**An Overview Paper Being Offered to the
Editor-in-Chief of **

**Elsevier's Journal of Petroleum Science &
Engineering **

**for Peer Review and Possible Publication.**

**By Walter Rose **

**(edited on 12/5/2003)**

**Table
of Contents **

**ABSTRACT**

**key words**

**INTRODUCTION**

**Background Ideas**

*DISCUSSION*

**Ad-hoc Theory for Simple Systems**

**Formulating More Complicated Cases**

‘**Unfinished’ Capillary Imbibition Algorithms**

**CONCLUDING REMARKS**

**NOTATIONS**

**AKNOWLEDGEMENT**

**REFERENCES**

**APPENDICES ADDED IN PROOF **

**1. Equation Road Map**

**2. Ad-Hoc Algorithms**

**ABSTRACT
**

** The
content of the watershed paper of Buckley & Leverett (1942) ^{6}
is being revisited here to further resolve some algorithm formulation
difficulties examined by the present Author some 15 years ago
(1988)^{19}. Questions long-overdue are addressed again
about the possibility of further upgrading the basic classical
Buckley/Leverett Algorithms so that their use will better yield
intended trustworthy predictions of future reservoir states. In a
nutshell, our simple aim here is to search for practical ways to
facilitate the monitoring of certain types of petroleum recovery
transport processes. Specifically, these occurrences are ones that
typically transpire during production of oil from subsurface
reservoirs which exist as scattered local features associated with
many worldwide regional aquifers. In particular, we shall be
justifying our ideas by considering plausible transport process
models that can show how (and better yet also show why)
entering reservoir upstream formation and/or injected waters can
efficiently displace and replace in-situ pore space oil
ganglia so that they coalesce and move naturally towards the
downstream production wells.**

**According to the classical ways of thinking, however, the proof
that rationally-based reservoir performance algorithms are being
employed is adequately confirmed when eventually definitive field and
laboratory experiments have been undertaken. And these are the ones
where the measured experimental output data turn out to be good
history-matching predictors of actual subsequently observed field
performance events. **

**This logical way to proceed is illustrated in what follows by
focusing on somewhat simplified but representative reservoir cases.
Here, for example, we start by dealing with an attempt to rationally
model certain idealized irreversible two-fluid phase flow transport
processes. Specifically we have in mind ones where wetting liquids
like brines spontaneously happen (say, because of prevailing ambient
artesian conditions) to replace and displace into production wells a
portion of the non-wetting oil phase fluids that originally were
present in the pore space of particular petroleum reservoirs. These
are events that occur in reservoir systems such as those found in
stratigraphically bounded sand lenses and/or in distributed
up-structure entrapments like anticlines. **

** The
classical approach to deal with these kinds of occurrences is: (a) To
first postulate the applicability of a plausible theory based on what
can be identified as modified Buckley-Leverett dogmatic
ideas ^{4,6,8,10,12.19};
and then (b) To confirm whether or not the data obtained in
subsequently scaled physical model (and/or Gedanken)
experiments appear to be sufficiently supportive of the selected
theory. Thereafter, it then becomes a job for reservoir engineers and
their managers to guard against accepting any surprising non
sequitur ambiguities that seem to surface if and when
contradicting laboratory model data are obtained.**

**Lastly, we call attention ^{20-30}
to what seems to be new in our present 'revisiting' of earlier
advocated algorithmic presumptions, which include: (a) Our
reemphasizing of the advantages coming from adopting coupled rather
than traditional Darcian flux-force relationships so that
irreversibility effects are better taken into consideration; and (b)
Our willingness to favor the use of computational algorithms that
stochastically yield good forecasting information even when the
theoretical justifications remain obscure.**

*Key Words: *Idealized Petroleum Reservoirs; Transport
Process Models; Simplistic Buckley-Leverett Algorithms; Viscous
Coupling Effects; Anisotropic Media Properties; Capillary Imbibition
Fluxes and Forces.

**INTRODUCTION**

**We start by recalling that in 1988 an overdue attempt was made ^{19}
to upgrade and extend the usefulness of what were only popular
vintage Buckley-Leverett algorithms^{2,8.10,31}. Unlike
Patek's recent 'revisiting' style (2002)^{13
}to confirm the correctness of the content of the
original 1855 Fick paper on "Liquid Diffusion", our
approach here has been to focus on simple (but not necessarily fully
tested) plausibly rational approaches, and even on grossly simplified
ones^{26.31}, whereby at least partial proof of the
applicability of some of them could be prospectively accepted and
applied in practical ways for the modeling of coupled multiphase
porous media transport processes. This means, of course, that
verbalizations of generalized rules must be concocted that take into
account how modern ideas about coupling phenomena control various
representative transport process outcomes of general interest^{12-23}.
In particular, it is this kind of information that is wanted to
facilitate the modeling (and hence the accurate forecasting) of
future field production outcomes that arise because of the underlying
causative nature of the transport processes that are assumed to be
involved. **

**Under consideration here to provide background for the topics to
be discussed are the ideas expressed in six key
monographs ^{3,4,5.8.11.12}, and also in a lot of the here
cited (but perhaps less-read journal papers by the present Author.
These citations in turn further point to a vast collection of
scattered international papers which in turn likely point to other
documentation of special interest to curious readers.**

**However, while many perplexing and disputed issues still remain
unresolved even after the elapse of more than a half Century
following the publication of the cited 1942 watershed B/L paper ^{6},
only some of them have been critiqued persuasively enough to earn
general acceptance. Accordingly, it should not be expected that what
written on these pages will conclusively settle all remaining issues
of disagreement.**

**The fact of the matter is that the full proof of either the
original or the modern-day opinions about the viability of underlying
Darcian-based Buckley & Leverett dogmatic presumptions actually
can not be fully assessed until coherent experimental proofs of at
least some of the many postulated, practiced, and published
contentions have been fully confirmed. What will be found in the
following text, therefor are mostly suggestions of new ways to
significantly rephrase some of the questions that can be asked today
about those innocent positions taken in bygone times. In particular,
upon accepting the preamble statements that appear in the cited Rose
(1988) paper ^{19},^{ } our modest aim now is simply
to verbally offer without full proof some alternative
analytically-appearing algorithms. And we take this approach as being
a prospective Cartesian way for future workers (particularly the
modern experimentally-minded modern ones) to improve and perhaps
confirm the use of what seems to be acceptable alternative ad-hoc
ways of thinking. And indeed that is why in what follows we
prospectively choose to imply as our principal thesis that strictly
linear models perhaps adequately describe many two-phase,
isothermal. and low-intensity flows of immiscible Newtonian fluid
pairs in homogeneous, isotropic, consolidated, water-wet porous rock
samples that are perturbed because of superimposed viscous coupling
and superimposed spontaneous capillary imbibition effects.**

* Background
Ideas: *As in the earlier Rose (1988)

**In any case, and indeed without apology, we accept the sense of
the Buckley-Leverett mass conservation theorem as phrased in
Equations (1) to (4) below, but as will be seen, we categorically
adopt the modern viewpoint due to enlightened thinkers pioneered by
DeGroot and Mazur (1962) ^{ 7}
that proper flux-force relationships to describe entropy-producing
coupled (hence irreversible) transport processes so far have not so
far been indicated by experiment to be Darcian in character as
originally was (and in some quarters still is) wishfully presumed.**

**Eqn.
Box 1**

**In Equations (1), for example, it is indicated that we are
prepared to be dealing (say in a finite element mode) with transport
due to capillary and gravity as well as to mechanical forces acting
in three-dimensional space, where fluxes of mass/energy laden fluid
particles and displacements occur (as Tribus ^{32}
uniquely mentioned a long time ago). **

**Figure 1, for example, is a not-to-scale schematic cartoon
depicting how transport is thought to occur, for example in two-phase
petroleum reservoir systems that are imbedded in regional aquifers.
Depicted there are upstream t container rock system ^{3}.o
downstream pore network domains with source and sink termini to
topographically complex in-series and parallel interconnected pore
paths. These in turn lie within contiguous macroscopic representative
volume elements (so-called RVEs) occupied spaces which in total
constitute the porous medium reservoir system space that is made up
of the fluid-filled interstices which both surround and are
surrounded by ^{4}
the solid mineral-occupied constituents of the container rock system
network.**

**Figure 1**

**Specifically, what is wanted, of course, are laboratory ways to
test whether the transport process theory that is under consideration
can be authenticated by examining the data obtained in scaled model
physical experiments. For example, here we shall be advocating the
use of Rose's (1997) ^{24}
recently described laboratory methodology that appears conceptually
and uniquely to be well-suited for the intended data-gathering
purposes (e.g. as described in Appendix 2). Before further
emphasizing this critically important contention, however, we choose
first to introduce the idea that the ad hoc transport process
theory to be considered here is made evident by the sense of the
following algorithmic formulations that appear in the 19 Equation
Boxes that are scattered throughout this paper (and also further
referenced in Appendix 1). As will be seen, verbally the major cited
equations assert that: (a) Whenever and wherever low intensity
diffusive fluxes are involved, linearity between conjugate fluxes and
forces are to be expected. This supposition, for example, is
elegantly suggested by Bear (1972, cf. §4.4)^{3};
and (b) Even so, simple mathematical relations and clever laboratory
measurement methodologies likely (and luckily) also seem to be
involved that avoid the need to search for and employ possibly
non-existent reciprocity relationships between either the diagonal or
cross transport coefficients that otherwise might be wanted in order
to facilitate determining values for them from the experimental data.**

** **Equations
(1) and (2) above, for example, are phrased to facilitate focusing on
** ad hoc** ways to
describe certain both steady- and unsteady-state petroleum reservoir
transport processes. The example selected ones are those which are
clearly based on mass and/or energy conservation principles.
Particular attention will be limited for simplicity to important
cases where the reservoir pore space domains at all times are
represented as being completely saturated by two essentially
incompressible and immiscible Newtonian liquids such as: (a)

Accordingly, that is why Equation
(3) then indicates that the local time-changing flow transport
processes under consideration are ones that involve vector operations
like addition gradients, multiplication of vectors by scalars, scalar
and vector products of two vectors, gradients and divergences of
scalar and vector fields, and the Laplacian of scalar and vector
fields as deal with compactly by the Bird et al (2002)** ^{5
}**classic text on
Transport Phenomena.

