205. Unified Formulation of Enhanced Oil-Recovery Methods

Abstract

En la actualidad las técnicas de recuperación mejorada de petróleo (EOR, por sus siglas en inglés) son esenciales para mantener los suministros de petróleo del mundo. A su vez, los modelos matemáticos y computacionales de los métodos EOR son fundamentales para su aplicación y perfeccionamiento. Debido a la gran diversidad de procesos que ocurren en la EOR, es valioso contar con procedimientos generales unificados para la construcción de los mismos, que puedan aplicarse fácilmente independientemente de la complejidad del sistema considerado. El objetivo de este trabajo es presentar un modelo matemático unificado, que incluya ambos: el sistema gobernante de las ecuaciones en derivadas parciales y las condiciones de choque. Éste se basa en una formulación axiomática, pues las formulaciones axiomáticas son el medio más eficaz para lograr generalidad, sencillez y claridad. En el enfoque que se propone aquí, la construcción del modelo matemático es en gran medida automática, todo lo que se requiere a fin de definir las ecuaciones en derivadas parciales y las condiciones de choque que constituyen un modelo básico consiste en identificar las fases y las propiedades extensivas que participan en el sistema de la EOR. Este modelo básico proporciona una base muy firme para incorporar la fenomenología. El procedimiento se ilustra derivando los modelos matemáticos más utilizados en la tecnología EOR; en particular, se derivan los modelos de petróleo negro y los composicionales, tanto en su versión isotérmica como sus variantes térmicas, en las cuales es indispensable incluir el balance de energía. Para ilustrar la aplicación de la formulación axiomática a modelos con discontinuidades, se realiza también una descripción exhaustiva de los choques que pueden ocurrir en el Modelo de Petróleo Negro.
doi: https://doi.org/10.22201/igeof.00167169p.2011.50.1.124

Full text

Geofísica internacional (2011) 50-1: 85-98
Resumen
En la actualidad las técnicas de recuperación mejo-
rada de petróleo (EOR, por sus siglas en inglés)
son esenciales para mantener los suministros
de petróleo del mundo. A su vez, los modelos
matemáticos y computacionales de los métodos
EOR son fundamentales para su aplicación y per-
feccionamiento. Debido a la gran diversidad de
procesos que ocurren en la EOR, es valioso contar
con procedimientos generales unicados para la
construcción de los mismos, que puedan aplicarse
fácilmente independientemente de la complejidad
del sistema considerado. El objetivo de este trabajo
es presentar un modelo matemático unicado,
que incluya ambos: el sistema gobernante de las
ecuaciones en derivadas parciales y las condicio-
nes de choque. Éste se basa en una formulación
axiomática, pues las formulaciones axiomáticas
son el medio más ecaz para lograr generalidad,
sencillez y claridad. En el enfoque que se propone
aquí, la construcción del modelo matemático
es en gran medida automática, todo lo que se
requiere a n de denir las ecuaciones en deri-
vadas parciales y las condiciones de choque que
constituyen un modelo básico consiste en iden-
ticar las fases y las propiedades extensivas que
participan en el sistema de la EOR. Este modelo
básico proporciona una base muy rme para
incorporar la fenomenología. El procedimiento se
ilustra derivando los modelos matemáticos más
utilizados en la tecnología EOR; en particular,
se derivan los modelos de petróleo negro y los
composicionales, tanto en su versión isotérmica
como sus variantes térmicas, en las cuales es in-
dispensable incluir el balance de energía. Para
ilustrar la aplicación de la formulación axiomática a
modelos con discontinuidades, se realiza también
una descripción exhaustiva de los choques que
pueden ocurrir en el Modelo de Petróleo Negro.
Palabras clave: EOR, recuperación mejorada,
modelación matemática, modelo del petróleo ne-
gro, modelo composicional, procesos térmicos.
Abstract
At present enhanced oil recovery (EOR)
techniques are essential for maintaining the oil
supplies of the world. In turn, mathematical
and computational models of the processes that
occur in EOR are fundamental for the application
and advancement of such methods. Due to the
great diversity of processes occurring in EOR, it
is valuable to possess unied general procedures
for constructing them, which can be easily applied
independently of the complexity of the system
considered. The leitmotiv of this paper is to
present a unied mathematical model, including
both: the governing system of partial differential
equations and shock conditions. It is based
on an axiomatic formulation, since axiomatic
formulations are the most effective means for
achieving generality, simplicity and clarity. In
the approach here proposed, the construction
of the mathematical model is to a large extent
automatic; all what is required in order to dene
the partial differential equations and the shock
conditions that constitute such a basic model is
to identify the phases and extensive properties
that participate in the EOR system. Such a basic
model supplies a very rm basis on which the
phenomenology is incorporated. The procedure is
illustrated by deriving the mathematical models
of the most commonly occurring EOR models:
black-oil, compositional and non-isothermal
compositional. An exhaustive description of the
shocks that may occur in black-oil models is also
included.
Key words: enhanced oil recovery, mathematical
models, black-oil model, compositional model,
non-isothermal compositional model.
85
Unied formulation of enhanced oil-recovery methods
Ismael Herrera and Graciela S. Herrera*
Received: June 7,2010; accepted: October 6, 2010; published on line: December 17, 2010
I. Herrera
Instituto de Geofísica
Universidad Nacional Autónoma de México
Ciudad Universitaria
Del. Coyoacán 04510
Mexico D.F.
e-mail: iherrera@geosica.unam.mx
G. S. Herrera*
Instituto de Geofísica
Universidad Nacional Autónoma de México
Ciudad Universitaria
Del. Coyoacán 04510
Mexico D.F.
*Corresponding author: ghz@geosica.unam.mx
oriGinal paper

