“revisiting” the enduring buckley–leverett ideas

www.elsevier.com/locate/petrol

Journal of Petroleum Science and E

bRevisitingQ the enduring Buckley–Leverett ideas

Walter Rose*, Dean Michael Rose

Illinois Technologists International, P.O. Box 2424, Champaign, IL, 61825, USA

Received 18 November 2003; accepted 25 August 2004

Abstract

For more than 60 years, the ideas imbedded in the watershed paper of Buckley and Leverett [Buckley, S.E., Leverett, M.C.,

1942. Mechanism of fluid displacement in sands. Trans. AIME, 146, 107–116.] have been employed by geo-scientists of

various persuasions to forecast what specifically might happen when fluids are either produced from and/or injected into

subsurface porous rock domains through systems of wells drilled for that purpose that connect near surface petrochemical

facilities with subsurface transport source and/or sink interface locations. Example cases of related applications are those when

and where the accessed local pore space already contains desired quantities of producible valuable fluid-phase species (e.g.,

such entities as potable water; certain and mineral-rich brines; petroleum liquids and gases; other valuable gases such as helium,

LPGs, CO2; semi-solids like bitumens, tars and methane hydrates; fluid and/or entrained solid Waste Disposal Materials; etc.)

which, for example, can hopefully be produced and/or economically injected into repositories and/or subsurface basin-wide

storage strata and/or along subsurface transport paths.

In Rose [Rose, W., 1988. Attaching new meanings to the Equations of Buckley and Leverett. Journal of Petroleum Science

and Engineering 1, 223–228.], however, it was suggested that the Darcian-based algorithm originally and even currently

employed by many traditionalist reservoir transport process simulation authorities only poorly models actual reservoir transport

process events. The thought behind this presumption has to do with the fact that viscous and/or other related coupling effects for

dynamic multiphase-saturated media are not quantitatively accounted for in Darcy’s law that more often than not is seen to

empirically only describe low-intensity single-phase flow data for Newtonian fluids. Accordingly, an algorithmic form was

prospectively adopted that was based on the theorems of non-equilibrium thermodynamics variously referenced in Truesdell

and Toupin [Truesdell C., Toupin, R.A, 1960. Classical field theories. Handbuch der Physik III/1, 226–793.], in the DeGroot

and Mazur [DeGroot S.R., Mazur P., 1962. Non-Equilibrium Thermodynamics, North-Holland Publishing, Amsterdam.]

rendition of Onsager [Onsager, Lars, 1931. Physical Review 37, 405–426. Physical Review 38, 2265–2279.] dogma, by Rose

[Rose, W., 1969. Transport through interstitial paths of porous solids, METU (Turkey). Journal of Pure and Applied Science 2,

117–132.] as applied to porous media transport phenomena, similarly by Bear [Bear, J., 1972. Dynamics of Fluids in Porous

Media, American Elsevier, New York.], and in many other places.

Accordingly, in this paper, our goal in revisiting the ideas of Buckley and Leverett one more time is to search for

modified schemes to conduct coherent reservoir process simulation studies that involve less computational and parameter

0920-4105/$ - s

doi:10.1016/j.pe

* Correspon

E-mail addr

ngineering 45 (2004) 263–290

ee front matter D 2004 Elsevier B.V. All rights reserved.

trol.2004.08.001

ding author. Fax: +1 217 359 9289.

ess: [email protected] (W. Rose).

W. Rose, D.M. Rose / Journal of Petroleum Science and Engineering 45 (2004) 263–290264

measurement work than is required, for example, by the standard procedures given in the definitive Bear and Bachmat

[Bear, J., Bachmat, Y., 1990. Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic

Publishers.] monograph that refers to aspects of the famous (Onsager, [Onsager, Lars, 1931. Physical Review 37, 405–426.

Physical Review 38, 2265–2279.], et. seq.) related schemes that are only somewhat akin (but not identical) to the several

unique methodologies we shall be proposing here that includes the so-called APTPA formulations of Rose and Rose [Rose

W., Rose, D., 2004. An upgraded porous medium coupled transport process algorithm. Transport in Porous Media,

Reference # TIPM2. (in press). See also Rose, W., Gallegos, R., Rose, D., 1988. Some Guidelines for Core Analysis

Studies of Oil Recovery Processes. Journal of the Society of Professional Well Logging Analysts (SPWLA) 29 (May–June

Issue), 178–183.].

D 2004 Elsevier B.V. All rights reserved.

Keywords: Idealized petroleum reservoirs; Transport process models; Simplistic Buckley–Leverett algorithms; Viscous coupling effects;

Anisotropic media properties; Capillary imbibition fluxes and forces

1 See Buckley and Leverett (1942), Babchin and Yuan (1997),

Bear and Bachmat (1990), Dullien (1992), Hadad et al. (1996),

Marle (1981), Rose (1988, 1990a, 1991a,b, 1995b, 1997, 2000b,

2001a); Rose et al. (1999), Siddiqui and Lake (1992); Yuan et al.

(2001).

1. Introduction

The content of the watershed paper of Buckley

and Leverett (1942) is being revisited here to further

resolve some algorithm formulation difficulties

timidly examined by the present author some 16

years ago (1988). Questions long overdue are

addressed about the possibility of further upgrading

the basic classical Buckley and Leverett algorithms

so that their use can time-wise and quality-wise

better yield improved predictions of future reservoir

states. In a nutshell, our simple aim here is to search

for practical ways to facilitate the monitoring of

certain representative types of petroleum recovery

transport processes. Specifically, we shall intend to

focus on occurrences that commonly transpire

during production of oil (and other fluids as well)

from those subsurface reservoirs which exist as

scattered local features associated with many world-

wide regional aquifers. In particular, we shall be

justifying our ideas by considering what we take to

be plausible transport process models that can show

how (and better yet also show why) entering

immiscible fluids such as waters (e.g., reservoir

upstream formation and/or injected waters) can

efficiently coalesce, displace and replace in situ

pore space oil and/or other immiscible fluid ganglia

so that the latter can move naturally towards the

downstream production wells that drain into surface

gathering systems.

According to the classical ways of thinking,

however, the proof that rationally based reservoir

performance algorithms actually are being employed

is best demonstrated and confirmed by undertaking

definitive field and laboratory modeling experiments.

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 production events.

This logical way to proceed is illustrated in what

follows by focusing on several somewhat simplified

but still representative reservoir cases. Here, by way

of 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 the characteristic cases of where

wetting liquids like brines spontaneously happen

(say, because of prevailing ambient artesian con-

ditions) to replace and displace a portion of the

nonwetting oil-phase fluids that originally was

present in the pore space of typical petroleum

reservoirs. These are events that can occur in

reservoir systems such as those found in strati-

graphically 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 applic-

ability of a plausible theory based on what can be

identified as modified Buckley–Leverett dogmatic

ideas1 and then (b) to confirm whether or not the data

obtained in subsequently scaled physical model (and/

or Gedanken) experiments appear to be sufficiently

W. Rose, D.M. Rose / Journal of Petroleum Science and Engineering 45 (2004) 263–290 265

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 to what seems to be new in

our present drevisitingT of earlier advocated algorith-

mic presumptions, which include (a) our reemphasiz-

ing of the advantages coming from adopting coupling

affected rather than traditional Darcian flux–force

relationships (viz. so that irreversibility effects are

better taken into consideration) and (b) our willing-

ness to favor the use of computational algorithms that

stochastically yield good forecasting information even

when the theoretical justifications remain obscure.2

We start by recalling that in Rose (1988), an

overdue attempt was made to upgrade and extend the

usefulness of what were only popular vintage Buck-

ley–Leverett algorithms such as given by Bartley and

Ruth (1999). Unlike Patek’s (in press) recent

drevisitingT style to confirm the correctness of the

content of the original 1855 Fick paper on bLiquidDiffusionQ, our approach here has been to focus on

simple (but not necessarily fully tested) plausibly

rational approaches and even on grossly simplified

ones such as Rose (2000a), 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 such

as given by Marle (1981) and Rose (1995a). In par-

ticular, 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 implied nature of the transport

processes that are prospectively and perhaps intui-

tively assumed to be involved.

