION Multiphase describing In both

mixtures

can contain

such porous

media

other hand,

miscible

in petroleum

In this paper, media

are derived

is appropriate

oil recovery

immiscible

from continuum

for cases such as reservoir

fluids should

be considered

engineering

two formulations

are presented.

and/or

fluids.

could be based on thermodynamic

cases, the formulations

fluid description

miscible

Emphasis

for miscible is placed

mixture

equations.

or unsaturated

with pollutant

phase

equations

leads

and immiscible

two-fluid

on the comparison

to the coupled

assumptions are discussed afterwards. Illustrations shown in the final section of the paper.

2. MATHEMATICAL The

mathematical

from conservation

formulations equations

transport,

them

enhanced

of the applicability

porous

at different

steps

of the mixture can and how combining

&ystem of equations.

on continuum

of mass and momentum

The immiscible model previously proposed medium and is based on the continuum theory

soils. On the

flow in deformable

between

FORMULATION

are based

theories. Immiscible

grouting.

during development and application. First, it is shown how each constituent be considered in an appropriate way depending on the miscibility hypothesis, the elementary

formulations

or averaging

conservation

simulation

when dealing

or chemical

Theoretical

The

constitutive

of each formulation

are

OF MODELS

mechanics.

for both models

Equations

are derived

considered.

by the authors [l] treats a three-phase porous of mixtures. Starting from mass and momentum

*Current address: Colenco Power Engineering AG, Department of Safety and Nuclear Technology, Mellingerstr. 207, CH-5404 Baden, Switzerland. Part of this work was funded by the Swiss NSF, Grant 21-50769.97, and by the Board of Directors of the Swiss Federal Institutes of Technology. 0895-7177/03/s - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: sos95-7177(03)00050-5

Typeset

by A_,&-‘$$

572

G. KLUBERTANZ

balance

equations

variables

[2], a highly

solid grains

are considered

ideal gas law. immiscible relation elasticity,

elasticity,

encompasses

and an equilibrium solid displacement,

The general

i.e., linear

focus is placed adsorbed

medium

upon

grout,

as a whole.

grout

fluid phase, The

basic

and specific deposit.

quantity,

as obtained

injection. grout

field variables

by Bear and Bachmat Up to now, existing generally

fluid flow and the porous coupling

and nonlinear

during

(water

system,

mass

requiring

matrix. The

specific

numerical

any

to exhibit

the proposed

mathematical

[3]

models

disregarding In order

transport,

and grout).

are

This model is based

equations,

miscible

The

solution)

postulates.

mass balance

balances

equation

density,

for a mixture’s

in the following) law, the following

+

where

??

model

formulation

techniques.

reads

(1)

0,

and pc the interaction

supply

rate density

case.

for all constituents, applicable.

equations

constituent

v [pv]- pc =

v the phase’s velocity,

are established

(grout,

Combining

to this model,

Equation

Darcy’s

Darcy’s

The two

with constitutive

which is zero in the immiscible

Immiscible

fluid phases.

on the transport

law for the fluid phase

where p is the intrinsic These

special

of an extensive

g

term,

by the

pore pressure-saturation

are associated

concentration,

of the hydromechanical

Balance

of t,hcx

to be governed

combined

coupled

field

densities

have focused

between

Darcy’s

in a highly

Mass

grout

equation

technique,

coupling

results

is concerned,

of the porous

mechanical

is assumed

and a nonlinear

behaviours

The principal

The specific

to be valid for both

phase)

equation

displacement

comprises

law is assumed (no static

of mass (solid phase,

of miscible

is obtained.

and the gas phase

equations

balance

using an averaging

formulation

and gas pressure.

and elasto-plasticity.

model

fluid pressure,

upon the general

nonlinear pressure,

types of constitutive

As far as the miscible

the importance

Darcy’s

flow freely

Several

nonlinear

liquid

to be constant

Furthermore,

fluid phases

is used.

model

coupled,

are solid displacement,

et al

i.e., solid, water,

This yields,

including

gas, and miscible

various

assumptions

agent such as

for the two models.

formulation solid and water

mass conservation

law, one finds, after some rearrangement,

for the water-solid n as” _-+__ 9 ape

balance;

and solid and air mass conservation the following

expression

and using

[l]:

and

n apa ~a apa >

apa at

n asw apw --sa ap=

at

+vovS+~-

1 1 -SW

k”K pa

VeVp” (3)

for air-solid

balance.

