Transport in Porous Media 3 (1988), 163-171. 163 �9 1988 by Kluwer Academic Publishers.

Measuring Transport Coefficients Necessary for the Description of Coupled Two-Phase Flow of Immiscible Fluids in Porous Media

W A L T E R R O S E Illini Technologists International, P.O. Box 2430, Station A, Champaign, IL 61820, U.S.A.

(Received: 30 March 1987; revised: 27 November 1987)

Abstract. In the simplist cases of coupled two-phase flow of immiscible fluids in porous media, the governing equations usually are written to show that there are four independent transport coefficients that implicitly have to be separately measured. The analysis presented here accordingly indicates that two types of known experiments involving two measurements apiece are needed at each fluid saturation condition in order to provide the necessary and sufficient information by which the unsteady as well as the steady states of ensuing transport processes can be ~ established and charac- terized. Apparently, however, the fact that methodologies are already available for the required laboratory work either is not widely appreciated or it is being overlooked. For this reason and others, mention is made of the surprising fact that the experimental difficulties to be confronted in the actual study of coupled transport processes are no greater than those that have already been dealt with by the advocates of classical relative permeability theory (i.e. the traditionalists who simplistically model two-phase flow as though no coupling effects are involved).

Key words. Porous media flows, coupled transport processes, relative permeability, Hassler-Richards methodology.

O. Nomenclature

Roman Letters (/, j = 1, 2 to designate wetting/nonwetting fluid phases). Dq

f g G, K, L M , N

Pi ~p Pc qi qr 0, Ri

transport coefficients defined by Equations (2.1), m3-s/kg fractional porosity (pore volume to bulk volume ratio) gravity vector, m/s 2 gravitational force per unit volume (PN), N / m 3

Darcian mobility term (permeability/viscosity), m3-s/kg lab. specimen sample length, m scale multipliers (cf. Equations (3.4)), m3-s/kg local fluid pressure, N/m 2 pressure drop, N / m 2

capillary pressure (P2- pl), N / m 2

seepage velocity vector, m/s total flow rate per unit area (ql + q2), m/s that part of qi due to gravitational forces (cf. Equations 4.2), m/s lumped transport coefficient (cf. Equations (2.6)), m3-s/kg

164 WALTER ROSE

Si fractional saturation per unit pore volume x, t independent variables of the spaceltiine reference frame, m, s Z another lumped transport coefficient (cf. Equations (2.6)), m3-s/kg

Greek Letters @i mechanical energy per unit volume of fluid particles, N/m 2 pi local value for density of fluid particles, kg/m 3. Ap local density difference (Pl- P2), kg/m 3

1. Introduction

In a recent paper (Whitaker, 1986) it is emphasized that experimental data are needed to establish how much coupling (if any) occurs when two or more immiscible fluids are flowing side-by-side through the interstices of porous solids. Coupling, which is an important feature encountered in the general study of transport phenomena such as thermodiffusion, should also be a factor to be considered whenever there is a possibility that the motion of one fluid will exert a viscous drag on contiguous immiscible fluids across proximate fluid-fluid inter- faces of contact. With theory indicating that a coupling effect should be noticed, it is left for the experimenters to design laboratory procedures by which it can be determined if the effect will be of first- or second-order importance in specific cases,

On the issue under discussion, Whitaker's assertion has to be accepted as valid, namely that traditional schemes of relative permeability measurement which are constrained to be steady, uniform flows in the absence of gravitational inter- ferences, do not yield the data by which the uncertainties can be resolved. The relevance of this position is not an issue however, since it can be shown that it is just as easy to conduct steady, uniform flow experiments in the presence (as well as the absence) of gravitational effects. The consequence of this is that a previously ignored way is now provided to obtain the data that will clearly reveal whether or not coupling due to the continuity of viscous stress across fluid-fluid interfaces occurs in specific cases to any significant extent.

2. Analysis

Limiting attention here to isothermal, low Reynolds Number and steady flow of two homogeneous and incompressible fluids of unequal density through isotropic, r!gid, and chemically inert porous media, we find it convenient to model the expected linearity between fluxes and forces in the familiar form (Bear, 1972):

ql -- -D11 grad ( I D 1 - - D12 grad qb2, (2.1)

q2 = -D21 grad ~1 - D22 grad ~2.