Under consideration here to provide background

for the topics to be discussed are the ideas expressed

2 See Rose (1963, 1990a,b, 1991a,b, 1995a,b, 1997, 1999a,b,

2000a,b, 2001a,b), Rose and Robinson (2004) and Rose and Rose

(2004).

in the key monographs of Bear (1972), Bear and

Bachmat (1990), Bird et al. (2002), Dullien (1992),

Kaviany (1995), Marle (1981) and also in a lot of the

here-cited (but perhaps less read) early journal papers

by the present author. These citations, in turn, further

point to a vast collection of scattered international

papers which likewise can point to other parallel case

documentation’s of possible 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 Buckley and Leverett paper

(loc. cit.), only some of them have so far been

critiqued persuasively enough to earn general accept-

ance. Accordingly, it should not be expected that what

is written on these pages will conclusively settle all

remaining issues of controversy and disagreement.

The fact of the matter is that the full proof of either

the historical or the modern-day opinions about the

viability of underlying Darcian-based Buckley and

Leverett dogmatic presumptions actually cannot 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 therefore

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,

our modest aim now is simply to verbally offer

without full proof some alternative analytically

algorithms. And we take this approach as being a

reasonably Cartesian way for future workers (partic-

ularly the modern experimentally minded ones) to

improve and perhaps confirm the use of what to this

author seem to be acceptable alternative ad hoc and

fruitful ways of thinking. And indeed that is why in

what follows, we prospectively choose to imply as our

principal sufficient thesis, namely, 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

with and/or without superimposed spontaneous capil-

lary imbibition effects.

Fig. 1.


1.1. Background ideas

As in the earlier Rose (1988) paper, we start by

affirming the relevance of the nicely phrased con-

servation of mass statement by Buckley and Lever-

ett, but reject their overblown Darcy-based

statements because of the arguments given by Rose

(1999a,b) and Rose (2000b).But where Buckley and

Leverett in the 1940s left matters ruminating now for

more than 60 years, and indeed where many

contemporary scholars like Hadad et al. (1996) in

one way and Siddiqui and Lake (1992) in other ways

also continue to do so, remains only justified as

being an interesting historical example of early-day

mistaken exaggerations.

In any case, we accept as an opening statement the

sense of the Buckley–Leverett mass conservation

theorem as phrased in Eqs. (1)–(4) below, but as will

be seen, we categorically adopt the modern viewpoint

due to enlightened thinkers such as DeGroot and

Mazur (1962) that proper flux–force relationships to

describe entropy-producing coupled (hence irreversi-

ble) transport processes so far have not been shown by

experiment proof to be Darcian in character as

originally was (and in some quarters still is) wishfully

presumed.

Equation Box I

3 As a complication to be ignored in this abbreviated account

said RVEs 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 centra

pore spaces. And as for other things that are topologically both

inside and outside of themselves, consider a snake swallowing its

tail to create a portion that is both inside and outside of itself, or

think about the air-filled pockets of the lung which are spaces within

the body of animal that connect directly with the air that sorrounds

the body that is not immersed is a tub of water!.

f BS1=Bt½ �ð x;y;z þrS J1f gÞ ¼ 0 where ð1Þ

Since S1 þ S2Þ¼1 u rS J1gÞ¼ rS J2f gð Þ:fððð2Þ

ulogically rS J1f gð Þ¼h J1þJ2f gS rXf gð ÞþrPcþrGf gS rWf gð ÞþW rSrPcf gð Þ: i

ð3Þ

whereu X ¼ a a þ bð Þ=

and W ¼ bX;

with a and b;and also c; d; e; �to be defined by

Eqs: ð7aÞ ð7cÞ j: ð4Þ

��

In Eq. (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 driving forces acting in three-dimen-

sional space, where fluxes of mass/energy laden fluid

particles and displacements occur as Tribus (1961)

uniquely mentioned a long time ago.

Fig. 1, for example, is a not-to-scale schematic

cartoon depicting how transport can be thought to

occur, for example, in two-phase petroleum reservoir

systems that are imbedded in regional aquifers.

Depicted there are upstream to downstream pore

network domains with source and sink termini to

topographically complex in-series and parallel inter-

connected pore paths. These, in turn, lie within

contiguous macroscopic representative volume ele-

mental [so-called representative volume elements

(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 the solid mineral-

occupied constituents of the container rock system

network.3

Specifically, what is wanted, of course, are

laboratory ways to test whether the transport process

,

l


theory that is under consideration can be authenti-

cated by examining the data obtained in scaled

model physical experiments. For example, here we

shall be advocating the use of Rose’s (1997)

recently described laboratory methodologies that

appears conceptually and uniquely to be well-suited

for the intended data-gathering purposes (e.g., as

described in Appendix B herein). 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 A).

As will be seen, verbally, the major cited equations

assert that: (a) At least 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. Section 4.4); 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 observed field and laboratory exper-

imental data.

Eqs. (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 explicitly or

implicitly based on mass/energy conservation prin-

ciples. Particular attention will be limited for

simplicity, however, to important cases where the

reservoir pore space domains at all times are

represented as being completely saturated by as

many as two essentially incompressible and immis-

cible Newtonian liquids such as: (a) In situ connate

water together with any injected or otherwise

invading aqueous liquid phases; and (b) Any

previously unproduced (i.e remaining) original oil

in place. Each of the selected RVE macroscopic

domains of the representative reservoir depicted in

the paper’s Fig. 1 will be chosen also for simplicity

to be characterized by possessing locally constant

porosity magnitudes that can be predetermined.

Moreover, the parameter subscripts {1, 2}={W, N}

for the process variables are employed simply to

make it clear which wetting and nonwetting fluid

phase will be found when and where during thc

course of the ensuing reservoir transport processes.

Thereby, appearing in those relationships (viz. where

the local positive or negative accumulations of each

of the two fluids are being referenced) the first term

on the left-hand side of Eq. (1) will be seen to be

exactly equal to the dinflow minus outflowT con-

vergences of the designated fluid phases as

expressed by the second left-hand term of the same

equations. Hence, Eq. (2) shows that the sum of the

companion two fluid-phase flux divergences locally

will be exactly equal to zero.

Accordingly, that is why Eq. (3) then indicates

that the local time-changing flow transport pro-

cesses under consideration are ones that involve

vector operations like addition, gradients, multi-

plication 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) 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) inter-

faces which can be both stationary or microscopi-

cally moving more or less parallel to the

macroscopically seen motions of the overlying

flood-front positions. 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 to indicate the fact that prevailing capillary

driving forces also may contribute to the local

convergences and associated divergences of the two

companion fluid phases. In any case, Eq. (3) shows

that when flow transport locally is to some extent not

directed horizontically, this means in consequence that

the superimposed gravity forces cannot safely be

ignored (N.B. Tribus, 1961, p. 520 ff). Finally, with

respect to Eq. (4), when the transport processes under

study are ones where so-called capillary pressure


gradients can be ignored, we have the special case

result that:

Equation Box IA

Jx1 ¼ A Xx

1

� �þ B Xx

2 Xx1

� �þ C xx

2

� �� ¼ ax Xx

1

� �� þ bx Xx

2

� �� ð5Þ

Jx2 ¼ D Xx

1

� �þ E Xx

2 Xx1

� �þ F Xx

2

� �� ¼ cx Xx

1

�g þ dx Xx

2

� �� ð6Þ

j a3u A Bð Þ ¼ J1=X1ð Þb2u Bþ Cð Þ ¼ J1=X2ð Þc3u D Eð Þ ¼ J2=X1ð Þd2u E þ Fð Þ ¼ J2=X2ð Þe4u Aþ Cð Þ ¼ J1=Xð Þu4u Dþ Fð Þ ¼ J2=Xð Þ

j ð7Þ

*J1 ¼ K1!X1

and

J2 ¼ K2!X2

+ð1aÞ

and�X ¼ K1=K1 þ K2

W ¼ K2Xð Þ

ð4aÞ

with {i, j}={1, 2} andP

Si=1, it follows that

DS1

Dt

�x

¼ DJ1

Dt

�t

where Ki ¼ Ki Sið Þ ð1bÞ

This the primative no-coupling Darcian case.

