isehunwaIJMA13-16-2014

Int. Journal of Math. Analysis, Vol. 8, 2014, no. 13, 635 - 648

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2014.4263

Mathematical Analysis of Oil Displacement

by Steam in Porous Media

S. O. Isehunwa and G. K. Falade

Department of Petroleum Engineering, University of Ibadan, Nigeria

Copyright 2014 S. O. Isehunwa and G. K. Falade. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

Steam flooding is an important thermal enhanced oil recovery technique for light and

viscous crudes. However, the processes involved are complex and not yet very well

understood with a consequent impact on effective design and evaluation. In this study,

the general transport equation describing the non-isothermal flow of fluids through

porous media is formulated and solved using the method of characteristics. The

resulting ordinary differential equations were solved analytically to obtain the profiles

of specific fronts. The final results show that steam flooding is a generalized non-

isothermal fluid transport phenomenon which incorporates a steam (gas) drive, hot

water and cold water flooding to achieve oil recovery. There is also a separate

temperature front which should not be neglected.

Keywords: Steamflooding, front tracking, enhanced oil recovery, method of

characteristics, Fluid transport equations

1.0 Introduction

Steam f1ooding which has been widely recognized as a viable enhanced oil recovery

method, accounts for a significant percentage of oil being produced in the industry. It

636 S. O. Isehunwa and G. K. Falade

has proved particularly useful in the recovery of heavy oils which constitute a large

share of world oil resources (Xiao-Hu and Hui-Qing, 2012), but was for long

overlooked because of the difficulties and costs involved in the recovery processes

(Ebrahimi, 2012). However, increasing oil prices and advances in technology have

made improved oil recovery techniques like steam flood to become more economical

(Wilkey, 2012). In China, more than 90% of heavy oil recovered depends on steam

stimulation or steam drive (Wang, 2012). By taking advantage of its adaptabi1ity to a

wide variety of reservoirs, ease of control, favourable oil mobility and induced gas

drive, many steam flood projects have been successfully implemented (Hoffman and

Kovscek, 2004, Van Dijk, 1972, De Hank and Shenk, 1969).

Numerical and experimental studies have demonstrated that steam flooding gives lower

residual oil saturations and higher recoveries than cold or hot water floods

(Mousarimirkalaei, et.al 2011, Lashanizadegan et al, 2008). Egbogah et al (2003)

opined that steam flooding is the most effective enhance oil recovery method currently

in use and offers higher oil recovery than the less expensive cyclic steam injection.

They also listed the important factors affecting the efficiency of steam operations to

include high oil saturation, high reservoir thickness, high injection rates and pressure.

Steam flooding involves a complex interplay of thermodynamic and physical changes in

the reservoir which enhances the volumetric sweep efficiency and displacement of

reservoir oil. The essential features include thermal expansion of oil, improved mobility

due to viscosity reduction, steam distillation effects, reduction of residual oil saturation,

in-situ solvent action, gas drive and dragging, relative perneabi1ity reduction and

gravity segregation (Dutt and Mandal, 2012). However, operational challenges that

must be addressed to ensure a successful steam flood project include considerations of

geometry and wettability effects (Guzman and Fayers, 1997; Gonzalez and Araujo,

2002) as well as steam and hot water channeling (Lau, 2012; and Zheng et al, 2013).

Experimental correlations, pilot field tests, physical (vacuum) models, and several

analytical and numerical models have been established as useful techniques for the

process design, understudy and reservoir engineering evaluation of steam flooding. A

very practical approach to the mathematical description and evaluation of the process is

to apply the Buckely and Leverett (1942) frontal advance theory. This approach, when

combined with the Welge (1952) analysis method could produce very simple but

competent and reliable technique of tracking system variables and predicting

performance of a steam flood.

Several attempts have been made to develop analytical models for evaluating steam

floods (Marx and Langenheim, 1959; Landrum et al, 1960; Boberg and Lantz, 1966;

Mandl and Volek, 1969; Neuman, 1975; Myhill and Stagemeier, 1978; Gomaa, 1980;

Mathematical analysis of oil displacement 637

Jones, 1981 and Chandra, 2005). Unfortunately, most of these methods considered

direct steam drive of oil and neglected steam condensation and the very important hot

water zone. The analytical approach for three-phase Buckley-Leverett problems

presented by Guzman and Fayers (1997) also neglected temperature effects.