Multiphase

S” is the degree respectively), velocity,

of saturation

and n porosity.

Darcy’s

permeability Ic:, the geometric defined as (p” - p”). Miscible

of the phase

under

consideration

and p” the pore-pressure.

p” its density

law, incorporated

permeability

573

Flow

(n = w, a for water

g denotes

the gravity

in the above equations,

K, and the fluid viscosity

and air,

vector,

vs the solid

contains

the relative

pLn. pc is the matric

suction

formulation

The miscible

model

eraging

rules,

develop

the mass

in addition

here to express general

conservation

the consideration and filtration,

equation

water

concerning

under

is composed

the grout

dimensional

dispersion

laboratory

of a phase,

and interstitial

a continuous

coefficient

experiments,

processes

averaging

dominant

medium

of interest,

process surface,

and avto

is important and miscibility

[4]. Starting

the proposed

from the

model

effects, hydrodynamic

allows

dispersion

effects.

of grout

and water,

The

before injection

quantity

varies from 1 to 0. Besides miscibility

of the longitudinal

the physical

present

convective

fluid flow, and consolidation

of a porous

on the solid skeleton

initially

of an extensive

of grout transport

between

model

for each component.

the effects of deposition

and interstitial

The fluid phase centration

Bear’s conceptual

assumptions

equations

correctly

the grout

using

to other

balance

between

surface

is obtained

of a definite

zone is observed

effects, which were proved

aL (see constitutive

grout

Instead

water.

transition

adsorption

interface

where grout

con-

by the measurement

assumptions,

below)

takes place between

during

one-

the fluid phase

and

the solid skeleton. Regarding

constitutive

permeability

function

assumptions, (usually

filtration

a generalisation

laws all rely on Darcy’s of the Kozeny-Carman

law with equation)

a modified to take

into

consideration the alteration of the porous medium structure induced by the adsorbed grout mass deposition [5]. Regarding the filtration term, a linearity assumption and Fick’s law are employed to express mass transfer surface is a definite diffusion.

Dispersive

the growth propagation

equation

surface,

grout-pure

is obtained

Indeed,

grout can only be adsorbed

flux is formulated

of the pure [3].

The flow equation continuity

from the fluid to the solid phase.

material

because

using Bear’s hydrodynamic

water

transition

dispersion

zone as injection

by adding the fluid phase continuity

and combining

with constitutive

assumptions

the fluid-solid

interface

on the solid skeleton’s tensor

length

equation (Darcy’s

surface by to express

increases

during

to the solid skeleton law, filtration

law) [6]

v') ( (Vpq$J) )-A~($--f); (4) +V* 1-q$ 1=-k$ ($ +VpS.vS psV b vs + np”

& g

+ /3, n)

+ np” (&VP.

vs + p,Vc.

,ofl and ~1” are the fluid density and viscosity, and ,&, and /?, are the compressibility terms defined below. The two terms appearing on the right-hand side, new with respect to a previous formulation including phase. porous

only viscosity

effects [7], are due to filtration.

p is the pressure

in the whole fluid

This flow equation governs the fluid phase transient flow within a saturated deformable medium, under grout concentration-dependent fluid density and viscosity variations.

Mass balance to display

for the grout is written,

hydromechanical

cV@vS+n

coupling

dc Z +Vcav”

(

and then combined

and filtration

*

with the solid skeleton

+ V 0 [-n&Vc]

--c -$ (Vp - pflg)

+ V 0 -

> =_A.,_;;

[ -E/L)

mass balance:

terms

[!Z$+vps.vsj,

I

\J)

I=\

574

G. KLUBERTANZ

some of the terms

in the above equation

merit

et al.

detailed

discussion

nD,jVc, Dij is the hydrodynamic

dispersion

the ambient water, leading of the microscopic velocity the grout

particles

hydrodynamic featured

are dispersed

dispersion

in equation

to correlate medium’s

the growth

of the transition

characteristics,

is the solid skeleton coupling

a term

PS

for the fluid flow equation,

for the grout

and the porous (see constitutive

dispersion

(in a linear

theory),

equation

expressing

and dispersion

and expresses

[3]

hydrome-

k+Ac(-p),

and

w) -A+$),

(

equation,

density

of the solid skeleton

hydrodynamic

rate

(

are due to filtration.

