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A NON-ISOTHERMAL FLOW AND

TRANSPORT MODEL

FOR A SINGLE FRACTURE IN A POROUSMEDIUM AT CORE SCALE

Martn Daz-Viera1 and Rafael Cabrera-Gutierrez1

1)Instituto Mexicano del Petrleo

[email protected]

9th North American Workshop on

Applications of the Physics of Porous

Media, 2011

October 26-29, 2011, CICESE, Ensenada, Mxico


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10/28/11 9th North American Workshop on Applications of the Physics of Porous Media, 2011 2

Outline

Background

MotivationIntroduction

Review of discrete fracture models

General Methodology

Conceptual Model

Mathematical Model

Numerical Model

Computational Model

Numerical Implementation in COMSOL

Water Coreflooding Case Study

Remarks and Future Work

References


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Background

Naturally fractured reservoirs containa significant portion of proved reserves

worldwide.

In Mexico historically such oil fieldshave contributed with the greatest

percentage to the total oil production in

the country.


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Background

Currently, several of these reservoirshave entered in a declination period of

its production because they have

exhausted the natural energy they

possessed during their primary stage.

The application of secondary recovery

methods, such as gas injection, have evenbecome ineffective to restore or maintain

the production pressure.


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Background

However, the reservoirs still containconsiderable reserves that could be

recovered through application of

enhanced oil recovery (EOR) methods.

For these purposes, at the present time

it is being considered the application of

thermal methods such as air injectioninto the reservoir for an in situ

combustion recovery process.


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Motivation

The application of EOR methodsrequire highly sophisticated laboratory

studies

to obtain the optimal parameters that

control the recovery process

to establish the optimal design strategyfor its implementation in each specific oil

field.


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Motivation

In this regard, the models that describeflow and transport in fractured

reservoirs become a fundamental

research tool to take the mostappropriate decisions


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Introduction

In recent years there is a growinginterest in modeling enhanced recovery

processes in greater detail and accuracy

for fractured reservoirs[1-4].

A series of alternative models which,

unlike the classic models of dual porosity

[5],fractures are modeled explicitly [6-7]

have been proposed .


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Introduction

These models are called discretefracture models.

Here, starting from a review of

previously published discrete fracture

approaches the main issues and

challenges are presented


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Introduction

A general non-isothermal flow and transportmodel for a single fracture in porous media is

derived, considering the fracture as an

interface betweenA numerical implementation is proposed

using a standard finite element formulation.

An application for a simple recoveryexperiment at core scale and laboratory

conditions is discussed


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The discrete fracture model is a geometrical simplified,

single porosity model.In the discrete fracture model, fractures are representedusually by (n-l)-dimensional elements in n-dimensionaldomain.

For example, the line elements are used to representfracture in 2-D while two-dimensional elements or surfacesare used in 3-D.

One of the earliest papers using the discrete fracture modelto examine fluid flow in a fractured porous medium was

published by Wilson and Witherspoon (1974). They studiedthe steady-state seepage in a fracture system beneath a damusing two finite element models.


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Gureghian (1975) presented a finite element model for

three-dimensional fluid flow in a fractured porousmedium. In his work the triangles that represent thefractures are made to correspond to the faces of aselected matrix element, represented by tetrahedrons.

Noorishad et al. (1982) studied two-dimensionaltransient flow in a fractured medium using an upstreamfinite element method to avoid oscillations inconvective dominated flow.

A similar approach was also used by Baca et al. (1984)in the study of two-dimensional single phase flow withheat and solute transport.


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For multiphase flow, Bourbiaux et al. (1999)introduced a discrete fracture model based on a finitevolume discretization method. They applied a joint-element technique to represent the fracture networks in

a two-dimensional problem.Kim et al. (2000) used an approach similar to that ofNoorishad et al. and Baca et al. to develop a paralleltwo-phase black oil model.

Karimi-Fard et al. (2001) adopted the same concept todevelop an IMPES two-phase black oil model, whereasKim et al. used the fully implicit method.


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Yang (2003) and Fu (2005) developed a control-

volume finite element based discrete fracture model fortwo-phase, two-dimensional and two-phase, three-dimensional block oil simulation, respectively. Both ofthem used flux-based upstream schemes to ensure local

flux continuity.Several finite difference-based unstructured discretefracture models have also been proposed. For example,Karimi-Fard et al. (2004) used two point flux

approximation and introduced a connectivitytransformation called "star-delta" to eliminate controlvolume at the fracture intersection, which causesnumerical instability and small time step.


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General Methodology

The general procedure for a model development includes the

following four stages:

Conceptual Model: The hypothesis, postulations andconditions to be satisfied by the model.

Mathematical Model: The mathematical formulation of theconceptual model in terms of equations.

Numerical Model: The discretization of the mathematicalmodel by the application of the appropriate numerical methods.

Computational Model: The computational implementation of

the numerical model.


