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International Journal of Multiphase Flow 28 (2002) 1659–1686

www.elsevier.com/locate/ijmul?ow

Modeling foamy oil ?ow in porous media

D.D. Joseph

a

a,*

, A.M. Kamp b, R. Bai

a

University of Minnesota, Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, 110 Union Street. S.E., Minneapolis, MN 55455, USA b PDVSA Intevep, P.O. Box 76343, Caracas 1070-A, Venezuela Received 9 February 2001; received in revised form 19 June 2002

Abstract Certain heavy oils which foam under severe depressurization give rise to increased recovery factor and an increased rate of production under solution gas drive. These oils not only stabilize foam, but also stabilize dispersion of gas bubbles at lower volume ratios. The way this phenomenon is related to the chemistry of the oil and its viscosity is presently not understood. We present here a mathematical model of reservoir ?ow of foamy oil which depends only on the velocity through Darcy?s law, the pressure and the dispersed gas fraction. The theory governs only in situations in which the bubbles do not coalesce to produce the percolation of free gas. In this theory the bubbles move with the oil as they evolve. The main empirical content of the theory enters through the derivation of solubility isotherms which can be obtained from PVT data; modeling of nucleation, coalescence, bubble drag laws and transfer functions are avoided. The local pressure di?erence and dispersed gas fraction are in equilibrium on the solubility isotherm. In a pressure drawdown the time taken for the system to return to equilibrium is described by a rate law characterized by an empirical relaxation time (rate constant). The resulting systems of equations can be reduced to a coupled pair of non-linear PDEs for the dispersed gas fraction and pressure di?erence, which can further be reduced in the equilibrium case to a second order evolution equation for the pressure di?erence. This system of equations can also be derived from usual theory of two-phase ?ow in a porous media based on relative permeability under the assumption that the bubbles and oil move in lock step. We propose a reformulation of the conventional theory in which the concept relative permeability of the porous media is replaced with the more familiar concept of an e?ective phase viscosity. The equations of our relaxation theory are solved numerically, and the mixture viscosity function and relaxation time are selected to match the sandpack experiments. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Porous media ?ow; Foamy oil; Foam ?ow

*

Corresponding author. Tel.: +1-612-625-0309; fax: +1-612-626-1558. E-mail address: joseph@aem.umn.edu (D.D. Joseph).

0301-9322/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 9 3 2 2 ( 0 2 ) 0 0 0 5 1 - 4

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1. Introduction In this paper a model is presented that is motivated by the need to explain anomalous features associated with production from reservoirs of so-called foamy oils. These oils are described by some of their properties of response to pressure declines; it is noted that they nucleate dispersed gas bubbles and display obvious foaminess in well-head samples produced by solution gas drive in which oil and gas are produced by the drawdown of pressure (Huerta et al., 1996; De Mirabal et al., 1996). When compared with the response of conventional oils, the response of foamy oils to drawdown of pressure is more favorable; primary recovery factor (percentage of the oil in the reservoir which can be recovered), the rate of production, the volume ratio of oil to gas which is recovered and the length of time that a given pressure gradient or rate of production can be maintained are all increased substantially; the reasons for the favorable response of foamy oils in solution gas drive are not well understood and tentative explanations which have been put forward are controversial (see Maini, 1996; Pooladi-Darvish and Firoozabadi, 1997; Sheng et al., 1999a for recent reviews). Foamy oils carry considerable amounts of dissolved gases in the condensed state. The relevant thermodynamic property for this is ‘‘gas solubility’’; a function of temperature and pressure at equilibrium which gives the volume ratio of dispersed gas from the crude oil by outgassing. Tables of solubility of methane, carbon dioxide and other gases in various Canadian crude oils have been given by Svrcek and Mehrotra (1982), Peng et al. (1991) and others. The oils considered to be foamy evidently cavitate small dispersed bubbles which, under some conditions, are believed to move with the crude oil in which they are dispersed. Experiments done by Pooladi-Darvish and Firoozabadi (1997) have shown that bubbles which arise from depressurization of silicone oil and heavy crude of equivalent viscosity are very di?erent; the bubbles in the silicone oil are larger and much more mobile than the ones in crude oil. Viscosity alone might not be enough to demobilize dispersed gas; it is desirable to look at other properties like surface tension and surface active agents. Possibly there are surface active agents which are present naturally in foamy crudes which allow them to stabilize foam, but the precise agents, their composition and the mechanics by which they are released apparently have not been studied. A ‘‘foamy oil’’ is a heavy oil which foams under rapid depressurization. Maini (1996) notes that ‘‘. . . the term foamy oil is often used to describe certain oils produced by solution gas drive which display obvious foaminess in well-head samples. The primary production of heavy oil from several reservoirs in western Canada is in the form of an oil continuous foam. This foam resembles a chocolate mousse in appearance and often persists in open vessels for several hours . . .’’ Foaming at a well head is a kind of foam formation analogous to the head on beer. To get such a head the pressure decline must be su?ciently severe to allow the bubbles which rise from outgassing of condensed gas in the bulk to accumulate at the free surface faster than they collapse. Perhaps there are surfactants in foamy oil which stabilize the ?lms between the bubbles preventing collapse, promoting the buildup of the head. Foam stability measurements in the laboratory have shown that the foaminess of crude heavy oils is comparable to aqueous foams used for steam ?ooding applications (Sheng et al., 1996). The outgassing of condensed gas will not lead to foam at the well head if the rate of depressurization is too low.