I. Herrera and G. S. Herrera
86 Volume 50 number 1
Introduction
Generally, three stages of oil recovery are
identied in the production life of a petroleum
reservoir: primary, secondary and tertiary
recovery (Lake, 1989). Primary recovery refers
to the production that is obtained using the
energy inherent in the reservoir due to gas under
pressure or a natural water drive. At a very
early stage the reservoir essentially contains a
single uid such as gas or oil (the presence of
immobile water can be usually neglected) and
often the pressure is so high that the oil or gas
is produced without any pumping of the wells.
Primary recovery ends when the oil eld and
the atmosphere reach pressure equilibrium. The
total recovery obtained at this stage is usually
around 12-15% of the hydrocarbons contained
in the reservoir (OIIP: oil initially in place).
The technique of waterooding used to be
considered as an enhanced oil recovery method,
but nowadays secondary recovery most frequently
refers to waterooding. In this approach, water
is injected into some wells (injection wells) to
maintain the eld pressure and ow rates, while
oil is produced through other wells (production
wells). In secondary recovery, if the oil phase
is above the bubble point, the ow is two-
phase immiscible with water in one phase and
oil in the other one. In such a case, there is no
mass exchange between the phases. When the
pressure drops below the bubble point, due to
oil production, the hydrocarbon component of
the system separates into two phases: oil and
gas. In this case the black-oil model applies; in
such a model the oil and gas phases exchange
mass while the water phase does not. Secondary
recovery yields an additional 15-20% of the OIIP.
After secondary recovery has been completed,
50% or more of the hydrocarbons often remains
in the reservoir. The more advanced techniques
that have been developed for recovering such
a valuable volume of hydrocarbons are known
as tertiary recovery techniques or by the more
generic term enhanced oil recovery (EOR).
The terms primary, secondary and tertiary
may be confusing. For example, water injection
(a secondary recovery strategy) is often
implemented from the start in the North Sea
and cyclic steam injection is also often applied
from the start in heavy oil reservoirs. Actually,
EOR methods are employed both to obtain
additional yields from a reservoir after secondary
recovery procedures have been applied and also
to treat non-conventional elds which, due to
their difcult characteristics, require advanced
methods from the start. In this respect, the term
EOR (also called improved oil recovery; IOR)
is more adequate. A suitable denition of EOR
is: “enhanced oil recovery processes are those
methods that use external sources of materials
and energy to recover oil from a reservoir that
cannot be produced economically by conventional
means”.
The most important EOR methods can be
grouped as follows:
1. waterooding (conventional, water-
alternating-gas, or WAG, polymer ooding);
2. miscible gas injection: hydrocarbon gas,
CO2, nitrogen, ue gas;
3. chemical injection: polymer/surfactant,
caustic and micellar/polymer ooding; and
4. thermal oil recovery: cyclic steam
injection, steam-ooding, hot-water drive, in-
situ combustion.
At present a large part of the oil reserves
of the world are located in mature oil-elds
whose production is declining and for which
the only possibility for expanding their yield is
by application of EOR techniques. Another large
fraction of oil reserves are non-conventional
oil-elds that are very difcult to exploit,
either because of the characteristics of their
hydrocarbons -such as very large viscosity- or
because of the characteristics of the soils and
rocks in which they are contained. In both cases,
such reservoirs can only be exploited applying
EOR methods. Therefore, today, EOR is an
important strategy for sustaining the oil supply
of the world (Lake, 1989; Chen et al., 2006).
On the other hand, mathematical and
computational modeling (MCM) of oil reservoirs is
fundamental for the development and application of
EOR techniques, because it permits predicting and
understanding the behavior of a reservoir when it is
subjected to the complicated and varied processes
that occur in EOR techniques (Lake, 1989; Chen
et al., 2006). Among other important capabilities
supplied by MCM, applying it, it is possible to
evaluate the different production strategies and
choose that which maximizes oil recovery; also,
to correct deviations from the reservoir expected-
behavior when they occur, as well as to estimate
its production life and overall yield.
Processes to be modeled
In primary production, the processes to be
modeled are the motion of one phase or of two
phases at most (Chen et al., 2006), without mass
exchange between the phases. In secondary
production, generally the motion of a three-
phase uid system has to be modeled; the phases
being water, oil and gas. Mass exchange between