Here, the two equations (Eqs. (1a) and (4a)) are

intended to describe low-intensity uncoupled two-

phase fluid flow in isotropic media as though Darcy’s

law holds for two-phase systems, and as such they are

equivalent to Eq. (20) below. Also, clearly, for the

following Eqs. (1a) and (4a) necessarily to be

derivatives of the first ones so far have not been

proven experimentally. . .but only innocently pre-

sumed by some early and later workers according to

Rose4 as well as by other investigators from Yuster

(1951) to those mostly modern workers (for example,

as cited in the body of this present paper).

In passing, however, it is also very much worth

noting that Eqs. (1a) and (4a) as they stand can also be

employed to describe low-intensity flow of single-

phase Newtonian fluids in two-dimensional aniso-

tropic 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 think-

ing, we look at Eqs. (5) and (6) shown below as

polynomials where the coefficients of linear propor-

tionality are those dozen somewhat redundant and

symmetrically interrelated saturation-dependent terms

that appear in Eq. (7). These are seen to express the

not unexpected and logically plausible linear transport

4 See Rose (1966, 1969, 1972a,b, 1974, 1976, 1999a,b, 2000b).

relationships that seem to properly model why and

how the causative thermostatic and thermodynamic

forces give rise to the consequent fluxes and displace-

ments by which natural non-equilibrium systems

irreversibly approach final end states.

Equation Box II

In passing. we notice that both Eqs. (5) and (6)

display flux vs. 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 [a, b, c,. . .]). Eq. (7), andthen Eq. (7a) then show the simple mathematical

relationships and notational equivalencies between the

six Greek and six Latin letter transport coefficients.

And, as seen in both Eqs. (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 Eq. (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:

Equation Box IIA

Given Dij=Dij(Si);P

Si=1; {i, j}={1, 2}={W,

N};

SinceD11 D12

D21 D21

� u

a b

c d

�

uA Bð Þ Bþ Cð ÞD Eð Þ E þ Fð Þ

� then Eqs. (5) and (6) can be rewritten in matrix

form as

�J1J2

� ¼ D11 D12

D21 D21

� !

X1

X2

� ; hence

ð5a; 6aÞ

X ¼ D11 þ D12XDij

;

W ¼ D11 þ D12XDij

!D11D12 D12D21ð Þ ð7bÞ

where Dij

¼ Dij S1ð Þ and whether DijuDji

� �or not:

ð7cÞ

Clearly, to monitor and numerically describe the

steady and unsteady states of the transport processes

which are called for by Eq. (1) above, we start with the

above two equations given in the Eq. (5a, 6a) 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 Dij to other laboratory experimentally obtained

flux and conjugate force data. Then with the four Dij

relationships all known and established, values for the

{X, W) terms as functions of saturation can be

inserted in Eq. (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 Eqs. (7a),

(7b), (7c), is that with the superscript (x=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, ( Jr, Xr) where {r, s}={1, 2}.

Specifically, this possibility can be seen in the

interpretation of Eqs. (8) below which presents in

matrix form the senses of Eqs. (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 equa-

tions to extract numerical values for the two sets of six

transport coefficients of proportionality as identified

by Greek and Latin characters in Eq. (7). And,

remarkably as a special feature of the unique

experimental method being employed contagiously

by Dullien and Dong (1995), Rose (1951a,b, 1985,

1987, 1996, 1997, 2001b), and Zarcone and Lenor-

mand (1994), 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 Jr fluxes

and displacements and Xs measured force data values

will tend to be differ during the conduct of each

experiment type.

Equation Box III


Here, it is seen that Eqs. (8a) and (8b) 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 the

fourth experiment, the two driving forces are con-

strained to be equal to each other nor equal to zero.

Other independent sets of experiment could be the

ones where the companion travel direction cold be set

to be counter-linear rather than collinear.

X1

0

�or

0

X2

�!

D11 D12

D21 D22

�¼ J1

J2

��ð9aÞ

D11 ¼J1ð ÞX1ð Þ ; and D12 ¼

J1ð ÞX2ð Þ ; ð9b and 9cÞ

D21 ¼J2ð ÞX1ð Þ ; and D22 ¼

J2ð ÞX2ð Þ : ð9d and 9eÞ

where Dik ¼ Dik Sið Þ! ð9f to 9iÞ

2. Discussion

In these connections and as alert readers can

carefully note, Rose (1997) 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 contempora-

neously) by systems of carefully located 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, refin-

ery process installations, storage tanks, waste disposal

facilities, etc. 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 Eqs. (8a) and (8b).

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 Eqs. (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 Dik transport coefficients.

2.1. Ad hoc theory for simple systems

As it turns out, Aitken (1939, Chapter II) describes

(as did other authorities both in antiquity and in

modern times) the classical ways to solve independent

simultaneous equations which while conceptually

straight-forward is occasionally computationally com-

plex. Accordingly, we abandon here employing the

popular method of determinants, but more simply

consider the ideas developed in the recent disclosures

of Rose and Robinson (2004) and Rose and Rose

(2004). We do this first for the particular case

presently being considered for illustrative purposes,

namely, where we start out only having the indeter-

minate set of two governing equations with four

unknowns, but then we are left dealing with the need

to find the equivalent of two other independent

relationships so the four equations can be solved

simultaneously.

In these connections, of course, for Onsager

diffusive flux cases where theoretically justified

Onsager Reciprocity Relationships can be postulated

to hold, the availability of additional relationships

such as those having the form of the (DijuDij)

equality are redundantly needed statements, for

example, in those cases where Gabriella et al.

(1996) show there is no need to invoke the Principle

of Microscopic Reversibility. This is because, specif-

ically, 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 Eq. (9a), one then ends up

with four independent Eqs. (9b and 9c) (9d and 9e)

and the four Eq. (9f to 9i) clearly follow, or:

Equation Box IV


Fig. 2, copied with permission from the indicated

Rose (1997) TiPM paper, schematically shows how

the measurements called for by the Eqs. (9a), (9b and

9c), (9d and 9e), (9f to 9i) 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

and Dong (1996), and before that by Zarcone and

Lenormand (1994), and indeed before that when Rose

(1976, cf. Appendix) was organizing a petroleum

engineering curriculum at the Institute of Technology

at Nigeria’s University Ibadan.

The Fig. 2 apparatus arrangement which has been

referenced in a number of the author’s recent

publications is also further critiqued in Appendix B

of this paper.

A proper legend for Fig. 2 implies the following.

Here are schematic depictions of Case 1 and Case 2

scenarios shown in the left-hand diagrams, respec-

tively, 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 nonwetting fluid

contained in the A and B siphon reservoirs where

{(q2gDH2)=X2}N0, while the imposed driving force in

the wetting fluid contained in the C and D siphon

reservoirs is zero since {(q1gDH1)=X1}=0. Similarly,

during the second set of experiments, then in the Case 2

Fig. 2. (After Ro

experiments, the imposed gravity free-fall driving force

is zero in the nonwetting fluid since in the A and B

siphon reservoirs, it is {(q2gDH2)=X2}=0, while the

driving force imposed by the elevation difference of the

wetting fluid contained in the C and D reservoirs is

finite since it now follows that {(q1gDH1)=X1}N0.

As shown, filling and spilling fluid reservoir tanks

A and B are for the nonwetting fluid, while C and D

are for the wetting fluid. In the right-hand graphs of

pressure, p, vs. elevation, z, curves daT and dbTaccordingly display the hydrostatic gradients when

the wetting and nonwetting fluids are stationary, while

curves dcT and ddT display gradients to be expected in

the mobile nonwetting and wetting phases, respec-

tively, that are present and caused to match those of

curves daT and dbT for the corresponding stationary

fluids.

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

se, 1997).