It is now widely accepted that during continuous steam injection, three separate fluid

fronts are created: a cold water front, hot water and distillate front and the steam front

(Bruining and van Duijn, 2000; Hoffman and Kovscek, 2004). This study applied the

front tracking technique to the steamflood problem and demonstrates that the hot water

zone also makes a significant contribution to oil recovery during steamfloods.

2.0 Theoretical Framework

The proper extensive property of a system can in general be defined in terms of an associated intensive property or local volumetric density parameter, as:

(1)

Where,

dv is some elemental volume of the system.

While the function is a positive-definite constant for the system, the function may

change with time and position within the system. For example, if the function

represents the total mass or total energy, then is the local density function or

enthalpy function of the system respectively. In any transport system, the proper

extensive property is fully accounted for by equating the net rate of accumulation at any

point in space to the net convergence of the flux and the net source/sink strength at that

point. This is the conservation theory which can be expressed in terms of the density

parameter as:

(2)

In equation (2), v is the velocity vector of a particle characterized by the property ,

while H is the local source/sink strength density.

A large variety of transport processes in porous media can be described by a form of

equation (2) and can be solved using the method of characteristics to give solutions

known as characteristics solutions or front tracking by describing the variation of an objective variable along a specified path within the domain of interest.


2.1 Application to Steam Flooding

The generalized transport equation (2) can be expressed in a more compact form as:

(3)

These functions may be functions of a dependent vector U(z, t) which is continuous in

the (z, t) domain and equation (3) can be expressed in terms of the solution vector U(z,

t) by completing the indicated differentiation, as:

(4)

Where, A0 and are the Jacobian matrices given as:

(5)

and,

(6)

If and are positive-definite and therefore have inverse, then equation (4)

can be arranged to yield:

(7)

Where,

and,

There is always a computational advantage in ensuring that the characteristics matrix A

is not only positive-definite, but is also symmetric. Equation 6 can be converted to the

homogenous form by defining a new independent variable x given as:

(8)

Thus equation (7) becomes:

(9)

The steam flood process in oil reservoirs can be considered as a three-phase, non-

isothermal fluid-flood problem. Thus, a conservation equation can be formulated in


terms of both the mass and energy balance and simplified to give the matrix-vector

equation:

0

DD x

UA

t

U (10)

Where, U is the vector of the dependent variables.

D

w

g

T

s

s

U

A is the 3 x 3 coefficient matrix, defined as:

wg21

wg21

D

w*

w

w

w*

w

g

w*

w

D

g*

g

w

g*

g

g

g*

g

ss

ff00

T

fY

s

fY

s

fY

T

fY

s

fY

s

fY

A

Equation (10) can be used to describe the steamf1ood process in porous media, and can

be solved using the method of characteristics (Wingard and Orr, 1994). From the theory

of generalized simple waves, it can be shown that the eigenvalues , of the characteristic matrix A, expressed in the form of equation (11) describes flow velocities

along the path (x,t):

dt

dx (11)

Solving equations (10) and (11) using the method of characteristics, i are obtained as:

w

g

g

w

wg

w

w

w

g

g

g

w

w

w

g

g

gs

f

s

fY

s

f

s

f

s

f

s

f**

2

****

1 Y4YYYY2

1 (12)

g

w

w

g

wg

w

w

w

g

g

g

w

w

w

g

g

gs

f

s

fY

s

f

s

f

s

f

s

f**

2

****

2 Y4YYYY2

1 (13)


wg

wg

ss

ff

21

21

3

(14)

Equations (12) and (13) can be further simplified by approximating respectively to:

w

w*

w

g

g*

g1s

fY

s

fY

(15)

and,

02 (16)

Thus conceptually, the 1-characteristics (equation 12), corresponds to a strong wave while the 2-characteristics (equation 13) represents a very weak wave along the same direction.