MEASURING TRANSPORT COEFFICIENTS 165

Here the mechanical energy gradients (per unit volume) for flows that macro- scopically are one-dimensional, will be given by

grad q~i = (dpi/dx) + Pig (2.2)

and the idea of conservation of mass is expressed by

[(OSJOt) + div qi = 0. (2.3)

Since the fluxes and forces of Equations (2.1) can be experimentally controlled, imposed, and regulated, some four separate experiments will have to be per- formed at each saturation level of interest in order to obtain empirical measures of the saturation dependency for the four transport coefficients. This is because any apriori presumption that an Onsager-like (reciprocity) relationship holds, such as D12 = D21, clearly is to be avoided unless and until a laboratory verification is obtained. The fact of the matter is, however, that values for all of the coefficients actually are not individually needed in order to have the data necessary for an adequate description of the limited class of coupled flow processes under consideration here. As shown below, in fact, the governing transport equations can be put in a form which only three lumped parameters appear whose dependencies on saturation become the crux of the matter for laboratory investigation.

Thus, for the cases ~vhere the flows are steady state, Equations (2.1) reduce in one-dimensional form to

qa = ({qrR1 + R2Z[(OPc/Ox) + Apg]}/(R1 + R2)), (2.4)

where qT is simply a notation for the sum (q~ + q2), and Pc likewise is simply a notation for the local pressure difference between the nonwetting and wetting fluid phases. Equations (2.3) and (2.4) can now be combined to provide for a description of three-dimensional and unsteady-state flows in the form

f(OS1/Ot) + div({qrR1 + R2Z[(grad Pc) + Apg]}/(R~ + R2)) = 0, (2.5)

where the three lumped parameters of Equations (2.4) and (2.5) are defined by

R1 = (Dn + Da2), R2 = (D21 + D22), (2.6)

Z = (DuDzz - DlzDz1)/(D2, + Dz2).

In passing, it is to be noted that if in Equations (2.6): (a) D n is replaced by K1; (b) D22 is replaced by K2; and (c) D12 = Dax are set identically equal to zero, we then would be dealing with the limiting case of no coupling. In such a case, Equations (2.4) and (2.5) would reduce without change in mathematical form to their Buckley-Leverett theory counterparts (cf. Rose, 1987 and 1988 in press) with Z = Rx reducing to K1, and with R2 reducing to Ka. In other words, it would appear that the numerical integration of the governing transport equations (2.5) involves the same finite difference and finite element methods whether or

166 WALTER ROSE

not coupling is present, and involves the same sort of specification of initial conditions (e.g. with respect to the time-zero value for the saturation gradient) and boundary conditions (e.g. with respect to qr and grad Pc). Furthermore, and as shown in the next paragraphs below, the same sorts of experiments must be performed for coupled as well as uncoupled flow processes involving the same sorts of laboratory difficulty, with the only difference being that four separate experiments are required in the former (coupled) case while only two are required in the latter (uncoupled) case at each level of saturation to be investigated.

3. Hassler-Richards Reference Frame

Hassler (1944) was the first to propose a methodology for what today are referred to as relative permeability measurements whereby and wherein uniform satura- tion distributions are obtained and sustained during flow according to the ideas that Richards (1931), in the very earliest days, had identified and prescribed. Since the specifics of the Hassler-Richards approach are well known (cf. Rose, 1987), and since improved procedures for practical laboratory work are disclosed in a recent patent (Rose, 1985), the main point to make here is the fact that without any modifications whatsoever, the Hassler-Richards scheme apparently can be directly utilized to establish the hysteretic saturation dependencies both of the R1, R2 and Z parameters that appear in Equations (2.4), (2.5) and (2.6) - and indeed also those of the four Dij transport coefficients that appear in Equations (2.1).

Two prerequisite conditions must be met when experimental measurements are being made. The first one is that at the onset, relevant saturation configurations must uniformly be established throughout the sample space. This is because of the hysteresis (viz. which makes it possible to produce many distinctly different types of pore space partitioning between wetting and nonwetting fluids for each macroscopic level of saturation that is to be studied). The second prerequisite requirement is that the uniform saturation conditions, as initially attained, shall not be inadvertently altered during the ensuing dynamic steady states as labora- tory measurements are being made. By employing semipermeable (capillary) barriers for the influx and efltux of the wetting fluid (i.e. so that well-defined drainage/imbibition sequences can be followed to reach relevant saturation configurations, and So that uniform conditions will persist until the data are obtained from which values for the transport coefficients can be calculated), the Hassler-Richards methodology ingenuously makes it possible to meet the two aforementioned prerequisite condi{ions.

