Transport in Highly Heterogeneous Porous Media-From Direct Simulation to Macroscale Two Equation or Mixed Model

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Transport in Highly Heterogeneous PorousMedia: From Direct Simulation to Macro-Scale

Two-Equation Models or Mixed Models

ARTICLE in CHEMICAL PRODUCT AND PROCESS MODELING · JUNE 2008

DOI: 10.2202/1934-2659.1130

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Debenest Gerald

Institut de Mécanique des Fluides de Toulo…

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Michel Quintard

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Retrieved on: 17 March 2016

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Transport in Highly Heterogeneous Porous Media: From Direct

Simulation to Macro-Scale Two-Equation Models or Mixed Models

G. Debenest1 and M. Quintard

*,1

1 Institut de Mécanique des Fluides de Toulouse*Corresponding author: Institut de Mécanique des Fluides de Toulouse, 4 allée C. Soula, 31400 Toulouse

([email protected])

Abstract: Flows in highly heterogeneous mediaare found in many practical fields, such as

hydrology, petroleum engineering, chemical

engineering. The case of two-region

heterogeneous media (fractured media, catalytic

beds, etc.) plays a fundamental role. The

different questions associated to this specificcase are illustrated in this paper for two different

kinds of transport: (i) flow of a slightly

compressible fluid, (ii) dispersion of a tracer.

Many different theoretical models are

implemented under COMSOL, using many of

the original features of the software. These

models correspond to direct simulation and

macro-scale or large-scale models such as fully

averaged models or mixed models.

Keywords: porous media, heterogeneous

system, double-porosity, mixed models

1. Introduction

Flow in highly heterogeneous media have

received a considerable interest in the scientific

literature since all natural media, such as the one

appearing in petroleum reservoir engineering or

hydrogeology, features more or less importantheterogeneity. Heterogeneity is known to lead to

abnormal transport mechanisms. Here, the word

abnormal means that classical convection-

dispersion equations cannot be used reliably [1].

A particular class of heterogeneous systems

shows easily this abnormal behavior: two-region

systems, often called also double-porosity

systems. Examples of such systems are given in

Fig. 1. The left picture corresponds to a fractured

system in which the ω-region permeability isseveral orders of magnitude lower than the

permeability of the η-region. Advection is themajor transport mechanisms in the fractures,

while diffusion is dominant in the matrix blocks.

These systems are also called immobile-mobile

systems. If the ratio of permeability is closer

than 1, some advection may occur also through

the ω-region, and the resulting systems are calledmobile-mobile system.Of course, the Darcy-scale equations relevant to

macro-scale description may be solved directly

to capture the transport phenomena within such

systems. This poses specific numerical problems

especially in the case of fracture systems forwhich the volume fraction of the η-region,

denoted φ η , is small. The major problem,however, for real-scale applications, is that direct

simulation is limited to a small number of matrix blocks and fractures.

ω

η

Figure 1. Example of double-porosity systems

Alternative models must be sought, which can be

divided into two different classes. The first class

corresponds to fully averaged models, or two-

equation models. Transport equations in both

regions are homogenized over a representative

volume of the heterogeneous system [2]. Hence,

the numerical model requires only two large-

scale grids, for each homogenized region, whichis efficient in terms of computation. An

exchange term must be implemented to connect

the two sets of large-scale equations. One

understands that this single exchange terms may

have difficulties in representing all the detailed

behavior associated to the exchange between the

fractures and the many blocks within a numerical

cell. If one wants to keep some of these details,

an alternative model, called mixed model, may

be used. It is based on a fully averaged equation

associated to the η-region, coupled to a smaller-scale model with the original equations (hence

Debenest, G. and Quintard, M., 2006. Transport in Highly Heterogeneous Porous Media:

From Direct Simulation to Macro-Scale Two-Equation Models or Mixed Models, COMSOL

2006. COMSOL, Paris, pp. 1-5.


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the name mixed model). Those different

strategies are schematically represented in Fig. 2.

{{

{

Direct model

Two-equation model

Mixed model

Figure 2. Schematic representation of the threedifferent models

The resulting PDEs have many different and

interesting features that may be solved nicely

using the different approaches offered in

COMSOL. This is illustrated in this paper.