In these connections, notice that
the adjacent ganglia of wetting and nonwetting fluids as seen at the
pore level frame-of-reference in general will display curved (locally
convex or concave) interfaces in turn have both stationary and moving
microscopic curved fluid-fluid interfaces. And the topology of the
angle of approach they make along the lines where wetting and
nonwetting fluid elements approach the surface of the bounding solid
pore space walls is such that indicates that implies that prevailing
capillary driving forces also may contribute to the local
convergences and associated divergences of the two companion fluid
phases. In any case (3) shows that when flow transport locally is to
some extent directed in non-horizontal directions, this means in
consequence that the superimposed gravity forces cannot safely be
ignored (N.B. see Tribus (1961, p. 520 ** ff.**)

Here the two Equations (1a) are
intended to describe low-intensity uncoupled 2-phase fluid flow in
isotropic media as though Darcy's Law holds for two phase systems,
and as such they are equivalent to Equations (20a) in the text. Also
clearly for the following Equations (1a) necessarily to be
derivatives of the first ones so far have not been proven
experimentally … but only innocently presumed by some early
and later workers according to Rose** ^{14-18,25,27}
**as well as by other
investigators from Yuster to those mostly modern workers (for
example, as cited in the body of this paper

In passing, however, it is also
very much worth noting that Equations (1a) as they stand can also be
employed to describe low intensity flow of single phase Newtonian
fluids in two-dimensional anisotropic media under conditions shown in
a recent Letter to the Editor of ** Transport in Porous
Media **by Rose (1996)

** To
continue our analysis of how to upgrade and modernize traditional
Buckley-Leverett ways of thinking, we look at Equations (5) and (6)
shown below as polynomials where the coefficients of linear
proportionality are those dozen somewhat redundant and symmetrically
interrelated saturation-dependent terms that appear in Equations
(7). These as seen to express the not unexpected and logically
plausible linear transport relationships that seem to properly model
why and how the causative thermostatic and thermodynamic forces give
rise to the consequent fluxes and displacements by which natural
non-equilibrium systems irreversibly approach final end states.**

**Eqn.
Box 3**

In passing. we notice that **both
Equations (5) and (6) display flux versus force relationships for two
water-oil fluid phase saturated systems that however are written in
two inherently equivalent ways for cases where only a single coupling
type (e.g. like viscous coupling) is involved. These two disparate
equation forms are: (a) Either where the fluxes are shown to be
expressed with three terms on the equation right-hand side (i.e.
where the transport coefficients appear as upper case Latin letter
notations like [A,B,C,….]); or (b) More commonly, compactly
and usefully with only two terms on the equation right-hand side
(i.e. where the transport coefficients appear as lower case Greek
letters notations like [α,β,γ, …]).
Equations (7), and then (7a) below then show the simple mathematical
relationships and notational equivalencies between the six Greek and
six Latin letter transport coefficients. And, as seen in both (5) and
(6) transport equations for each fluid, the simple coupled transport
processes involve the same two identical driving force terms as shown
below in Equations (5a,6a) as written below in matrix form. These
latter Equations will be seen to apply to cases where the pore space
is saturated with two immiscible fluids, and where only a single type
of coupling has to be considered, and that ^{ }involves two
fluxes and conjugate forces. And so we write:**

**Eqn. Box 4 **

^{
}

Clearly, to monitor and numerically
describe the steady and unsteady states of the transport processes
which are called for by Equations (1) above, we start with the above
two equations given in the (5a,5b) matrix form which independently
provide relationships since the various Dij are independently given
as functions of saturation are experimentally observable and simply
measured in terms of the available flux-force data. Implicitly then,
two additional independent relationships will be needed that also
explicitly relate the D** _{ij}**
to other laboratory experimentally obtained flux and conjugate force
data. Then with the four Dij relationships all known and
established, values for the {Ω,Ψ) terms as functions of
saturation can be inserted in Equations (1) to generate the wanted
relevant forecasts of steady and unsteady state segments of
anticipated future reservoir process events.

** Implied
more generally, for example by Equations (7), is that with the
superscript (ω=1.2.3.4), we can then designate the four
semi-redundant experiments that can be performed in order to compute
values for the 12 overlapping transport coefficients from the
measured values of the conjugate flux/forces pairs, (J_{r},X_{r})
where {r,s}={1,2}. Specifically, this possibility can be seen in the
interpretation of Equations (8) below which presents in matrix form
the senses of Equations (5) and (6) when applied to the several
distinct experimental cases. For example, in the first experiment,
both driving forces are non-zero and also not equal to each other. In
the second and third experiments one of the driving forces is set
identically equal to zero, but the other one not. And in the fourth
experiment the two driving forces are equal to each other. And this
means in effect that we have eight independent relationships which
are sufficient by the method of simultaneous equations to extract
numerical values for the two sets of six transport coefficients of
proportionality as identified by Greek and Latin characters in
Equations (7). And, remarkably as a special feature of the unique
experimental method being employed ^{9.24.35}, the capillary
pressure (and hence the local saturation) can be held fixed and
constant during the course of the course of the ensuing suites of the
four experiments to be conducted at each system saturation level (and
this even though the J_{r} fluxes and displacements and X_{s}
measured force data values will tend to be differ during the conduct
of each experiment type.**

**Eqn.
Box 5**

** Here
it is seen that the first two of Equations (8) are to be conducted by
observing the 1 and 2 fluxes by imposing the two driving forces to
be unequal with each other and not equal to zero. Then in the second
and third sets of experiment first one and then the other driving
force is set equal to zero, while in he fourth experiment the two
driving forces are constrained to be equal to each other nor equal to
zero. Other independent sets of experiment could be ones where the
companion travel direction cold be set to be counter-linear rather
than collinear.**

**DISCUSSION**

** In
these connections and as alert readers can carefully note, the Rose
reference in (1997) ^{24 } shows explicitly how for two-phase
(say water-oil) liquid saturated porous rock, a novel laboratory
procedure is made available by the unique instrumentation
characteristics of an apparatus system by which the validity and
utility of the computerization algorithm that is chosen for various
specified applications.**

**We
are now ready to address the fact that to forecast what happens when
a horizontally oriented anticline reservoir that has been
more-or-less lying dormant over long periods of geologic time (i.e.
before eventually they are first accessed contemporaneously) by
systems of special upstream and downstream injection and production
wells. These are both to provide entry for the displacing aqueous
fluids, and to permit commercial lifting of available oil to the
surface collecting facilities such as separators, gathering and
distribution pipelines, refinery process installations, storage
tanks, waste disposal facilities, and the like. Here as we consider
the petroleum recovery process, we now will be assuming that it in
part will be at least approximately modeled by idealized
relationships such as the first two of Equations (8). To proceed,
however, we must cope with the problem associated with the fact that
for the present case now under consideration where there are two
transported entities (i.e. n = 2), and where implicitly
we will need two more independent relationships so that with
Equations (8) we now will be dealing with four then. It is then that
by solving 4 by 4 matrix simultaneously that we can thereby extract
values for the needed four D_{ik}
transport coefficients. **

* Ad-hoc Theory
for Simple Systems:* As it turns out, Aitken (1956, Chapter
II)

**Specifically,
by arranging for the two driving forces in two successive
experiments to alternately and sequentially have the values implied
by the four line vectors shown on the left-hand side of Equation
(9a), then the four Equations (9b) to (9e) and the four Equation
(9f) clearly follow, or: Eqn.
Box 6**

**In other words, Equations (9a) can serve as the four relationships
necessary to solve for the four transport coefficients in terms of
experimental measurements. These are seen to be calculated from the
experimentally measured force to flux ratios that are the saturation
dependent parameters as Equations (9b) to (9i) indicate. Specifically
all of Equations (9) represent the form which the parent Equations
(1) and (2) in Box 1 will have for the particular n = 2 case
under discussion.**