D11 ¼Jpushed1

� �L

q1gDH1

and D21 ¼Jdragged2

� �D11

Jpushed1


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 Eqs. (10a), (10b), (10c)

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 Fig.

2. These, as seen, are consistent with indications of the

earlier general relations already anticipated by Eqs. (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:

Equation Box V

when DH2 ¼ 0; but j p1 p2ð Þ ¼ 0;

u jSi ¼ 0 u D22 ¼J total2 J

dragged2

� �L

q2gDH2ð Þ

whenDH2p0 jSip0

unless DH1 ¼ DH2

� where DH1 ¼ Ljp1½ �= q1g½ �N0 ð11aÞ

J1 ¼ D11

q1gDH1ð ÞL

þ D12

q2gDH2ð ÞL

ðJ total1 Þ ¼ Jpushed1

� �þ J

dragged1

� �ð10aÞ

J2 ¼ D21

q1gDH1ð ÞL

þ D22

q2gDH2ð ÞL

ðJ total2 Þ ¼ Jdragged2

� �þ J

pushed2

� �ð10bÞ

Note that XiuqigDHi

LuZ

dpi þ qig

Zdz

� �

Also Dij

¼ D11 D12

D21 D22

� �where D12 bD21!!!

ð10cÞ

D22 ¼Jpushed2

� �L

q2gDH2

and D12 ¼Jdragged1

� �D22

Jpushed2

� �when DH1 ¼ 0 but r p1 p2ð Þ u rSi ¼ 0;

however u K11 ¼J total1 J

dragged1

� �L

q2gDH2

whenDH1p0 u rSip0

unless DH1uH2

� where

DH2 ¼ Lrp2½ �= q2g½ �ð ÞN0 ð11bÞ

being characterized by Eqs. (9a), (9b and 9c), (9d and

9e), (9f to 9i) and (10a), (10b), (10c). To be noted in

passing, Eqs. (10a), (10b), (10c) also apply to the

laboratory configuration for the measurement scheme

illustrated in Fig. 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 arrange-

ment 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) remarkably

are identically zero during the steady-state two-phase

flow episodes. Hence, we now can accept Eqs. (11a)

and (11b) as shown below. Hence, for the Case 2

formulae, we have:

Equation Box VI

And for the Case 1 formulae, we have:

Equation Box VII

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

½J ai ¼ D

a;bi;j

�!X b

j

� ið12aÞ

and

Xequal

Da;bi;j

� �¼ or D

b;aj;i

� �not pX

266664

377775 ð12bÞ

{where [a, b]=[W, N] are mass/energy phases and

[i, j]=[x, y] are locations}.


first of Eq. (11a) to the last of Eq. (11b), and likewise

by equating the last of Eq. (11a) to the first Eq. (11b).

And more than that, by forming a ratio between the

second of Eq. (11a) to the second of Eq. (11b), one

can assess whether or not the Phenomenological

Equations of Onsager (cf. as given by DeGroot

and Mazur) asserting that (Dij=Dji) perhaps may

or may not apply for the particular transport

processes being characterized by Eqs. (9a), (9b

and 9c), (9d and 9e), (9f to 9i) and (10a), (10b),

(10c).

2.2. Formulating more complicated cases

Finally, we close this overview discussion by

employing the ideas of Rose and Robinson (2004)

and also the related ideas to some extent already

illustrated in other recent publications such as Rose

(2001a,b). And as above in the previous subsection

to this paper, we extend our discussion of rationales

for on occasion employing generalized ad hoc (rather

than fully justified empirically and theoretically

based algorithms) as sensibly accurate and economic

ways to easily model even more complicated cases

of coupled irreversible fluid-phase transport pro-

cesses. These, in fact, are those important ones,

which are known to commonly occur in porous

sediments.

The ones under consideration, however, do not

seem to correspond except perhaps superficially to

those other important special case processes like

thermodiffusion which have been described by the

followers of Onsager with such acuity by DeGroot

and Mazur (1962) 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 pro-

cesses are seen to be changing with time. Alter-

nately, from the Lagrangian point of view, one may

think of such RVE locations as being occupied by a

ddroplet-trainT 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 mo-

tions 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 Eqs. (12a) and (12b) in

the form previously presented by Rose (1995a,b) for

particular miscellaneous transport process of interest.

Thus, we hazard to suggest that the situation being

monitored:

Equation Box VIII

It is in Eq. (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

aforementioned classical Onsager relationships. For

example, in coupled thermodiffusion systems, the a, bsuperscripts stand for thermal and chemical energy

fluxes, but in Eq. (12a), they can stand for the {W, N}

terms that designate the two-phase immiscible wetting

and nonwetting pore fluid phases. . .while the {x, y}

subscripts stand for the Cartesian two-dimensional

spatial locations.

jwxjwyjwxjnj

2664

3775¼

hDw;wx;x i Dw;w

x;y Dw;nx;x Dw;n

x;y

Dw;wy;x hDw;w

y;y i Dw;ny;x Dw;n

y;y

Dn;wx;x Dn;w

x;y hDn;nx;x i Dn;n

x;y

Dn;wy;x Dn;w

y;y Dn;ny;x hDn;n

y;y i

2664

3775

xwxxwyxnxxny

2664

3775;

ð13aÞ*i; j ¼ x; ya; b ¼ W;N

Da;bi;j i

jai

xbj

+; ð13bÞ

ðDW ;Nx;y Þ ¼ Dy; x

� �! DN ;W� ��

; ð13cÞhence where�

Da;að ÞN Da;b� �

u Db;a� ��

Dx;x

� �N Dx;y

� �u Dy;x

� �� ð13dÞ


On the other hand, the ambiguous Eq. (12b)

formulation clearly lacks the definiteness of the

classical Onsager Reciprocity Relationships. Accord-

ingly, we find ourselves now left facing the heretofore

somewhat neglected task of inventing and authenti-

cating 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 nondiffusive

transport process cases, we now can jump to consid-

ering 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 (1995a) paper, the

above Eq. (12a) display with a simplified notation

the possibly probable linear relations between fluxes

and forces which are predicted if and when the

indications of Eq. (12a) are to be believed to apply

to the case of two-phase flow in 2D anisotropic

media. And Eq. (13b) which appears below seems

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 Eq. (12b), Rose

(1995a,b, 1996) have suggested that relationships

displayed by the 16 Dijab transport coefficients of

Eq. (12a) might, in some cases, experimentally

prove to be:

DWWXX NDWW

XY where DWNXX N?bDWw

XY :

The equivalence of them, however, to the 16 Dijab

diffusive flux transport coefficients given in Equa-

tions Boxes VII-VIII [i.e., where Eqs. (11a), (11b),

(12a), (12b), (13a), (13b), (13c), (13d), (14a), (14b),

(14c), (14d), (14e), (14f to 14i), (15) are located] at

this point so far has not been established; hence,

the existence of reciprocal relations between the

terms of Eq. (13b) remains an open question, but

not one necessarily 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 Eqs. (12a), (12b), we can write

Eqs. (13a), (13b), (13c), (13d) as:

Equation Box IX

Here, Eq. (13a) presents four scalar equations.

containing 16 initially unknown transport coefficients,

and Eq. (13b) shows 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 deter-

mined, and this without the need to verify in advance

which (if any) of the reciprocity relationships postu-

lated by the Eqs. (13c) and (13d) relationships need to

be experimentally verified.

For example, we may consider an innovative

device presented by Rose (1976) and revisited again

in Rose (2001a,b) to uncover more than 16 independ-

ent 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) 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 Eqs.

(12a), (12b) and (13a), (13b), (13c), (13d) can be


extended to other more simple as well as more

complicated cases of nondiffusive 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:

Equation Box X

a

b

�¼ A E

B F

�e

f

��ð14aÞ

Ae=að Þ Ef =að ÞBe=bð Þ Ef =bð Þ

� �¼ 1

whereeu aF bEð Þ= AF EBð Þ½ �fu bA aBð Þ= AF EBð Þ½ �

� ð14bÞ

B

N

b

i?