Although Hassler-Richards relative permeability procedures are usually pic- tured as though the immiscible fluids are being caused to colinearly flow in the horizontal direction, the fact of the matter is that there are no serious procedural difficulties to overcome if and when the flows are conducted in the vertical direction. Of course, if the transport process under study is being modelled as

MEASURING TRANSPORT COEFFICIENTS 167

though no coupling is occurring, no advantage will be gained by obtaining data from vertically-directed flows. On the other hand, when coupling is present it can be shown that the logical way to proceed is to conduct two experiments where ql and q2 are measured at each saturation level as a function of the pressure drop for the horizontal flow case, followed by two more experiments where q~ and q2 are measured as a function of both the pressure and gravity forces for the vertical flow case.

To see what is involved, it is necessary to rewrite Equations (2.1) and (2.4) to conform to the sense of how the Hassler-Richards measurements are to be made, o r

qa = R~(Ap/L), q2 = R2(Ap/L) (3.1)

for the case of linear horizontal flow where grad q~l = grad q~2 = -(zXp/L), and

ql(R1 + R2) = (qTR0 + (ZR2 2~pg) (3.2)

for the case of linear vertical flow, and where again grad~l = grad~2 is a constant, hence (OPc/Ox) = 0. Clearly then, the two experiments performed in the sense of Equations (3.1) involving the measurement of the two fluxes which are proportional to the same fixed pressure drop in each phase, become the indicated way to measure the transport coefficients, R1 and R2. Similarly, with R~ and R2 thus established, the data obtained from the following two experiments conducted so that the flows will be vertically directed, lead to the fixing of the value for Z by rearranging Equation (3.2) as

Z = (qaR2 - q2R~)/(R2 Apg) (3.3)

In fact, the state-of-the-art Hassler-Richards procedures are easily seen to be the ones to be followed, firstly to obtain R1 and R2 by separate measurement of the fluxes and forces of Equations (3.1), and then to obtain Z by separate measurement of the fluxes and density difference values of Equation (3.3). In these connections, it is worth noting that in the vertical flow case the observed pressure drop can be made to be positive, negative, or zero at the option of the experimenter. Inspection shows, however, that in order to minimize experimental error associated with the subtracting of two large numbers (i.e. as appear in the numerator of Equation (3.3)), one can be advised to choose pressure drop values that are slightly negative so that q2 will itself be more-or-less zero, while even so ql will be directed downwards as a consequence of the gravity free-fall effect. This, in fact, appears to be the logical way to minimize errors in the estimate of Z by Equation (3.3).

In any case, it will be clear that the first two experiments conducted in the sense of Equations (3.1) by themselves will not reveal any information about whether or not a coupling due to viscous drag is occurring. This conclusion is consistent with the one already reported by Whitaker (1986) and others. In the affirmative case, the R~, R2 coefficients will have the sense assigned to them in

168 WALTER ROSE

Equations (2.6), but otherwise they will simply take on the meaning of being the classical Darcian mobilities (effective permeability to viscosity ratios, such as K1 and K2) of the respective fluids. More than that, performing the second two experiments conducted in the sense of Equation (3.3) will only provide redundant information in the case of no coupling (i.e. because the parameter, Z, in this case also reduces to the same Darcian mobility ratio already referred to as the limiting value for R1). By comparing the data coming from the two horizontal and two vertical flow experiments, however, enough information is obtained to fully discriminate between the opposite cases of coupling versus no coupling (or negligible coupling).

To illustrate how powerful schemes can be devised to ascertain whether or not coupling is occurring from an inspection of experimental data, the following simplistic calculations will be made. Let it be assumed the flow situations are being considered where the following assumptions will at least facilitate the drawing of qualitative inferences

Dll --- S12M, 022 -- (1 - S1)M, 312 = D21 -- Sl(l - S1)N. (3.4)

Here the scaling parameter, M, has the conceptual and dimensional sense of a so-called mobility ratio (i.e. an effective permeability that might be observed in an uncoupled flow situation, divided by the fluid viscosity), and the scaling parameter, N, determines the magnitude of the coupling effect to be expected as a function of saturation. Equations (3.4) in fact are similar in form to the less approximate ones commonly employed for the calculation of uncoupled relative permeability curves (cf. Rose, 1987). Thus, the dimensionless ratio, (M/N), approaches infinity whenever the coupling effect is negligible or nonexistant, while it approaches zero when the coupling effect is large especially when low permeability media are involved.