**Below in Figure 2 copied with permission from the indicated Rose
(1997) ^{24} TiPM
paper, schematically shows how the measurements called for by the
Equation (9) algorithms actually can be made. Proof of this
contention as given below also implicitly follows from considering
the experimental methodology described earlier in the works by
Dullien & Dong (1996)^{9}, and before that by Zarcone &
Lenormard (1994)^{35, }, and indeed before that when Rose
(1976, cf. Appendix)^{17 }was
organizing a petroleum engineering curriculum at the Institute of
Technology at Nigeria's University Ibadan.**

**The Figure 2 apparatus arrangement which has been referenced in a
number of the Author's recent publications ^{24.,25,27.28,29,,38,39
} is also further critiqued in Appendix 2 of this paper. **

*Figure 2* - (After Rose in TiPM, 28: 221-231 (1997)^{24}

A proper legend for Figure 2 implies the following. Here
are schematic depictions of Case 1 and Case 2 scenarios shown in the
left-hand diagrams, respectively, for where in Case 1 the imposed
gravity free-fall driving force is proportional to the difference in
elevation of the free surfaces of the non-wetting fluid contained in
the A and B siphon reservoirs where *{(ρ _{2
}g H_{2})
= X_{2} } > 0*, whilst the
imposed driving force in the wetting fluid contained in the C and D
siphon reservoirs is zero since

**In these connections, notice that for the Case 1 experiments, the
oil is being pushed up by an amount indicated by the
oil siphon flow meter because A is always kept filled while the
excess oil is spilled from the B reservoir. And the water also is
dragged upwards by the viscous coupling effect since C
is kept filled and D spills at the rate indicated by the dragged
water flow-meter. And then opposite things happen during the Case 2
experiments because the water is pushed down at a rate indicated by
the water siphon flow meter since the level in D is greater than that
in C, while the oil is dragged down by the viscous coupling effect at
a rate measured by the dragged oil flow meter because the A and B oil
fluid levels are kept at the same levels.**

**Clearly, the following Equations (10) relationships, formulated
specifically to apply for manipulating the data obtained when
conducting flow experiments for the two ingenious Cases 1 and 2
configurations illustrated on the right-hand side of Figure 2. These,
as seen, are consistent with indications of the earlier general
relations already anticipated by Equations (1) and (2). Here we
notice that the two fluxes in general will be given by the sum of the
pushed portion caused by the imposition of an imposed
driving force plus the dragged portion caused by the
viscous coupling effect. In other words, we have:**

**Eqn. Box 7**

To be noted in passing,
Equations (10) also apply to the laboratory configuration for the
measurement scheme illustrated in Figure 2. The latter has been
purposely designed so that operationally in the successive
experiments that first one and then the other driving force terms are
set equal to zero. In such cases, the only finite external driving
force that remains is that provided by the gravity drainage action of
a siphon which results in the companion force to be
positive-definite. In fact, it is this unique configuration
arrangement which makes it possible during each set of the successive
experimental episodes, to have two phase flows occur where the
capillary pressure gradients (and hence the associated saturation
gradients) re,arkably are identically zero during the steady-state
two-phase flow episodes. Hence we now can accept Equations (11a) and
(11b) as shown below. **Hence for the Case 2 Formulae we have:**

**Eqn. Box 8**

**And for
the Case 1 Formulae we have:**

**Eqn.
Box 9**

**To be noted here is the interesting fact that redundantly the
validity of the above relationships to some degree can be obtained by
equating the first of Equations (11a) to the last of Equations
(11b), and likewise by equating the last of (11a) to the first
(11b). And more than that, by forming a ratio between the second of
(11a) to the second of (11b), one can assess whether or not the
Phenomenological Equations of Onsager (cf. as given by DeGroot &
Mazur ^{7}) asserting that (D_{ij} = D_{ji})
perhaps may or may not apply for the particular transport processes
being characterized by Equations (9) and (10).**

__ Formulating More Complicated Cases__: Finally, we
close the
overview
discussion by employing the ideas of Rose and Robinson (2004, in
press)

**The ones under consideration, however, do not seem to correspond
except perhaps superficially to those other important special-case
processes like thermo-diffusion which have been described by the
followers of Onsager with such acuity by DeGroot & Mazur (1962) ^{7}
and many others by invoking the Principle of Microscopic
Reversibility. **

**In these
connections, it will be remembered that when one has in mind
representative volume elements (i.e. RVEs) viewed as continuums in
which multiphase transport processes are occurring, then from the
Eulerian point of view one may think of them as fixed averaged
spatial locations where extensive system quantities in unsteady-state
processes are seen to be changing with time. Alternately, from the
Lagrangian point of view, one may think of such RVE locations as
being occupied by a ‘droplet-train’ succession of
traveling fluid particles (following one after the other along
tortuous streamline paths) where each one contains a fixed amount of
some extensive quantity of the mass/energy phase under consideration.
And then there may be various associated state variables that happen
to be aboard and dragged along with the moving fluid particles. **

**Specifically,
the macroscopically observable motions seem to occur due to the
action of prevailing mechanical and/or internal energy driving force
energy gradients. Analytical expressions for these motions are
given below as Equations (12a) and (12b) in the form previously
presented by Rose (1995) ^{25} for particular miscellaneous
transport process of interest. Thus we hazard to suggest that the
situation being monitored:**

**Eqn. Box 10**

**It is in Equations (12a) where the summation rule applies, that we
find ourselves now considering what we are calling ad-hoc
relationships that interestingly enough are superficially similar in
appearance to the early afore-mentioned classical Onsager
relationships. For example, in coupled thermo-diffusion systems the
α,β superscripts stand for thermal and chemical energy
fluxes, but in Equations (12a) they can stand for the {W,N}
terms that designate the two-phase immiscible wetting and non-wetting
pore fluid phases … while the {x,y} subscripts stand for the
Cartesian two-dimensional spatial locations. **

**On the other hand, the ambiguous (12b) formulation clearly lacks
the definiteness of the classical Onsager Reciprocity Relationships.
Accordingly, we find ourselves now left facing the heretofore
somewhat neglected task of inventing and authenticating what amounts
to plausible intuitively-based rationalizations for these
relationships.**

**Anyhow, as a way to facilitate and expedite our search for closure
to the nagging questions about how to model the dynamics of
questionably non-diffusive transport process cases, we now can jump
to considering the interesting but severely complicated cases of
two-phase isothermal flow of single component and incompressible
fluids in anisotropic media systems whenever viscous coupling effects
in addition are a prominent feature to be considered. **

** According
to the cited Rose (1995) ^{25} paper, the above Equations
(12a) display with a simplified notation the possibly probable linear
relations between fluxes and forces which are predicted if and when
the indications of Equations (12a) are to be believed to apply to the
case of 2-phase flow in 2D anisotropic media. And Equations (13b)
which appears below seem to indicate plausible symmetry relationships
that may be cautiously applied if needed. But to be trusted they must
be verified by experiment. In
such a case, for example, in Equations (12b), Rose (1995)^{23}
and (1996)^{38}
has suggested that relationships displayed by the sixteen
transport coefficients of (12a) might in some cases experimentally
prove to be:**

**<>
>
where
>?<
**

**The
equivalence of them, however, to the sixteen D_{ij}^{αβ}
diffusive flux transport coefficients given in Equations\ Boxes
7 to 13 [i.e. where Equations (11) to (15) ae located] at
this point so far has not been established, hence the existence of
reciprocal relations between the terms of Equation (13b) remains as
an open question, but not one necessary to be addressed here. As will
be seen, this is because of the ease with which other independent
relationships can be formulated to render the necessary matrix
relationships determinant. **

**Upon expanding Equations (12) we can write Equations (13) as:**

**Eqn. Box 11**

**Here Equations (13a) present four scalar equations containing 16
initially unknown transport coefficients, and Equations (13b) show in
the classical manner how at least 12 independent definitions for the
non-diagonal ones provided enough additional relationships so that
the matrix problem becomes unambiguously determined, and this without
the need to verify in advance which (if any) of the reciprocity
relationships postulated (by the (13c) and (13d) relationships need
to be experimentally verified.**

**For example, we may consider an innovative device presented by
Rose (1976) ^{18} and revisited again in (2001a,b)^{28,29},
to uncover more than 16 independent polynomial equations where the
elements of the defining matrix can be defined in terms of knowable
laboratory measured functions of the corresponding scalar elements of
the observed flux and force vectors. The proof of this is in fact
supplied by the formulated in Rose and Robinson (2004)^{30}
in ways described below and again in the Appendices that follow.**

**To close
this particular discussion, we may briefly illustrate the obvious
fact that the application of Equations (12) and (13) can be extended
to other more simple as well as more complicated cases of
non-diffusive transport processes. One of these is case of two phase
viscous-coupling affected flow in isotropic media where we seek by
the methods of determinants to obtain expressions for the four
transport coefficients for this case. And we shall show this can be
done without the need to a priori presume the existence of
symmetry conditions.**

**For the two phase cases
we shall be dealing with, in order to simplify to simplify the
argument, wee adopt the symbolic notation for the flux, force and
coefficient terns which are employed below, or: **