2.2 Characteristic Relations

In studying the characteristics curves in the (Sg, Sw, TD) domain, the left-hand eigen-

vectors i of the matrix A corresponding to the eigen-values i can be determined, and expressed as:

i

D

w*

w

i

g

g*

gi

g

w

D

g*

w

*

g

g

g*

gi

g

w*

w

321

T

fY

s

fY

s

f

T

fYY

:1:

s

fY

s

fY

:: (17)

Using equations (12), (13) and (17) along with the coherence or regularity condition of

the form:

0d

Udi

(18)

We obtain the characteristic relation:

0

*

*

**

*

*

D

i

D

w

w

i

g

g

gi

g

w

D

g

wg

wg

g

g

gi

g

w

w

dTT

fY

s

fY

s

f

T

fYY

dsds

s

fY

s

fY

(19)


Equation (19) is a non-linear ordinary differential equation with three variables.

However, it can be simplified and solved analytically using the front-tracking method

which is known to give accurate resolution of discontinuities (Lia and Juanes, 2004):

2.2.1 Characteristic Directions corresponding to 1 and 2 Along a constant water saturation front, dSw = 0, hence equation (19) reduces to:

g

w*

w

g

g*

gi

i

D

w*

w

i

D

g*

g

sD

g

s

fY

s

fY

T

fY

T

fY

dT

ds

w

(20)

Similarly, along a constant steam saturation front, dSg = 0, so that the equation (19)

simplifies to

g

g*

gi

g

w*

w

i

D

g*

g

i

D

w*

w

sD

w

s

fY

s

fY

T

fY

T

fY

dT

ds

g

(21)

Equation (21) is very instructive in that at a front of zero steam saturation,

i

D

w

0sD

w T

f

dT

ds

g

(22)

where,

w1

w1

s

f

(23)

Equations (21) and (22) describe the steam condensation front which in practical terms

is the hot water front and are identical to those obtained by Fayers (1962) for the hot-

water flood case.

Furthermore, TD is a function of the domain variable (x, t). Therefore,


D

D

D

D

D

D

D

D

t

T

t

x

x

T

dt

dT

(24)

The variation of saturation with time can thus be deduced by modifying equation (20)

with equations (19) and (24) to obtain:

D

D

g

w*

w

g

g*

gi

D

w*

w

D

g*

g

D

g

dx

dT

s

fY

s

fY

T

fY

T

fY

dT

ds

(25)

Similarly, the saturation history of the constant water saturation front ahead of the

condensation front can be obtained by combining equations (19), (22) and (24) to yield:

D

D

D

w

D

g

t

T

T

f

dt

ds

(26)

2.2.2 Characteristic Directions corresponding to 3

Along this characteristic path, 3 = and therefore using equation (19) we have:

0wS

g

D

ds

dT (27i)

and

0Sgw

D

ds

dT (27ii)

Similarly, the time-space description gives:

0D

g

dt

ds (28)

and

0D

w

dt

ds (29)

Equations 27(i) and (ii) imply that the characteristic curves lie on constant temperature

planes. Furthermore, the composition route describing the relationship between Sw and

Sg for any constant temperature plane is derivable from equation (19) as:


i

g

g

g

g

W

W

T

g

w

s

fY

s

fY

ds

dsD

*

*

(30)

Equation (30) was first presented by Pope (1980) for three component isothermal

processes, where .1** gW YY

2.3 Shock Formation

Since the system of equations (Equation (10)) is hyperbolic, the solutions will exhibit

shock characteristics (Fayers, 1962) just like the solution of the norma1 water flood

system. These shocks or discontinuities in Sw, Sg and TD are determined by mass (and

energy) conservation considerations. Thus, using the method of Fayers by applying

mass conservation across the shock fronts yields the Rankine-Hugoniot relations

(Courant and Friedrichs, 1948) as:

;*

gg

gg

g

D

g

ss

ffY

dt

d for the gas phase (31)

and

;*

WW

WW

W

D

W

ss

ffY

dt

d for the water phase (32)

While the energy conservation considerations similarly give the temperature shock as:

D

T

dt

d (33)

or,

WW

o

w

g

o

s

WW

o

w

g

o

s

D

T

ssss

ffff

dt

d

g

g

11

11

(34)

Where the +s are values ahead of the shock while the s are values just behind the shock (Courant and Friedrichs, 1948, Fayers, 1962).