PS

the surface

are available

and fluid flow, while

aps

(1 - n) ~

and adsorbed

tensor

of a fluid phase under convection

x+vp

PS

ible) grains

fluid properties dispersion

law, nDtJVc

tensor

here. to the classical

(1 - 7%) dp” dt+vp

transport

follows from solid skeleton

with

This effect is called

diffusion-like

dispersion

zone to the flowing

deformation

(1 - n)

-~ a term

and miscibility

medium.

accepted

for the hydrodynamic

soil consolidation --

dilution

the porous

by a commonly

of a component

volumetric

between

grout

but Bear’s hydrodynamic

are new with respect

the fluid flow and the transport

chanical

when flowing through

[3], and is expressed

below) will be retained

Some terms

which expresses

(5). Several expressions

geometrical

assumptions,

tensor,

to the development of a transition zone. Due to the heterogeneity field and the tortuosity of the path followed by the grout particles,

grout.

dp” at+vps*VS

(

variations,

The first component

)

as the solid skeleton

In the last term, per unit volume

Xc, represents of porous

comprises

initial

the mass of grout

medium.

(incompressdeposition

on

Last, 4

expresses the variation of the fluid phase density under variations of the grout component concentration, since the assumption of an ideal tracer, usually encountered in environmental or computational fluid dynamics literature meaning a fluid phase whose density and physical are not altered by the presence of distinct. components, is not retained here. Momentum The general

properties

Balance momentum

balance

neglecting

inertia

terms

reads

(6) The effective

stress is expressed

as

immiscible

Uij = 0:j - S”p”bij

miscible

Oij

=

flaj

-

pbij.

- (1 - Sut) pa6ij,

(7) (8)

Multiphase

aij

and aij

mixture

are the total

density

and effective

stresses,

575

Flow

respectively,

gi is the gravity

vector

and p the

with

immiscible

p = (1 - n)p” + nS”p”

miscible

p = (1 - n)p” + n/Y.

+ 72(1 - SW) pa,

(9) (IO)

densities of solid grains, water and air, pfl and c designate PS, PW, and pa are the real (intrinsic) (defined as the mass of grout per unit volume the fluid phase density and the grout concentration of fluid phase). Combining the above equations, immiscible

v

one finds

??

(“ij - SwpVij

- (1 - S”)p”rQ)

miscible

v

??

(11) (12)

+ pgi = 0,

(“:j - p%J

+ pg = 0.

3. COMPARISON BETWEEN MISCIBLE AND IMMISCIBLE FORMULATIONS Constitutive Common

quantities

The following different

Assumptions

quantities

parameters,

constitutive

appear

yielding

assumptions

in both formulations

different

k:’ = kF(n,c)

viscosity:

It should

be noted

that

in the miscible

is, nevertheless,

similar

way, but involving

sometimes

The most important

or

k:

(n,S*) ,

p = /-J(c),

fluid density:

plicable

in a similar

in both formulations.

are

permeability: dynamic

nonlinearities

(13)

P* = P”(P,C). some of the above quantities

case only. The mathematical

are a function and numerical

of the concentration treatment

c, ap-

of these quantities

for both formulations.

Expressions for the miscible formulation In particular,

the following

expressions

are necessary

for the miscible

formulation:

pfl = pfl(c,P)

holds, and one defines the coefficients

a=-$!, as the fluid compressibility. involves

the density

with

Note that respect

in the immiscible

(14) formulation,

to its own phase pressure

and that

for the water and one for the gas phase. For the miscible formulation, continuous domain with no interstitial tension (see Laplace equation, pressure

within

the fluid phase.

In a similar

the corresponding there

term

are two terms,

one

the fluid phase is one [9]), implying a unique

way, one defines

i apfl PC=---, pfl ac as the fluid concentration coefficient, which expresses the variation a function of the concentration of the grout concentration.