**Eqn.
Box 12**

**As shown by the Equations (14), two independent experiments are
performed by letting the driving forces alternately be set first at
some finite measured value and thereafter set equal to zero as
indicated by noting that Equations (9d) and (9e) together provide
four equations for calculating the four unknown {A,B,E,F} transport
coefficients from the flux and force measured data. The paper of Rose
(1997) ^{24} describes a measurement methodology by which the
required number of experiments to be conducted can be suitably
performed.**

**For readers who think it is a waste of time to conduct so many
complicated experiments to measure the transport coefficients needed
to conduct computerized simulations of particular reservoir transport
processes, it is the opinion of this writer to caution that it is a
false economy to try to minimize the expenditure of laboratory time
and expense when the consequence is that only flawed misinformation
will be the result! And the same is true, or course, when trying to
save computer time and expense by employing less costly computational
algorithms which are an insidious guaranteed to obtain faulty
calculations.**

__‘Unfinished’
Capillary Imbibition Algorithms__

* *One
reason to deal with what to many is a perplexing capillary imbibition
subject matter is because of the fact that quite frequently various
petroleum recovery process cases are encountered that involve the
replacement of hydrocarbon fluids with initially resident or invading
aqueous phase ganglia that become entrained in bounded subsurface
sedimentary pore space. This is a subject not only touched on below
as an 'unfinished' (meaning not well-understood) topic of commercial
as qwell as scientific importance, but also one worth revisiting by
reading in Appendix 2 to follow how the topic needs unraveling and
unscrambling and extrication before reservoir engineers of the 21^{st}
ebtury can say that the art of constructing truly coherent algoritmic
forecasts of petroleum reservoir behavior. For example, the ordinary
water-flooding process after-all is a paradigm example affirming the
relevancy of developing clear understandings about this subject now
being discussed. Here, however, it is to be agreed that the
formulation of coherent reservoir simulation algorithms involves
complications that heretofore have not always been widely or wisely
treated.

** For
example, one class of difficulties has to do with the fact that
sometimes it is the inherent complexity of the attending transport
process coupling effects that must be taken into account. These arise
because of the dual way the invading aqueous phases in general can be
caused by mechanical and/or as well as by free surface energy driving
forces. The latter is where the former are imposed and/or imparted
because of accompanying fluid injection processes, while the latter
are the consequence of inherent capillary actions that automatically
give rise to spontaneous imbibition of the wetting fluid. **

** **

** Accordingly,
herein an effort is made to deal with the attending problems of
describing the complex nature of the commonplace petroleum recovery
processes where inbibition in one form or another occurs. In
particular, three general cases therefore will be at least partially
considered, namely: (a) Where the attending driving forces alone
involve mechanical energy gradients acting on the bulk fluid phase
elements; (b) Where in addition, there are also free surface energy
gradient driving forces existing, and this because of local wetting
phase saturation variations in time and space that give rise to
spontaneous capillary imbibition effects; and (c) Where no resulting
saturation gradients that are finite in magnitude exist within the
pore space domains being investigated, and this even when
steady-state conditions are finally reached.**

** Some
30 years ago it was Jacob Bear (1972) ^{3}
who was the earliest one of several later monograph authors like
Marle (1982)^{12},
Bear & Bachmat (1990)^{4},
and Dullein (1992)^{8}
who took the trouble to recognize that Yuster's 1951 watershed
paper^{34}
provided a foundation upon which certain rational analytical
algorithms modeling non-equilibrium coupled transport processes
involving diffusive fluxes possibly could logically be based on
Onsager's famous 1931 Reciprocity Relationships for which a Nobel
Prize finally was finally awarded in 1968 (cf. Rose in 1969^{15}).
Here, however, for the non-diffusive cases currently being
considered, only a few minor amplifications of the general theory can
be informatively restated here. This is because the relevancy of
that subject matter for capillary imbibition applications so far is
by no means fully established. The intention as mentioned in
previously given commentary is to follow economical ways to increase
practical understandings about the oil field applications, rather
than to get overly immersed in considering obtuse second-order
theoretical matters. **

** **

** In
other words, the intention of what is being written now is to
modestly uncover further clarifications that still are needed so that
the ideas being dealt with in current works can more fruitfully be
employed by investigators who are engaged in reservoir simulation
studies of hybrid spontaneous versus induced capillary imbibition and
related coupled processes.**

** Specifically to be dealt with in what follows
here is the nature of the transport processes that commonly occur in
porous sediments when saturated by pairs of immiscible fluids for
those common cases where one of them usually can be considered to
preferentially ‘wet’, hence adhere to the pore space
surfaces more strongly than any other of the contiguous immiscible
fluid phases. And in our analysis, but with only minor loss of
generality, the fluids can be idealized as being homogeneous,
chemically inert, incompressible, and possessing a Newtonian
rheology. Furthermore, the porous medium for its part will be taken
to be uniform, usually isotropic, rigid, insoluble, and chemically
non-reactive. Finally, and for simplicity, attention will be limited
to isothermal transport processes of low intensity (viz. so the fluid
flows will be laminar in character). And the underlying theme of the
remarks to be made below quite naturally will deal with the
perplexing question about why so many past and present authors in
general have seemed to think that spontaneous capillary imbibition
effects can be treated as though they are caused exclusively by the
action of mechanical energy driving forces even for cases where
common sense alone makes it clear that such processes inherently are
the result of the action of free surface energy gradient driving
forces.**

** The description of the transport processes of
the two-phase systems as to be described here, for example, already
was first given in an approximate way by the Author in two of his
recent disclosures (cf. Rose 2001a,b) which are both based on a
timely revisiting of an earlier Sabbatical Leave (1963) ^{28a}
paper by W. Rose here cited in two current papers^{28,29}.
The canonical forms appear in Equation Boxes 13 and 14 a
Equations (15) and (16) which
apply to the general case of where there may be two fluid flow fluxes
and/or one parallel and accompanying free surface energy flux. These
presumed nondiffusive fluxes, for example were shown to be driven
respectively by two conjugate mechanical energy gradient forces, but
also occasionally by an associated and superimposed free surface
energy gradient force term as discussed previously in an enlightened
way by Tribus (1961)^{32}. Thus we have:**

*Eqn.
Box 13*

and also:

*Eqn.
Box 14*

In the above Equations (15) and
(16), the subscripts, {i,j} = **{1,2} respectively
designate the wetting (say W = aqueous) and the non-wetting (say N =
hydrocarbon) pore fluids; hence the Ji are the so-called
Darcian approach velocity vectors for the two fluids which are
locally at measurable pore space saturation levels, S_{i},
and where respectively the X_{j} for {j =1,2}
are the conjugate mechanical energy gradients (per unit mass) acting
as forces to give rise to the ensuing two phase flow processes. Note
that these are shown above to be identically equal as the intended
way to achieve a quasi-zero zero dynamic capillary pressure gradient
… and hence a uniform (or at least a steady-state saturation
condition) during the flow measurements (cf. Rose, 1997)^{24}.
On the other hand, these equations X_{E} and J_{E}
respectively, refer to free surface energy gradient driving forces
and fluxes which are manifested by spontaneous capillary driving
force effects of those special sorts perceptively mentioned by
Tribus (1961)^{32} where he draws a distinction between how
non-equilibrium thermostatic and thermodynamic processes should be
separately considered. **

** **

** For
example, in Equations (16) notice is to be taken of the fact that in
total there are nine transport coefficients, D_{ij
}, which each are functions of how geometrically the pore
space of the sediment is partitioned throughout time and space
between the two locally present immiscible fluid phases. Clearly
experiments will have to be performed to establish the quantitative
form of these linear dependencies. Moreover, for starters there are
the three flux-force relationships and the three Onsager Reciprocity
Relationships that are given by Equations (16). Also notice should be
taken of the fact that in the experimental work to be performed,
imposed values of the mechanical energy gradient driving forces, X_{W}
and X_{N}, can be selected such as those shown in the
right-hand matrices of Equations (17) for what are being illustrated
as the logically chosen four separate experimental measurement cases
that need to be performed. And, in passing it is to be also noticed
that the fluid flux terms, J_{W} and J_{N}
are ones that easily can be measured with conventional flow meter
instrumentation for each of the four (A,B,C,D) laboratory test cases.**

**Specifically, in Equations (15) or
(16) there are admittedly a large number of independent, dependent
and disposable variables to be dealt with.. And what is to be sought
are a similar number of independent relationships to evaluate their
characteristic variations with respect to time and position.
Logically, four separate steady-state experiments with the same
ambient saturation values and interstitial saturations distributions
locally are to be held constant for each of them, as indicated by the
chosen decision to separately undertake the experiments prescribed by
the four sets of Equations (17) as being the logical ones to be
performed. This choice, of course, is so that numerical values can be
obtained for the transport coefficients as explicit functions of
saturation, spatial location and temporal time during the ensuing
recovery--processes.**

