SPE SPE 11017 SOCiety of Petr'Olelm EngineeI's of A 1M E

Multiphase Flow in Porous Media: I. Macroscopic Observations and Modeling

by Stephen J. Salter and Kishore K. Mohanty, ARCO Oil & Gas Co.

Members SPE

Copyright 1982, Society of Petroleum Engineers of AIME This paper was fJ'tJDtJl ItJll at the 57th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, held in New LA, Sept. 26-29, 1982. The material is subject to correction by the author Permission to copy is restricted to an abstract of not more than 300 words. Write: 6200 N. Central Expressway, PO. Drawer 64706, Dallas, Texas 75206.

ABSTRACT

Tracer displacements were run during steady­state, two-phase flow, where tracers were added to both the wetting and nonwetting phases. Such experi­ments were conducted in the primary drainage, imbibi­tion, and secondary drainage cycles over a wide range of fractional flows. The effluent tracer profiles from these experiments were fit with a four-parameter, capacitance-dispersion model, which divides each phase into three fractions: flowing, dendritic, and iso­lated. Mixing in the flowing fraction is described by a dispersion coefficient, and communication between the flowing and dendritic fraction is represented by a single mass transfer coefficient. This model is shown to adequately fit tracer effluent data over a wide range of conditions.

In some areas the model is nonunique--thus, more than one solution fits the experimental data equally well. Additional experiments were conducted to dif­ferentiate mechanisms in these regions. The effects of core wettability were investigated and shown to be important. The results of all of these experiments were used to formulate a mechanistic picture of steady, two-phase flow in strongly wetted porous media.

I. INTRODUCTION

In all enhanced recovery techniques, it is impor­tant to understand the nature of multiphase flow. Traditional 'black oil' modelling assumes a number of continuum level properties (e.g., permeability, rela­tive permeability, capillary pressure, dispersion co­efficients, etc.) to describe stable macroscopic fluid flow. Such parameters can and have been mea­sured in laboratory experiments. A few attempts have been made to make such measurements in the field. Unfortunately, the results of laboratory and field measurements frequently are not in agreement. Thus, an important question arises regarding the proper scale-up of laboratory measurements. To begin such a study, we have looked into the mechanisms of multi­phase flow and have attempted to correlate pore level mechanisms with macroscopic laboratory observations.

"";;; ... <: ... eu ..... <;;;o and illustrations It !nd If paper.

The work will be reported in a sequence of papers. This paper reports the results of laboratory experiments and macroscopic level modelling of the mixing phenomena observed in the wetting and nonwet­ting phases during steady-state flow. The second paper 1 uses a pore level, network model to describe the mechanisms behind the observations reported in this paper. The results are consistent pore level and macroscopic models which not only show what happens but why it happens.

Most of the work was done on a single porous medium; strongly water wet, fired-acidized Berea sand­stone. Relative permeabilities and capillary pressure curves were measured and a series of tracer displace­ments conducted during two-phase, steady state flow. Similar experiments have been performed and reported in the miscible flooding literature for displacements in the nonwetting phase. 2- 7 Transport during flow in the wetting phase has been examined by soil scien­tists investigating groundwater movement in soils partially saturated with air. 8- 1S What sets our work apart is 1) we have run tracer displacements in both the wetting and nonwetting phases in the same porous medium, 2) we have investigated in detail a broad range of fractional flows and 3) we have observed these phenomena in the primary drainage, imbibition, and secondary drainage cycles.

A macroscopic model was formulated based on a Coats-Smith16 approach which allows us to break down both of the phases into three fractions: flowing, dendritic, and isolated. The flowing fraction repre­sents the fluid in that network of pores through which a phase flows in at least one throat and out at least one. These pores contribute to the permeability of the phase at that saturation. The dendritic fraction of a phase is connected to the flowing fraction of that phase but does not exhibit flow itself. These structures have often been attributed to so called 'dead-end' pores. 17 However, in these experiments, the phenomenon is actually caused by the morphology (distribution) of the phase within the pores rather than the structure of the pores themselves. The fluid in these pores can be recovered by diffusion into the flowing fraction. The isolated fraction of a phase is completely surrounded by the other phase through which no diffusion can occur. This material

2 MULTIFHASE FLOW IN POROUS MEDIA: I. MACROSCOPIC OBSERVATIONS AND MODELLING SPE 11017

is therefore not recoverable. At this point we shall define Xf ¢, Xd¢ , and Xi ¢ as the flowing, dendritic and isolated fractions of phase ¢, the sum of which is always equal to one.

After describing the methods used in the experi­ments, information is given describing the core in which the experiments were conducted. This is fol­lowed by a description of the observed results. These results are then quantified using the macroscopic model. Some of the uncertainties inherent in this type of modelling are discussed. A discussion of the nature of steady-state flow is given which utilizes the results of these experiments. Finally, a series of other experiments are then discussed which were designed to clarify some of these uncertainties and elucidate the mechanisms which cause the observed phenomena.

II. EXPERIMENTS

There were three tasks involved in running these experiments. The tasks were: the establishment of steady-state, 2-phase flow; the measurement of the pressure drop across the core (to allow phase permea­bility calculation); and finally to put tracers in both the wetting and nonwetting phases and measure their effluent profiles. Many people have run such experiments in the past Z- 1S from which Raimondi and Torcaso were among the earliest. A typical experi­mental setup is shown in Fig. 1. This figure shows parallel systems for the wetting and nonwetting phases. For each phase a constant flow rate, pulse­free pump sends fluid either into the core or into the receiving side of a piston cylinder, depending on the position of valves lA and B. Initially, the fluids are pumped into the core. The core effluent is col­lected in the separating vessel and recirculated into the pump_ Saturation is determined by material balance from changes in height of the interface in the separating vessel.

Floods were started from 100% brine saturation (primary drainage), connate water (imbibition), or residual oil (secondary drainage). Oil and brine were recircclated for many pore volumes until the satura­tion in the core and the pressure drop across the core had not changed over a period of time (usually 2 or more days). At this point the pressure drop was re­corded and valves 1A and/or B switched to allow tracer injection. The effluent selector valve was also switched to allow the collection of samples. In prac­tice, both mixing experiments were never done simulta­neously, and frequently the apparatus used varied somewhat from the one pictured in Fig. 1. In many cases, the piston cylinders were eliminated by pumping immiscible fluids through a sequence of vessels. Also this system is not well suited for operation at either very high or very low fractional flows. For example, one experiment was run with a brine flow rate of 0.25 cubic centimeters per hour while the oil rate was set at 10 cubic centimeters per minute. Sampllng was con­ducted during the brine tracer flood for a period of over 3 weeks. Obviously, if samples had been collect­ed as shown, huge amounts of oil would have been re­quired.

Great care was taken to minimize system (non-core pore volume) dead volume. Such volume would allow for fluid mixing outside of the core. Valves 1A and Band the effluent selector valve were mounted directly on

the core end butts. The inlet end butt passed each fluid independently to the core face through two con­centric spiral grooves cut in its face. The tubing running from the effluent selector valve to the frac­tion collector was of extremely small inside diameter and was kept as short as possible. Both the inlet and outlet end butts had centrally mounted tubes which ran through them and extended 1/4 inch into the core. These tubes were sealed into holes drilled in the ends of the core using rubber O-rings. These tubes were connected to a differential pressure transducer which read out continuously to a chart recorder, thus any pump fluctuations or experimental problems could be detected. The core was mounted in a rubber-sleeved Hassler system, and confining pressure was kept at least 500 pounds per square inch above the inlet core pressure.

The fluids used were sodium chloride solutions of different salinities, various pure hydrocarbons with matched viscosities, and in some cases a 17.8 centi­poise viscosity mineral oil. The brines were usually analyzed for chloride content, but analyses showed that sodium worked just as well over the range of concentration in which we worked. The hydrocarbons were analyzed using gas chromatography. The precision of the brine and hydrocarbon analyses were 0.2 and 0.1% of the full concentration change, respectively.

Most of the floods were conducted at a constant total injection rate, oil rate plus brine rate, equal to 10 cubic centimeters per minute. This rate was selected to minimize capillary end effects, as de­scribed by Kyte and Rapaport,18 even under the most extreme fractional flow conditions.

In cores such as the ones used in this study attempting a connate water drive with a low viscosity hydrocarbon is a difficult task. An alternative would be to drive the core to connate water with a more viscous oil and then displace this viscous oil with the hydrocarbon. As our standard hydrocarbon analysis procedure does not detect mineral oils, we determined this to be a risky proposition. We chose instead to run the connate water drives with the pure hydrocarbon at a rate of 600 cubic centimeters per hour for seven days. This drive was followed by an approximately three pore volume pulse of hydrocarbon injected at 1000 pounds per square inch pressure. This final pulse generally produced one to two saturation percent of end effect, trapped water. During the connate water drive with inviscid hydrocarbon, oil would break through when the core was at an oil saturation of be­tween 45 and 50%. This saturation would rise quickly to ~ 60 percent over the next 5 to 10 pore volumes of injection. The saturation would then rise at an ever decreasing rate until after a week (~ 200 pore vol­umes) the saturation was generally between 70 to 75 percent. We show later the importance of achieving as high an oil saturation as possible during connate water drive.