3775

2664E ð14cÞ

0 Ef =að Þ0 Ef =bð Þ

�¼ 1

�ð14dÞ

Ae=að Þ 0

Be=bð Þ 0

�¼ 1

�ð14eÞ

u

A ¼ a=eð Þ; B ¼ b=eð Þ; E ¼ a=fð Þ;F ¼ b=fð Þ ð14f to 14iÞwhere hA, B, E, Fi are fixed constants of fluid

saturation.

As shown by the Eqs. (14a), (14b), (14c), (14d),

(14e), (14f to 14i), 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 Eq. (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) 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 compu-

terized 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 employ-

ing less costly computational algorithms which are an

insidious guaranteed to obtain faulty calculations.

2.3. dUnfinishedT capillary imbibition algorithms

Finally, we consider it to be a matter of great

importance to call attention to the formulation of

viable algorithms which in a rational way can be

predictors of what we descriptively call forced vs.

spontaneous capillary imbibition reservoir processes.

There are at least four interconnected reasons why

this should be true as follows. (a) Involved in many

(if fact almost all cases of ordinary reservoir usages)

immiscible multiphase fluid phases are seen and

caused to flow variously out of, through, and/or into

the pore space of subsurface reservoir rock domains.

(b) The associated transport processes which are

involved are comparatively complex and not well

understood in terms of the microscopic quantum laws

that only loosely can be employed to explain

observed macroscopic behavior. (c) The fact is that

almost all oil and gas subsurface accumulations co-

exist with and/or are contiguous to bottom and edge

water aquifers influxes and sometimes also to

injected surface waters. (d) These displacements

and replacements often are subject to the complex

and often dominant action of important capillary

forces that affect the movement of the resident

subsurface hydrocarbon fluids.


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 dunfinishedT (meaning not

well-understood) topic of commercial as well as

scientific importance, but also one worth revisiting

by reading in Appendix B to follow how the topic

needs unraveling and unscrambling and extrication

before reservoir engineers of the 21st Century can say

that the art of constructing truly coherent algoritmic

forecasts of petroleum reservoir behavior. For exam-

ple, the ordinary water-flooding process after all is a

paradigm example affirming the relevancy of devel-

oping 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 varia-

tions 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 Bear (1972) who was the

earliest one of several later monograph authors like

Marle (1981), Bear and Bachmat (1990), and Dullien

(1992) who took the trouble to recognize that Yuster’s

(1951) watershed paper 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, 1969). 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 imbibi-

tion 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 clarifica-

tions 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 vs. 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

dwetT, 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 chemi-

cally non-reactive. Finally, and for simplicity, atten-

tion will be limited to isothermal transport processes

of low intensity (viz. so the fluid flows will be

JW

JN

JE

0B@

1CA ¼

DWW DWN DWE

DNW DNN DNE

DEW DEN DEE

0B@

1CA

!

X1 ¼ j

Z Z Zdp1

q1

þ g

� � �

X2 ¼ j

Z Z Zdp2

q1

þ g

� � �X3 ¼ jAþ coshð ÞjA½ �

0BBBBB@

1CCCCCA

ð16Þ

where DNWuDWN, DWEuDEW, DNEuDEN and

Dij=Dij(Si).


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) paper

by W. Rose here cited in two current papers. The

canonical forms appear in Equations Boxes XIII and

XIV. Here, Eqs. (15) and (16) are seen to 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). Thus, we have:

Equation Box XI

Ji ¼Xji

Dij!Xj

� �" #and

!

r

� �¼

Xji

Ji!Xið Þ" #

N0

when i; jf g ¼ 1; 2; 3f g: Assume that DijuDji:

Then for i; jf g ¼ 1; 2f Þ;

X1 ¼X2 ¼ qi j

Zdpi

qi

þ g

Zdz

dx

� �for Ji ¼ Jw

or Jn: But for

if g¼ 3f g; X 3cFcwn jAnw þ coshð ÞjAsn½ �; also

J3cBAsn

Bt

coshk

� ��therefore D33 ¼

J3

X3

where perhapsjAsnc0; also Dij ¼ Dij Sið Þ andXSi ¼ Sw þ Snð Þ ¼ 1 ð15Þ

and also:

Equation Box XII

In the above Eqs. (15) and (16), the subscripts,

{i, j}={1, 2}, designate the wetting (say W=aqueous)

and the nonwetting (say N=hydrocarbon) pore fluids,

respectively; hence, the Ji are the so-called Darcian

approach velocity vectors for the two fluids which are

locally at measurable pore space saturation levels, Si,

and where, respectively, the Xj 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 measure-

ments (cf. Rose, 1997). On the other hand, these

equations XE and JE, respectively, refer to free

surface energy gradient driving forces and fluxes

which are manifested by spontaneous capillary

driving force effects of those special sorts percep-

tively mentioned by Tribus (1961), where he draws a

distinction between how non-equilibrium thermo-

static and thermodynamic processes should be

separately considered.

For example, in Eq. (16) notice is to be taken of the

fact that in total there are nine transport coefficients,

Dij, which each are functions of how geometrically

the pore space of the sediment is partitioned through-

out time and space between the two locally present

CJWCJNCJE

0B@

1CA ¼

0 DWN DWE

0 DNN DNE

0 DEN DEE

0B@

1CA!

0

b

Fc

0B@

1CA

YVCVhtest all forces are finiteN0i ð17cÞ

DJWDJNDJE

0B@

1CA ¼

0 0 DWE

0 0 DNE

0 0 DEE

0B@

1CA!

0

0

Fc

0B@

1CA

YVDVtest where a ¼ b ¼ 0

ð17dÞ

whereXa; Xb; Xcare thethermodynamic

thermostatic

� � FORCES


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 Eq. (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, XW and XN, can be selected such as

those shown in the right-hand matrices of Eqs. (17a),

(17b), (17c), (17d) for what are being illustrated as the

logically chosen four separate experimental measure-

ment cases that need to be performed. And, in passing

it is to be also noticed that the fluid flux terms, JW and

JN 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 Eqs. (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 sepa-

rately undertake the experiments prescribed by the

four sets of Eqs. (17a), (17b), (17c), (17d) 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 satura-

tion, spatial location and temporal time during the

ensuing recovery processes.

Equation Box XIII

AJWAJNAJE

0B@

1CA ¼

DWW 0 DWE

DNW 0 DNE

DEW 0 DEE

0B@

1CA!

a

0

Fc

0B@

1CA

YVAVtestwhere b ¼ 0 ð17aÞ

BJWBJNBJE

0B@

1CA ¼

0 DWN DWE

0 DNN DNE

0 DEN DEE

0B@

1CA!

0

b

Fc

0B@

1CA

YVBVtestwhere a ¼ 0 ð17bÞ

Accordingly, Eqs. (17a), (17b), (17c), (17d) 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 Eqs. (18a), (18b) and (18c) as

also given below, a display can be given for the three

dependent and six independent (Di) material response

transport coefficients that can be assessed from the

given input values of [Xi={a, b, c}] 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 J3 and X3 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 they are in the slightly

different cases, where the multiphase flow caused

by mechanical energy gradients is coupled with

unambiguous concentration and/or temperature gra-

dient forces which give rise to heat and mass

transfer effects.

For example, however, by manipulating with

the terms of Eqs. (17a), (17b), (17c), (17d),


auxiliary useful relationships can be developed such

as:

Equation Box XIV

h AJi� �

BJi� �

þ CJi DJi� ��

¼ 2 aJi� �

u MJi� �

AJi� �

BJi� �

CJi� �

þ DJi� ��

¼ 2 bJi� �

u NJi� �

alsoAJi� �

þ BJi� �

CJi� ��

¼ cJi� �

iu OJi� � i

ð18aÞ

D11 ¼aJ1� �

a; D12 ¼

bJ1� �

b¼ D21 ¼

aJ2� �

a;

D22 ¼bJ2� �

b; D23 ¼

cJ2� � c

¼ D32 ¼bJ3� �

b;

D33 ¼cJ3� �þ c

; D13 ¼cJ1� �þ c

¼ D31 ¼aJ3� �

a

ð18cÞ

In Eq. (18a), the flux terms that are wanted are the

nine (j=abcJi=1, 2, 3) and these essentially the ones that

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 FcJi flux terms caused by the Fc driving

forces.