*Eqn.
Box 15*

** Accordingly, Equations
(17) above have been presented in
order to identify the flux-force conditions that apply to the four
{A,B,C,D} cases of separately independently undertaken experiments.
The objective of this experimental
approach, of course, is so that with Equations (18a,b,c)
as also given below, a display can be given for the three dependent
and six independent (D_{ij} ) material response
transport coefficients that can be assessed from the given input
values of [Xi ={,β,γ}]
together with the observed values of the [Ji ={1,2, and
sometimes 3}] measured flux data. **

** The
problem of combining the Equations above that are presented in order
to obtain useable coupled capillary imbibition algorithms, however,
clearly lies in the fact that the J_{3}_{ }
and X_{3} terms which refer to the spontaneous
capillary imbibition flux and driving force parameters sometimes
will not be as clearly observable and/or easily and directly
measurable as the are in the slightly different cases where the
multiphase flow caused by mechanical energy gradients is coupled with
unambiguous concentration and/or temperature gradient forces which
give rise to heat and mass transfer effects.**

** For
example, however, by manipulating with the terms of Equations (17),
auxiliary useful relationships can be developed such as: **

**Eqn.
Box 16**

_{}

**In
Equations (18a), the flux terms that are
wanted are the nine (^{j=αβγ}J_{i=1,2,3}).
And these essentially the ones that themselves are not directly
measured during the (A,B,C,D) experimental tests. That is, it
is mainly the (i = 1,2) flux elements that can be easily
observed for these (A,B,C,D) experimental tests since each of
the (A,B,C,D) fluxes are summations involve a somewhat
un-measurable ^{±γ}J_{i} flux
terms caused by the ± γ driving forces.**

** **

** Combining
Equations (17) with (18a), then one can easily obtain definitions
for all of the transport coefficients, as:**

*Eqn.
Box 17*_{}

** The
matrix Equations ( 18b), of
course, define how the {A,B,C,D} experiments provide the data
needed to calculate the values in space and time for the nine
material response transport coefficients, D_{ij},
that is by dividing line-by-line the elements of the left-hand flux
matrix by the corresponding directly measurable force elements of the
right-hand matrix. Thus even for cases where γ is finite in
magnitude (hence giving rise to possible spontaneous imbibition
effects), we have the ratio values for the non-zero D_{ij}
elements of the central matrix of Equations (18c), such as:**

*Eqn.
Bo**x
18*

_{}

**and so
forth. In these connections, however, it must be mentioned in
fairness that the indicated values for the ambiguous diffusivity
element, D _{33 }shown in the ninth position of the third
column of the central matrix, is perhaps questionable, but
unfortunately at the present time no other clever way so far has
been discovered to remove this uncertainty except by observing
real-time experimental data.**

** To
summarize the senses of what has just been described above, notice
can be taken of the agreeable fact that by sequentially (but
separately) performing the four {A,B,C,D} experiments defined
by Equations (17), but for the desired result to be obtained, of
course. this must be done in a way where reference values for the
local fluid phase saturation's and interstitial saturation
distributions are held fixed and constant until all of the four
experiments have been performed in each sequence of interest. That
is, the investigator will be obtaining laboratory data which when
combined: [a] with input data about the three driving force terms,
X_{j} where these will be X_{1} = α
or zero; X_{2} =
β or zero, and X_{3} = plus or minus γ
always (1), and [b] with observational data about the two measurable
flux terms, J_{1} and J_{2},
then eventually the nine transport coefficient (material response)
terms as defined by Equation (18b) and (18c) can be unambiguously
computed either uniquely or even sometimes redundantly, as shown
below. Thus, it follows that:**

** [1] By
sdtarting with Experiment D, and by observing {^{D}J_{1}
} and {^{D}J_{2 }= - ^{D}J_{1 }}
and by knowing gamma, γ, one
can compute {D_{13} = D_{31} } and {D_{23}
= D_{32} } (and/or also by making use of presumed
analogs of the Onsager Reciprocity Relationships) as functions of
Saturation and Saturation distribution. On the other hand, if one
wishes to verify the applicability of the definition given in
Equations (18) above as independently and explicitly providing a
correct value for D_{33}, one then must seek still
other means to deduce and know values for the four {^{A}J_{3}},
{^{B}J_{3 }}, {^{C} J_{3} }, {^{D}J_{3
}) _{ }terms. **

** [2] Then
by performing Experiment A, and by observing
{^{A}J_{1} and ^{A}J_{2} } and
applying the plausible reciprocity Relationships, and
additionally by knowing the input magnitude of α
one now can compute values for the {
D_{11} } and {D_{21} = D_{12} } terms.
**

** [3]
Then by performing Experiment B, and by observing { ^{B}J_{1}
} and { ^{B}J_{2} } and again applying the
available reciprocity Relationships, and additionally by knowing the
input magnitude of {α}, one now can finally compute redundantly
the { D_{12} = D_{21}} and the { D_{22}
} term.**

**Eqn.
Box 19**

_{}

**As previously shown elsewhere ^{6,10,19,20,31}, those who
have and continue to adhere (blindly or otherwise) to
Buckley-Leverett dogma, would say that for two phase flow where the
driving forces are only mechanical energy gradients, it is clear that
the following simple definitions for the Ψ and Ω parameters
of Equation (19) that apply to the Darcian modeling case will be
those that appear on top line of Equations (20), while those that
apply to the coupling cases appear on the bottom line, as follows:**

**Eqn. Box 20**

_{}

Clearly the algorithm on the first
line of Equations (20) applies to outdated Buckley-Leverett
assumptions dogma which have been discredited to some extent in
certain current papers^{25,27,36}^{,},
while the algorithm on the second line is to be employed to take
viscous coupling effects into account** ^{20}**
And Equations (21) below, however, are offered to ’fly’
in the face of those who are puzzled by the idea that has been
proposed and questioned recently

**Eqn. Box 21**

_{}

**To be noted in connection with Equations (21a) and (21b), however,
is the important fact that spontaneous capillary imbibition effects
can only occur in those domain regions of the system pore space
where the X _{3} (Capillary Driving Forces) are of finite
(i.e. non-zero) magnitude. Clearly, such a condition, of course,
only holds whenever and wherever locally and for whatever reason the
wetting phase saturation is changing with time. In contrary cases,
the 3rd, 6th and 9th rows of the matrix Equations (18b) are
eliminated and disappear, leaving the simplex Equations (20) rather
than the complex Equations (21) to serve as the algorithms that will
properly describe those partially unsteady-state processes dealt with
earlier^{20} for
cases where capillary driving forces are not involved because imposed
conditions are such that finite saturation gradients are somehow
everywhere avoided throughout time and space when the D_{ij}
terms are being measured by the laboratory procedures described
and/or simply referenced by Rose^{24}.**

**In conclusion, if a comparison were to be made between the early
papers by the Author on the dynamics of capillary-controlled
reservoir processes with those that have followed up to the present
time , the reader might wonder why it has taken more than a
half-Century for him to be now composing still another newly based
one. The fact of the matter is that some writers tend to be thinking
faster than they write while others engage in the opposite. Even so,
the Author will find it reassuring if at least some of the reader of
this paper agree that a rational algorithm to **

**model
capillary imbibition processes necessarily must be one that will
enable undertaking reservoir process simulations if and when the
following propositions are found to be true. ^{5}**

** If the
displacement mechanisms under study are ones where viscous coupling
or analogous effects may possibly occur, then this is reason enough
to employ Equations (21) with Equations (1) as a safe way to
formulate trustworthy ways to forecast future reservoir performance.
More than that, if the displacement mechanisms are ones where
spontaneous imbibition possibly is occurring, this will provide a
second and even more compelling reason to employ the algorithm of
Equations (21) over the simplistic ones imbedded in Equations (20)
when reservoir simulation computations are being undertaken. But
finally, if because of the possible intervention of burdensome time
and cost factors, some still think that there are persuasive reasons
to employ short-cut reservoir simulation methodologies, such ‘wishful
thinkers’ should keep in mind that prudence alone may dictate
that the value of these questionable ways of thinking must be
proportional to the conclusions arrived at by conservatively
undertaken parallel cost-to-benefit ratio assessments.**

**CONCLUDING REMARKS**

(a) If the displacement mechanisms under study are ones where viscous coupling or analogous effects alone may possibly occur, then this is reason enough to employ Equations (20) as a precautionary way to formulate trustworthy ways to forecast future reservoir performance.

(b) If the displacement mechanisms are ones where spontaneous imbibition possibly is occurring, this will provide a second and even more compelling reason to employ the algorithm of Equations (21) in spite of incurring burdensome time and cost factor inconveniences , some still think that there are persuasive reasons to employ short-cut reservoir simulation methodologies, such ‘wishful thinkers’ should keep in mind that prudence alone may dictate that the value of these questionable ways of thinking must be proportional to the conclusions arrived at by conservatively undertaken parallel cost-to-benefit ratio assessments.