Generally, tracer floods were run until the pro­duced fluids were at the injected fluid concentra­tions, or very near them (99.8%+) for over several pore volumes. In the brine displacements this was usually accomplished within 3 to 5 brine pore volumes (core pore volume times brine saturation) of injec­tion. In the oil phase, however, some of the low oil fractional flow experiments required injection of up to 25 oil pore volumes of injection. These volumes

SPE 11017 S. J. SALTER and K. K. MOHANTY

were desired in order to carefully discriminate between dendritic and truly isolated oil. Upon com­pletion of each flood, the core was extracted with - 5 pore volumes of isopropyl alcohol. This alcohol ex­tract was then analyzed for water and hydrocarbons by gas chromatography. These analyses provided indepen­dent checks of the mass balance saturation values and the sum-of-the-samp1es hydrocarbon phase recoveries.

III. THE CORE

Nearly all of the experiments being reported here were run in a single core. This core was a 2 inch diameter (5.08 em), four foot long (121.5 em) cylin­drical piece of Berea sandstone which had been fired and acidized according to the procedure given pre­viously.19 The purpose for firing and acidizing is discussed there and has no relevance to the experi­ments being reported here. SEM and thin section anal­ysis of this core material indicate that it is a well­sorted, highly connected, fine-grain sandstone. The porosity of the rock is 23.9%, its brine permeability is 1000 millidarcies, the specific surface area is 0.65 m3/gm, and the rock density is 1.96 gm per cubic centimeter. These data are summarized in Table I. From these data the average hydraulic radius is deter­mined to be 4.91 microns. Photographic image analy­Sis, as discussed in Paper II of this series,l yields an average pore size of 7.8 microns.

Nine core plugs were cut from the same block of sandstone, fired and acidized similarly, and subjected to mercury injection capillary pressure determination. The results of these tests shown in Fig. 2 indicate that the nine plugs were identical to within experi­mental error. The results of such experiments are related to the size distribution of pore throats which are accessible to the surface of the core through channels of continuous nonwetting phase (mercury). The correlation between throat size distribution and capillary pressure will be discussed in detail in Paper II of this series, and will not be discussed further here. Nine other plugs were also cut from the same rock sample and subjected to mercury injection capillary pressure determination without being fired or acidized. Comparing the throat size distribution for the treated and untreated rock indicates that the average throat radius increases with treatment, gen­erally as a result of the increase in the size of many of the very small pores. Since firing and acidizing removes cementing materials and clays, this increase in size is not surprising. The porosity of the core material also increases by from 1 to 2 percent.

Relative permeability was determined during the tracer displacements. As discussed above the connate water and residual oil saturations depended on the viscosity of the oil used in the experiment. When 17.8 centipoise mineral oil was used, oil saturations at the end of five days of connate water drive (~ 125 pore volumes) averaged about 82 percent. The permea­bility to oil under these conditions was 1000 mill i­darcies (equal to the brine permeability at So = 0). The endpoint residual oil saturations when inviscid hydrocarbons were used were generally between 28 to 30 percent. The permeability under these conditions was 200 millidarcies. The permeabilities to oil at connate water when inviscid hydrocarbon was used and to brine at residual oil when mineral oil was used were both somewhat lower. The flood endpoint satura­tions for these two cases averaged So = 75 and 35

percent respectively. The primary drainage, imbibi­tion, and secondary drainage relative permeabilities are shown in Fig. 3. Within the accuracy of the meth­od used, no hysteresis was observed between the imbi­bition and secondary drainage relative permeability curves. The U-shaped curves above the relative per­meability curves represent the total permeability (sum of both phases). The minimum in these curves repre­sents the saturations (one each for each cycle) at which the two flowing fluids most interfere with one another. The majority of the mixing experiments were run with the inviscid (0.4 centipoise) hydrocarbon. Fig. 4 represents the fractional flow curves (a. pri­mary drainage and b. imbibition and secondary drain­age) for an oil with this viscosity. The nature of these curves is of course quite different when the 17.8 centipoise mineral oil is used.

Figs. 5a and b are plots of the effective vis­cosities (reciprocal total relative mobilities [~(k ./~)l-l) for steady-state oil banks as they de­p~ndr~n saturation for each of the two oils. The data in Fig. Sa for the inviscid hydrocarbon show that in the secondary drainage and imbibition cycles, as sat­uration approaches residual oil the mobility of the brine (wetting phase) dominates. Thus, the minimum in total mobility occurs very near residual oil, and also near the minimum in total permeability. For the mineral oil, the viscosity of the oil causes the min­imum mobility condition to be shifted to higher oil (nonwetting) phase saturations. If mobility require­ments such as demonstrated by Gogarty20 apply to an oil bank displacing another oil bank with a different saturation, then the direction of oil saturation change (imbibition versus secondary drainage) and the saturations of the two oil banks relative to the max­imum in effective viscosity may become important. If one does a Buckley-Leverett analysis (which assumes stable displacement) and determines both the shape of the saturation transition zone, and then determines the relative total mobility at each point in the transition, one sees immediately the possibility of zones of stable and unstable displacement both in saturation shocks as well as finite zones of satua­tion change. Mobility requirements are not the topic being addressed here, however, it would be surprising if the fluid distribution among the pores at a given saturation did not depend on the path from which that saturation was approached.

N. RESULTS

This paper reports the results of over 100 tracer displacements, 25 for single-phase flow and 79 for two-phase flow. It is obviously impractical to attempt to display effluent concentration profiles for each of them. The effluent profiles for 100% satu­rated coreS are well known. 21 The results of these experiments will only be discussed as they relate to the data for the partially saturated runs.

The two-phase experiments are broken down into 6 groups according to the core condition at the start of the flood and which phase (wetting or nonwetting) is being considered. Floods started at So = 0, with the nonwetting phase saturation increasing, are called primary drainage floods. These floods are further differentiated by the fractional flow imposed at steady state [fo ~ Qo/(Qo + Qw), where Q is the pump injection rate] and the corresponding oil saturation. The fractional flow and saturation are, of course,

3

4 MULTIPHASE FLOW IN POROUS MEDIA: I. MACROSCOPIC OBSERVATIONS AND MODELLING SPE 11017

related by the fluid phase viscosities and the rela­tive permeabilities of the porous medium. In the limit, when fractional oil flow is 1, the core is eventually driven to its irreducible water condition, connate water. Floods started at connate water, with a given fractional flow, naturally have their nonwet­ting phase saturations decreasing. These are called imbibition floods. The limiting case of the imbibi­tioncycle, fo = 0, results in the establishment of the irreducible, or residual oil saturation. Finally, floods started at residual oil, with oil saturations increasing, are called secondary drainage floods. Rigorously, the above discussion should have been written in terms of the wetting and nonwetting phases. Since the core of primary interest here is strongly water wet, oil is the nonwetting phase. Later, in Section VI-C, we shall discuss results for a core which has been made oil wet. For this core the drainage cycles have water (nonwetting phase) satura­tions increasing and the imbibition cycle has the oil (wetting phase) saturation increasing. As will be shown, the distribution of the phases among the pores--and the macroscopic transport parameters which arise as a result of these distributions--depend not only on the saturation/fractional flow conditions at a given location and time, but also on the direction from which this condition is approached (i.e., satura­tion history). When attempting to model transport for a given process, it is important to determine what type of saturation history is relevant to that pro­cess. Transport properties measured for that type of process should be applied.

In this section we will discuss qualitatively the results for the nonwetting phase; first in primary drainage, then in imbibition, and finally in secondary drainage. Then we will discuss the wetting phase results for the three cycles in the same order. We will describe the nature of the effluent profiles and discuss the ultimate recoveries of the floods, as well as how these properties depend on the conditions under which the floods were run. In the next section we will attempt to quantify these observations using an ana­lytical model. The conditions under which each of the floods was run are given in Table II.

Nonwetting (Oil) Phase Tracer Floods

Effluent profiles for 4 of the nonwetting phase -primary drainage floods are shown in Fig. 6. All such figures presented here show how the normalized concen­tration (tracer concentration of the effluent divided by the injected concentration) varies with the amount of fluids injected (time), which is expressed in terms of pore volumes of the phase being considered, i.e.

Dimensionless Time

volume of phase produced x 100

total pore volume x phase saturation

In each figure, only the first 3 pore volumes of fluid produced are shown. As stated before, in most cases where tailing was pronounced 15 to 30 pore vol­umes were injected. All of this effluent data were used for analysis procedures, but only the first 3 pore volumes are plotted. From Fig. 6 and the data given in Table II, certain trends are evident. Tracer breakthrough comes earliest for the lowest fractional flow (and saturation) experiments. The concentration climbs rapidly to a breakover concentration, whereupon

the effluent concentration rises slowly to that of the injected fluid. This breakover concentration also in­creases monotonically with fractional flow. The time necessary to achieve C ~ 1 decreases dramatically as the fractional flow increases. Thus, the amount of tailing and the quantity of fluid injected necessary to complete the experiment are decreased. Recoveries of the nontracer fluid are about 75%, even at the lowest fractional flows. Recovery stays at this level until about fo ~ 0.5 and So ~ 30, and then climbs to 100% as fo and So are increased.

At this point some mention needs to be made of the nature of nonwetting phase flow during primary drainage. As will be discussed later, the pores through which the nonwetting phase is flowing are 'selected' according to whether the local capillary pressure is sufficient to cause the nonwetting phase to push through a given pore throat. At conditions below those for which the recovery rises dramatically [fo ~ 0.5], there are always a number of pore throats which are right on the verge of being invaded. Thus, even the smallest perturbations to the system (e.g., pump pulsing) cause extra pores to be invaded and therefore, nonwetting phase saturation to rise above the level it should be for that fractional flow. We have observed such phenomena in several, ill-fated experiments. The effluent concentration temporarily either drops or levels off and the recoveries are abnormally high. Also, the pressure drop across the core increases. In cases where this happened, the experiments were terminated and rerun.