(15) of the fluid phase density

as

576

C:. KLUBEKTANZ

The filtration

rate appears

et al.

only in the immiscible

case as

x = X(0,12). This law can be considered mass of grout first-order

adsorbed

during

form of the relation

on the solid skeleton’s

linearisation

into the formulation,

as a special

was retained

as the simplest

it proved sufficient and the real-scale

The hydrodynamic

dispersion

three

coefficients

a~ the transversal

designates

remarks

Effective

stress

Cvfl>i (““)j VI

in the isotropic

dispersion

the fluid’s velocity

General

terial

fluid phase

laws, to several

about

agreement

formulation

In the immiscible

complicates problems

concerning

the miscible

single fluid phase saturating component,

here, are mainly

or between

grout convection,

evidence,

those

interactions

solid phases other than ciple of effective stress.

relation

derived

(17)

aL the longitudinal

D the molecular

due to capillary to induced

must be considered

separately

case, the necessity and leads,

Regarding

dispersion

diffusion

coef/vfl /

coefficient;

combined

component

to induce

coupling, between

effects. momentum

to bring

to that of a

the fluid phase

and the solid skeleton physical

in ma-

the immiscible

corresponds

The interactions

and deposition

are not assumed

out to be difficult

formulation

in the fluid phase’s

Terzaghi

the pressure

with more complex

hydromechanical

the latter

to the classical

to incorporate

It also turns

void space.

the grout

dilution,

the classical Concerning

alent with respect

properties

considered (density

and

Due to a lack of experimental coupling

between

fluid and

hydromechanical coupling corresponding to Terzaghi’s printhe immiscible model formulation, momentum transfers are

pressure

and surface tension

mechanical

stresses

in the effective

effects; the two phases

on the solid skeleton,

are not equiv-

and their

contributions

stress principle.

of saturation

The absence difference.

observed

Dbi,,

case is taken to be similar

one because

due to the heterogeneity

under

Degree

most of the features

by the commonly-used

consistency.

the interstitial

viscosity)

more complex

[G]. A effects

[8].

+ UT 1~1’ 6ij1 +

and

the expression

with some observations.

encompasses

and the grout

medium

deposition

norm.

stress formulation.

the second

injection

case, namely

coefficient,

Note that the effective stress in the miscible effective

of porous surface

[3]

Dij= (UL - UT) and requires

to reproduce

D, is expressed

tensor

a,f = cy,r(c, g, n) = Xc, aSfT being the

per unit volume way to introduce

experiments

by Bear and Bachmat

ficient,

surface

and because

the laboratory

I I(jl

of an explicit

degree of saturation

Its role is in some respect

assumed

in the miscible by the concentration

formulation

is another

c. Nevertheless,

treatment concerned

must take this into account, particularly where the momentum balance and the dependence of pfl and pfl on c has to be accounted for carefully.

Number

of phases

major

numerical equation

is

involved

As can be seen from the equations,

the fluid-grout

mixture

has only a unique

pressure,

while in

the immiscible formulation, there are two pressures for the gas and water phases. The miscible formulation has, consequently, less phases, even if the number of constituents is the same as that in the immiscible case.

Multiphase

When

looking

at the number

one, in the sense that

of phases,

the immiscible

it allows the consideration

soils. On the other hand,

by focusing

577

Flow

model is more flexible than

of the unsaturated

on a singular

conditions

the miscible

prevailing

in most

fluid phase (air or water or the mixture

between

grout and water), the miscible model allows the consideration of as many fluid components required by the degree of refinement necessary to describe the physical processes of interest. The miscible

model can indeed

for each component same way that Numerical

be extended

to multispecies

transport

by writing

of the fluid phase which can be viewed as a separate

the grout

transport

equation

a transport

chemical

equation

species,

in the

was derived ‘[4].

integration

Both formulations of the numerical

are integrated

formulation

in a finite element

and the strategies

code developed

used to obtain

by the authors.

field variables

Discussion

can be found in [6,9].

4. PERFORMANCE OF THE MODELS WITH RESPECT TO EXPERIMENTAL RESULTS As there is no analytical integration

is obtained

and immiscible), presented

different

for the developed

comparison

laboratory

tests were performed

CASE. A 2D laboratory

(A)

IMMISCIBLE

the experimental a constant

set-up

shown

water table

test

in Figure

was imposed

and 0.15m

below the surface,

respectively,

the bottom,

in the left third

of the set-up.

wide.

formulations,

the validation

with experimental

results.

of the numerical

For both

by the authors,

cases (miscible

and some are briefly

here.

with

problem:

solution

through

The emphasis

and air pressures

of the experiment

were monitored

was carried

at the LMS-EPFL reproduced

at the left- and right-hand

and a pore water

pressure

the experiment,

development

laboratory,

a slope stability

sides of the box at 0.2

of 1.6 kPa

The box was 1.5 m in length,

lay on pressure

during

out

1. The experiment

was applied

1 m high,

and phase

at

and 0.25 m

coupling:

water

as well as solid displacements.