**Finally. It is worth suggesting that
negative as well as positive points of possible special
interest to practical reservoir engineers might include the
following:**

**(a) Exactly (it may be asked) how can one conduct experiments
and obtain as many independent linear force-flux relationships as
there are numbers of the initially unknown transport process
coefficients of proportionality? This question clearly needs further
quantitative study. After all, it is necessary to fully confirm the
fact that the latter indeed can be unambiguously assessed by the
standard simultaneous equation solving methodologies as applied to
standard models of the various local transient saturation level
changes that model the accompanying imbibition and drainage cycles
that will be occurring. This complex matter will surely occupy the
attention of future investigators. **

**(b) So far as whether a comprehensive theory for describing
spontaneous capillary imbibition phenomena can ever be fully
developed, a weakness is to be anticipated and acknowledged about the
use of a simplistic capillary pressure gradient term for the
driving force instead of a surface energy gradient term.**

**(c) With reference to the {α, β, γ, δ, ε,
φ } composite transport coefficients of proportionality seen in
the second of Equations (5) and (6), and related to the {A, B, C, D,
E, F} transport coefficients as seen in Equation (7), and also the
D _{ij } related transport coefficient terms seen in Equations
9, 10 and 11 are consequences of the mind-boggling fact that it is
implied that {α+β-ε}= 0 = {γ+δ-φ}!
The deeper meanings of this curious result clearly needs further
study.**

**(d) The point is to be emphasized that although the cost of
conducting careful multiple experiments is high (and sometimes
prohibitory so), it makes sense to spend money and time to get the
right answer than to cheaply adopt simplistic methodologies that end
up with the wrong information.**

**(e) And finally the fact that a lengthy paper of some 10,000
words has been written based largely on the content of some 40
literature citations of which almost half have been written by the
Author himself, and indeed by a person who has been a student of the
subject for almost 60 years while in the company of acknowledged
gigantic innovators of the past and present, would seem indeed to
imply the obvious. On this the readers will surely have there own
opinions about which one!
**

**NOTATIONS**

*(A,B,C,D,E,F) =transport coefficients *in Equations (1) to
(9).

*A _{sw}, A_{sn }=interfacial surface
energy per unit are.*

*A _{αβ }=interfacial surface
area per unit volume.*

*D _{Ij } =transport coefficients* in
Equations (9) to (11).

*E _{1,2,3,… } =denoting locations of
macroscopic RVE *in Figure 1.

*f =local fractional porosity of reservoir rock.*

**g =acceleration due to gravity.**

**H =siphon fluid level heads in the Figure 2 flow
meters. **

**i,j,k =counter for fluid phase, fluxes, forces **

**J =local macroscopic approach flux displacement
rates .**

**L =core sample length. **

**n =number of mass/energy phase.**

**p, Pc =local fluid phase pressures and capillary
pressures/**

*S _{1 or 2 }=_{ }local
fractional pore space fluid saturation's . *

*U, D =Figure 2 Upstream and Downstream reservoir
locations *

**X =thermodynamic/thermostatic driving forces. **

**x,y,z,t =3-D space and time independent variables.**

**μ,ρ,γ =fluid phase viscosities, densities,
& interfacial tensions.**

*Ώ,Ψ =functions of the X _{1} and
X_{2}driving forces in Eq. (7).*

**λ =average pore perimeter)/(pore volume
in local RVEs**

*(α,β,γ,δ,ε,φ)
=transport coefficients* in Equations (4) to (7).

**Θ =advancing contact angle.**

*σ =entropy.*

*ω*
*=experiments needed for data to calculate the D _{ij }.*

↑** ↑ **

**SYMBOLS INTERPRETATIONS**

**ACKNOWLEDGEMENT**

**In this Author's case it has been assorted students and strangers
alike who have been my most treasured teachers both about Life
and about Petroleum Reservoir Engineering even as I was trying
to do some mentoring myself while together we seeking to surmount
those language and cultural barriers which world-travelling
peripatetic professors and other vagabonds in the end find so
charming to endure! **

**REFERENCES**

^{1 } A. C.
Aitken (1939) *Determinants and Matrices, *9^{th}
Edition, Oliver & Boyd, Edinburgh

^{2 } J. T.
Bartley and D. W. Ruth (1999), "Relative Permeability Analysis
of Tube Bundle Models", *Transport in Porous Media, *__36__:
161-187.

^{3 }J. Bear
(1972), *Dynamics of Fluids in Porous Media, *American Elsevier
(New York).

^{4 }J. Bear &
Y. Bachmat (1990), *Introduction to Modeling of Transport Phenomena
in Porous Media*,* *cf. §5.3.5 in particular, (Kluwer
Academic Publishers).

^{5
}R. Byron Bird, Warren Stewart, Edwin Lightfootl (2002),
*Tr/ansport Phenomena,* 2^{nd} Edition, Wiley,
New York.

** **

^{6} S. E.
Buckley and M. C. Leverett (1942), "Mechanism of Fluid
Displacement in Sands", *Transactions AIME*, __146:__
107-116.

** ^{7
}S. R. DeGroot and P. Mazur (1962), Non-Equilibrium
Thermodynamics, (North-Holland Publishing Company. Amsterdam).**

** ^{8
}F. A. L. Dullien (1992) in Porous Media: Fluid Transfer and
Pore Structure, 2^{nd} Edition, (Academic Press. New
York)**

^{9} Dullien &
Dong (1995) "Experimental Determination of the Flow Transport
Coefficients in the Coupled Equations of Two-phase Flow in Porous
Media", *Transport in Porous Media*, __25__: 97-120.

** ^{10
}A. Hadad, J. Benbat, H. Rubin (1996), "Simulation of
Immiscible Multiphase Flow in Porous Media. A Focus on the Capillary
Fringe" in Transport in Porous Media, 12: 229-240.**

^{ 11 }M.
Kaviany (1995) in * Principles of Heat Transfer in Porous Media*.*
*2^{nd} Edition, (Springer Verlag.
Berlin).

^{ 12
}C. Marle (1981) in *Multiphase Flow in Porous Media, *(Gulf
Publishing Company).

^{ }

^{13} C. W.
Patek (2002, in press), "Fick's Diffusion Experiments Revisited"
A*rchive for History of Exact Science*.

^{14} W. Rose
(1966), "Reservoir Engineering, Reformulated" in the
*Bulletin of the Penn State Engineering Experiment Station,
*Circular 71: 23-68.

^{ }

^{15} W. Rose
(1969), "Transport through Interstitial Paths of Porous Solids",
*METU (Turkey) Journal of Pure & Applied Science, *2:
117-132.

^{16 }W. Rose
(1972), "Reservoir Engineering at the Crossroads. Way of
Thinking and Ways of Doing" *Ibid*, 46: 23-27.

^{17} W. Rose
(1974), "Second Thoughts on Darcy's Law", *Bulletin of
the Iranian Petroleum Institute,* 48: 25-30.

^{18} W. Rose
(1976), "Darcy's Law Revisited (cf. Appendix therein)",
*Journal of Mining Geology (Nigeria), * 13: 38-44.

^{19} W. Rose,
(1988), "Attaching new meanings to the Equations of Buckley and
Leverett", * Journal of Petroleum Science & Engineering,*
__1__: 223-228.

^{20 } W. Rose
(1990), Lagrangian Simulation of Coupled Two-phase Flows",
*Mathematical Geology,*__ 22__: 641-654.

^{21} W. Rose
(1991a), "Richards Assumptions and Hassler's Presumptions",
*Transport in Porous Media*, 6: 91-99.

^{22 } W. Rose
(1991b), "Critical questions about the coupling hypothesis",
*Journal of Petroleum Science & Engineering,* 5: 299-307.

^{23 } W. Rose
(1995a), "Ideas about Viscous Coupling in Anisotropic Media",
*Transport in Porous Media,* 18: 87-93 cf. W. ros (1995b) cf.
^{23a }W. Rose
(1905b), "Generalized Description of Multiphase Flow in
Anisotropic Porous Media", * *Extended Abstract, pp. 483-
489.*.*

^{25} W. Rose
(1999), "Relative Permeability Ideas … Then and Now",
buried in the *Proceedings of the SPE Eastern Division Regional
Meeting,* SPE Paper 42718, pages 115-141.

^{26} W. Rose
(2000a), ” A Commentary on the Bartley/Ruth paper",
*Transport in Porous Media,* __40__: 355-358.

^{27 }W. Rose
(2000b), "Myths About Later-day Extensions of Darcy's Law",
in* J. of Petroleum Science & Engineering,* __26__:
187-198.

^{28 }W. Rose
(2001a), "Modeling *Forced* versus *Spontaneous
*Capillary Imbibition Processes Commonly Occurring in Porous
Sediments*"*, *Journal Of Petroleum Science &
Engineering*, __30__: 155-166.

^{28a }W. Rose (1963) titled "Aspects des Processus
de Mouillage dans les Solides Poreaux" in *l'Institut
Française du
Petrole.* XVIII: 1571-1590.

^{29} W. Rose
(2001b), "Theory of Spontaneous versus Induced Capillary
Imbibition", *Transport in Porous Media, Technical Note*
44: 591-598.

^{30} W. Rose
and Derek Robinson (2004, in press), "Transport Processes in
Fluid-saturated Porous Media" in a __Letter-to-the-Editor__
format, *Transport in Porous Media*. N.B. Downloaded Preprint
available through Kluwer Academic Publisher's HomePage.