In the imbibition experiments (see Fig. 7) recov­eries at high oil saturations are nearly 100%. The recovery decreases slowly with saturation at first and then falls precipitously between So ~ 50 and 40%. Below 40 percent oil saturation the recovery remains essentially constant at ~ 40%. In the same manner, the tracer concentration front arrives earlier as oil saturation decreases, with most of this shift occur­ring in the So = 40 to 50 percent range. Little change occurs between residual oil and So 40%. The amount of tailing increases over the same saturation range paralleling the trend toward earlier break­through.

Fig. 8 shows a sampling of the nonwetting phase effluent profiles for the secondary drainage experi­ments. The changes in these profiles and the respec­tive recoveries mirror the imbibition behavior with one exception. In the range of oil saturations be­tween residual oil and 37.5%, the recoveries actually decrease, from 50% very near residual oil saturation (see run 8D2, Table II) to 40% at the minimum. The recoveries then increase along the same trend as the imbibition floods. The secondary drainage floods at very near residual oil saturation appear to have tracer breakthroughs very similar to those of their imbibition counterparts. The secondary drainage floods exhibit a greater amount of tailing in this region which accounts for their greater recoveries.

As mentioned above, most of the published data of this kind for nonwetting phase displacements are in the miscible flooding literature. We are not aware of any of this data for similar porous media which exhibit trends which are inconsistent with ours.

SPE 11017 S. J. SALTER and K. K. MOHANTY

Wetting Phase Tracer Floods

The first subject to be addressed for these floods is ultimate recovery. In the wetting phase, independent of cycle, the recovery is always equal to 100 percent within experimental error regardless of the fractional flow or saturation. This result has been obtained numerous times in the past. 9 ,22,23 Recently, in our laboratory, we have measured an even stronger indication of the nature of wetting flow. 24 We have conducted ion exchange preflush experiments in partially saturated Berea cores. The results of these experiments indicate that all of the ion exchange sites are contacted even with oil present. This result is a much more sensitive test of the nature of water transport in the oil containing pores because, whereas the volume of water they con­tain is small (because they are largely occupied by oil), their surface areas are still the same. Thus, under all conditions for a strongly wetted medium, the wetting phase has access to and through all of the pores.

Figs. 9, 10 and 11 show typical effluent profiles for wetting phase tracer displacements in the primary drainage, imbibition, and secondary drainage modes respectively. The trends in all three cycles are the same. At low fractional oil flows a symmetric, dispersion-shaped effluent profile with C = 0.5 at one brine pore volume is produced. As the oil fractional flow increases, the brine profile shifts to the left and tailing becomes apparent. This shift continues until fo ~ 0.5, whereupon the profile shifts back to the right. At extremely high oil fractional flows, the brine profile again looks symmetric and dispersion-like. Such behavior has been observed in primary drainage, wetting phase displacements for water drainage through partially saturated, unconsol­idated soils and sand packs. 10 ,13,14 The earliest breakthroughs generally correspond to regions of intermediate saturation (moisture content).

V. ANALYSIS

A. Analytical Methods

In order to express quantitatively the results described in the previous section, it becomes con­venient to define the recoverable fraction Xr , which equals the sum of the flowing and dendritic frac­tions. As we have seen, when tracer breaks through somewhat earlier than one pore volume, very asymmetric profiles generally result, and the outlet concentra­tion does not reach that of the inlet for quite some­time. This effect is due to diffusion into and out of the dendritic fraction of the phase. Coats and Smith16 proposed a model to describe such effects in single phase flow through a linear porous medium. This model uses a single mass transfer term for trans­port into and out of the dendritic fraction Xd' The following equations describe the model

a2e ae ae ac+ K - u f + (I-f) (la)

ax2 ax at at

ae+ (1 - f) M (C - c+) (lb)

at

where f is the fraction of the accessible fluid that flows, u and K are the interstitial velocity and dis­persion coefficients in pores occupied by the fluid and M is the mass transfer coefficient between the dendritic and flowing fluid. e(x) can either be the average in-situ concentration, Ci(x), or the average flowing concentration, Cf(x), of a component defined by

K ae i Cf ci - (2a)

u ax

as shown by Baker. 25 C+(x) stands for the concentra­tion in the dendritic fluid; it can be either e+i or e+f , which are related by an equation similar to Eq. 2, i.e.

K ae+i e+f e+i (2b)

u ax

However, the boundary conditions used to solve Eq. 1 must correspond to the definitions of C and C+ used. In most experiments, the flowing concentration Cf at the end of the core is monitored. Hence, it is con­venient to use the latter definition, ef , of C(x) in Eq. 1 (cf Brigham26) which we have done here.

When all of the phase flows (i.e., Xd = 0), Eqs. la and b reduce to

a2c ae ae K __ - u (3)

ax at

the well known convection-dispersion equation which, when coupled with the appropriate boundary and initial conditions to describe a step change in tracer concen­tration, yields the error function/probability paper solution. 21

Eq. 1 can also be used for two phase flow if the definitions of K, u, and f are based on the recover­able fraction of the phase and no interphase mass transfer occurs. For a given phase, ~, then f = Xf~/(Xf~ + Xd~)' Using this new definition of f, which allows for an isolated phase fraction, Eq. 1 can be rewritten in dimensionless form as

a2e ae ac ae+ Xk - f + (1-f) ( 4a)

ay2 dy aT aT

ac+ ~ (e - c+) (1 - f) (4b)

aT

where Xk uL/K, Xm ML/u, T ut/L, y x/L and L is the length of the core. Xk is called the Bodenstein number or sometimes a Peclet number, though for a true Peclet number L should be replaced by a characteristic pore size. The rate group Xm represents the ratio of mass transfer to convection.

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6 MULTIPHASE FLOW IN POROUS MEDIA: I. MACROSCOPIC OBSERVATIONS AND MODELLING SPE 11017

A step change in input concentration of a compo­nent into a core can be expressed mathematically as

C (y, 0)

C+(y,O)

C (0,.)

0, y ) 0

0, y ) 0 ( 5)

1.0, • ;;. 0

0, • .., 0

Eqs. 4 and 5 can be solved by Laplace transform. Baker 25 gives the solution in the complex frequency domain of the Laplace/transform. If L(s) is the Laplace transform of C(y,T) defined by

L(s) - J °

then

e-ST C(y, T) dT ( 6)

4s ( (1 - f) X )~! 1 + f + m (7) Xk s(l - f) + Xm

A Laplace transform with purely imaginary 8 is a Fourier transform. This transform is valid for all values of s with non-negative real parts. The efflu­ent concentration profile at any y, C(y,T) can be obtained from L(8) by the use of Cauchy Integral theorem.

a + joo

J L(s)e ST ds C(y,T) ( 8) 2Tfj a - joo

where s ~ a + bj, j = I=I and a > O.

Given a set of parameters, f, Xk and Xm, one can calculate the effluent concentration profile c(l,T) resulting from a step input at the inlet of a core. The effect of these parameters on the concentration profiles is shown in Figs. 12 to 14. As f decreases, a smaller fraction of the accessible fluid flows; hence breakthrough of the displacing component is earlier and the tail at the back is longer. As Xk decreases effluent concentration gets more dispersed. At low values of Xm, the dendritic fluid behaves like isolated fluid. The breakthrough time is only slight­ly affected by the mass transfer group.

From an experimentally measured concentration profile, it is possible to determine the set of pa­rameters that theoretically predicts such a profile. In any experiment, however, one measures the effluent concentration profile C(l,t). Note that T ut/L and u is the superficial velocity in the recoverable fraction of the phase, i.e.

u (9)

where Q is the metered injection rate of the fluid into the core. Xr can be obtained from the ultimate recovery of a displaced component. Hence one can calculate Cexp(l,T) from Cexp(l,t).

In order to best fit the experimentally measured effluent data we have used a method like Baker's,25 which involves minimization of error in the frequency space of the Laplace or Fourier transform. This in­volves computation of the Laplace transform of the experimental data using Eq. 6, least squares error minimization, and determination of the best set of f, Xk, and Xm• Taking the Laplace transform of an efflu­ent concentration curve generates data over the entire frequency domain. Computationally, it is impossible to minimize error over the entirety of this domain. After trying a number of alternatives, we selected minimization of error at a geometric sequence of points along the purely imaginary axis because the error there seemed more sensitive to changes in f and Xk. Error minimization was accomplished using a modified Levenberg - Marquardt algorithm. This method proved to be a factor of 60 more efficient computa­tionally than error minimization in y, t-space. To calculate the effluent profile C(l,t) from the deter­mined values of f, Xk, and Xm, Eq. 8 was integrated numerically along a straight line S = 1 + wj from w = -100 to w = 100.

B. Model Limitations

As demonstrated in Figs. 12, 13, and 14, the model has the capability to generate effluent profiles with a variety of shapes. The model has the most dif­ficulty fitting floods with either very low f's or fls very near unity. The problem associated with low flowing fractions is related to the ability of a 4-parameter, dispersion-capacitance model to accurately describe mixing. This model includes only one mass transfer term. Physically, as the flowing fraction decreases and the proportion of the recoverable frac­tion which is dendritic increases, there are a series of denrites of varying sizes and shapes. Thus, there should be a series of mass transfer terms. In such cases the model frequently predicts the steep, break­through portion quite well, implying proper evaluation of f, underestimates the middle portion where the easily diffusible dendrites are dominant, and predicts effluent concentrations which are higher than observed at longer times. This limitation could be reduced by subdividing the dendritic fraction,S adding two more adjustable parameters for each new subdivision. This was not attempted here.