Combining Eqs. (17a), (17b), (17c), (17d) with Eq.

(18a), then one can easily obtain definitions for all of

the transport coefficients as:

Equation Box XV

aJ1bJ1cJ1aJ2bJ2cJ2aJ3bJ3cJ3

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

¼

D11 0 0

0 D12 0

0 0 D13

D21 0 0

0 D22 0

0 0 D23

D31 0 0

0 D32 0

0 0 D33

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA!

abFc

1A ð18bÞ

0@

where it may be the case that D31uD13 because

probably/possibly bJ3uaJ3 !

The matrix Eq. (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, Dij,

i.e., 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 c is finite in magnitude

(hence giving rise to possible spontaneous imbibi-

tion effects), we have the ratio values for the non-

zero Dij elements of the central matrix of Eq. (18c)

such as:

Equation Box XVI

etc. In these connections, however, it must be

mentioned in fairness that the indicated values for

the ambiguous diffusivity element, D33, shown in

the ninth position of the third column of the central

matrix, are perhaps questionable; unfortunately,

however, at the present time, no other clever way

so far has been discovered to remove this uncer-

tainty 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 Eqs. (17a), (17b), (17c), (17d), 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 saturations 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, Xj, where these will be

X1=a or zero, X2=b or zero, and X3=plus or minus

c always (1), and [b] with observational data about

W ¼ D1= D1 þ D2ð Þ and X ¼ D1D2ð Þ= D1 þ D2ð Þ

when X3 ¼ D12 ¼ D21 ¼ 0; but

W ¼ D11 þ D12ð ÞXij

Dij

and X ¼ D11D22 D12D21ð ÞXij

Dij

only if X3 ¼ 0 ð20Þ


the two measurable flux terms, J1 and J2, then

eventually the nine transport coefficient (material

response) terms as defined by Eqs. (18b) and (18c)

can be unambiguously computed either uniquely or

even sometimes redundantly, as shown below. Thus,

it follows that:

[a] By starting with Experiment D, and by observing

{DJ1} and {DJ2=DJ1} and by knowing

gamma, c, one can compute {D13=D31} and

{D23=D32} (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 Eqs. (18a), (18b) and (18c) above as

independently and explicitly providing a correct

value for D33, one then must seek still other

means to deduce and know values for the four

{AJ3}, {BJ3}, {

CJ3}, {DJ3) terms.

[b] Then by performing Experiment A, and by

observing {AJ1 and AJ2} and applying the

plausible reciprocity Relationships, and addi-

tionally by knowing the input magnitude of a,one now can compute values for the {D11} and

{D21=D12} terms.

[x] Then by performing Experiment B, and by

observing {BJ1} and {BJ2} and again apply-

ing the available reciprocity Relationships, and

additionally by knowing the input magnitude

Equation Box XVII

fBS1

Btþj!J1

� � �¼ f

BS2

Btþj!J2

� � �¼ 0;

with J1 þ J2ð Þ ¼ CONSTANT; then"f

BS1

Bt

� þ h J1 þ J2ð Þ! jWð Þi þ X

Dj2jc

E

þ*�

jjc þ DG

! jXð Þ

+#¼ 0 ð19Þ

where S1=S1(x, y, z, t) and (S1+S2)=1 every-

where, hence jd S1=jd S2; also, [jp2jp1]=jjc and X1=X2 necessarily if q1=q2.

of {a}, one now can finally compute redundantly

the {D12=D21} and the {D22} term.

As previously shown elsewhere (for example, see

Hadad et al., 1996; Siddiqui and Lake, 1992; Rose,

1988, 1990a,b), where some, including the author of

this paper, have on occasion continued to adhere

(blindly or otherwise) to the original Buckley–

Leverett dogma that for two-phase flow one can

suppose that the driving forces are only mechanical

energy gradients, it is clear that the following simple

definitions for the W and X parameters of Eq. (19)

that apply to the Darcian modeling case will be those

that appear on top line of Eq. (20), while those that

apply to the coupling cases appear on the bottom line,

as follows:

Equation Box XVIII

Clearly, the algorithm on the first line of Eq. (20)

applies to outdated Buckley–Leverett assumptions

dogma which have been discredited to some

extent in certain current papers such as Rose

(1999a,b; 2000b) and Ayub and Bentsen (1999),

while the algorithm on the second line is to be

employed to take viscous coupling effects into

account as shown by Rose (1990a). And Eqs.

(21a), (21b), however, are offered to dflyT in the

face of those who are puzzled by the idea that

has been proposed and questioned recently by

Bentsen (2001) and Rose (2000a), namely, to

the effect that spontaneous capillary imbibition

perhaps can be the result of mechanical as well

as surface energy driving forces! Thus, when

with X3N0, the view is held here that when

spontaneous capillary imbibition processes will be

involved along with viscous coupling effects in


this case, then one should consider making use

of the following more general final relationships:

Equation Box XIX

W ¼ D11 þ D12ð Þ þ D13ð Þ X3=X1ð ÞXij

Dij

!þ D13 þ D23ð ÞX3

X1

0BBBB@

1CCCCA ð21aÞ

X¼ D11D22D12D21ð Þ K13K31þD23D32ð Þ X3=X1ð ÞXij

Dij

0@

1Aþ D13 þ D23ð ÞX3

X1

0BBBBBB@

1CCCCCCA

ð21bÞwhere, necessarily again, X1=X2N0 is an imposed

condition.

5 Petrach, the highly regarded poet and philosopher of the 14th

Century A.D., said in his Epistol& de Rebus Familiaribus [XXII.v]

bThat Simulation which aids Truth cannot be regarded as a Lie!QAnd this opinion was given at a time when it was common belie

that b. . .unfortunately many of the most dangerous lies were

thought, by those disseminating them, to be assisting some large

truth!Q as reported by Bergen Evans in his Dictionary Quotations

published by Bonanza Books, New York, in 1968.

To be noted in connection with Eqs. (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 X3

(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 what-

ever reason the wetting-phase saturation is changing

with time. In contrary cases, the third, sixth and ninth

rows of the matrix Eq. (18b) are eliminated and

disappear, leaving the simplex Eq. (20) rather than the

complex Eqs. (21a), (21b) to serve as the algorithms

that will properly describe those partially unsteady-

state processes dealt with earlier in Rose (1990a,b) 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 Dij terms are

being measured by the laboratory procedures

described and/or simply referenced by Rose (1997).

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

Eqs. (21a), (21b) with Eq. (1) as a safe way to for-

mulate 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 Eqs. (21a), (21b) over the simplistic

ones imbedded in Eq. (20) when reservoir simu-

lation 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 dwishful thinkersTshould 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.

3. Concluding remarks

Some conclusions to be drawn are the following. If

a comparison were to be made between the early

papers by the author on the dynamics of capillary-

controlled reservoir processes (such as Rose 1948,

1949, 1951a,b) with those that have followed up to

:

.

f

r

,

Notations

Symbols Interpretations

(A, B, C,

D, E, F)

Transport coefficients in Eqs. (1), (2), (3), (4),

(5), (6), (7), (9a), (9b and 9c), (9d and 9e),

(9f to 9i)), (1), (2), (3), (4), (5), (6), (7), (9a),


the present time (e.g. Ayub and Bentsen, 1999;

Bentsen, 2001; Bear and Bachmat, 1990; Yuan et

al., 2001), the reader might wonder why it has taken

more than a half-century for him to be now compos-

ing 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 the

readers of this paper agree that a rational algorithm to

model capillary imbibition processes necessarily is

one that will enable undertaking reservoir process

simulations if and when the following propositions are

found to be true. These conditions are:

(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 Eq. (20) as a precautionary

way to formulate trustworthy forecasts of 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

Eqs. (21a), (21b) in spite of incurring burden-

some time and cost factor inconveniences. While

some still think that there are persuasive reasons

to employ short-cut reservoir simulation method-

ologies, such dwishful thinkersT 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 readers may

discover both as many negative as well as positive

points of possible special interest to be considered by

practicing reservoir engineers such as:

(9b and 9c), (9d and 9e), (9f to 9i)

Asw, Asn Interfacial surface energy per unit area

Aab Interfacial surface area per unit volume

Dij Transport coefficients in Eqs. (9a), (9b and 9c),

(9d and 9e), (9f to 9i), (10a), (10b), (10c), (11a),

(11b)

E1, 2, 3,. . . Denoting locations of macroscopic RVE in Fig. 1

F Local fractional porosity of reservoir rock

G Acceleration due to gravity

H Siphon fluid level heads in the Fig. 2 flow meters

(a) Exactly (it may be asked) how can one in the

worst cases conduct experiments and obtain as

many independent linear force–flux relation-

ships as there are numbers of the initially

unknown transport process coefficients of pro-

portionality? 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 drain-

age 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 phe-

nomena 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 precisely

phrased surface energy gradient term.