Making use now of the approximations implied by Equations (3.4) together with the definitions given by Equations (2.6), is how it becomes possible in an illustrative way to compare the value of Z that would be measured by experi- ments conducted in the sense of Equation (3.3) versus the value of R1 that would be measured by experiments conducted in the sense of the first of Equations (3.1). Since the (Z/R1) ratio so measured will be unity only when there is no coupling im:olved in the transport process under study, this ratio clearly serves as a useful indicator to discriminate between cases where coupling can versus cannot be neglected.

In the accompanying table, (Z/R1) ratios at $1 = 0.5, for example, are seen to range from low values such as 0.167 that clearly are reflective of a large coupling effect, to values which are so close to unity such as 0.985 that (because of experimental error) could either be indicative of miniscule coupling or indeed no coupling. On the other hand, Table I also indicates that at low values of saturation for a fixed intermediate value for N (such as N = 0.1), the (Z/RO ratios are reduced (meaning that they can serve as more reliable indicators of

MEASURING TRANSPORT COEFFICIENTS

Table I. A simplified numerical example where M = 1

169

Equation (3.4) Equation (2.6) Saturation S 1 M / N D11 D22 D12 D21 R1 R2 Z Z/R1 a

0.1 10 0.010 0.900 0.009 0.009 0.019 0.909 0.010 0.516 0.3 10 0.090 0.700 0.021 0.021 0.111 0.721 0.087 0.782 0.5 1 0.250 0.500 0.250 0.250 0.500 0.750 0.083 0.167 0.5 10 0.250 0.500 0.025 0.025 0.275 0.525 0.237 0.861 0.5 100 0.250 0.500 0.003 0.003 0.253 0.503 0.249 0.985 0.5 infinity 0.250 0.500 0 0 0.250 0.500 0.250 1.000 0.7 10 0.490 0.300 0.021 0.021 0.511 0.321 0.457 0.893 0.9 10 0.810 0.100 0.009 0.009 0.819 0.109 0.742 0.906

~Equations (2.6) can be combined to yield: (Z[Ri)=(D11Dz2-Da2D21)](DllD22+D12D21+ DlzDzz+D21D11) from which can be derived the intriguing result that O<(ZIRO<I for (DHD22) > (D12Dz i )>0 , where the latter inequality independently is simply a reflection that the entropy production is positive-definite for irreversible transport processes. The values reported in this table for (Z/RO are those where the Dq are assumed to have magnitudes as given by Equations (3.4). Here it is seen that (Z/Ra) at each saturation level becomes larger when larger (M/N) values are selected, as they do also for each fixed value selected for (M/N) as saturation increases. Note, (M/N) 2 >" (1 - S 1 ) necessarily.

coupling), while conversely at high values of saturation, the tendency is for the uncertainties about thd importance of coupling to be increased, since the (Z/R1) ratio values will again be approaching unity. This, of course, is because the third of Equations (3.4) has been cast logically in a form so that the coupling c o e f f i c i e n t s , D ] 2 = D21, approach zero at both low and high saturation levels - indeed just as will be the case for the interfacial specific surface area whose magnitude and extent will surely determine the amount of coupling to be expected in particular cases.

4. Discuss ion

What has been shown above is that when coupled flow processes are under study, experimental means are needed to obtain estimates of the three parameters, (RI, R2 and Z), in order to be able to deal with finding solutions for the equations of transport (2.4) and (2.5). Equations (3.1) and (3.3) were proposed for these assessments. Another approach can be taken, however, which also has the benefit that it directly leads to explicit values for the four Dq transport coefficients.

For the two horizontal flow experiments, Equations (3.1) can be rearranged in the form

RI = (D11 + D12) = ql(L/Ap) , R : = (D21 + D22) = q2(L/Ap) (4 .1 )

while for the two vertical flow experiments it follows from Equation (2.1) that

170 WALTER ROSE

G I D 1 1 + G z D 1 2 = q l - R I ( A p / L ) = O l ,

G 1 D 2 1 + GzD22 = q2 - R 2 ( A p / L ) = Q 2 . (4.2)

Here the Q~ may be thought of as that part (if any) of the fluxes, % which are due solely to the fact that gravitational forces in the vertical flow cases add to the pressure forces in causing the fluid transport. Clearly, Equations (4.2) reduce to (4.1) in the limit where G1 = p i g = G2 = p 2 g = g = 0 . In any case, these four equations can be solved simultaneously (Rose 1988) to yield

D n = (Ol - G 2 R 1 ) / ( G 1 - G 2 ) ,

D21 = ( O 2 - G 2 R 2 ) / ( G 1 - G2),

D12 = ( G I R 1 - Q 1 ) / ( G1 - G2),

D22 = ( GIR2 - 02)/(G1 - G2). (4.3)

With the Dq values thus obtained, the useful lumped parameters, (R1, R2 and Z), can be calculated directly with the aid of Equations (2.6).