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Conceptual Model

Fractures are considered as

Case A: n-dimensional porous subregions (including aperture)

Case B: (n-1)-dimensional porous subregions (excludingaperture)

Figure A Figure B

Porous medium

Porous medium

Fracture

1

2

1

mp

2

mp

f a 1n n 2n

1mp

Porous medium

2

mp

Porous medium

Fracture

1n n


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Mathematical Model

Case A: The same mathematical continuous model is applied tomatrix and fracture, but with different properties

Case B: A specific (n-1) dimensional model for fractures isderived from the n dimensional model for the matrix

Case A and B: Additional interface (boundary) conditionsbetween fracture and matrix are considered


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Mathematical ModelFluid flow model in a homogeneous porous medium with discontinuities

B t

M t d x (1)

where poro tot alV V is the porosity and is the fluid density.

Global balance equation

( ) ( , ) ( , ) ( , ) B t B t t

d M t g x t d x x t nd x g x t d x

dt

(2)

Local balance equation

;t u g x B t (3) v ;u n g x t (4)

Where vu - is the Darcy velocity


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Mathematical Model

If we consider that there is no diffusion ( 0 ) and that the interface is still ( v 0 )

in the case when the fluid and porous media are slightly compressible the above

equations can be rewritten as follows

;tp

c u g x B t t

(1)

;u n g x (2)Where tc - is the total compressibility

For constant density results

;t pc u g q x B t t

(3)

;u n g q x (4)


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Mathematical Model

Figure A Figure B

Porous medium

Porous medium

Fracture

1

2

1

mp

2

mp

f

a

1

n n 2n

1

mp

Porous medium

2

mp

Porous medium

Fracture

1n n


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Mathematical Model

Mass balance equations for porous regions (MP)

; 1,2i

ii i i mp

t i

pc u q x i

t

(1)

; 1,2i

i i

iu n q x i (2)

where ; 1,2

i

i i mp

iku p x i

(3)

Mass balance equations for fracture region (F)

;

ff f f f f

t

pc u q x

t

(4)

where ;

f

f f fk

u p x

(5)


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Mathematical Model

The following additional conditions are considered:

a) Continuity of the flow through interfaces

0; ; 1,2f ii

iq x u n u n i (1)

b) Continuity of pressure through interfaces

; 1,2f i p p i (2)

The equation for fracture can be rewritten as

;f

f f f f f f

t n

pc u u q x

t

(3)

Where n

and

are the normal and tangential divergence operators, respectively.


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Mathematical Model

If we integrate the last equation in the normal direction of the fracture, it is

2 1

;f

f f f f f f

t

Pa c u n u n U Q x

t

(1)

where2

1

2

af f

aa

P p d n

,2

2

af f

aU u d n

y2

2

af f

aQ q d n

We can rewrite velocity in the fracture in terms of the normal and tangentialcomponents:

f f f

nu u u (2)

where

f

f fnn n

ku p

normal and

f

f fk

u p

tangencial velocities (3)

Integrating tangential velocity in the fracture in the normal direction the Darcy law

for a fracture in (n-1) dimensions is obtained ;

f

f fk

U a P x

(4)


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Mathematical ModelSummary of single-phase flow through a fracture

; 1,2i

ii i i mp

t i

pc u q x it

(1)

; 1,2

mp

i i mpii

ku p x i

(2)

1 2 ;f

f f f f

t

Pa c U u u n aq x

t

(3)

;

f

f fk

U a P x

(4)

11 2

1 21

1 2

1 1;

2 2

f f f u n p u n P x

(5)

22 1

2 12

2 1

1 1;

2 2

f f f u n p u n P x

(6)

; \ 1,2i i mp

i p p x i (7)

;f fP P x (8)


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Numerical Model

Case A: Structured (SDFM) or unstructured (UDFM) mesh

refinement in the fracture region


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Numerical Model

Case B: Structured discretization (SDFM) using finite

difference method. For example, Lee et al. (2001) presented ahierarchical modeling of flow in fractured formations.

Case B: Unstructured discretizations (UDFM), there are twomain approaches: finite-element, Baca et al. (1984) and finite-volume (or control volume finite-difference) methods, Karimi-Fard et al. (2004).


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Computational Model

Case A: The same code could be used for both the matrix and

the fracture regions, but additional coding for coupling fractureand matrix interactions are required.

Case B (SDFM): the code for matrix simulation could bemodified to include the fracture effects as source term similar tothe concept of the wellbore productivity index (PI) introduced by

Peaceman (1978) to derive the transport index between matrixand fractures in a grid cell.

Fluid flow is formulated as a well-like equation inside thefracture and a source/sink term between fracture and matrix.

The source/sink term allows for coupling multiphase flowequations in fractures and matrix.

The pressure is assumed to vary linearly in the normal directionto each fracture.