2.4 Oil Recovery

Oil recovery calculations are carried out by applying the velocity equations (equations

12 and 13) using the Welge analysis technique. Thus, before breakthrough, oil recovery

is equal to the same pore volume of steam injected and can be expressed as:

W

W

W

g

g

g

i

s

fY

s

fY

W

**

1 (35)

At hot water breakthrough, the oil recovery becomes:

bt

W

W

W

g

g

g

pbtwi

s

fY

s

fY

N

**

1 (36)

as compared to a conventional water flood where,

bt

W

W

pbtwi

s

fN

1 (37)

3.0 Discussion of Results

Equation (10) and the subsequent characteristic solutions and relations (equations (12)

to (17)) show that steam flooding is a generalized three-phase, non-isothermal fluid

transport phenomenon which incorporates most other specialised fluid-flood cases such

as the hot water flood and cold water flood. The equations should help us to gain more

insight into the steam flood process. Specifically, equations (14) and (15) describe the

characteristic paths of temperature and of strong material wave propagation

respectively. Equation (15) can be used to generate the saturation distance profiles and

is also significant in that it shows that contributions to oil recovery in a steam flood

process derive from both "gas drive" and "water drive", a fact which is well known but

seldom shown quantitatively in the literature. When equations (14) and (15) are

considered together, they show that whenever the temperature shock front is behind the

fluid shock front, a separate cold water front distinct from the hot water front should be

expected.

To be able to use equation (15), the terms *gY and *

wY must be estimated. For processes

without interphase mass transfer these terms strictly assume values of unity. However,


where there is interphase mass transfer, like the steamflood processes, 0 < *iY < 1 and

the values can be estimated empirically.

In general, three regions are represented in a steamflood process: the steam, hot water

and cold-water regions. This observation confirms several reported experimental and

pilot studies (Bruining and Duijn, 2000; Hoffman and Kovscek, 2004; Lashanizadegan

et al, 2008). This observation also emphasizes the fact that the hot water zone

contributes significantly to oil recovery in a steamflood process. Thus, the steam drive

analysis models presented by researchers that such as Landrum et al (1960) that

neglected the contribution of the hot water zone are inadequate.

It is obvious from equation (36) that oil recovery from steam flooding is driven by both

water drive and gas drive, the relative contribution of each drive will be determined by factors such as the reservoir geometry and the nature of the crude, whether light or

viscous. Increases in recovery with steam temperature rise are only marginal hence it is

doubtful if the use of superheated steam can be more economical than using saturated

steam.

Finally, it is obvious that the established Welge-type analysis technique used for

reservoir engineering evaluation of normal water floods can be employed for

steamfloods. This approach no doubt offers a very fast, simple and reliable method of

systematic reservoir evaluation of steam floods.

4.0 Conclusion

From the foregoing, the following conclusion can be made:

1. The analytical model of the steam flood process in oil reservoirs can be considered as a generalized three-phase, non-isothermal fluid-flood model

which is amenable to analytical solution and the front-tracking analysis

technique just as has been widely used for cold and hot water flooding.

2. Mathematical analysis shows that oil recovery during steam flooding is driven essentially by both a gas or steam drive and water drive. The contribution of the

displacement by water could be significant and should not be neglected in the

analysis.

3. There is a temperature front that is clearly established during steam flooding and this should not be neglected as has been done by several researchers.


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NOMENCLATURE

H = Local source/sink density strength

T = Temperature

Wi = Pore volume of steam injected, dimensionless.

X = Distance [L]

Y,Yi* = Interphase mass transfer modification parameters

, 1 , 2 = Parameters defined by eqn.

, 1 = Parameters defined by eqn.

1, 1 = Parameters defined by eqn. (14)

I = Parameter defined as i cpi

= Characteristic path

= Characteristic path

= Density [ML-3]

= Eigenvalue, characteristic velocity [LT-1]

= Porosity, dimensionless

= Viscosity [ML-1T-1]

i = Left hand eigenvectors.

= Extensive Property

= Intensive Property

Subscripts

tb = Breakthrough

D = Dimensionless

F = Front

g = Gas, steam

o = Oil

W = Water

s = Steam

r = Reservoir

R = Rock

Received: February 21, 2014

isehunwaIJMA13-16-2014

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