The crux of the matter, of course, is to see developed reliable procedures by which the measurements will unambiguously be made. While over the years many methodologies have been proposed and described for the measurement of relative permeability, it would appear (cf. Rose, 1987) that the Hassler-Richards scheme is the only one now being practised that guarantees the achievement of rationally and uniformly distributed saturation configurations when flow measurements are being made. Indeed, those who accept this judgement are forced to concede that the vexing measurement problem referred to by earlier students of coupled flow processes, now takes on the status of being the proverbial nonproblem.

In other words, a first important step has been taken, namely by demonstrating as done herein that Hassler-Richards experimental measurements are ones that are capable of yielding precisely the kinds of data about the transport coefficients defined by Equations (2.1) and (2.6) that appear in the governing process descriptions (viz. Equations (2.4) for the steady states, and Equation (2.5) for the unsteady states).

The second important step will be taken as soon as experimenters learn how to cope in the laboratory with the attending instrumentation and data-processing requirements involved in the measurement and control of parameters such as low flow rates, pressure drops and pressure differences, saturation changes, density differences, viscosities, and the like. After all, workers have known about the sense of the Hassler-Richards methodology requirements for more than 50 years, but even so, essentially no data have been published to confirm that practical procedures have ever been developed, even for the study of the simple un- coupled cases. In fact, the recent patent disclosure cited above (Rose, 1985) has more to say about the nature and extent of the unsolved measurement difficulties which still need to be faced, than it does about recommended procedures that are ready to be followed in cookbook fashion.

Finally, the third important step will have been taken when finally the intuitive judgements such as those voiced a long time ago by the perceptive J. R. Philip

MEASURING TRANSPORT COEFFICIENTS 171

(1972), can be reviewed and reevaluated on the basis of conclusive empirical evidence. After all, the issue is not one of whether or not some early worker was right or wrong about the importance of taking coupling into account, but on the contrary, the issue is to have the matter settled once and for all in the conclusive way of letting the case rest on carefully obtained experimental data.

5. Conclusions

Whitaker 's paper (1986) provides all the proof that will be needed to support the content ion that very high levels of theoretical understanding about the nature of porous media flow processes already have been achieved. In fact, it may even turn out to be the case that an interim limit has been reached, so that a new era of data-gathering will now be fostered. By focusing on observations to confirm, explain, apply, and indeed occasionally contradict the indications of present theory, an opportunity may eventually recur again when it will make sense to resume theoretical studies to reach still higher levels of understanding.

Accordingly, in this paper the conclusion is offered that the famous Hassler- Richards methodology provides the framework as well as the vehicle to resolve the nagging issues about the importance of coupling effects in practical cases of multiphase flow, and that it can also provide the missing links which will eventually serve to stimulate new theoretical thinking.

Specifically, it is shown in this paper (apparently for the first time) that for each uniform saturation condition which is established, experimental measurements can be made under horizontally and then vertically directed flow conditions in order to generate the information needed for the evaluation of the three principal (lumped) parameters defined by Equations (2.6), and also of the four transport coefficients defined by Equations (2.1) and (4.3).

Reterences

Bear, Jacob, 1972, Dynamicx of Fluids in Porous Media, Elsevier, New York~ Hassler, G. L., 1944, Methods and apparatus for permeability measurements, U.S. Letters Patent No.

2,345,935. Philip, J. R , 1972, Flow in porous media, Proceedings XIII International Congress of Theoretical and

Applied Mechanics, Moscow. Richards, L. A., 1931, Capillary conduction of liquids through porous media, Physics 1, 318-333. Rose, Walter, 1972, Some problems connected with the use of classical descriptions of fluid/fluid

displacement processes, in Fundamentals of Transport Processes in Porous Media, Elsevier, Am- sterdam, pp. 229-240.

Rose, Walter, 1985, Aparatus and procedures for relative permeability measurements, U.S. Letters Patent No. 4,506,542.

Rose, Walter, 1987, Relative permeability, Chapter 28 in Handbook of Petroleum Reservoir Engineer- ing, Society of Petroleum Engineers, Dallas.

Rose, Walter, 1988, Attaching new meanings to the equations of Bucktey and Leverett, J. Petrol. Sci. Eng. 1.

Whitaker, Stephen, 1986, Flow in porous media. II. The governing equations of immiscible, two-phase flow, Transport in Porous Media 1, 105-125.