At flowing fractions near one, a problem of an­other sort arises. The model generates mUltiple solu­tions each of which match the experimental effluent profiles to within their experimental accuracies. This problem arises because very easy (high Xm) mass transfer generates effluent profiles centered near one pore volume which are symmetric and look like disper­sion profiles. In order to demonstrate this fact, we used the model to generate two sets of synthetic 'experimental' data; 1) f = 0.5, Xk = 25, and Xm = 1, and 2) f = 0.96, Xk = 25, and Xm = 1. We then used the model to calculate the error in the frequency domain between the 'data' and model effluent curves generated with various values of f, Xk, and Xm• To make visualizing the situation easier, we calculated errors for given values of f and Xk, where we let the model pick the best Xm• These errors are plotted as contours on Figs. 15 and 16 for the f = 0.5 and 0.96 data sets respectively. For Fig. 15 (f ~ 0.5) an error contour of 0.01 generates \Xk - 25\< 4 and \f - 0.5\< 0.02. On the other hand, the 0.01 error contour in Fig. 16 bounds an extremely large region.

SPE 11017 S. J. SALTER and K. K. MOHANTY 7

All sets of parameter values in this region generate effluent profiles which are identical to within exper­imental accuracy. From the data presented in the previous section, one realizes that this problem is minimal for the nonwetting phase displacements. In the wetting phase displacements, however, the problem was substantial and required additional work, which will be discussed later, to elucidate which mechanisms were at work. The nonuniqueness of the high f solu­tion can be shown analytically for a limiting case. When the dispersion coefficient is very low (relative to the mass transfer), Eq. 1 becomes

ae - u ~

ax

ae f _ + M (C-c+)

at (10)

Klinkenberg23 has shown that when the mass transfer term becomes large, the solution of this equation approaches an error function in the limit. The error function is also the solution of the convection­dispersion equation (Eq.3), thus in this limit the two equations (phenomena) generate identical solutions (effluent profiles).

The analysis of tracer profiles in which C = 0.5 occurs very near one pore volume of production using our modified Coats-Smith model is thus very sensitive to small perturbations in the data. Parameters calcu­lated for these floods should be viewed as having error bars larger than those for the floods where f is not near 1. For floods where f was equal to one, least squares fits were made to the error function solution. These floods should be viewed as having smaller error bars.

C. Experimental Data Analysis and Discussion

The initial experiments were brine displacements in oil-free cores. The experiments were performed at interstitial velocities, u = Q/A¢ , from 8.83 x 10-4 centimeters per second (2.55 feet per day) to 9.71 x 10-1 centimeters per second (2750 feet per day). In general, the effluent profiles were symmetric with concentrations equal to 0.5 being produced at one pore volume of production. In a few cases there was some slight tailing at the high concentration region. It is our feeling that this tailing is an experimental artifact. But, in any case, the amount of it is quite small which indicates that under fully saturated con­ditions no appreciable amount of dead-end pores l7 exists in this porous medium and that the existence of a bimodal pore-size distribution is unlikely. The effluent profiles were analyzed by least squares fit­ting the error function solution as described above. The dependence of the interstitial velocity on the dispersion coefficients thus calculated is shown in Fig. 17. Over the entire range of velocities studied, the relationship

K = 0.269 u1.16 (11)

fits the data quite well, where K is the longitudinal dispersion coefficient in square centimeters per sec­ond and u is the interstitial velociy in centimeters per second. The range of Reynolds numbers studied (4.25 x 10-5 to 4.67 x 10-2) are all within the limits of creeping flow. These experiments span the range of Peclet Numbers from 0.13 to 144, based on diffusion and average pore radius. The lower Peclet number

should be very near the transition region between dispersion and diffusion domination.

All of the two-phase tracer floods were analyzed in the manner described above. For each case, Laplace space error contour plots (like Figs. 15 and 16, only for experimental data) were generated to aid in the evaluation of the fit. The results of these analyses are tabulated in Table II. These numbers, and the figures to be discussed presently, express quantita­tively the observations made in Section IV. Here we will discuss the various phase fractions first, the dispersion coefficients second, and finally the mass transfer coefficients.

The data for the phase fractions for both phases during each of the three cycles is shown in Figs. 18 through 23. For the nonwetting phase, the amount of isolated oil goes to zero as connate water is approached. This trend is to be expected because more and more of the pores contain oil, and the oil which became isolated has become reconnected. Note also that the flowing and dendritic fractions do not appear to be going to zero at fa = O. This observa­tion is also logical because the phase fractions go to zero at residual oil discontinuously. When all of the oil is nonflowing, there are no continuous oil paths from the core inlet to the outlet. If one thinks in terms of the imbibition cycle, in the most extreme limit there is one pathway for oil to flow from the injection face of the core to the outlet. Because of the extreme conditions under which this pathway was created, it is highly tortuous (see Paper II) and has a multitude of dendrites extending from it. Suddenly, due to changes in condition, this pathway is fissured and oil flow stops. Thus, the volumes of this flowing pathway and its connected dendrites are transferred, all at once, into the isolated fraction. Cessation, or initiation, of oil flow occurs at a percolation threshold.

In the nonwetting phase, dendrites are created due to the nature of the process by which oil finds its way through the core. When the fractional flow of oil is increased, the additional oil flow can occur by two mechanisms; either more flow can go through the same pores, or additional pores must be invaded to create new flow channels. The relative importance of these two mechanisms is evident from the fractional flow curve. In some saturation regions the fractional flow changes faster than the saturation, and in others it changes slower. In a region where new flow chan­nels are being created, oil displaces water from pores because the local capillary pressure exceeds the value necessary for the oil to pass through a pore throat. Imagine a core at 100 percent brine saturation as the first oil reaches the inlet core face. The local capillary pressure at the core face rises until it becomes sufficiently high so that the largest pore throat on the inlet face is invaded. A potential flow channel has been created. This channel will continue to grow, down the pressure gradient, until it reaches a throat that it can not get through--due to insuffi­cient capillary pressure. As oil injection continues, the pressure rises until either the throat where the original oil channel had stopped is invaded or, some other throat accessible to oil (either on the inlet core face or somewhere along the original channel) is penetrated. This process continues until at some pressure (often called breakthrough capillary pres­sure)7 a continuous channel from the inlet to the

8 MULTIPHASE FLOW IN POROUS MEDIA: I. MACROSCOPIC OBSERVATIONS AND MODELLING SPE 11017

outlet end of the core is created, through which the nonwetting phase can flow. At this point, all of the blind alleys of oil which are connected to this flow­ing channel are dendritic, and communicate with it by diffusion.

During the maze-solving problem of the oil find­ing its way through the core, the flowing wetting phase has continually had to reorient its flow paths. Sometimes in doing this, the wetting phase has had to reenter a pore already occupied by oil, thus cutting in two an oil channel. This process has now stranded one or the other of the oil segments--making it into isolated oil. This process is also important as the oil fractional flow is decreased. Finally, at steady­state, it is necessary to have capillary pressure equilibrium everywhere. In order to accomplish this, wetting phase may be sucked by capillarity into pore throats creating an oil-water interfacial configura­tion which is unstable. The water will then coalesce, breaking apart the oil channel and choking off one of the sides of the oil filament from its connection to the flowing channels. Thus, more isolated oil has been created. Paper II of this series will discuss these mechanisms in detail and attempt to quantify them.

Shelton and Schneider6 state, "At a given k level, the saturation difference between the draI~age and imbibition kro curves, represents the trapped oil saturation." Here, their drainage curve is for pri­mary drainage and their "trapped oil" corresponds to our isolated oil. Actually, permeabilities are re­lated to the pores through which flow is occurring. Thus, in opposition to Shelton and Schneider, we would contend that kro(SoPd) = k (S im) actually implies d d . . ro 0 that XfP SoP ~ Xf

lm Solm. Examination of Figs. 3, 18, 19, and 20 show that this is approximately true.

The phase fraction curves for the wetting phase, Figs. 21, 22, and 23, indicate that there are two kinds of brine flow. When flowing oil is present in the core, the existence of these flow channels creates a group of water-filled pores which are 'almost isolated'. That is, they communicate with the flowing water only through the nooks, crannies, and films of water that exist in the oil filled pores because water is the wetting phase. Water transport through such restricted passages is more difficult and thus, a den­dritic water fraction is observed. As the nonwetting phase saturation increases, the number of water-filled pores decreases. A larger and larger fraction of the transport in the wetting phase takes place in the nooks, crannies, and films, and fewer and fewer 'almost isolated' water-filled pores exist. The wet­ting phase permeability drops dramatically--reflecting the relative size of the nooks, crannies, and films relative to the pores. As connate water is approach­ed, all wetting phase transport is taking place in this highly connected set of minute channels and the flowing fraction is unity.