I. Herrera and G. S. Herrera
Geofísica internacional
January - march 2011 87
the oil and gas phases must be included. The
standard computational model to mimic such
a system is technically known as the black-oil
model. In Enhanced Oil Recovery, the processes
to be modeled are extremely varied; among
them, multiphase-multispecies transport in
two modalities, isothermal conditions and non-
isothermal conditions, and many more. A process
especially complex is in-situ combustion, which is
further complicated by the fact that sharp fronts
(shocks) may need to be included.
Generally, when constructing a simulator for
the processes occurring in EOR the following
models have to be developed successively: a
mathematical model, a numerical model and a
computational model (Lake, 1989; Chen et al.,
2007). The mathematical models of petroleum
reservoirs consist of a system of partial
differential equations, usually non-linear, whose
numerical solution by computational means
permits predicting the reservoir behavior. Due to
the great diversity of processes occurring in EOR,
it is valuable to possess a unied mathematical
model that can be applied independently of the
complexity of the system considered. The leitmotiv
of this paper is to present such a unied model.
Furthermore, this unied model includes
the systematic manner of treating shocks
(i.e., discontinuous solutions). The petroleum
engineering community has been aware of the
occurrence of shocks in oil-reservoir models
since Buckley-Leverett reported them (Buckley
and Leverett, 1942). Double discontinuity
shocks that occur in Black-Oil models were more
recently reported and studied (Herrera, 1996;
Herrera and Camacho, 1997). Furthermore, an
exhaustive account of those that may occur in
Black-Oil models was presented by I. Herrera et
al. (1993). As it is well known, shocks are useful
approximations to abrupt changes of the values
of physical properties that may occur in the real
systems. In EOR technology, they may valuable
for example when modeling combustion fronts
(Akkutlu and Yortsos, 2003).
Extensive properties and balance equations
Axiomatic formulations are the most effective
means for achieving generality, simplicity and
clarity. In particular, in what follows the use of
an axiomatic approach will permit us to construct
the unied general model for the very varied
processes that occur, or may occur in the future,
in Enhanced Oil Recovery.
There are two approaches to the study of
matter and its motion: the microscopic and the
macroscopic approaches. The former studies
molecules, atoms and elemental particles, while
the latter studies and models large systems.
Oil reservoirs constitute macroscopic physical-
chemical systems. The theoretical foundations of
macroscopic models lie on Continuous Mechanics.
The axiomatic method of Continuous Mechanics
was established in the second half of the XX
Century by a group of scholars and researchers
two of whose most conspicuous leaders were Noll
(1974). Recently, Allen et al. (1988), summarized
the results that are essential for the axiomatic
formulation of the basic mathematical models.
A revised and, in some respects, improved
presentation of the subject, in which we base
the developments of the present paper, is due to
appear soon (Herrera and Pinder, to appear).
Firstly, an abstract framework that is essential
in the axiomatic approach will be introduced. The
main thrust of this paper lies on the applications;
so, when explaining the axiomatic formulation that
will be applied to EOR processes, the theoretical
foundations are explained as briey as possible;
details are included just enough to make them
understandable and without sacricing clarity.
We start with a denition: An ‘extensive property’
is a body property such that, for every body
B
and every time t, can be expressed as a volume
integral over the domain, B (t)(see Figure 1), of
the physical space occupied by the body; that is,
E ( , t) ≡
∫
y
(x, t)d x (2.1)
B(t)
The integrand
y
(x, t) which represents the
extensive property per unit volume, is said
to be the intensive property associated with
the extensive property E. The main extensive
properties considered in EOR models are the mass
of different components, as well as the energy,
contained in each phase of an EOR system.
A model of a macroscopic system is said to
contain a shock when there is a surface,
S
(t), across
which at least one of the intensive properties of
the model is discontinuous. In our notation,
S
(t)
will be the union of the surfaces across which
the intensive properties are discontinuous. In
particular, if the model does not contain a shock,
S
(t) is void. The kind of discontinuities that may
occur across
S
(t) are ‘jump discontinuities’,
exclusively. By denition, a jump discontinuity of
a function is one in which the limits from each
side of
S
(t) exist, but they are different from each
other. Furthermore, the surface
S
(t) is oriented
arbitrarily (i.e., a positive and a negative side
are dened) and a unit normal vector, n, pointing
towards the positive side is taken on it. The jump
on
S
(t) of any function, f(x), is dened to be
(2.2)
B

