The Effect of the Fractal Dimension on Saturation Trajectories in Multi-phase Flow

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FISICA DEL PETROLEO REVISTA MEXICANA DE FISICA49 SUPLEMENTO 3, 1416 NOVIEMBRE 2003

The effect of the fractal dimension on saturation trajectories in multi-phase flow

M. Gonzalez V. and M. Araujo F.

Modeling and Reservoir Simulation Department, PDVSA Intevep

Recibido el 13 de diciembre de 2001; aceptado el 19 de sptiembre de 2002

Multiphase flow properties are frequently affected by the saturation history followed by the fluids up to their actual condition. In this work

we evaluate the effect of rock wettability and its geometry on fluid saturation trajectories for imbibition and drainage processes. The reservoir

rock is described by a fractal pore type model. By using the method of characteristics, we determine the region in the saturation space where

the different displacement sequences take place. A strong dependence between the area associated to the saturation trajectories, the wetting

condition of the rock, and its fractal dimension is found. These results allow us to quantify the impact of rock geometry and wettability on

displacement hysteresis phenomena as observed in natural porous media.

Keywords:Wettability; fractal dimension; relative permeability; multi-phase flow.

Las propiedades asociadas al transporte de fluidos en flujo multifasico son frecuentemente afectadas por la historia de saturacion que siguen

los fluidos hasta su configuracion actual. En este trabajo se evalua el efecto de la mojabilidad de la roca y su geometra sobre la trayectoria

de saturacion de fluidos en procesos de imbibicion y drenaje. La roca de yacimiento se describe con un modelo tipo poro fractal. Se modelan

procesos de imbibicion y drenaje en sistemas bifasicos y trifasicos, y se determina la region del espacio de saturacion asociado a las diferentes

secuencias de desplazamiento que pueden obtenerse para cada tipo de flujo a partir del metodo de las caractersticas. Se encuentra una fuerte

dependencia entre el area asociada a las trayectorias de saturacion, la condicion de mojado de la roca y su dimension fractal. Los resultados

obtenidos permiten cuantificar el impacto de la geometr a de la roca y la mojabilidad sobre los fen omenos de histeresis observados en medios

porosos naturales.

Descriptores: Mojabilidad; dimension fractal; permeabilidad relativa; flujo trifasico.

PACS: 47.55M

1. Introduction

The simultaneous flow of three phases - oil, water and gas

- occurs in a variety of displacement processes in oil and

gas reservoirs. To describe the fluid behavior under theseconditions, flow properties known as three-phase relative

permeabilities are needed. For the case of two-phase flow

(oil/water, gas/oil or gas/water) there are only two different

saturation paths since the saturation of one phase may ei-

ther increase or decrease. These are called imbibition and

drainage processes. The three phase flow case is more com-

plicated since there may be six different displacement paths

in which the saturation of one phase may either increase or

decrease. Thus the modelling of three-phase relative perme-

ability poses a particular challenge. There are three main ap-

proaches to describe three-phase flow in porous media [1]:

Use of steady-state experimental data for displacementsimulations;

Use of two-phase data to calculate three-phase param-eters by empirical models such as Stones model [2,3].

Network models that predict macroscopic parame-ters, such as relative permeabilities, directly from the

knowledge of the pore structure and the physics of the

displacement. A network could be constructed with

parameters tuned to match two-phase data, and then

used to predict three-phase relative permeabilities for

any type of displacement [4, 5].

In this work we present an extension of a simple fractal

pore model known in the literature as IFPs model [1] which

allows to introduce wettability effects on three-phase correla-

tions and derive analytically three phase flow properties. The

derived equations are used with the characteristics method toobtain the saturation trajectories of a three-phase system with

a particular geometry and wetting condition. We apply this

methodology to study pore geometry and wettability effects

over saturation space.

2. Method description

2.1. Method of characteristics

We apply the method of characteristics (MOC) by calculating

the displacement efficiencies for a three-phase (water (i= 1),

oil (i = 2), gas (i = 3)) flow problem [6]. Fractional flowfunctions f= (f1, f2, f3)are assumed to be a function of sat-uration only S= (S1, S2, S3). We neglect dissipative effects- capillary pressure and pressure-dependent fluid properties.