Dispersive mixing occurs due to the variance in arrival times at the core outlet of a number of tracer particles each of which entered the core together. The distribution of arrival times is characterized by both its mean and its variance. If get) is the dis­tribution of arrival times normalized such that

J get) dt ::: a

(12)

then the variance of this distribution, V, is

V f g(t)t 2dt a

[J g(t)tdt]2 (13) o

In our model the variance in arrival times is repre­sented by the longitudinal dispersion coefficient, K. For single-phase flow in a totally saturated core, we have seen (see Fig. 17) that the dispersion coeffi­cient depends on the interstitial velocity of flow through the pores. Under partially saturated condi­tions, in which a phase may have isolated, dendritic, and flowing fractions, dispersion represents the vari­ance in arrival times due to transport in the flowing fraction. The disperSion coefficient should therefore be correlated with the interstitial velocity in the flowing fraction, vf, which is defined as

(14)

The values of K and vf determined from model fitting each of the tracer displacements are tabulated in Table II. The relationship between K and vf is shown for the nonwetting and wetting phases in Figs. 24 and 25 respectively, where the solid line represents this dependence for single-phase, 100 percent saturated cores (see again Fig. 17).

Examination of Fig. 24 shows that flow in the nonwetting phase can be broken into two regions. For oil fractional flows below ~ 0.35, flowing fractions remain approximately constant and the dispersion co­efficients show a variation with flowing fraction interstitial velocity very similar to that of the 100 percent saturated core. The values of the dispersion coefficient over this range are, however, 50 (primary drainage) to 175 (imbibition and secondary drainage) times higher than those in the 100 percent saturated core at equal velocity. Over this range of fractional flows, the flowing oil saturation, Xf So, is changing slowly relative to fa. Thus, more oil is being forced through essentially the same pores. It is, therefore, logical that vf rises and K(So,Vf)/K(O,vf) remains about constant in this region.

As oil fractional flow rises above 0.35, the flowing oil saturation (XfoSo) rises rapidly--first due to rapid increase in Xf and then to rapid in­creases in So' In this region more of the oil is becoming connected and the flowing phase interstitial velocity remains almost constant. It should be remem­bered that the total flow rate, oil plus brine, is constant in these experiments. Thus, as the frac­tional oil flow is increased, both the effects of reconnecting of the oil so as to use a larger fraction of its volume for flow, and the replacement of flowing water paths by flowing oil paths are felt.

When the ratio, K(5,vf)/K(O,vf), is plotted ver­sus saturation--as in Figs. 26 and 27 for the nonwet­ting and wetting phases respectively--the effects of velocity are removed so that it is possible to see the effects of saturation independently. Such saturation effects arise, as will be shown in Paper II, due to differences in the nature of the fluid distribution of the flowing phase fraction. As Bear28 points out, the value of tortuosity depends on how it is defined or measured. At high phase saturation many intersections

SPE 11017 S. J. SALTER and K. K. MOHANTY 9

between flowing pathways exist. Thus, a given tracer particle is likely to take a fairly straight­forward path down the pressure gradient, i.e., the tortuosity of the average path will be less. The lowest tortuosity would exist for the fully satu­rated core. As the saturation decreases, fewer and fewer pathway intersections exist and the paths tend to wander more in directions perpendicular to the macroscopic pressure gradient. Thus, the tortuosity increases as the saturation decreases. The average value of the flow weighted tortuosity is related to the phase permeability. Burdine29 defined tortuosity ratio as the tortuosity of a phase at a given satura­tion divided by the tortuosity of the fully saturated medium. He then used an empirical relationship for the dependence of tortuosity on saturation to extend a pore size distribution calculation of total permeabil­ity to relative permeabilities. The tortuosities he used came from core resistivity measurements. Flow weighted tortuosities are more highly weighted by pore radius than those measured by resistivity. Thus the numbers one would calculate by his method would be somewhat in error.

The dispersion coefficient, as a measure of the variance in tracer particle arrival times, depends on the variance in flow weighted tortuosity. Initially, as the saturation of nonwetting phase decreases, its tortuosity increases and the variance in tortuosity increases. Thus, the dispersion coefficient in­creases. However, as nonwetting saturation decreases still further, the tortuosity continues to increase-­but--the number of flowing oil pathways is decreasing! At some point the decrease in the number of pathways overtakes the effect of the increase in tortuosity, and the variance in tortuosity begins to decrease. In the limit where there is only one flowing oil pathway, the variance in tortuosity must become identically zero. Thus, one would anticipate nonwetting phase dispersion coefficient ratio curves which go to zero at zero oil phase saturation, rise to a maximum as phase saturation increases, and then decrease to unity as the core becomes fully saturated. The importance of tortuosity on the dispersion coefficient has been demonstrated for a randomly oriented capillary tube model by Wilson and Gelhar. 30 Examination of Fig. 26 shows that the nonwetting phase exhibits a monotonic increase in dispersion coefficient ratio as its phase saturation decreases. Apparently the number of flow­ing pathways even at these very low fractional flows is still quite large. In a core such as the one used for these experiments, each cross section has roughly 6 million pores. Under the most extreme conditions investigated here (fo ~ 0.01) there are still more than 220,000 flowing oil pathways. Thus, the antici­pated maximum in dispersion coefficient ratio is experimentally unattainable.

For the wetting phase a somewhat different situa­tion exists. As discussed above, wetting-phase flow occurs in two different regimes. At high wetting­phase saturation, flow through pores dominates--while at low-wetting phase saturations, flow through nooks, crannies, and films dominates. Thus, as the satura­tion of the wetting phase decreases, the tortuosity of the flowing channels increases. However, at suffi­ciently high saturation such that the flow is being transferred to the nooks, crannies, and films--the tortuosity begins to decrease again. In the limit near irreducible water saturation, a network structure of these nooks, crannies, and films exists which,

although these conducting channels have much smaller cross sections, looks very similar to the network structure for the whole porous medium. Thus, it is not surprising at all that the dispersion coefficient ratio curves for the wetting phase, see Fig. 27, have maxima in the regions where the transition between the flow regimes occurs (see also Figs. 21, 22, and 23).

A number of authors 2 ,30 have attempted to corre­late the saturation dependence of the dispersion coef­ficient with changes in pore size distribution in par­tially saturated flow. We show1 that the variance in pore-size distribution is, of course, maximum when the core is fully saturated. Furthermore, the assumption that the nonwetting phase occupies only the largest pores, regardless of cycle, is relatively poor except near connate water and the distribution in the wetting phase is always much broader than would be derived from this assumption.

The mass transfer coefficient, M, is a measure of the rate at which diffusion between the flowing and dendritic fractions occurs. The values calculated by fitting each of the tracer displacements are tabulated in Table II. For the nonwetting phase, the tabulated values reflect two phenomena. One, associated with the ease with which tracer can diffuse into and out of the dendritic pores, is proportional to the diffusion coefficient and the cross-sectional area of the dendrite--and inversely proportional to the length of the dendrite. The second effect is again associated with the interstitial velocity in the flowing frac­tion. Thus, by dividing M by vf a quantity which reflects the dendrite morphology is obtained (since the diffusion coefficient is a constant). In Fig. 28 the dependence of this ratio on saturation is shown for the nonwetting phase. As expected, M/Vf increases with saturation implying that the dendrites become shorter as the core becomes more saturated. This conclusion is necessary on purely geometrical grounds. No explanation for the amount of scatter in the pri­mary drainage data is obvious. The curve drawn through the data for this case is quite arbitrary. The concept of mass transfer in the wetting phase is not as clear cut as for the nonwetting phase. Cer­tainly, under some conditions, we are modelling slow transport of brine through some pores as mass trans­fer. Also the range of saturations over which a sig­nificant amount of dendritic water exists is quite limited. Finally, the insensitivity of the model when f 1 should be recalled. Thus, the value for M/vf for the wetting phase averages 5.2 ± 1.9 cm- 1 and shows no consistent variation with saturation. On the average, mass transfer was more than twice as fast in the wetting than the nonwetting phase.

VI. ADDITIONAL EXPERIMENTS

A. Cause of Early Breakthrough in the Wetting Phase

As discussed above~ the maximum shift in the wetting-phase tracer effluent profile (minimum in flowing fraction) occurs in run SD 11, where fa 0.6 and So 39.4. These conditions happen to coincide with both the minimum in total permeability and the maximum in effective viscosity for this system. The minimum in total permeabilty always occurs at this saturation, however, the saturation corresponding to the maximum in effective viscosity depends on the oil viscosity. Without rerunning all of the tracer displacements with a different viscosity oil, it is

10 MULTIPHASE FLOW IN POROUS MEDIA: I. MACROSCOPIC OBSERVATIONS AND MODELLING SPE 11017

impossible to tell if this maximal shift corresponds to the maximum viscosity. We did, however, conduct brine tracer displacements with a mineral oil with a viscosity ~ 45 times higher than that of the hydrocar­bon under such conditions as to approximately repro­duce the saturation (run SD 21 with fa = 0.025) and the fractional flow (runs s022 and SD23 with So's 56.5 and 59.3%). The effluent profiles for these runs are shown in Fig. 29 and the model fit parameters are tabulated in Table II. It is clear from the results of these experiments that the shift in the brine effluent profile is due to the fractional flow. Under the low fo conditions, the brine displacement is still taking place in brine-filled flow paths. In the intermediate fractional flow regime, when a signifi­cant quantity of simultaneous flow is occurring, the flowing oil has cut off a considerable amount of 'nearly isolated' brine, which is fit by our model as a fairly large (- 37%) dendritic fraction.