(c) With reference to the {a, b, c, d, e, u} composite

transport coefficients of proportionality seen in

the second of Eqs. (5) and (6), and their relation-

ship to the {A, B, C, D , E, F} transport

coefficients as seen in Eq. (7), and also the Dij

related transport coefficient terms seen in Eqs.

(9a), (9b and 9c), (9d and 9e), (9f to 9i), (10a),

(10b), (10c) and (11a), (11b), are these the

consequences of the mind-boggling fact that it

can be implied that the relationship:

a þ b ef g ¼ 0 ¼ c þ d ff gThe deeper meanings of this curious result

clearly needs further study.

(d) The point is to be emphasized that although the

cost of conducting careful multiphase flow

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

S1 or 2 Local fractional pore space fluid saturations

U, D Fig. 2 upstream and downstream reservoir locations

X Thermodynamic/thermostatic driving forces

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

l, q, c Fluid-phase viscosities, densities, and interfacial

tensions

VX, W Functions of the X1 and X2 driving forces in

Eq. (7)

k (Average pore perimeter)/(pore volume in local

RVEs)

(a, b, c,d, e, u)

Transport coefficients in (4), (5), (6), (7)

H Advancing contact angle

r Entropy

x Experiments needed for data to calculate the Dij


experiments is high (and sometimes prohibitory

so), it makes sense to spendmoney and time to get

the right answer rather than to cheaply adopt

simplistic methodologies that end up with provi-

ding the wrong information. Or so it would seem!

Appendix A. Predictive transport process

algorithms

Traced here in a sense is the way the ideas

analytically imbeded in the various equation boxes

of this dBuckley–LeverettT paper have evolved.

In what follows, the more than 100+ sometimes

somewhat redundant equations are cited and para-

phrased as follows:

(a) In Boxes I and IA, Eqs. (1), (2), (3), (4) are

presented that embody the here-unchallenged senses

of the original Buckley–Leverett (1942) mass con-

servation related contentions that were applied by

Rose (1988) to describe simple unsteady- and steady-

state coupled transport processes. Specifically, Eq. (4)

give definitions for the important {X, W} terms that

first appear in Eq. (3). Then Eqs. (1a) and (4a) follow

to show that when the divergence of the J3 flux is

adequately expressed by setting the second and third

terms on the right-hand side of Eq. (3) equal to zero

and then by only considering as finite the first term.

This contention, for example, is implied in the

analysis of Rose and Robinson (2004).

(b) In Boxes II and IIA, the following Eqs. (5), (6),

(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, Eq. (7a) defines

the one-to-one relationships between the various

notational Latin and equivalent Greek lettered trans-

port coefficients as made clear by the cartoon in Rose

(1996).

(c) In Box III, four sets involving eight flux–force

Eqs. (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 informa-

tion from the measured laboratory data so that needed

values of the transport coefficients as functions of

local saturation in the macroscopic RVEs of Fig. 1

can be computed. This possibility is addressed further

in the (d) to (f) paragraphs that follow, and in the text

of Appendix B below.

(d) In Box IV, Eqs. (9a), (9b and 9c), (9d and 9e),

(9f to 9i) show that the calculated Dij have the

notational form of being certain experimentally

measured flux to force ratios that can be uniquely

measured as recommended by Rose (1997).

(e) In Box V with Eqs. (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 dpushingT caused by the driving force acting

directly on the fluid phase that is then being observed,

and also by a tangential ddraggingT across fluid–fluidinterfacial 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 inter-

pretations by Bartley and Ruth (1999) and for the sake

of argument was later adopted by Rose (2000a). Then

in Eq. (10c), it is suggested that the Onsager type

reciprocity referred to by DeGroot and Mazur (1962)

does not always need to be additionally presumed

because of the arguments given in Appendix B below.

(f ) In Boxes VI and VII, Eqs. (11a) and (11b)

show that the same pushing and dragging effects also

naturally will be encountered when laboratory model-


ing experiments are being undertaken of the method-

ology sorts that have been recommended by various

current investigators such as Dullien and Dong (1995),

Rose (1997) and Zarcone and Lenormand (1994).

(g) In Box VIII, the important Eq. (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 Box I, but now the special cases being

considered are for where two or more categories of

coupling are simultaneously occurring (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

dpushT and ddragT effects such as were studied recentlyby Rose et al. (1999).

(h) In Box IX, for example, Eq. (13a) shows 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 Aitken (1939) 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 prop-

erly 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 B below.

(i) In Box X, we are confronted with the

expansive Eqs. (14a), (14b), (14c), (14d), (14e),

(14f 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 Eq. (14b). Then we can show the equivalent unit

matrix from which we can form two important

additional relationships for the two driving forces,

{e, f }, as explicit but complex functions of the two

corresponding fluxes {a, b} together with the four

aforementioned transport coefficients. Then given Eq.

(14c) which asserts that no apriori assumptions

actually do not always have to be made about

concocting reciprocity existing between the diagonal

cross coefficients. This is because explicit calculated

values for the four transport coefficients just as well

can be obtained in the form of ratios of exper-

imentally determined flux to force data that are

obtained during the conduct of the four experiments

specified above by Eq. (13b). . .and now again by

Eq. (14f to 14i).

( j) In Boxes XI and XII, Eqs. (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 (1961) and later by Richards

(1931) then timidly by Rose (1963). . .but with more

conviction in Rose (2001a,b). Notice in passing that

the existence coupling due to Onsager-like reciprocity

is being presumed to occur.

(k) In Box XIII, we have Eqs. (17a), (17b),

(17c), (17d) which include four sets of possible

experiments that can be easily (if laboriously)

performed under controlled laboratory conditions.

These are labeled as follows: (a) the dAT test whereonly the b driving force is set equal to zero; (b) the

dBT test where only the a driving force is set equal to

zero ; (c) the dCT test where none of the driving forces

are set equal to zero; and, finally, (d) the dDT test

where the c driving force is the only one that is not

equal to zero.

(l) In Boxes XIV, XV and XVI, Eqs. (18a),

(18b) and (18c) appear and show in an evolutionary

way how the nine Dij terms can be evaluated by

taking ratios of particular super and subscripted fluxes

{( JA, B, C)a, b, c} divided by the appropriate {a, b, c}driving force.

(m) In Box XVII, the aim is to have an independent

way with Eq. (19) to model the performance of certain

petroleum reservoir systems in order to obtain quanti-


tative values for the nine transport process coefficients

referred to in the previous 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 aforemen-

tioned Onsager-like reciprocity relationships that

appear as the last equations of Eq.(16) given above.

Exactly how this magic is achieved is indicated in the

last two boxes given below.

(n) In Boxes XVIII and XIX, we have definitions

for the important {X,W} terms given for Eq. (20)

cases, where either {X3=D12=D21=0} or only

{X3=0}, and is also given for the more general Eqs.

(21a), (21b) cases, where {X1=X2N0} (cf. Rose and

Robinson, 2004).