Load&y Pump-----,___

____---

Forcesensors

Figure 1. Sketch of the experimental

The pore water pressure the first failure

at the bottom

. .

_

setup for the immiscible fluid experiments

was applied

of the lower part of the slope occurred.

at time t = 0. Soon after, at about The slope continued

t = 80 s,

to fail by backward

erosion and outflow appeared at the lower part shortly afterwards. Therefore, the FE calculation based on a continuum assumption is no longer valid after about t = 90s. At about that time, the numerical solution becomes unstable and tends to diverge [lo].

G

KLUBEKTANZ

et al

Figure 2. Displacements: shaded zones: measured values. Lines: calculated values.

0

0

20

40

60

80

100

Time (s) Figure 3. Pore water pressure at several locations.

Displacements As shown in Figure 2, the overall displacement pattern can be fairly well reproduced, even if the measured displacement magnitude is underestimated by almost a factor of two with respect to the calculations. The location of the maximum displacement, where failure was observed soon afterwards, is reproduced very accurately. This is also valid for the slip zone. The starting failure of the slope is clearly visible and the displacement pattern is still correctIySreproduced! even if the magnitude Water

of the measured

and calculated

results

continue

to differ.

pressures

Calculated and measured water pressures are reported in Figure 3 for various locations (Pt,l (0.4/0.2), Pt2 (0.4/0.4), Pt3 (0.4/0.6), Pt4 (O.S/O.S), Pt6 (0.9/0.4)). It may be seen that the calculated and measured is given in Figure 4.

pressures

coincide

reasonably

well. The predicted

degree of saturation

(B) MISCIBLE FORMULATION. For the validation of the miscible model, a real-scale injection experiment under axisymmetric conditions was conducted at the laboratory. Displacement and

Multiphase

Figure 4. Calculated

Flow

degree of saturation

pore pressure

were monitored

tions are illustrated

in Figure

at time t = 90s.

_,_..“....“...,.

._--,-~-..-“~-..“““-“““~~

Figure 5. Experimental reported in mm.

579

set-up for the miscible fluid experiments.

during

the experiment.

??

‘All

lengths are

The model geometry

5. The grout was injected

and transducer

from a finite strainer

length

loca-

and the axis

of the grout injection pipe coincided with the model axis. The model was filled with Leman sand (medium dense sand) and saturated with water, before being injected with micro-fine cement (Spinor

A12-Sika).

samples

of the hardened

Agreement

The grout

between

distribution

was analysed

at the end of the experiment

soil. the experimental

and numerical

pressure

values is quite good, except

a small region near the injection zone, where the experimental pressure as attested to by Figure 6, representing pore fluid pressure distribution process.

by taking

Such a discrepancy

may be due to a partial

occlusion

of injection

gradient during

over

is very high, the injection

holes, or to hydraulic

loss between the injection tube and the sand (screen mesh). Another explanation could be the linearity of the model chosen for the filtration effects. The filtration coefficient X is taken as a constant here, due to a lack of experimental filtration data (adsorbed mass density), but it is more

580

G. KLUBERTANZ et al.

t...t...m...m...m 0

2

.

4

8

8

..I.

,,I,

IO

,,

12

I,,

14

,,

16

Time (lo2 s) Figure 6. Pore fluid pressure, comparison

between numerical results and experiments.