^{31} F. Siddiqui and Larry Lake (1992), "A Dynamic
Theory of Hydrocarbon Migration", *Mathematical Geology*
24: 305-328.

^{32 }M. Tribus (1961), *Thermostatics and
Thermodynamics (*D. Van Nostrand, see Chapter 15, p. 519 ff.).

** ^{33}
Jian-Yang Yuan. Dennis Coombe, David Law Alex Babchin (2001).
"Determination of the Relative Permeability Matrix
Coefficients", Proceeding of the Canadian International
Petroleum Conference, Calgary (June 12-14, 2001), Paper 2001-02.**

^{34} S. T.
Yuster (1951), "Theoretical Considerations of Multiphase Flow in
Idealized Capillary Systems", *Proceedings of the Third World
Petroleum Congress, *II: 437-445.

^{
34a }**W.
Rose (1951), Discussion of the Watershed Yuster Paper",
Proceedings of the Third World Petroleum Congress, II: 437-445;
ibid, on page 444.**

^{
34b }**W.
Rose (1951). A related companion paper "Some problems of
Relative Permeability Measurement", ibid., on pages
446-459.**

^{35} Zarcone
& Lenormand (1994), "Determination experimentale de couplage
visqueux dans les ecoulements diphasiques en Milieu poreux", *C.
R. Acad. Sci. Paris Serie II* , 1429-1438.

^{36 }Ayub, Muhammad and Bentsen, Ramon, (1999),
“Interfacial viscous coupling: a myth or reality?”,
*Journal of Petroleum Science & Engineering, *23: 13-26.^{
}

^{
37 }**Bentsen, R. G. (2001), “The Physical
Origin of Interfacial Coupling in 2-Phase Flow in Porous Media”,
Transport in Porous Media, 44: 109-122.**

^{
38}** W. Rose (1996), "Letter to the Editor of
TiPM", Transport in Porous Media, 22: 359-360.**

^{
39}** W. Rose, A. Babchin & J. Y. Yuan (1999).
"Coupled Transport in and through Fractured Rocks",
Proceedings Lawrence Berkeley National Laboratory, #42717, Extended
Abstract pp, 210-223.**

^{
40 }**L. A. Richards (19331), "Capillary
Conduction of Liquids through Porous Media", Physics, 1:
318-333.**

^{
41 }**A. Babchin and J-. Y. Yuan (1997), "On
the capillary coupling between two phases n a Droplet rain Model",
Transport in Porous Media, 26: 226-228.**

**APPENDICIES
ADDED IN PROOF**

**APPENDIX
ONE: Equations Road Map**

Traced here is the evolution of ideas analytically

imbedded in the 23 equation boxes of this

'Buckley-Leverett' Revisiting Paper

**(JOPSAE Paper Under
Review 11/2003)**

**In what
follows the more than 100+ sometimes somewhat redundant equations are
cited and paraphrased as follows:**

Eqns. (1) to (4) are presented that embody the here-unchallenged senses of the original Buckley-Leverett (1942)__In Boxes I and Ia__^{6}mass conservation related contentions that were applied by Rose (1988)^{19 }to describe simple unsteady- and steady-state coupled transport processes. Specifically, Eqns. (4) give definitions for the important {Ω,Ψ} terms that first appear in Eqns. (3). Then Eqns. (1a) and (4a) follow to show that when the divergence of the J_{1}flux is adequately expressed by setting the second and third terms on the right-hand side of Eqns. (3) equal to zero and then by only considering as finite the first term. This contention, or example. Is implied in the analysis of Rose and Robinson^{30}.

*In BoxII and IIa***the following Eqns. (5) to (7) are shown to be (respectively somewhat silly then sensibly serious) ways to model linear flux-flow relationships. These are ones that involve the two-fluid phase force of immiscible fluids where only viscous coupling and like effects singly have to be taken into account. More than that, Eqns. (7a) define the one-to-one relationships between the various notational Latin and equivalent Greek lettered transport coefficients as made clear by the cartoon in (1996) Rose**^{38}.

four sets involving eight flux-force Eqns. (8) are presented and shown to be useful for describing the senses of the four separate independent experiments that can be conducted. Of course, with them, the aim is to hopefully secure enough information from the measured laboratory data so that needed values of the transport coefficients as functions of local*In Box III*__saturation__in the macroscopic RVE's of Figure (1) can be computed. This possibility is addressed further in the (d) to (f) paragraphs that follow, and in the text of Appendix 2 below.

Eqns. (9a to 9i) show that the calculated D*In Box IV*_{ij }have the notational form of_{ }being certain experimentally measured_{ }^{ }flux to force ratios that can be uniquely measured as recommended by Rose (1997)^{24}.

*In Box V*_{ }with Eqns. (10a) and (10b), it is indicated firstly that the measured fluxes of the particular {1,2}={W.N} fluids are made up both by a '__pushing__' caused by the driving force acting directly on the fluid phase that is then being observed, and also by a tangential '__dragging__' across fluid-fluid interfacial boundaries that arise because of the parallel existence of driving forces acting within the adjacent immiscible fluid phase ganglia. This ancient way of thinking already had been held with modern interpretations by Barley and Ruth (1999)^{2}and for the sake of argument was later adopted by Rose (2000)^{26}. Then in Eqns.(10c) it is suggested that the Onsager type reciprocity referred to by DeGroot and Mazur (1962)^{7}does not always need to be additionally presumed because of the arguments given in Appendix 2 below.

Equations 11a and 11b show that the same pushing and dragging effects also naturally will be encountered when laboratory modeling experiments are being undertaken of the methodology sorts that have been recommended by various current investigators*In Boxes VI and VII*^{9,24.35}.

the important Eqns. (12a) are to be thought of as generalized flux-force relationships shown for the case of two immiscible fluid phase transport systems. This was also the case for the generalized mass balance statements already referred to in__In Box VIII__, but now the special cases being considered are for where two or more categories of coupling are simultaneously occurring (*Box I**viz*. instead of just a single one as was mentioned before). Example cases now being considered are those where viscous coupling is occurring (however, with or without coupling interference from simultaneous parallel Fourier and Fick Law thermodiffusion transport processes that are modified by Dufor and Soret 'push' and 'drag' effects such as were studied recently in 1999 by Rose*et al*^{40}).

, for example, Eqns. (13a)*In Box IX*^{.}show that now perhaps four fluxes and conjugate driving forces are being indicated for the cases just mentioned above, but they also show that investigators who agree with Aikens (1939)^{1}will understand that implicitly more than four independent equations will be needed when solving for any larger number (say, 16) unknown transport coefficients. Moreover here it is being further supposed that unlike some historical (and occasionally hysterical) contentions that properly-obtained laboratory data themselves sometimes will indicate that at least a minimal number of reciprocity relationships will be required as a convenience to deal with stubborn ordinary needs. Debunking this presumed thesis is explained in Appendix 2 below.

we are confronted with the expansive Eqns. (14a) to (14i). Here starting with two governing flux-force relationships when shown in matrix form will display the four transport coefficients, {A,B,E,F}, as in Eqns. (14b). Then we can show the equivalent unit matrix from which we can form two important additional relationships for the two driving forces,*In Box X**{e,f},*as explicit but complex functions of the two corresponding fluxes*{a,b}*together with the four aforementioned transport coefficients. Then given (14c) which asserts that no*apriori*assumptions actually do not always have to be made about concocting reciprocity existing between

*In Boxes XI and XII*Eqns. (15) and (16) also refer to two phase immiscible fluid flow as has also been the case for the other examples cited above. Now, however, it is three (rather than two) flux and conjugate driving force terms which appear and hence nine or more (rather than merely four) transport coefficients that will be considered. Specifically, the cases now being dealt with are specifically those where there are two coupled fluxes are which driven by conjugate mechanical energy gradient forces (namely that give rise to viscous coupling effects), and then at least a third flux-force pair that takes into account*spontaneous (viz. as opposed to induced)*capillary imbibition fluid displacements of the sorts referred to by Tribus^{32}and later by Richards (in (1931)^{40 }then Rose timidly in (1963)^{28 }… but Rose with more conviction in (2001)^{28a,29}. Notice in passing that the existence coupling due to Onsager-like reciprocity is being presumed to occur.

we have Eqns. (17) which include four sets of possible experiments that can be easily (if laboriously) performed under controlled laboratory conditions. These are labeled as follows: (a)*In Box XIII*__The 'A' Test__where only the*β*driving force is set equal to zero; (b)__The 'B' Test__where only the*α*driving force is set equal to zero ; (c)__The ' C' Test__where none of the driving forces are set equal to zero; and finally (d)__The 'D' Test__where the γ driving force is the only one that is not equal to zero.