B. Mechanism for Wetting Phase Transport at High fo

As stated in Section V-B, an entire family of solutions fit error function shaped effluent profiles with C = 0.5 at one pore volume of production equally well. It would be easy to assume that hydrodynamic dispersion is the relevant mechanism in such regions. A region where this assumption could be easily ques­tioned, and evaluated, is for the wetting-phase dis­placements very near irreducible water saturation. Such an experiment is SD IB, which was run with miner­al oil (So ~ 7Z.Z and fa ~ 0.9967). The effluent profile for this run is fit well by an error function solution (f 1) with K ~ 1.29 x 10-4 cmZ/sec and u = 4.24 x 10-4 cm/sec. The effluent data are also fit equally well by; f = 0.465, K = 0, M = 0.372 x 10-3 sec -I, and vf = B.B9 x 10-4 cm/sec. If one examines Eq. 4, it becomes apparent that the relevant dimen­sionless groups (Xk = uL/K and Xm = ML/u) have very different dependencies on interstitial velocity, assuming M is noz strongly dependent on u. As shown above, K a u 1• 16 and thus Xk a uO. 16 , or almost in­dependent of velocity. Xm on the other hand varies inversely with the interstitial velocity, u. Thus, if the total flow rate is raised by a factor of 3, Xk goes up just slightly (less than 10%) - while Xm goes down by a full factor of three. Fig. 30 shows the predicted effluent profiles for the base case (labeled "I"), and the three times higher flow rate effluent profiles based on the dispersion mechanism prediction (labeled "K") which almost overlays the base case-and the mass transfer mechanism case (labeled "M"). These profiles would clearly be different by more than ex­perimental error.

As a result of this analysis, we ran flood 1MB (QTOTAL = 28 ml/min, fo 0.9988 and So 60%). The results for this flood, see Table II, clearly indicate that the shape of the effluent profile under these fo' So conditions is only very slightly dependent on interstitial velocity as would be predicted by the dispersion mechanism. Thus, as connate water is approached, the transport mechanism involves flow through the nooks, crannies, and films.

C. Effect of Core Wettability

The dramatic differences between the results for the wetting and nonwetting phases indicate that core wettability must be an important factor. Tiffin and Yellig31 have shown that recoveries in COZ displace-

ments run under a variety of water to gas ratios in oil-wet cores are equal to or greater than those run under similar conditions in strongly water-wet cores. Stalkup5 predicted this result previously. In the experiments discussed so far, we have shown that the ultimate recovery of the wetting phase (in a strongly wetting rock) is always 100%. Our experiments were run in highly connected Berea; however, in a less connected porous medium, perhaps with some dead-end pores, one might expect the time necessary to reach this ultimate recovery to be considerably longer. But, nonetheless, recovery should be complete. To see if the behavior of the wetting and nonwetting phases remained the same, we converted a fired­acidized Berea core to oil wet.

A cylindrical core four feet long and two inches in diameter, which had been fired and acidized as dis­cusssed above, was selected. One phase (So - 0%) brine dispersion floods, run at several rates, indi­cated that the dispersion coefficient and its rate dependence were the same for this core as those mea­sured for the core discussed previously to within experimental accuracy. The brine was then displaced by deionized water, which was in turn displaced by dry nitrogen. The core was then heated to 140°C and dry nitrogen passed through for 24 hours. The core was then evacuated and cooled to room temperature. The core was then saturated with hexane. The hexane was displaced (carefully) with ~ 5 pore volumes of a 7% solution of dichlorodiphenylsilane. This was in turn displaced by ~ 5 pore volumes of hexane. This hexane was displaced (again carefully) by ~ 5 pore volumes of chlorotrimethylsilane, followed by another ~ 5 pore volumes of hexane. The final hexane hexane was tested for silane by mixing it with water and finally by UV. At this point, several miscible (So = 100%) tracer floods were conducted. The results of these floods show that the dispersion coefficient and its rate dependence were not affected by the silane treatment. From this result we infer that the pore structure of the core (pore-size distribution, connectivity, etc.) was not altered.

An identical silanization procedure was carried out on a number of core plugs which were subjected to imbibition wettability tests along with a number of unsilanized plugs. The results of these imbibition tests indicate that the unsilanized plugs were strongly water wet--however not so strongly as ones which have not been fired and acidized. We speculate that the firing/acidizing procedure removes the car­bonate cementing material and clays, thus enlarging some of the smaller pores which tend to imbibe water most strongly. This speculation is further substanti­ated by the observed changes in the capillary pressure curves. The silanized cores were very strongly oil wet. Over several weeks they did not imbibe any water (drain oil). Upon being driven to residual oil saturation (by centrifuge) they were subjected to oil imbibition. These cores imbibed So ~ 50% almost immediately, with the final imbibition being over 60% saturation. For the unsilanized plugs, the connate water saturations averaged 7% while the residual oil saturations averaged 16%. The silanized plugs had connate water saturations,of 18% and residual oil saturations of 4%. (Note; these saturations were obtained in a centrifuge).

The four foot, silanized core was waterflooded from So = 100%. This flood was marked by early water

SPE 11017 s. J. SALTER and K. K. MOHANTY Il

breakthrough (So; 43%) compared with the unsilanized cores. Water breakthrough was followed by a long period (30 pore volumes injection) of two phase, high water-cut flow which eventually was terminated at So = 19.4%. All of this information indicates that the core was oil wet. Our thanks go to Norm Morrow of New Mexico Tech and Tom Burchfield and Phil Lorenz of DOE for helpful suggestions in developing this proce­dure.

Three dual wetting!nonwetting tracer floods were then carried out in the silanized core at fractional flows of 10, 50 and 90%. These floods were run in the secondary drainage mode (starting from connate water in this case). The results of these floods are given in Table III and the effluent profiles shown in Figs. 31a and b. These effluent profiles were fit with our modified Coats-Smith model as discussed above. The results of these experiments compare well with those run in the water-wet core when the respective wetting phase and nonwetting phase displacements are compared, with one exception (see below). Wetting phase recov­eries (oil) in each case were 100%. Nonwetting phase recoveries (brine) decreased as the saturation (frac­tional flow) of nonwetting phase decreased. Comparing the flowing, dendritic, and isolated fractions for these floods with those run in the water-wet core is somewhat tricky. It is possible to assume that since the wettability has been reversed the relative permea­bility curves (wetting and nonwetting) will stay the same. Unfortunately, the oil to water viscosity ratio is ~ 0.4 and this ratio does not change when the wet­tability switches. Thus, the fractional flow curves are vastly different. In the first-order approxima­tion, if these differences are ignored, the phase fractions for the fa = 10% and 50% floods still com­pare reasonably well with those run in the water-wet core. In fact, they compare quantitatively if a satu­ration shift of ~ 3% is made [Snwow + 3 = SnwWWj • The recovery in the nonwetting phase (water) in the oil­wet core in the fo ~ 10% flood is considerably lower than the corresponding flood in the water wet core. The flowing fraction in the wetting phase is also much lower for this flood. At this point we can only guess that since establishing a well-waterflooded condition in this oil-wet rock is so difficult, that we may have run this flood before final equilibrium conditions were established. The wetting and nonwetting phase dispersion coefficients compare quite well with those with the same nonwetting phase saturation run in the water-wet core. Also, the nonwetting phase mass transfer coefficients compare well--the value for the fa 0.1 flood again having the greatest error.

D. Importance of the Connate Water Drive

A new fired-acidized core was set up as described in Section II. Brine dispersion floods with no oil present were run at three injection rates to verify that the dispersion coefficient and its velocity dependence were the same as the original core. They were to within experimental accuracy. Three secondary drainage floods were then conducted with oil frac­tional flows of 10 percent. The difference between these three floods was the degree to which the connate water drive was carried out. In the first flood, the connate water drive was carried out as before (7 days, 200 pore volumes, pressure pulse at 1000 psi). In the second flood, the connate water drive was run for 4 pore volumes at 10 cubic centimeters per minute injec­tion rate. In the third flood, the connate water

drive was run at 0.5 cubic centimeters per minute until oil broke through. During the experiments only oil tracers were injected, no samples were collected, only oil recovery was determined by extraction with isopropyl alcohol.

Table IV lists the results of these experiments. As can be clearly seen, the percentage of recoverable oil in place at the start of the tracer flood in­creases dramatically when the connate water drive is stopped prematurely. The longer and more highly pres­sured the connate water drive is, more oil is forced through more smaller pore throats into pores where it can become isolated during the waterflood. Conditions do not exist during the displacement to reconnect much of this oil, and it is left unrecoverable.

VII. CONCLUSIONS

Based on the results presented here, we have made the following conclusions about the nature of steady­state flow in strongly wetted porous media:

1) A four-parameter capacitance-dispersion model is shown to model accurately transport in each phase. This model breaks each phase into three fractions; flowing - which is contained in pores through which flow is going on, dendritic - which is contained in pores which are connected to the flowing fraction but through which no flow is occurring, and isolated - which is contained in pores, which are completely surrounded by the other phase. The recoverable fraction is the sum of the flowing and dendritic fractions.

2) Recovery in the wetting phase is always complete, whereas the recovery in the nonwetting phase generally decreases as the fractional flow of that phase decreases. The recovery of the non­wetting phase approaches a nonzero value as its fractional flow goes to zero. The nonwetting phase recovery goes to zero discontinuously at zero fractional flow.

3) As nonwetting phase saturation increases, the fraction of the wetting phase which is flowing at first decreases to a minimum, but this begins to increase with a further increase in nonwetting phase saturation (see Figs 21, 22 and 23). The fraction of the nonwetting phase which is flowing decreases as the saturation of that phase de­creases, and then levels off at a nonzero value as zero fractional flow is approached (see Figs. 18, 19 and 20). The flowing fraction of nonwet­ting phase goes to zero discontinuously at zero fractional flow.