12

10

6

6

4

2

0

oW

2

4

6

6

10

12

14

41 16

Time (102s) Figure 7. Displacement,

comparison

between numerical results and experiments.

likely that X depends on deposition rate and concentration field x(n, (T,c) because only a finite number of sites are available for adsorption on the solid skeleton surface, and after completion of those sites, filtration may stop or a more critical deposition mode may begin, leading to a localised increase of pressure within the porous medium, such as that observed near the injection pipe. As may be observed in Figure 7, the calculated displacement evolution is very similar to the calculated pressure evolution; the increase in displacements follows the pressure field increase: in accordance with the linearity of the model (Hooke’s law was used to compute displacements). Computed and experimental displacements show reasonable agreement, and attest to the hydromechanical coupling, as displacements here were induced by the miscible grout propagation within the saturated, deformable porous medium. Water injection experiments were performed before the grout injection experiment, and displacements recorded were constant and very low, while for the grout injection the solid phase shows a transient *behaviour, typical of suspensiontype flow under filtration effects. Nevertheless, the measured displacements are quite low, mainly

581

Multiphase Flow due to over-restrained venting

boundary

any radial

and bottom

displacement

ant. not seem to be r The results of these tests influence

show that

aspects.

doubt that viscosity

dependence

while the grout

means

of Darcy’s

law).

increase

pattern,

pressure parameters within

filtration

front

speed

On the contrary,

laboratory deposit

injection

&rout

process

of practical

and pressure its influence

correlated

does

in the fluid physical distributions

much more fields leaves no

from the beginning

to the fluid pressure

effects are complementary,

shown

(by

by the fluid

amount.

several numerical simulations with varying as grout permeates = 2.5~~~~~~) is instantaneous

location

as soon as the grout

(X = 1.5 x lo4 as identified

becomes

a sufficient

manifests

is directly

the predominant

Although

importance

during

transient

the filtration

term in (4) or (5), it h as a cumulative

to the convective long-term

filtration

experiments),

has reached

and variations

of the top

variations

by conducting

of viscosity

the soil mass, and ceases at a certain

proportions.

displacement

on pressure

the concentration

and filtration

as observed

variations

and concentration

on grout concentration

Viscosity

[6]. The influence

between

model walls were rigid, pre-

and any vertical

features

fluid pressure,

Coupling

of the experiment,

surface,

injection

of solid displacement

solid displacement,

than pure consolidation

(the cylindrical

of the lateral

and the influence

plan

properties

conditions

is present during process

rate is quite

effect, which explains

the whole duration

in sufficient

one-dimensional when

the mass

low as compared that

filtration

is a

of the experiment.

5. CONCLUSIONS a. Immiscible Different

Formulation comparisons

show that

they

pore pressure effective that

carried

are often

in both elastic

between

In particular,

the evidence

similar.

the fluid and solid phases was seen.

stress formulation,

nonlinear

out by the authors

are found to be important

or unsaturated

conducted

experiments.

be robust

and viable,

elasto-plastic

and experimental

of coupling

Material

to obtain

laws, and consequently,

viable

the proposed

quite substantial

results

and the influence

results.

laws yield the best results

As an overall appreciation, but requires

numerical

formulation

computational

of the

Experience

shows

with respect

to the

has proven itself to

resources,

even for relatively

small problems. b. Miscible

Formulation

The proof of any coupling evident

between

consolidation

on the basis of the outlined

laboratory

and miscible study.

First,

grout propagation it should

is by no means

be recalled

that

a linear

elastic law was used for the skeleton’s deformation, and when the permanent pressure regime is reached the coupling disappears [6]. Further developments of the model comprise experimental identification real-scale

of the coefficients

injection

phenomenological realistic

in the formulation.

will be undertaken The consolidation

including

elasto-plastic

next, model

Comparison

with laboratory

in order to validate will also be extended

the model

and from a

to include

more

behaviour.

Remarks

Two coupled a miscible

perspective.

solid behaviour,

c. General

introduced

experiments

models were proposed

one. Similarities

basis may be rather the different and validity, transport or Concerning general one,

similar,

for porous

and differences the macroscopic

media containing

pore fluids:

of those models were shown. formulation

is different,

an immiscible

and

Even if their microscopic

as must be expected

due to

physical processes modelled. Each model has its particular domain of applicability such as reservoir simulation or unsaturated soils in the immiscible case, pollutant enhanced oil recovery in petroleum engineering in the miscible case. the description of the fluid phase only, the miscible representation is the more in the sense that it allows the consideration of several components of the fluid phase,

582

G. KLUBERTANZet al.

and the monitoring

of the

flow of the fluid phase components

or between

distribution

as a whole. a particular

of the component Phenomena

component

of interest

such as chemical

during

the

reactions

and the solid skeleton

surface

macroscopic

between

the fluid

(adsorption

process!