I. Herrera and G. S. Herrera
88 Volume 50 number 1
Finally, in Eq. (2.3), the function
t
(x, t)•n at the
point x of the body external-boundary,
∂
B(t), is
the amount of extensive property that enters the
body there at time t (Figure 1), per unit area per
unit time. In EOR models,
t
(x, t) represents the
ux due to diffusion (mainly, material dispersion-
diffusion or thermal diffusion).
A purely mathematical result (i.e., a mathema-
tical theorem; see, Allen et al.,1988 and Herrera and
Pinder, to appear) shows that Eq. (2.3) is fullled
by each body of a continuous system, if and only if,
the following ‘balance conditions’ are satised:
(2.4)
and
(2.5)
Eq. (2.4) will be referred to as the “balance
differential equation”, while Eq. (2.5) is the “jump
condition”. The systems of differential equations
that constitute the mathematical models will be
derived from Eq. (2.4), while the treatment of
shocks will be based on Eq (2.5).
A unied model of EOR systems
Most EOR systems are multiphase systems, albeit
there are a few that are one-phase systems. In
any event, EOR systems are particular cases of the
most general multiphase system treated in this
Section. In the axiomatic approach that permits
such a unied treatment any multiphase system
is characterized by the following axioms:
1. A family of phases. Each phase of this
family moves with its own velocity;
2. A family of extensive properties. Each
extensive property of such a family is associated
with one and only one phase; and
Here, f+ and f— are the limits on the positive
and negative sides, respectively.
A Basic Axiom is: “An extensive property can
change in time, exclusively, because it enters
into the body. In turn, it can enter into the body
through its boundary or directly in its interior.”
The mathematical expression of this Axiom is:
(2.3)
In these statements the amount that ‘enters’
may be positive or negative. With regards to
Eq. (2.3), in the nomenclature of the general
formulation of Continuum Mechanics, Allen et al.
(1988), g (x, t) is called the external (distributed)
supply of the extensive property and
t
(x, t) is
the ux of the extensive property. As for g
S
(x,
t), it is also an external supply, but this one is
concentrated on the surface
S
(t); see: (Herrera
and Pinder, to appear). The function g (x, t) is the
amount of the extensive property per unit volume
per unit time that enters the body at the point x
of the physical space at time t, while the function
g
S
(x, t) is the amount of the extensive property
per unit area per unit time that enters the body
at the point x of the surface
S
(t), at time t. The
inclusion of such concentrated sources is essential
in order to be able to treat shocks with sufcient
generality; in particular, this will permit us to
present in Section 5 an exhaustive description of
shocks that may occur in Black-Oil models, which
was originally reported in (Herrera et. al., 1993)
and includes Buckley-Leverett’s shocks (Buckley
and Leverett, 1942) as well as those reported
by Herrera and Camacho (1997) and Herrera et
al. (1993). In particular, the double-discontinuity
models of shocks occurring in variable-bubble-
point systems is an example in which, for some
of the extensive properties that constitute the
model, g
S
(x, t) ≠ 0 (Buckley and Leverett, 1942;
Herrera et al., 1993; Herrera, 1996; Herrera and
Camacho, 1997).
Figure 1. A uid body and a shock in it.

I. Herrera and G. S. Herrera
Geofísica internacional
January - march 2011 89
3. The basic mathematical model of the
multiphase system is constituted by the system of
partial differential equations and jump conditions
that is obtained when the balance conditions
for each one of the extensive properties of the
family is expressed in terms of the corresponding
intensive property.
Once the axioms of multiphase systems have
been described in this very brief manner, what
remains is to state them with greater precision in
mathematical terms, as we do next. In particular,
with respect to Axiom 2, the manner in which
extensive properties are associated with phases
requires being described in greater detail.
The number of phases and extensive properties
of these families will be denoted by M and N,
respectively. We observe that N≥M, necessarily.
It is assumed that at each time, at each point of
the physical space, there are M particles; each
one of them corresponds to one and only one
phase. Furthermore, given any domain of the
physical space, let B
x
(t) be the set of particles
of phase
x
(
x
may be 1,..., M) that are located
at such a domain at time t. A family of M bodies
{B1 (t),..., BM (t)} , is dened in this manner.
Then, a precise statement of the association
between extensive properties and phases referred
to in Axiom 2 follows. Given any extensive
property E
a
∈{E1,..., EN}, there is a unique
x
, to be
denoted by
x
(
a
), such that it has the property
(3.1)
Clearly, here the range of
a
is {1,..., N}, while
that of
x
is {1,..., M}. In Eq. (3.1),
ya
is the unique
intensive property associated with E
a
. Thereby, we
observe that corresponding to the family {E1,...,
EN} of extensive properties, we have a unique
family {
y
1 (x, t),..,
y
N (x, t)} of intensive properties,
since the correspondence E
a↔ya
,
a
= 1,..., N is
one-to-one. Therefore, every intensive property
is also associated with one and only one phase.
The global balance conditions for each
extensive property E
a
∈{E1,..., EN} can now be
expressed, as follows (see Figure 2):
Figure 2. A body in a multiphase system.
(3.2)
Here, it is understood that
x
=
x
(
a
), where
x
(
a
) corresponds to the unique phase associated
with extensive property E
a
. Furthermore, g
a
and
g
a
S
, are the ‘supplies’ of the extensive property
a
, the rst one distributed in B
x
(t) and the second
one concentrated on
S
(t), while
ta
is the ‘ux’
eld of the same extensive property. The supply
terms, g
a
and g
a
S
, will also be referred to as the
source terms. As was mentioned before,
S
(t) is
the surface where discontinuities of any of the
intensive properties of the model may occur;
we observe that this denition is independent of
the extensive property considered. However, for
many extensive properties to be considered we
will have g
a
S
= 0, on
S
(t).
As it was seen in Section 2, the balance equation
of any extensive property can also be expressed
in terms of its corresponding intensive property
by means of Eqs. (2.4) and (2.5). So, instead of
the system of Eqs. (3.2), in what follows we use
(3.3)
We recall that the jump conditions are
fullled on
S
(t), while the differential equations
everywhere, except at
S
(t) and the outer
boundary of the continuous system.
This system of partial differential equations
and jump conditions constitutes a unied mathe-
matical model of EOR systems, which is the
leitmotiv of the present article mentioned before;