4)

5)

Dispersion coefficients in two-phase flow should be compared with those in single-phase flow which have the same flowing fraction interstitial velocity. In the wetting phase, the ratio of these coefficients exhibits a maximum in their dependence on saturation (see Fig. 27). In the nonwetting phase, this ratio increases with de­creasing saturation for all experimentally at­tainable fractional flows (see Fig. 26).

Mass transfer between the dendritic and flowing fractions of a phase can be described in terms of a mass transfer coefficient. This coefficient is approximately constant in the wetting phase, but

12 MULTIPHASE FLOW IN POROUS MEDIA: 1. MACROSCOPIC OBSERVATIONS AND MODELLING SPE 11017

6)

7)

increases with saturation in the nonwetting phase (see Fig. 28).

When a significant amount of simultaneous flow is occurring, flowing oil has cut off a considerable amount of 'nearly isolated' brine. Early break­through is experimental evidence of this 'nearly isolated' brine, and it is modelled as a den­dritic fraction. The mechanism which causes early breakthrough in the wetting phase is, thus, due to simultaneous flow rather than saturation alone.

When considering transport in a phase, it is im­portant to know the wettability of that phase. If core wettability changes, the nature of trans­port will change.

8) The extent to which endpoint saturations are reached in laboratory experiments, affects the transport properties at all subsequent conditions (i.e. saturation history is important).

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12. Biggar, J. W. and Nielsen, D. R.: "Miscible Displacement, II. Behavior of Tracers," Soil Sci. Soc. Amer., Proc. (1962) ~, 125-12~

13. Krupp, H. K. and Elrick, D. E.: "Miscible Dis­placement in an Unsaturated Glass Bead Medium," Water Resour. Res. (1968) ~, 809-815.

14. Gaudet, J. P., J~gat, H., Vachaud, G. and Wierenga, P. J.: "Solute Transfer, with Exchange Between Mobile and Stagnant Water, Through Un­saturated Sand," Soil Sci. Soc. Amer. J. (1977) i!.. 665-671.

15. De Smedt, F. and Wierenga, P. J.: "Mass Transfer in Porous Media with Immobile Water," _J_._~~ ....... (1979) i!., 59-67.

16. Coats, K. H. and Smith, B. D.: "Dead-End Volume and Dispersion in Porous Media," Eng. J. (March 1964) ~, 73-84; _T_r_a_n_s.::.-..:.=:.::.=...

17. Fatt, I., Maleki, M. and Upadhyay, R. N.: "De­tection and Estimation of Dead-End Pore Volume in Reservoir Rock by Conventional Laboratory Tests," Pet.. J. (Sept. 1966) 206-212.

18. Kyte, J. R. and Rapoport, L. A.: "Linear Water­flood Behavior and End Effects in Water-Wet Porous Media," J. Pet. Tech. (Oct. 1958) 213, 423-426; Trans. AIME, 213.

19. Meyers, K. O. and Salter, S. J.: "The Effect of Oil to Brine Ratio on Surfactant Adsorption from Microemulsion," Soc. Pet. J. (1981) ll, 500-512.

20. Gogarty, W. B., Meabon, H. P. and Milton, H. W., Jr.: "Mobility Control Design for Miscible-Type Waterfloods Using Micellar Solutions," Pet Tech. (1970)~, 141-147.

21. Perkins, T. K. and Johnston, O. C.: "A Review of Diffusion and Dispersion in Porous Media," Soc. Pet J. (March, 1963) 1., 70-84.

22. Russell, R. G., Morgan, F. and Muskat, M.: "Some Experiments on the Mobility of Interstitial Waters," AIME (1947) 170, 51-61.

23. Brown, W. 0.: "The Mobility of Connate Water During a Water Flood," Trans AIME (1957) 190-195.

24. Meyers, K. O. and Sal ter, S. J.: "Concepts Per­taining to Reservoir Pretreatment for Chemical

SPE 11017 S. J. SALTER and K. K. MOHANTY

Flooding," to be published.

25. Baker, L. E.: "Effects of Dispersion and Dead­End Pore Volume in Miscible Flooding," Soc. Pet.

(June 1977) 12, 219-227.

26. Brigham, W. E.: "Mixing Equations in Short Labo­ratory Cores," ~~_~:..-.----!><~J. (Feb. 1974) ..!.i, 91-99.

27. Klinkenburg, A.: "Numerical Evaluation of Equa­tions Describing Transient Heat and Mass Transfer in Packed Solids," I EC (1948) ~, 1992-1994.

28. Bear, J.: Dynamics of Fluids in Porous Media, American Elsevier (1972) pp 106-118.

29. Burdine, N. T.: "Relative Permeability Calcula­tions from Pore Size Distribution Data," Trans. AIME (1953) 198, 71-78.

30. Wilson, J. L. and Gelhar, L. W.: "Dispersive Mixing in a Partially Saturated Porous Medium," Report No. 191, Ralph M. Parsons Laboratory, Mass. lnst. of Tech., Cambridge (1974).

31. Tiffin, D. L. and Yellig, W. F.: "Effects of Mobile Water on Multiple Contact Miscible Gas Displacements," SPE/DOE 10687, SPE/DOE Third Joint Symposium, Tulsa (1982) 197-215.

TABLE I: DISPERSION CORE PROPERTIES

Length 121.5 cm (~48 inches)

Diameter 5.08 em (2 inches)

Pore Volume 589.6 cubic centimeters

Porosity 23.9%

Brine Permeability 1000 millidarcies

Rock Bulk Density 1.96 grams per cubic centimeter

Specific Surface Area 0.65 square meters per gram

Average Hydraulic Radius 4.91 microns

vJettabili ty strongly Water Wet

13

Table II: Experimental Results

jl)

JII(J 1.1'11 111 III .un 10 1,1 10

:\111 10 lill' ;\1

IIU III lill) 11)

10 III '1 II' ~ il I,j III ;1<\

III III \(1

IO() III III

1111) III

111(1 III

1111] 10

IUli 10 \111)

~ 0 ;Il!

III 1[ld

:0 lilU

III jill)

III \11",

III 10'1 10 Il)1

III III

'Iii, III 1" 10 ll1(1

101 III :Oll

II ~(l

TABLE III: RESULTS OF FLOODS RUN IN OIL WET CORE

FLOOD OWl mJ2 OW3 fo 0.1 0.5 0.9 So 38.7 53.3 66.0

PHASE BRINE (N~v ) OIL (W) BRINE (NVi1) OIL (W) BRINE (NW) OIL (W)

XI 0.1940 0 0.3530 0 0.6110 0

XF 0.6870 0.6642 0.4515 0.7821 0.2752 0.8799

XD 0.1190 0.3559 0.1955 0.2179 0.1138 0.1201

V(cm/sec) 0.0789 0.1460 0.0770 0.0442 0.0266 0.0577

K(cm 2/sec) 0.0288 0.0940 0.1348 0.0758 0.1080 0.0995

M(sec- l ) 0.473 x 10-3 0.320 x 10-4 0.387 x 10-3 0.117 x 10- 3 0.500 x 10- 4 0.124 x 10- 3

TABLE IV: FLOOD RESULTS FOR VARY ING COlHJATE WATER DR IVES

EXPEHIMENT VI-Dl VI-D2 ilI-D3

fo 0.1 0.1 0.1

72.4 56.3 42.5

S wf '0 30. 28.7 26.3

flood 37.5 34.2 30.1

Recovery (% ) 50.9 57.6 64.7

Recovery (t..8 0 ) 19.1 19.7 19.5

-(J) a.. --w a: ::J (J) (J)

ES w a: a.. >-

tf a: « ...J

SAMPLER ...J

a.. « ()

Fig. 1-The experimental apparatus.

100 0.1 --. "'C >-g I-

- - - SECON DARY ::J >- DRAINAGE (Q I- AND IMBIBITION ::J « w CD

I ~ « a: w 10 I 0.01 w ~ I a.. S a: W I W 0 a.. I > ...J

;:: LL W I (J) « ....J « I ...J « I I

w Z a.. a: 0 I ;:: 1.0 , 0.001 ()

I « a: LL

0.1 0.0001 0 20 40 60 80 100

OIL SATURATION (0/0)

Fig. 3-Experimental relative permeability curves.

103

102

10° 0

1.0

0.5

0

• _. -----_. --••

20

-.... -._ .

40 60

••• ••

80

• -100

WETTING SATURATION

Fig. 2-Experimental capillary pressure curve.

0.39 cp OIL

, I I ,

I I I I I I , , , ,

-- PRIMARY DRAINAGE , I

--- SECONDARY DRAINAGE I AND IMBIBITION I

0 50 100

OIL SATURATION (%)

Fig. 4-The fractional flow curve.

.-C. .£ >-t: (j)

a 0 (j)

> UJ > l-0 UJ LL LL UJ

" , I 0.39 cp OIL , I

8 I I I -- PRIMARY DRAINAGE I I --- SECONDARY DRAINAGE I AND IMBIBITION , , I I I ,

6 , I , I I I I I , , , \ I

4 \ I , I I

2

o ~----------------~------------------~ o 50 100

OIL SATURATION (0/0)

Fig. 5a-Effective viscosity curve-low viscosity hydrocarbon.

z a i= « 0: I- 0.5 Z UJ 0 z a 0

0.0 0.0 1.0

60

.-C. .£ >-t: (j)

a 0 (j) 40 :; UJ > i= 0 UJ LL LL UJ

20

17.8 cp OIL

... , I , I , : \ , ,

\ , \

I , , , , , , , , ,

\ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~

II.