Under the above restrictions fluid flow is described by a pair

of partial differential equations in the variables S1(x, t) andS : 2(x, t):

S1t+ f1x(S1 , S2) = 0, (1)

S2t+ f2x(S1 , S2) = 0, (2)


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THE EFFECT OF THE FRACTAL DIMENSION ON SATURATION TRAJECTORIES IN MULTI-PHASE FLOW 15

the second index indicates differentiation with respect to that

variable. The system of equations is written in the form of a

conservation law:

St+ ASx = 0, (3)

trivially, the Jacobian matrix A is given by

A= f11 f12f21 f22

. (4)

For the solution, we search for a curve in the(x, t)space suchthat

dx

dt|S = f11= f22= , (5)

whereS1 = S1(),S2 = S2(),x = x(), andt = t().f1and f2 are expressed by

fi = i3j=1 j

, i= 1, 2, 3, (6)

wherei is the mobility of theith phase, and is defined by

i kri(S)

i, (7)

kri being the relative permeability andi the fluid viscosity.We solve the eigenvalue problem and find

=1

2

(f22+ f11)

(f11 f22)2 + 4f12f21

, (8)

from these expressions, we are able to calculate one satura-

tion as a function of the other:

dS1dS2

= f11

f12, (9)

note thatS3= 1S1S2. Saturations paths for the systemstarting from an initial condition (SI) up to an injected con-

dition (SJ) are determined. This trajectory will depend on of

three-phase relative permeability correlation used in Ref. 7.

2.2. Three-phase relative permeability model for various

wettability conditions

The fluids are distributed in pores of different sizes according

to their wetting characteristics: for a water-wet solid, water

occupies the smallest pores, gas the largest ones and oil the

intermediate size pores. We describe the structure of porous

sample by a fractal pore type model [1] which consists of abundle of parallel capillary tubes with a fractal cross-section.

The cross-section of each tube is constructed by an itera-

tive process, dividing half perimeter of a circle into a cer-

tain number of parts, and replacing each one by half a cir-

cle. In this process, a fractal object is generated. The fractal

dimensionDL is related to the number of elements gener-ated at a given scale. This property can be determined from

macroscopic properties such as the capillary pressure using

Pc S11/(DL2). To describe in a more general case the

wetting condition of a sample, a wettability index is intro-

duced. This parameter has a value between 0 and 1 according

to the wetting characteristic to be modelled (m = 1 water-wet,m = 0oil-wet). Poiseuilles law is applied to each cap-illary of the bundle to calculate the relative permeability to

each fluid. The expressions found are

kr1(S1 ) = mS1

S1r

+ (1 m) (1 (1 + S1r S1 ))4

, (10)

kr2(S1 ,S2) = (1 m)2S2

S2r

+ m

(SL) (S1r+ S2r)

, (11)

kr3(S3) = Krgmax(1 (1 S3 S3r)

)4, (12)

whereS1r, S2r andS3r are the residual saturations of eachphase, and exponents and are related to the pore geom-etry, = 1/2 DL and = 2 1. This model maybe tested experimentally by using the wettability index I asderived from an Amott-Harvey test. SinceIhas a value inthe range[1, 1], we may usem= (I+ 1)/2.

3. Results

In Fig. 1, it is shown a comparison of the applica-tion of the derived three phase relative permeability

Eqs. (10)-(12) with Stone model, commonly used in

reservoir simulation for a water-wet sample. The sat-

uration paths are very similar in this case even though

the relative permeabilities are completely different.

A marked effect on the saturation paths is observed

when the wettability and pore geometry are varied(Fig. 2). For the same boundary conditions a depen-

dence of residual oil saturation on the wetting condi-

tion is observed. Residual oil saturation is larger as the

affinity of the surface for oil increases.

FIGURE 1 . Comparison of saturation paths for Stones model and

the extended IFP model presented here, for various flow boundary

conditions (from SIato SJand from SIbto SJ).

Rev. Mex. F s.49 S3(2003) 1416


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16 M. GONZALEZ AND M. ARAUJO

FIGURE2. Effect of wettability (a) and pore geometry (b) over saturation paths.

4. Conclusions

The fractal pore model introduced by Moulu et al. was ex-

tended to derive general analytical expressions of three phase

relative permeability for a porous structure with a wide range

of wettability conditions. The fluid affinity is described by

a wetting index with values in the range [0, 1]. The MOCmethod was used to study the behavior of saturation trajecto-

ries for samples of different geometries and wettability condi-

tions. The results found indicate that the saturation paths can

be used as a tool to quantify flow effects under three phase

conditions.

1. J-C. Moulu, O. Vizika, and F. Kalaydjian,SPE38891 (1997)499.

2. H.L. Stone,J. Pet. Tech. 22 (1970) 214.

3. H.L. Stone,J. Can. Pet. Tech.12 (1973) 53.

4. V. Maini and K.K. Mohanty,SPE Jour3 (1998) 238.

5. D.H. Fenwick and M. Blunt,SPE Jour3 (1998) 86.

6. L. Lake,Enhaced Oil Recover, 2nd ed (Ed. Hannover, 1999).

Rev. Mex. F s. 49 S3(2003) 1416

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The Effect of the Fractal Dimension on Saturation Trajectories in Multi-phase Flow