I. Herrera and G. S. Herrera
90 Volume 50 number 1
it includes exhaustively the mathematical models
of EOR systems. In the remaining of this article,
this general model is specialized to obtain the
most commonly used models of EOR technology.
In particular, we will obtain the models discussed
in the very comprehensive and complete treaty
of EOR models, authored by Chen et al. (2006).
For models of EOR systems, it is useful to
decompose the source terms g
a
(
a
= 1,..., N) into
two parts whose origins can be traced back to
the system exterior (g
a
e
) and the system interior
(g
a
I
), respectively. Thus, we write
(3.4)
The black-oil model
The basic mathematical model of a Black-Oil
system corresponds to the case when in the
unied EOR model of Section 3, the following
conditions are satised; the family of phases is
constituted by:
1. A water-phase,
2. An oil-phase (liquid), and
3. A gas-phase.
In Black-Oil and isothermal Compositional
Models the phase constituted by the solid matrix
of the oil reservoir is usually ignored because it
does not move and it does not exchange mass
with the uid phases.
There are three different substances (or
components): water, nonvolatile oil (usually
called ‘oil’) and volatile oil (usually called ‘gas’),
and only the volatile-oil may be in more than one
phase; namely, the oil and gas phases. The family
of extensive properties has four members:
1. The mass of water in the water-phase,
2. The mass of nonvolatile oil in the oil-
phase,
3. The mass of volatile-oil contained in the
oil-phase (‘dissolved gas’), and
4. The mass of volatile-oil contained in the
gas-phase.
Let
y
i (i = 1,2,3,4) the intensive properties
associated with each one of them. In this case
the function
x
(
a
) is:
x
(1)=1,
x
(2)=2,
x
(3)=2 and
x
(4)=3. Then, the most basic mathematical model
of Black-oil systems is given by Eq.(3.3); that
is:
(4.1)
The jump conditions are:
(4.2)
However, the jump conditions will not be
discussed in this Section; instead, in the next
one they will be used for making an exhaustive
analysis of shocks that may occur in Black-Oil
models.
Explicit expressions, as integrals, of these
extensive properties are:
(4.3)
So, the corresponding intensive properties are
(4.4)
Furthermore, in what follows, we shall replace
the superscripts 1,2,3,4, above, by w, Oo, Go and
g, respectively.
Therefore, the basic system of differential
equations of the Black-Oil model is constituted
by the following system of equations

I. Herrera and G. S. Herrera
Geofísica internacional
January - march 2011 91
(4.5)
In the Black-Oil model dispersion-diffusion
processes are neglected, so that
ta
= 0 (= w, Oo,
Go, g). As for the internal sources, g
a
I, all them
are zero except when
a
= Go, g. In the absence of
chemical reactions, as it is assumed in the Black-
Oil Model, the mass of volatile oil is conserved
and then:
(4.6)
Hence, Eq. (4.5) reduces to:
(4.7)
Generally, the external sources g
a
e are due to
wells; they are negative if extraction occurs and
positive if injection takes place. In particular, the
terms —g
w
e and —g
Oo
erepresent the mass of water
and non-volatile oil extracted from the water and
oil phases, respectively, while —g
Go
e and —g
g
e
represent the mass of nonvolatile-oil extracted
from the oil and gas phases, respectively.
Starting with Eq. (4.7) it is easy to derive the
standard forms of the mathematical model of
Black-Oil reservoirs; in particular, for comparison
we will use that presented in Chen et al. (2006).
To this end we introduce the Darcy velocity, for
each phase, which is dened by
(4.8)
Then, adding up the last two equalities of Eq.
(4.7), we get
(4.9)
together with
(4.10)
or
(4.11)
Eqs. (4.10) and (4.11) are not usually
mentioned in the literature, since they are a
sub-product of the methodology here applied
for deriving the basic differential equations.
However, they may be useful in some instances
since either one of them can be applied for
evaluating the nonvolatile-oil exchanged by the
oil and gas phases.
The rest of the derivation is standard. Eqs.
(4.9) can be transformed into (see Chen et al.,
2006):
(4.12)
This is achieved introducing the following
notation (further details of the symbols used
here are given at the end of this Section):
(4.13)
The formation volume factors, Bw, B0 and Bg
are dened by
(4.14)