can fully be described by the model when writing the mass balance for the component considered, supplemented with the expression of the rate of reaction. In the miscible model! emphasis is laid upon the species On the other fluid phases

mass conservation. hand,

occupy

the immiscible

the interstitial

phase in the immiscible phases,

model originate

so the main interest

immiscible

phases,

Another

point

saturation distribution

concerns

of interest

is continuously an optimal

except

over the transition

variable

diluted

but is present

model,

the fluid phases

is always

and several

prevailing

among

surface

grout

continuous

in each

the various

between

(meaning

concerned.

fluid

the various

are mixed,

that

present so that

present

void space.

after water

injection

and role.

the concentration variable

can be

constituent,

will be expelled. the grout

zone. On the other hand,

the interstitial

model)

play a similar

For a miscible

all of the fluid initially and water

within

(miscible they

from 0 to l), the saturation

inside the transition

are distributed

concentration

it was seen that

mechanism

air and water can be concurrently

be completely evacuated mechanical interactions

between

extent,

efficiency;

zone where

everywhere

phenomena

of a thin interface

in the medium

injection

per fluid phase,

fluid pressures

stress balance.

This is due to the underlying

this will ensure

for example,

from surface tension

To a certain

varying

one species

The different

is the correspondence

model).

the concentration

discontinuous.

considers

the existence

and the microscopic

(immiscible

but although

model

void space.

may be

in the immiscible

In unsaturated

soils,

since the air cannot

(irreducible saturation) due to the different fluid pressures and complex between the two immiscible phases. The injection efficiency (part of the

fluid initially present which is recovered) is much lower than for the miscible case. Last, by considering several fluid phases, and thus, several fluid pressures, a more general

model

was provided

of the

effective

stress

Nevertheless,

in the immiscible includes

case for studying

the simplest

it has been shown that

which allow for easier understanding gained

expression both

consolidation

used in the miscible models

and efficient

from one type of model can be applied

effects,

have comparable

treatment

of both

as the expression

case. variables models

and parameters

in parallel:

insights

to the other one.

REFERENCES 1. G. Klubertanz, L. Laloui and L. Vulliet, Numerical modelling of the hydro-mechanical behavlour of unsaturated porous media, NAFEMS World Congress 1997, Stuttgart, pp. 1302-1313, (1997). 2. K. Hutter, L. Laloui and L. Vulliet, Thermodynamically based mixture models of saturated and unsaturated soils, Journal of Mechanics of Cohesive-l+iction& Materials 4, 295-338, (1999). 3. J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomenon in Porous Media, Kluwer Academic, (1991). 4. S.S. Mohanka, Theory of multistage filtration, Proceedings of the ASCE, Journal Sanitary Engineerzng Division 95 (SA6), 1079, (1969). 5: F. Bouchelaghem, L. Laloui, L. Vulliet and F. Descoeudres, Numerical model of miscible grout propagation ni deformable saturated porous media, In 7 th Internatzonal Symposium on Numerzcal Models in Geomechamcs. pp. 243-248, NUMOG, Graz, (1999). 6. F. Bouchelaghem, L. Vulliet and L. Laloui, Mathematical and numerical filtration-advection-dispersion model of miscible grout propagation in saturated porous media, Accepted for publication, International Journal for Numerical and Analytical Methods in Geomechanics, (2001). 7. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, (1960). 8. F. Bouchelaghem, L. Vulliet, D. Leroy, L. Laloui and F. Descoeudres, Experimental study of a miscible grout through a deformable saturated porous media, and performances of advection-dispersion-filtration model, Int. Journal of Num. and AnaEy. Methods in Geomechanics 25, 1149-1173, (2001). 9. G. Klubertanz, L. Laloui and L. Vufliet, On the implementation and performance of a coupled multiphase model for porous media; Numerical studies and applications, In European Conference on Computatzonal Mechanics, Solids, Structures and Coupled Problems in Engineering, p. 19, Munich, (1999). 10. G. Klubertanz, Zur hydromechanischen kopplung in dreiphasigen porosen medien, Ph.D. Thesis No. 2027, Swiss Federal Institute of Technology, Lausanne, (1999).