In conclusion, the student who carefully and

considerately studies sequentially the various equa-

tions found in the various equation boxes, will notice

that this 1000+ 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!

6 I think it was Bertrand Russell who thus spoke when wearing

the mathematical hat of his youth, but maybe it was Voltaire!

Appendix B. Confirming modeling schemes

Here, we start by expanding Eqs. (9a), (9b

and 9c), (9d and 9e), (9f to 9i) with Eqs. (8)

that already has appeared in the main body of the

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 Eqs. (9a), (9b and 9c), (9d and 9e), (9f

to 9i) shows 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 repre-

sentative paradigm process interest. Among the other

relationships that also can be considered are those

where reciprocal reciprocity is postulated between

the {DijuDji} coupling coefficients for cases which

the Nobel Laureate, Professor Lars Onsager, in

1931, pronounced as a plausibly applicable. This

contention is based on invoking a so-called Princi-

ple of Microscopic Reversibility, which clearly

applies 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 and Mazur (1962) to Bear

and Bachmat (1990).

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 extension of

the underlying ideas also appears to be possible

according to Rose and Robinson (2004) 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) 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-match-

ing evidence that is obtained; and (c) to along the

way keep in mind the maxim of the philosopher6

who said b. . .all Generalities are False including This

One. . .Q, and which can be given the added meaning

that the 18th Century Voltaire was right when he said

in his ingenuous Candide XXX b. . .let us work


without theorizing (since) it is the only way to make

life endurable!Q7

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 con-

dition will not be reached when different method-

ologies are employed such as the #s I, II and III as

defined in the modified Eqs. (9a), (9b and 9c), (9d and

9e), (9f to 9i) 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 Fig. 2 herein, is

especially well suited for the intended purpose.

A satisfactory procedure to follow, for example,

will include sequentially taking the following labo-

ratory 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 strati-

graphic and/or structural entrapments to form future

7 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 b. . .general time-asymmetric behavior of macroscopic sys-

tems—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. . .Q. And remember, just

because Gertrude Stein in her Sacred Emily said b. . .Rose is a Rose,is a Rose, is a Rose!. . .Q does not prove that plausible generalities atbest are assumptions and at worst presumptions as Rose described

in (1991a) how fact and fancy sometimes can be confused!

petroleum reservoirs can occur. Thus, to model such

complex geophysical events, we proceed as follows.

(a) We select representative core samples of

reservoir rock that have been drestoredT 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 Siddiqui

and Lake (1992). These furthermore were taken to be

proximate to organic-rich sediment deposits that in the

passage of uncounted eons of time to have locally

become catalytically converted to coalescing ganglia

of proto-petroleum fluid phases because of prevailing

high temperature and pressure conditions in ambient

high specific surface area reactive clay environments.

Then because of the tectonic forces causing sediment

compaction during overburden growth, the chosen

sample pore space after first being cleaned, flushed

and saturated with a synthesized connate water.

Thereafter, and as is shown in the Case 2 config-

uration of the Fig. 2 cartoon where the driving force

siphon levels, {yH1N0} and {yH2=0}, have been

properly set so that both fluids are flowing down-

wards, but here it is the wetting fluid that is being

dpushed while it is the nonwetting fluid that is being

ddraggedT along. This procedure to be followed can be

thought of as somewhat equivalent to the classical

restored-state capillary pressure drainage (so-called

restored-state) experiment as described throughout

Chapter 9 and especially in Section 9.2 by Bear

(1972). 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 dpushedT downwards and doutT,and this causes the oil nonwetting phase to be

ddragged’ down ’into’ the system.

(b) The next laboratory step to take is that referenced

in the Eqs. (9a), (9b and 9c), (9d and 9e), (9f to 9i) box

above as involving the Exp. #2 data that is needed to

calculate values for the two Dij transport coefficients

which are needed when an algorithm is wanted predict

outcomes to be expected when conducting an

imbibition kind of capillary pressure process of the

sort characterized by the Case 1 conditions illustrated

in Fig. 2. Here, both fluids will be flowing upwards,

http://www.britanicca.com/article%3Feu%3D53830



but now the imposed boundary conditions are where it

is that {yH2N0}, and also where {yH1=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 Fig. 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

(viz. flow rates and displacements) and forces. Here,

we start by expanding Eqs. (9a), (9b and 9c), (9d and

9e), (9f to 9i) with Eqs. (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 Eqs. (9a), (9b and 9c), (9d

and 9e), (9f to 9i) 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 represen-

tative paradigm process interest. Among the other

relationships that also can be considered are those

where reciprocal reciprocity is postulated between the

{DijuDji} coupling coefficients for cases which the

Nobel Laureate, Professor Onsager (1931) pro-

nounced as a plausibly applicable a so-called Princi-

ple 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 and Mazur (1962) to Bear and Bachmat

(1990).8

In the paper to which this Appendix is attached,

however, an alternative way of thinking has been

8 A recent tribute to the memory of Professor Lars Onsager.

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 and Robinson (2004) for more complex and

perplexing cases where the need to postulate micro-

scopic reversibility can perhaps be regarded as

unnecessarily superfluous.

To avoid a transgression that even a guru like

Aitken (1939) 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 relation-

ships 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-match-

ing evidence that is obtained; and (c) to along the way

keep in mind the maxim of the philosopher9 who said

b. . .all Generalities are False including This One. . .Q,and which can be given the added meaning that the

18th Century Voltaire was right when he said in his

ingenuous Candide XXX b. . .let us work without

theorizing (since) it is the only way to make life

endurable!Q7

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 drunsT are made on the

same laboratory sample starting with the same initial

and boundary conditions, the final equilibrium con-

dition will not be reached when different method-

ologies are employed such as the #s I, II and III as

defined in the modified Eqs. (9a), (9b and 9c), (9d and

9e), (9f to 9i) 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 Fig. 2 herein, is

especially well suited for the intended purpose.

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 strati-

graphic and/or structural entrapments to form future

petroleum reservoirs can occur. Thus, to model

9 I think it was Bertrand Russell who spoke thus when wearing

the mathematical hat of his youth!

such complex geophysical events, we proceed as

follows:

(c) We select representative core samples of

reservoir rock that have been drestoredT 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 Siddiqui

and Lake (1992). These furthermore were taken to be

proximate to organic-rich sediment deposits that in the

passage of uncounted eons of time to have locally

become catalytically converted to coalescing ganglia

of proto-petroleum fluid phases because of prevailing

high temperature and pressure conditions in ambient

high specific surface area reactive clay environments.

Then because of the tectonic forces causing sediment

compaction during overburden growth, the chosen

sample pore space after first being cleaned, flushed

and saturated with a synthesized connate water.

Thereafter, and as is shown in the Case 2 config-

uration of the Fig. 2 cartoon where the driving force

siphon levels, {yH1N0} and {yH2=0}, have been

properly set so that both fluids are flowing down-

wards, but here it is the wetting fluid that is being

dpushedT while it is the nonwetting fluid that is being

ddraggedT along. This procedure to be followed can be

thought of as somewhat equivalent to the classical

restored-state capillary pressure drainage (so-called

restored-state) experiment as described throughout

Chapter 9 and especially in Section 9.2 by Bear

(1972). 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 dpushedT downwards and doutT,and this causes the oil nonwetting phase to be

ddraggedT down dintoT the system.

(d) The next laboratory step to take is that referenced

in the Eqs. (9a), (9b and 9c), (9d and 9e), (9f to 9i) box

above as involving the Exp. #2 data that is needed to

calculate values for the two Dij transport coefficients

which are needed when an algorithm is wanted predict

outcomes to be expected when conducting an

imbibition kind of capillary pressure process of the

sort characterized by the Case 1 conditions illustrated

in Fig. 2. Here, both fluids will be flowing upwards,



but now the imposed boundary conditions are where it

is that {yH2N0}, and also where {yH1=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 Fig. 2 in the body of the paper is

one where co- and counter-current flow conditions

can be imposed and monitored.

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“revisiting” the enduring buckley–leverett ideas

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