I. Herrera and G. S. Herrera
92 Volume 50 number 1
Eqs. (4.12) have to be complemented by
several constitutive equations relating the variables
occurring in them. In particular, Darcy’s law
(4.15)
the saturations identity
(4.16)
The capillary pressure equations
(4.17)
which relate the phase pressures. Both pcow and pcgo
are functions of other parameters of the reservoir
system, the saturations in particular, which must
be determined in advance, experimentally or
otherwise. The gas-oil ratio equation of state:
(4.18)
which is satised when Sg > 0; i.e., when the
reservoir is truly a three-phase system.
The following list of symbols complements
the notation already explained in the text of this
Section:
From the mere fact that the system consists
of three phases and one of them contains
two components, while each one of the other
two contains only one, without any additional
knowledge, we have derived the system of
equations given by Eqs. (4.1) and (4.2). Such
equations give a very rm basis for incorporating
all the scientic and technological knowledge
available about the system, as we have done, to
obtain as nal product the mathematical model
of Black-Oil systems.
Analysis of shocks that may occur in black-
oil models
To start with, we observe that Darcy’s Law
implies that the pressure is continuous and, if
the solid-matrix does not have abrupt changes
in properties, as in the contact between two
geological formations of different origin, the
porosity is also continuous. Then, the shock
conditions of Eq. (4.2) when the external source
terms vanish can be written as:
(5.1)
where Rs(p, T)is the gas-oil ratio dened by
r
Go
= Rs
r
Oo.
A. In a region where the gas-phase is
absent
The density
r
Oo is continuous across the shock
S
because so is the pressure po. When the gas-
phase is not present Sg = 0, so that Eq. (5.1)
implies gGo = gg = 0, and implies
(5.2)
where the notation is such that for any function
f, is the the average of the function f
across
S
.
From the rst equality in Eq. (5.1) it follows
that when , the shock velocity is
given by
(5.3)
On the other hand, from Eq. (5.2) it follows
that when , the shock velocity is given
by
(5.4)
S
g
S
Go

I. Herrera and G. S. Herrera
Geofísica internacional
January - march 2011 93
Generally,
(5.5)
Therefore, if these two kinds of shocks
coincide at any time they will separate, since
their velocities are different. Thus, at a shock
either , in which case and
the velocity of the shock is given by Eq. (5.3),
or , in which case and the
velocity of the shock is given by
(5.6)
B. At a gas front
For simplicity in what follows we only consider
the case when the water phase is not present.
At a front that advances into a region of under
saturated oil, we take the unit normal vector
pointing towards the under saturated region
(Figure 3). Adding up the last two equalities in
Eq. (5.1), one gets:
(5.7)
In the under-saturated oil region Sg = 0; i.e.
(Sg)+ = 0, on
S
.
According to the second equality in Eq. (5.1),
So(vo — v
S
) is continuous across the shock and
therefore, from Eq. (5.7), it follows that
(5.8)
Using the identity ,
it can be seen that
(5.9)
With
(5.10)
The physical interpretation of this result is
that when a gas front advances into a region of
under-saturated oil, it is retarded in its motion
by the ‘retardation factor’
e
(see Herrera, 1996,
and Herrera and Camacho, 1997).
C. In a saturated oil-region (Buckley-
Leverett theory)
This is the case of a biphasic system in which the
gas-phase and the liquid oil-phase coexist, which
was treated by Buckley-Leverett in their classical
theory (Buckley and Leverett,1942). In this case
, because the oil is saturated and the
pressure is continuous, Eq (5.1). reduce to
(5.11)
This is analogous to Eq. (5.2). Corresponding
to Eq. (5.3), we now derive the equation:
(5.12)
or,
(5.13)
A special case of this equation is the
immiscible and incompressible case considered
by the classical Buckley-Theory, for which Eq.
(5.13) becomes the well-known relation (see, for
example, Herrera and Camacho, 1997):
(5.14)
Figure 3. A gas-front that advances into a region of
under saturated oil.

I. Herrera and G. S. Herrera
94 Volume 50 number 1
Here,
(5.15)
The compositional model
In this Section we apply the general scheme of
Section 3 to obtain the basic mathematical model
of compositional oil reservoirs. The derivation
need not be as formal as in Section 4, since the
general protocol has already been illustrated
there.
The characteristic features of compositional
models are:
1. The family of phases is the same as before:
the water-phase, the oil-phase and the gas-
phase. The notation used for the velocities of
such phases is also the same:
i) vw is the velocity of the water-phase;
ii) vo is the velocity of the oil-phase; and
iii) vg is the velocity of the gas-phase.
2. Besides the water, we distinguish Nc chemi-
cal substances; these substances do not change
chemical composition through time. However,
through time, each one of them is exchanged by
the oil and gas phases. The family of extensive
properties consists of 2Nc + 1 members; namely,
i) The mass of the water-phase, denoted by
Mw;
ii) The mass of the i-th component in the oil-
phase, denoted by Mio(t), i = 1,...,Nc; and
iii) The mass of the i-th component in the
gas-phase, denoted by Mig, i = 1,...,Nc.
The notations used for the corresponding in-
tensive properties, supplies (i.e., mass-sources)
and ux-elds are:
(6.1)
3. The basic mathematical model of the
multiphase system is constituted by the system of
partial differential equations and jump conditions
of Eqs. (3.3); i.e.,
(6.2)
and
(6.3)
In Eq. (6.2), the source-terms have been
expressed as in Eq. (3.4).
Eqs. (6.2) and (6.3) constitute the basic
mathematical model of ‘compositional’ EOR sys-
tems. A model capable of predicting the system
behavior, when it is subjected to suitable initial
and boundary conditions, will be derived from it
by incorporating sufcient scientic and techni-
cal information about the actual system following
the protocol leitmotiv of this article.
The integral expressions for each one of the
2Nc + 1 extensive properties are:
(6.4)
Therefore,
(6.5)
Furthermore, in the Compositional Model
diffusion-dispersion processes of the components
are neglected; this implies that all the ux-elds
are zero; i.e.,
(6.6)