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Computational Model

Case B (UDFM): The USDFM uses the discretization approach

proposed by Karimi-Fard et al. (2004).The method is based on unstructured gridding and employs theso-called lower dimensional approach to DFM gridding wherethe rock matrix is modeled by 3D polyhedral cells and thefracture network is represented by a subset of the 2D interfaces

separating grid cells.The material balance for each control volume requires theknowledge of neighboring control volumes (a connectivity list)and the transmissibility associated with each connection in orderto compute fluid exchange between neighboring control

volumes.A two-point flux approximation is applied in the transmissibilitycalculations.


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Numerical Implementation in COMSOL

Biphasic oil-water flow model:

Two nonlinear coupled equations.

Pressure Equation (elliptic)

Saturation Equation (degenerate parabolic in general or

hyperbolic of first order, whenpcow=0)

;cow

w w w o

w

dp

k p k S q qdS

;w coww w w w

w

S dpk S k p q

t dS



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Numerical Simulations

Water Coreflooding Case Study

Water flooding experiment through a sandstone

core under laboratory conditions.

in

pg

outp

0.25 m



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Numerical Simulations

Water flooding experiment through a sandstone

core under laboratory conditions.Modified Brooks-Corey model for relative permeabilities

Corey model for capillary pressure

where characterizes the pore size distribution.

1

1

w rw

cow w t

rw ro

S S p S p

S S

0 0; ;1w on n

rw w rw e ro w ro ek S k S k S k S


Data for Numerical Simulations


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Data for Numerical Simulations


Property Value Description

L 0.25 [m] Core length

d 0.04 [m] Core diameter

muw 1e-3 [Pa*s] Water viscosity

muo 7.5e-3 [Pa*s] Oil viscosity

rho_w 1.0 [g/cm^3] Water densityrho_o 0.872 [g/cm^3] Oil density

p_out 10 [kPa] Outlet pressure

u_w_in 25 [cm/h] Inlet velocity

phi_M 0.2295 [1] Matrix porosity

k_M 326 [milidarcy] Matrix permeability

Swr_M 0.2 [m^3/m^3] Residual water saturation (matrix)

Sor_M 0.15 [m^3/m^3] Residual oil saturation (matrix)

p_t_M 0.1*1e5 [Pa] Entry pressure for Brooks-Corey Pc model (matrix)theta_M 2 Exponent for Brooks-Corey Pc model (matrix)

nw_M 4 Corey exponent for water (matrix)

no_M 6 Corey exponent for oil (matrix)

krw_0_M 0.1 End point Kr to water at residual oil saturation (matrix)

kro_0_M 1.0 End point Kr to oil at residual water saturation (matrix)

phi_F 1.0 Porosity (fracture)

k_F 1000 [milidarcy] Permeability (fracture)

Swr_F 0.0 [m^3/m^3] Residual water saturation (fracture)Sor_F 0.0 [m^3/m^3] Residual oil saturation (fracture)

p_t_F 0.0 [Pa] Entry pressure for Corey Pc model (fracture)

theta_F 2 Exponent for Brooks-Corey Pc model (fracture)

nw_F 1 Corey exponent for water (fracture)

no_F 1 Corey exponent for oil (fracture)

krw_0_F 1.0 End point Kr to water at residual oil saturation (fracture)

kro_0_F 1.0 End point Kr to oil at residual water saturation (fracture)


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Numerical Simulations: the mesh



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Numerical Simulations: a zoom of the mesh



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Numerical Simulations: k_F=1000 mD



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Numerical Simulations: k_F=10,000 mD

S


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Numerical Simulations: k_F=100,000 mD

R f


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References1. Alboin, C., Jaffr, J., Roberts, J. and Serres, Ch., Modeling fractures as

interfaces for flow and transport in porous media, Research Report N 4390, INRIA,2002.

2. Bogdanov, I. I., Mourzenko, V. V., Thovert, J.F. and Adler, M., Pressuredrawdown well tests in fractured porous media, Water Resources Research, Vol. 39,No. 1, 1021, doi:10.1029/2000WR000080, 2003.

3. Hoteit, H. and A. Firoozabadi, Multicomponent fluid flow by DiscontinuousGalerkin and Mixed methods in Unfractured and Fractured Media, Water ResourcesResearch, Vol. 41, 2005.

4. Huang, Chung-Kan, Development of a general thermal oil reservoir simulatorunder a modularized framework, PhD dissertation, Department of ChemicalEngineering, The University of Utah, May 2009.

5. Lemonnier, P. and Bourbiaux, B. Simulation of Naturally Fractured Reservoirs.State of the Art. Part 2: Matrix-Fracture Transfers and Typical Features of NumericalStudies. Oil & Gas Science and Technology Rev. IFP, Vol. 65 , No. 2, pp. 263-286,2010.

6. Martin, V. , Jaffre, J. and Roberts J., Modeling fractures and barriers as interfacesfor flow in porous media, SIAM J. Sci. Comput.,Vol. 26, No. 5, pp. 16671691, 2005.

7. Tatomir, Alexandru-Bogdan, Numerical Investigations of Flow through FracturedPorous Media, Master's Thesis, Master of Science Program, Universitt Stuttgart,November 29, 2007.


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Thank you

for your attention!

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