Eqns. (18a). (18b) and (18c) appear and show in an evolutionary way how the nine*In Boxes XIV, XV and XVI**D*terms can be evaluated by taking ratios of particular super and sub-scripted fluxes_{ij }*{(J*divided by the appropriate^{A.B.C})_{α,β,γ}}*{α,β,γ}*driving force.

the aim is to have an independent way with Eqns. (19) to model the performance of certain petroleum reservoir systems in order to obtain quantitative values for the nine transport process coefficients referred to in the previous*In Box XVII**Box XVI*where both viscous and capillary coupling effects are involved, but for cases where one does not feel justified or otherwise inclined to assume the validity of the three afore-mentioned Onsager-like reciprocity relationships that appear as the last equations of Eqns.(16) given above. Exactly how this magic is achieved is indicated in the last two boxes given below.

we have definitions for the important*In Box XVIII and XIX**{Ω.Ψ}*terms given for Eqns. (20) cases where either*{X*or only_{3 }=D_{12 }=D_{ 21}=0}*{X*and is also given for the more general Eqns. (21) cases where_{3 }=0},*{X*cf. Rose & Robinson (2004)_{1 }=X_{2 }>0},^{40}.

**In conclusion, the student who carefully and considerately studies
sequentially the various equations found in the 23 Boxes, will notice
that this 1,000+ word Appendix can be thought of as an abbreviated
but adequate summary of the content of the entire 10,000+ word paper
to which it is attached!**

*APPENDIX
TWO: Ad-Hoc Algorithm Fabrication*^{6}

** Here
we start by expanding Eqns. (9) with (8) that already has appeared in
the main body of this paper. This is being done in order to be able
to visually consider the possible value of the revision shown below
to reservoir engineers when they are engaged in forecasting outcomes
of future petroleum recovery processes. For example, the first of the
newly revised Eqns. (9) show that three independent experiments can
be simply (if laboriously) performed in order to obtain crucially
needed engineering data of economic importance. **

**In the first #I scheme we describe here the
governing flux-force equations of a simple representative paradigm
process interest. Amongst the other relationships that also can be
considered are those where reciprocal reciprocity is postulated
between the {D_{ij}≡D_{ji}}
coupling coefficients for cases which the Nobel Laureate, Professor
Lars Onsager, in 1931 pronounced as a plausibly applicable a
so-called Principle of Microscopic Reversibility which one
can invoke for cases where diffusive fluxes characterize the
transport processes of interest. After all, the validity and utility
of this way of thinking has been accepted and certified by many
authorities … for example from DeGroot & Mazur (1962)^{7}
to Bear and Bachmat (1990)^{4},
and beyond. **

^{7}

**In the paper to which this Appendix is attached, however, an
alternative way of thinking has been preferentially considered for
reasons which now will be illustrated by citing one simple
application case. And as will be suggested that plausible extensions
of the underlying ideas appear to be possible according to Rose &
Robinson (2004, in press) ^{30}
for more complex and perplexing cases where the need to postulate
microscopic reversibility can perhaps be regarded as unnecessarily
superfluous.**

**To avoid a transgression that even a guru like Aitken (1939) ^{1}
might occasionally accept (viz. to avoid the complexity and
indeed the possible absence of a real necessity for always dealing in
convincing ways with cases where there are fewer governing
relationships than there are unknowns to be assessed), we instead
elect here to proceed as follows, namely: (a) To postulate that since
we are dealing with low intensity transport processes, it is
reasonable (if not entirely rational) to suppose that the governing
relationships between fluxes and forces to a high degree of
approximation at least sometimes can be beneficially modeled as being
linear where the coefficients of proportionality are measurable when
laboratory model experiments are conducted of the sorts indicated by
the #II and #III schemes; (b) To be reassured that even
when employing short-cut methodologies, forecasting future
time-dependent events become acceptable to the extent that
predictions are more-or-less confirmed when consistency is displayed
by the history-matching evidence that is obtained; and (c) To along
the way keep in mind the maxim of the philosopher^{8}
who said "…all Generalities are False including This
One…", and which can be given the added meaning that
the 18^{th} Century Voltaire was right when he said in his
ingenuous Candide XXX "…let us work without
theorizing (since) it is the only way to make life endurable!"^{9}**

With the above points made, however, there is a crucially additional one to add. The missing link is to make the serious though subtle point that will be understood better by experimental than theoretical physicists, which

is
if repeat 'runs' are made on the same laboratory sample starting with
the same initial and boundary conditions, the final equilibrium
condition will not be reached when different methodologies are
employed such as the ** #s I, II and III **as
defined in the modified Eqns. (9) shown above. Specifically, the most
astute reservoir engineers will expect that the local wetting phase
saturation (and hence the saturation distributions) of the laboratory
samples will show magnitude variations which are specific for each
experimental procedure that is followed. To deal with this
inconvenience, we close this Appendix by describing how employing the
laboratory procedure described by Rose (1997, see Figure 2 herein)

A satisfactory procedure to follow, for example, will include sequentially taking the following laboratory steps to realistically model the migratory counter-flow causing ejection of hydrocarbon fluid phases that have catalytically originated in organic rich source bed sediments which subsequently are both being stressed by overburden forces and at the same time becoming receptive to the influx of invading and preferentially wetting aqueous fluids from downstream aquifer locations where stratigraphic and/or structural entrapments to form future petroleum reservoirs can occur. Thus to model such complex geophysical events, we proceed as follows:

We select representative core samples of reservoir rock that have been 'restored' as much as possible to their original physico-chemical states that more-or-less are indicated to have existed back in geologic time when they had been domain parts of some prehistoric aquifer say of the sort implicitly envisioned to have existed in earlier times by Siddique and Lake (1992)

These furthermore were taken to be^{31}.and*{ΔH*_{1 }> 0}have been properly set so that both fluids are flowing downwards, but here it is the wetting fluid that is being 'pushed while it is the nonwetting fluid that is being 'dragged' along. This procedure to be followed can be thought of as somewhat equivalent to the classical restored-state capillary pressure*{ΔH*_{2 }= 0},(so-called restored-state) experiment as described throughout Chapter 9 and especially in §9.2 by Bear (1971)*drainage*. In consequence the connected pore space that are at the sample top now will contain the lowest (e.g. perhaps approaching irreducible) levels of the wetting aqueous phase saturation. And during this process the aqueous wetting phase escapes from the system because it is being 'pushed' downwards and 'out', and this causes the oil nonwetting phase to be 'dragged' down 'into' the system.^{3}

^{ }The next laboratory step to take is that referenced in the Eqns. (9) Box above as involving the Exp.#2 Data that is needed to calculate values for the twotransport coefficients which are needed when an algorithm is wanted predict outcomes to be expected when conducting an*D*_{ij}_{ }kind of capillary pressure process of the sort characterized by the Case 1 conditions illustrated in Figure 2. Here, both fluids will be flowing upwards, but now the imposed boundary conditions are where it is that*imbibition*, and also where*{ΔH*_{2 }> 0}The experiment being conducted as referenced above in effect constitutes a basic type of waterflooding process of oil recovery. The end point of the model experiment undertaken for example can be set to occur when the output data indicate that only irreducible (i.e. immobilized) residual oil is left in that part of the reservoir being represented by the selected core sample. With ingenuity, of course, algorithms for processes of greater complexity than the one being treated here likely can be developed for reasons such as the following: (a) The experimental apparatus shown as Figure 2 in the body of the paper is one where co- and counter-current flow conditions can be imposed and monitored, More than that fluxes (*{ΔH*_{1 }= 0}.and forces*viz. flow rates and displacements)*

^{ }

2*Illini
Technologists International;* P.O.Box 2424, Champaign, 61825,
Illinois, USA; __wdrose@uiuc.edu__;
Facsimile <1.217.359.9289>.

3 As a complication to be ignored in this abbreviated account, said RVE's may enclose internally distributed so-called 'dead-end' pores functioning like internal sources and sinks which can add or remove fluid to and from the surrounding somewhat major central pore spaces.

4 i.e. Just as the alimentary tract extending from the mouth to the anus, and indeed the lung are both inside and at the same time outside of the human body!

**5**
Petrach, the highly regarded poet and philosopher of the 14th
Century A.D. said in his ** Epistolæ de Rebus
Familiaribus [XXII.v]:**
"That Simulation which aids Truth cannot be regarded as a
Lie!". And this opinion was given at a time when it was common
belief that " ... unfortunately many of the most dangerous lies
were thought, by those disseminating them, to be assisting some
larger truth!" as reported by Bergen Evans in his

6 Currentyly in preparation as of 12/06/2003.

7 A current tribute to the emory of ProfessorLars Onsager

8 I think it was Bertrand Russell who spoke thusly when wearing the mathematical hat of his youth!

**9
Those who think the Principle of Microscopic
Reversibility is sacrosanct rocket science
because it works can read:
http://www.britanicca.com/article?eu=53830,
but then weep when they hear "… general time-asymmetric
behavior of macroscopic systems---embodied in the second law of
thermodynamics---arises naturally from time-symmetric microscopic
laws due to the great disparity between macro and micro-scales. More
specific features of macroscopic evolution depend on the nature of
the microscopic dynamics. In particular, short range interactions
with good mixing properties lead, for simple systems, to the
quantitative description of such evolutions by means of
autonomous... :. And remember, just because Gertrude Stein in her
Sacred Emily said "…Rose is a Rose, is a Rose, is
a Rose!…" does not prove that plausible generalities at
best are assumptions and at worst presumptions as Rose described in
(1991a)^{21} how fact and fancy
sometimes can be confused !**