I. Herrera and G. S. Herrera
Geofísica internacional
January - march 2011 95
According to our assumptions , since no
matter is exchanged with the water-phase.
When all this is taken into account, the basic
mathematical model for compositional systems,
of Eqs. (6.2) and (6.3), becomes:
(6.7)
and
(6.8)
The contributions to g
a
e,
a
= w, io, ig, generally,
are due to extraction or injection through wells.
We observe that the following conditions are
fullled:
(6.9)
since the mass of each component must be
conserved.
Mass interchange between phases is determi-
ned by the mass distribution of each component
in the oil and gas phases through the condition
of stable thermodynamic equilibrium (Chen et al,
2006). This condition is expressed by means of
the equation
(6.10)
Here, fio and fig are the fugacity functions of the
component in the oil and gas phases, respectively.
For the application of the stable thermodynamic
equilibrium condition, the following notation and
nomenclature is relevant:
.
(6.11)
Here, Ww and Wi are the “molar masses”. On
the other hand
x
w,
x
io, and
x
ig are the “molar
densities” of the water, the i—th component in
the oil-phase and of the i—th component in the
gas-phase, respectively. Furthermore,
xa
is the
“molar density of phase
a
”. The “mole fraction
of component i in phase
a
”, is
c
i
a
. Since the
condition of stable thermodynamic equilibrium
is expressed in terms of the mole fraction of
component i in phase
a
, it is advantageous to
express the basic equations of the compositional
model in terms of such parameters. Then
(6.12)
Adding up the last two equalities appearing in
Eq. (6.12) and using Eq. (6.9), we get
(6.13)
As in the case of the Black-Oil model, the
last two equalities in Eq. (6.12), can be used
to evaluate the supplies gio
ig and gig
io due to
exchange. We recall that for saturated ows,
these equations are supplemented with the
saturation identity:
(6.14)
Other equations that are applied in order to
obtain a model capable of predicting the system
behavior are:
Darcy’s Law,
(6.15)
where kr
b
and
mb
are the relative permeability
and the dynamic viscosity of the
b
phase, and
is the absolute permeability.
Mole fractions identities,

I. Herrera and G. S. Herrera
96 Volume 50 number 1
(6.16)
Capillary relations,
(6.17)
here pcow and pcgo are the capillarity pressures of
oil and water and gas and oil, respectively, and p
b
is the pressure of the
b
phase. Equations (6.13)-
(6.17), together with the stable thermodynamic
equilibrium condition of Eq. (6.10) constitute
a 2Nc + 9 system of equations for the 2Nc +
9 dependent variables
c
io,
c
ig, u
a
, p
a
and S
a
,
a
= w, o, g, i = 1,...,Nc, which when subjected to
suitable boundary and initial conditions provide
a model capable of predicting the behavior of the
compositional system.
The non-isothermal models
As stated before, in Black-Oil and Isothermal
Compositional Models the phase constituted by
the solid matrix of the oil reservoir is usually
ignored because it does not move and it does
not exchange mass with the uid phases.
However, the solid matrix cannot be ignored
when formulating non-isothermal models since
then energy balances have to be included and
the solid phase plays a signicant role in them.
In turn, this is due to the fact that rocks and
other materials that form the solid matrices of oil
reservoirs have large heat capacities, frequently
larger than the uid phases participating in the
reservoir systems.
Thus, the basic mathematical model of the
non-isothermal compositional systems here
discussed, has the general features that follow.
The family of phases has four members: the
solid matrix, the water-phase, the oil-phase and
the gas-phase. The velocity of the solid phase is
zero, while the notation used for the velocities of
the other phases is the same that has been used
in previous Sections. The family of extensive
properties has 2Nc + 5 members. The rst 2Nc + 1
are the same as for the Compositional Model. So,
we only discuss here the other four; they are:
i) The total energy of the water-phase, denoted
by Ew (t);
ii) The total energy of the oil-phase, denoted
by, Eo (t);
iii) The total energy of the gas-phase, denoted
by, Eg (t); and
iv) The total energy of the solid-phase,
denoted by, ES (t)
For simplicity jump conditions will not be
discussed. The differential equations required to
complete the non-thermal compositional model
are obtained applying Eq. (3.3). The total energy
of each one of the phases is given by
(7.1)
Where U
a
is the specic internal energy (per
unit mass) of phase
a
. Due to the smallness of
the velocities occurring in ow of uids through
porous media, inertial effects are neglected and
the kinetic energy is taken to be identically zero.
Therefore, we take
(7.2)
Using Eqs. (7.1) and (7.2) it is seen that the
intensive properties that correspond to the total-
energies of the different phases are
(7.3)
Since we are dealing with total energy, the
energy sources should be decomposed into two
parts: heat sources and mechanical-energy
sources (Herrera and Pinder, to appear). A similar
decomposition applies to the energy uxes.
This is better understood making the analysis
in integral form. In such a form, the balance
equations for the extensive properties associated
with the different uid phases (
a
= w,o,g) are:
(7.4)
Here, the heat sources are:
f
S
ara
h
a
and q
a
L
.
The former represents the rate per unit volume
of the physical space at which internal energy is
supplied to the
a
phase by sources distributed in
the body-interior (due, for example, to exothermal
chemical reactions), while q
a
L
is the heat loss of the
a
phase to the overburden and underburden, also
per unit volume of the physical space. The term
—
gra
(
ua
)z (
g
is the gravity acceleration and (
ua
)z is
the component of
ua
in the z direction) represents
the mechanical work done by the gravity force
on phase
a
. Furthermore, we put together the
exchange of energy between the phases in the