PRIMARY DRAINAGE

SECONDARY DRAINAGE AND IMBIBITION

o~------------------~------------------~ o 50 100

OIL SATURATION (%)

Fig. 5b-Effective viscosity curve-mineral oil.

2.0 3.0

TIME (PV)

Fig. 6-Effluent profiles for oil phase-primary drainage.

z o i= « a: ~ 0.5 LU () Z o ()

0.0 l!...L._--L--I._~-'----'-___ -'--___ ",---__ ---L ___ ---'

0.0 1.0 2.0 3.0 TIME (PV)

Fig. 7-Effluent profiles for oil phase-imbibition.

z o i= « a: I- 0.5 Z LU () Z o ()

O.O~~~~--~---'-------~----'--------~----~ 0.0 1.0 2.0 3.0

TIME (PV)

Fig. a-Effluent profiles for oil phase-secondary drainage.

1.0 1--r----,--/7y~...-J::==:::::~::::;=-=-_,

z o i= « a: ~ 0.5 LU () Z o () P03

P08 P02

o.o~----~~~~~------~------~---~----~ 0.0 1.0 2.0 3.0

TIME (PV)

Fig. 9-Effluent profiles for brine phase-primary drainage.

z 0 i= « ex: 0.5 I-Z W 0 Z 0 ()

0.0 0.0 1.0 2.0

TIME (PV)

Fig. 10-Effluent profiles for brine phase-imbibition.

z o i= <t: ex: ~ 0.5 W () Z o ()

3.0

0.0 I.---__ --'---"-'''-''-''_-'--__ ---''--__ --'-_~--'-------'

0.0 1.0 2.0 3.0

TIME (PV)

Fig. 11-Effluent profiles for brine phase-secondary drainage.

1.0 r--,----,----s::;:::::::;:;::::.-.,.---=:::r::=====1

z 0 i= 1 2 3 « 0: 0.5 0.7 0.9 I- 0.5 Z 1 Xk 100 100 100 w ()

Xm 0.5 0.5 0.5 z 0 0

o.o~--~~~~-~--------'-------~------~----~ 0.0 1.0 2.0 3.0

TIME (PV)

Fig. 12-The effects of flowing fraction on effluent profiles.

f

z 0 I-« a: I-z w () z 0 ()

z 0 I-« a: I-z w () z 0 ()

1.0

0.5

0.0 0.0

1

3

1 2 3

0.7 0.7 0.7

Xk 10 100 300

Xm 0.5 0.5 0.5

1.0 2.0 3.0

TIME (PV)

Fig. 13-The effects of Bodenstein number on effluent profiles.

1.0.------.-------,------1I~----~------._----_.

1 2 3 0.5 1 0.7 0.7 0.7

Xk 100 100 100 Xm .05 .5 5

0.0 '--____ -""''"'''''''--____ ---' ______ ---' ______ --'-______ --'-_____ --1

0.0 1.0 2.0 3.0

TIME (PV)

Fig. 14-The effects of mass transfer coefficient on effluent profiles.

65r---------~~-----.~~---.-------, 100.--r--._--~--~._~------~----~

60

50

38L-----------'---------'-------~------~ 5 20 30 40 50

Fig. 15-Model fit error contours (exact solution: f", 0.5, Xk ~ 25 and Xm =1).

f

90

80

70

a a ~ a C\J

20 40 50

Fig. 16-Model fit error contours (exact solution: f=0.96, Xk 25 and Xm = 1).

[) 10-1 w CJ)

:::E o o CJ) -10-2 I-z W

U u:: LL W

8 10-3

z o U5 a: w a.. ~ 10-4

10Q%BRINE SATURATION DISPERSION COEFFICIENTS

10-5~------~--------~------~------~ 10-4 10-3 10-2 10-1

en z 0 i= () « a: u. W C/)

« J: a..

I NTERSTITIAL VELOCITY (CM ISEC)

Fig. 17-The effect of interstitial velocity on longitudinal dispersion coefficient.

0.5

OIL PHASE

IMBIBITION

FLOWING

o ~ ______________ L-~ ______________ ~

o 50 100

OIL SATURATION (0/0)

Fig. 19-Phase fractions for oil phase-imbibition.

C/)

Z 0 i= () « a: LL

W C/)

« I a..

en z 0 i= () « a: LL

W en « J: a..

0.5

OIL PHASE

FLOWING PRIMARY DRAINAGE

o L-______ ~ ________ ~ ________________ ~

o

0.5

50

OIL SATURATION (0/0)

Fig. 18-Phase fractions for oil phase-primary drainage.

OIL PHASE

SECONDARY FLOWING DRAINAGE

100

o ~ ______________ -LL-________________ ~

a 50 100 OIL SATURATION (0/0)

Fig. 20-Phase fractions for oil phase~secondary drainage.

11'\ J ~NDRITIC

(J) z o ~ () « a: 0.5 I­

U.

w (J) « :c a..

\11-- --

FLOWING

I

-

BRINE PHASE

PRIMARY DRAINAGE

-

a ~----------~~--~I--------------~ o 50 100

(J) z o i= () « a: 0.5 u. w en 4: :c a..

OIL SATURATION (0/0)

Fig. 21-Phase fractions for brine phase-primary drainage.

FLOWING

BRINE PHASE

SECONDARY DRAINAGE

o ~----------------~--------------~ o 50 100 OIL SATURATION (%)

Fig. 23-Phase fractions for brine phase-secondary drainage.

(J) Z o ~ () 4: a: 0.5 I­U.

-

W en 4: :c a..

6 UJ en

FLOWING

BRINE PHASE

IMBIBITION

a ~ ________________ ~I~'L-______________ ~

o 50 100 OIL SATURATION (%)

Fig. 22-Phase fractions for brine phase-imbibition.

10~------~--------~------~------~

NONWETTING PHASE DISPERSION COEFFICIENTS

• • • •

•

• •

•

. " .,. • • • •

• PRIMARY DRAINAGE

• IMBIBITION

• SECONDARY DRAINAGE

10-5~-------L ________ L-______ ~ ______ ~

10-4 10-3 10-2 10-1 1 FLOWING FRACTION INTERSTITIAL VELOCITY (CM I SEC)

Fig. 24-Effects of flowing fraction interstitial velocity on the oil phase dispersion coefficient.

:::;E ()

a CJ)

;=- 10-2 z L.U

U u::: U. L.U

8 10-3 z o U5 a: L.U a..

5 10-4 •

WETTING PHASE DISPERSION COEFFICIENTS

• •

• • •

• •

• PRIMARY DRAINAGE

• IMBIBITION

• SECONDARY DRAINAGE

10-5 L-______ ~ ________ ~ ______ ~ ______ ~

10-4 10-3 10-2 10-1 1 FLOWING FRACTION INTERSTITIAL VELOCITY (CM I SEC)

Fig. 25-Effects of flowing fraction interstitial velOCity on the brine phase dispersion coefficient.

--> -0 >11 o 0 en en --~ ~

100 r---------r---------,

10

IMBIBITION AND SECONDARY DRAINAGE

• PRIMARY DRAINAGE

• IMBIBITION • SECONDARY DRAINAGE

0.1 L...--_____ ..L....-_________ ----J

o 50 100

OIL SATURATION (%)

T'"" I

E .£

:2:1:;;-

1000 r------------.,------------.

100 \ '\ .~

--- ~ IMBIBITION AND '+-

> -- 6 ~ SECONDARY -> 10 DRAINAGE

0 0 • fQ fQ

\ ~ ~

PRIMARY ., DRAINAGE

• PRIMARY DRAINAGE • IMBIBITION

• SECONDARY DRAINAGE

0.1 0 50 100

OIL SATURATION (%)

Fig. 26-The effects of saturation on the oil phase dispersion coefficient ratio.

8

• 6 NONWETTING PHASE

• •

4 • • •• 2

o ~----------------~------------------~ o 50 100

OIL SATURATION (%)

Fig. 27-The effects of saturation on the brine phase disper­sion coefficient ratio.

Fig. 28-The effects of saturation on the oil phase mass transfer coefficient.

z o i= <t: a: ~ 0.5 W o Z o o

z o i= <t: a:

1,0 .------.-------.------~------_.------~------~

I-Z 0,5 W o Z o o

z o i= <t: a:

0.0 L..-__ ...LJ..L-..oiL-_--'-___ -L-___ "--__ --'-___ --.J

0,0 1 .0 2,0 3.0

TIME (PV)

Fig. 29-Wetting phase effluent profiles showing the cause of early breakthrough.

1.0 ,.-----,------,-----:......-:---,----,-------,-------,

~ 0,5 W o Z o o

0.0 0.0 1.0

TIME (PV) 2,0 3.0

Fig. 30-Proposed wetting phase effluent profiles for very high fractional oil flow.

z 0 i= <t: a: I- 0.5 Z W 0 Z 0 0

0.0 L..-__ --.&. ........... ~ ____ --..l. ______ __'_ ______ -.L ______ __'__ ____ ~ 0.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0

TIME (PV) TIME (PV)

Fig. 31a-Effluent profiles for oil phase in the oil wet core. Fig. 31 b-Effluent profiles for brine phase in the oil wet core.

3.0