2. Direct simulation

The problems posed by direct simulations are

two folds: (i) high contrast of properties, (ii)

small volume fraction of one of the region in the

case of fractured media. If one considers, for

instance, the flow of a slightly compressible fluid

over the system represented Fig. 3, the equationsto be solved are:

1 P c P

t

βη

η βη βη

β

∂

∂ µ

= ∇ ⋅ ⋅ ∇

K (1)

and a similar equation in the other region, with

continuity of flux and pressure over all interfaces

between matrix blocks and fractures. This

transport equation for the fractures can easily be

transformed into a PDE over a fracture seen as a

line (in 2D) or plane (in 3D), we have

exchange with surroundingmatrix blocks

1

1 .

f f

P

h c h P t

P

βη

η βη βη β

ηω βω βω

β

∂

∂ µ

µ

= ∇ ⋅ ⋅ ∇

+ ⋅∇n

K

K(2)

where derivatives are taken along the fractures.

This can be solved nicely in COMSOL using the

application mode “Weak Form, Boundary” (see

for instance the application to adsorption

problems in COMSOL manuals). An example of

the obtained pressure field at a given time is

shown Fig. 4.

Figure 3. Simulation over a fractured system.

Figure 4. Pressure field (a well is located at the

center)

In this paper, we consider the flow of one phase

together with the dispersion of a tracer. For now

on we focus on the tracer dispersion problem, i.e.

Darcy-scale equations of the type

( )*t

C C C η

η η η η η η ε ε ∂ + ⋅ ∇ = ∇ ⋅ ⋅∇

∂V D (3)

The velocity field, Vη, will be computedindependently from Darcy’s law.

As an illustration, we will consider the system

represented Fig. 5 (after [3]). Low permeability

nodules are scattered through a highly permeable

medium. The direct dispersion problem is solved

and the concentration field at a given time is

represented Fig. 5. The necessary PDEs are

directly available in COMSOL.


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We now discuss the two different large-scale

models that may be associated to this system,i.e., the two-equation model and the mixed

model.

Figure 5. Concentration field at t =10000s field(injection from the left, initial zero concentration)

3. Two-equation model

The two-equation dispersion model is

characterized by the following equations [2]

( )

( )

*

* * ** *

* **

*

* * ** *

* **

t

...

t

...

C C C

C C

C C C

C C

η

η η η η ηη η

η ω

ω

ω ω ω ω ωω ω

η ω

ε ϕ

α

ε ϕ

α

∂ + ⋅ ∇ = ∇ ⋅ ⋅ ∇ ∂

− +−

∂ + ⋅ ∇ = ∇ ⋅ ⋅ ∇ ∂

+ +−

V D

V D

(4)

in which the concentrations are those of a

representative volume of the heterogeneous

system. In the case of the problem represented

Fig. 5, the large-scale equations may be solved in

1D, i.e., the Darcy-scale 2D problem is replaced

at the large-scale by two 1D equations. These

equations can be solved easily in COMSOL.

A more interesting aspect for COMSOL

application is the calculation of the effective properties in Eq. (4) from a representative Unit

Cell of the porous medium. The PDEs to be

solved are given in the Appendix. In particular,

the mass exchange coefficient, α *, in Eqs. (4)requires the solving of an integro-differential

system for a r -field, which in facts links the

concentration deviation to the difference in the

large-scale concentrations, i.e., ( )* *C C η ω − .This is beyond the scope of this paper to give

theoretical details, and we refer the reader to [2]

for a complete explanation. Two computational

difficulties can be found in these problems: (i)

the use of periodicity boundary conditions, (ii)

the existence of an integral involving the

solution as a source term (the term α *). Thosetwo features are directly available from

COMSOL! Figs. 6 and 7 show an example of

solution (the unit cell is simply a single nodule

within the η-region) in the case of a diffusiveregime or a more convective regime.

Figure 6. r-field (diffusive regime)

Figure 7. r-field (convective regime)

4. Mixed model

The equations for the mixed model are the

following. The transport equation in the η-regionis homogenized, we have

( ) ( )*

* * ** *

*

exchange with ω

1 .

A

C C C

t

C dAV

ωη

η

η η η η η η η η

ηω ω ω ω

ϕ ε ϕ ε

ε ∞

∂+ ∇ ⋅ = ∇ ⋅ ⋅ ∇

∂

+ ⋅ ∇∫

V D

n D

(5)


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where we need to compute the flux exchanged

with the embedded nodules. For our particular

application, this equation will be solved in 1D.The exchanged flux will be calculated by solving

several independent Darcy-scale problems over

several nodules. These problems are given by

( )*t

C C ω ω ω ω ω ε ε

∂= ∇ ⋅ ⋅ ∇

∂D (6)

within each nodule, and a mixed boundary

condition

*

Darcy-scalelarge-scale

atC C Aη ω ωη = (7)

which connects the (here) 2D Darcy-scale

problem to the 1D large-scale problem.