I. Herrera and G. S. Herrera
Geofísica internacional
January - march 2011 97
term gE
a
I, which represents the total energy
that enters phase
a
from other phases. As for
energy uxes that enter the
a
phase through its
boundary, they are given by q
a
and
f
S
asa
v
a
. The
former is the heat ux, while the latter comes
from mechanical work done on the boundary of
a
bodies. Finally, we observe that
(7.5)
For the non-isothermal compositional model
to be developed, the mechanical work done by
the gravity force will be neglected (i.e.,
g
(u
a
)z 0).
Also, the work done by viscous forces will be
neglected and, so, the stress tensor acting on
phase
a
will be . Therefore,
(7.6)
and Eq. (7.4) becomes
(7.7)
This balance equation, when expressed in
terms of the intensive property, becomes
(7.8)
It can also be written as
(7.9)
By denition, the “enthalpy per unit mass” of
the uid phase H
a
(
a
= w, o, g), satises
(7.10)
Therefore, for
a
= w, o, g, Eq. (7.9) is
(7.11)
The balance equation for the extensive
property associated with the energy of the solid
phase is:
(7.12)
This balance equation, when expressed in
terms of the intensive property, is:
(7.13)
Adding up Eq. (7.13) and the four equalities
of Eq. (7.11), we get
(7.14)
Here, we have applied Eq. (7.5) and written
as well as
. They are refe-
rred to as total heat ux and overall heat source
term, respectively.
A very important assumption that is made
in the non-isothermal compositional model that
we are presenting is that, at each point of the
oil reservoir, the different phases reach thermal
equilibrium instantly, which implies that all the
phases have the same temperature, denoted by
T, at each point. Then, the total heat ux is given
by an overall Fourier Law:
(7.15)
This leads to the following form, of Eq. (7.14):
(7.16)
Here, qc is referred to as the overall heat
source term and it is dened by
(7.17)
Conclusions
Mathematical and computational models of the
processes that occur in enhanced oil recovery
(EOR) technology are fundamental for the
application and advancement of such methods.
At least the following three stages can be
=

I. Herrera and G. S. Herrera
98 Volume 50 number 1
distinguished in the development of EOR models:
construction of a mathematical, a numerical
and a computational model, respectively. In
particular, the construction of the mathematical
model is the starting point and base on which
the remaining construction is built. Due to the
great diversity of processes occurring in EOR, it
is valuable to possess a general and systematic
procedure for constructing their mathematical
models, which can be used as a unied protocol
when building the corresponding computational
simulators. In this paper we have presented such
a procedure, based on an axiomatic formulation
of Continuous Mechanics (Allen et al., 1988),
which is systematic, rigorous and easy to apply
independently of the complexity of the system
considered.
As it is here explained, mathematical models
of EOR processes are constituted by a system
of partial differential equations together with
a system of conditions, the jump conditions,
which model discontinuities must fulll when
and where they occur; albeit, in standard
treatments the jump conditions are not usually
discussed. When building the mathematical
model of an EOR system, the following stages
can be distinguished: rstly, a basic system of
partial differential equations and jump conditions
that only depend on the number of phases and
components in each phase are established –given
in Eq. (3.3)-; and secondly, the phenomenology
is incorporated into it. In this paper, the system
of partial differential equations and jump
conditions derived in the rst step is referred to
as the ‘basic mathematical model’. This supplies
a very sound and rm basis for a second step,
which consists in incorporating other scientic
and technological information that is required
to complete the mathematical model; this latter
purpose is achieved by means of certain number
of constitutive equations such as Darcy’s Law,
chemical laws, results of eld measurements
and many more. Specic applications of the
procedures here introduced have already been
made in the development of EOR projects such as
water-alternating-gas injection and air injection
methods.
In standard approaches, the system of partial
differential equations of the basic mathematical
model is derived by means of balances that are
carried out in cubes (or parallelepipeds) of the
physical space, putting together the different
phases of the system and the jump conditions
are not usually discussed. In the protocol here
proposed, on the other hand, such balances
in cubes are not required since instead the
starting point is the system of partial differential
equations and shock conditions of Eq. (3.3).
The procedure is systematic, rigorous and easy
to apply independently of the complexity of
the system considered. In Sections 4 to 6, we
have shown that when this approach is used for
constructing the basic mathematical models,
such construction is to a large extent automatic;
all what is required in order to dene the partial
differential equations and the jump conditions of
the mathematical model is to identify the phases
and species, as well the energy sources, that
participate in the EOR system.
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