Solution is carried out in the following manner,using the multi-domain feature of COMSOL,

and the possibility to exchange information

between the two domains. Eq. (5) is

implemented in a 1D domain, with source terms

coming from the flux calculation at the nodule

boundaries and Eqs. (6) and (7) are solved other

10 nodules, as shown in Fig. 8. The arrows

illustrate the links between the two domains.

Figure 8. The two-domain representation of the mixedmodel.

Fig. 9 shows the 1D concentration field for the

η-region, and the concentration fields within thenodules, at a given time.

Figure 9. One example of mixed model concentrationfields.

Finally, it is interesting for the physicist to

compare the results obtained from the different

models. As an example, we will consider thetime evolution of the exit concentration, as

shown in Fig. 10.

0.001

0.101

0.201

0.301

0.401

0.501

0.601

0.701

0.801

0.901

1 201 401 601 801

time (minutes)

C o n c e n t r a t i o n

direct model

two-equationmodel

mixed model

Figure 10. Exit concentration versus time for the

different models.

We see that the slightly more complex mixed

models offer a better representation than the two-

equation model. In terms of computational

resources, of course, the direct computation is

very heavy, the mixed model still requires 2D or

3D calculations, while the two-equation model

does not require detailed Darcy-scale

simulations. The choice between the different

possibilities will be made depending on the

required accuracy, and the computational

requirements.

7. Conclusions

In this paper, we have provided an example

of the use of COMSOL for a comprehensive

analysis of transport through two-region

heterogeneous porous systems. Many questions

are raised from a physicist point of view when

considering these problems. In particular, which

large-scale models may be used to avoid a heavy

direct computation of the Darcy-scale equations?Answering these questions requires a thorough

theoretical analysis. The analysis leads to several

sets of PDEs, which have many peculiarities

which are often a problem in terms of

computational implementation: integro-differential equations, periodicity conditions,

coupling between domains of different

dimensions, …

It is remarkable that all these difficulties

were solved readily with COMSOL through the


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available tools, without the need of special

developments.

8. References

1. Cushman, J.H. and Ginn, T.R., 1993. Nonlocal

Dispersion in Media with Continuously Evolving

Scales of Heterogeneity. Transport in Porous

Media, 13: 123-138.

2. Cherblanc, F., Ahmadi, A. and Quintard, M.,2003. Two-medium description of dispersion in

heterogeneous porous media: Calculation of

macroscopic properties. Water Resources Res.,

39(6): SBH 6-1:6-20.3. Zinn, B. et al., 2004. Experimental

visualization of solute transport and mass

transfer processes in two-dimensional

conductivity fields with connected regions of

high conductivity. Environ Sci Technol., 38(14):

3916-3926.

9. Acknowledgements

This work has received partial support from

INSU/CNRS, and Institut Français du Pétrole.

10. Appendix

The “closure problem” ([2]) to be solved inorder to calculate the effective properties of

the two-equation model is given for two

unknown fields r η and r ω by

( ) ( ) 1 *r r η η η η η ϕ α ∗ −∇ ⋅ = ∇ ⋅ ⋅ ∇ −V D

r r at Aη ω ηω= + 1 ,

atr r Aηω η η ηω ω ω ηω ∗ ∗⋅ ⋅ ∇ = ⋅ ⋅∇n D n D

( ) ( )1r r ω ω ω ω ω ϕ α ∗ − ∗∇ ⋅ = ∇ ⋅ ⋅ ∇ +V D

Periodicity:

( ) ( ) , ( ) ( ) , 1, 2, 3i ir r r r iη η ω ω + = + = =r r r rℓ ℓ

r r ηη

ωω

n s l q= =0 0,

where

( )* *1

A

r r dA

V ωη ηω η η η η α

∞

= − ⋅ − ⋅∇∫ n V D

Similar problems are available to calculate the

large-scale dispersion tensors.

Documents

Transport in Highly Heterogeneous Porous Media-From Direct Simulation to Macroscale Two Equation or Mixed Model