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To create foam in a reservoir or in a sandpack it is necessary to depressurize rapidly enough to produce close packed solution gas bubbles which can undergo a topological phase change to stable ?lms and plateau borders. This kind of in situ foaming of sandpacks has been achieved in the experiments of Maini and Sarma (1994). Gas and liquid move in lock step in these foams and lead to very high primary recovery factors. At the actual reservoir, gas fractions can be as low as 5–10%. At these gas fractions the oils cannot foam but are well dispersed and protected against coalescence, possibly by the same natural surfactants that might stabilize the foams. In solution gas drive of foamy oil the depressurization of the sample leads to cavitation of small dispersed bubbles. The volume ratio of dispersed gas increases the volume of our composite ?uid and it acts as a pump, gas coming out of solution pumps the ?uid outward. This pumping action is well described by the continuity equation (4.4) which implies that in a closed volume v with boundary S containing dispersed bubbles of volume fraction / I Z 1 D/ dv ? u ? n dS; ?1:1? v 1 ? / Dt S where n is the outward normal on S and u is the velocity of our composite ?uid. The relative velocity of dispersed gas is important; if the bubbles coalesce and move relative to the oil more gas and less oil will be produced. Good recovery is sometimes described by a critical gas saturation value; this is the volume fraction of gas at which the gas becomes connected and starts to move faster than the oil. Maini (1996) identi?es this critical saturation as a percolation limit, whilst Firoozabadi et al. (1992) and Pooladi-Darvish and Firoozabadi (1997) identify this even by visual observation of bubbles in a viewing window. The values given by Firoozabadi et al. are about ?ve times smaller than those given by Maini and his coworkers. Mastmann et al. (2001) wrote a paper ‘‘Predicting foamy oil recovery’’ which they applied to the Driza crude oils found in a large Albanian oil ?eld. This oil gives rise to huge amounts of stable foam under rapid depressurization. They say The basic mechanism of foamy oil behavior is related to the existence of what is commonly termed a ‘‘pseudo’’-bubble point. The true bubble point is the pressure at which, at the reservoir temperature, the ?rst small bubbles of free gas evolve from solution in the oil and nucleate as a distinct free gas phase. For most conventional oils having relatively low viscosity, the gas rapidly coalesces into large bubbles and evolves almost immediately from the oil to form a separate and distinct gas phase. In many reservoirs, this forms a secondary gascap during depletion operations and results in high gas–oil ratios at production wells due to the high mobility of a free gas phase in the porous media in comparison to the more viscous oil. For this reason, for conventional or non-foamy oils, the bubble point and the pseudo-bubble point have the same or very close to the same values. For foamy oils, due to the high viscosity of the oil, the gas bubbles cannot immediately coalesce together to form bubbles large enough to allow gravitational forces to separate from the oil. For this reason, the oil phase remains as a continuous dispersed gas–oil emulsion with a higher and higher concentration of increasingly larger bubbles trapped in a milkshake-like format within the oil as the pressure is reduced. The point for the foamy oil at which the bubbles of free gas can ?nally start to escape from solution as a distinct free gas phase is known as the pseudo-bubble point.

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Their use of the term ‘‘pseudo-bubble point’’ is slightly unconventional. We prefer to refer to this condition as a ‘‘percolation threshold’’. This threshold is a ?uid mechanics, possibly rheological property of the oil and the producing reservoir or sandpack, and not a thermodynamic property; it could depend on time. Under very rapid depressurization in a sandpack or at the well head in the reservoir, the gas is nucleated so rapidly that it closes packs and, in a foamy oil, forms a foam without gas percolation. In a reservoir, after long time evolution of gas the mechanics of coalescence is such as to lead to percolation. When the gas percolates, the good news about recovery is over; it is no wonder that all authors ?nd that the critical saturation values are about the same as the primary recovery factors (which is the fraction of oil recovered by solution gas to oil in the reservoir) even when they disagree about de?nitions. A few models of foamy oil ?ow have been put forward; each emphasize some special feature. One of the most recent models by Sheng et al. (1996, 1999a) are multiphase theories based on conservation laws with transfer from solution gas to dispersed gas and from dispersed gas to free gas. Their theory does not seem to follow the curve of experimental values. Sheng et al. (1996) say that Published models include the psuedo-bubble point model (Kraus et al., 1993), the modi?ed fractional ?ow model (Lebel, 1994) and the reduced oil viscosity model (Claridge and Prats, 1995). Maini (1996) gave a detailed review and discussion of some of these models. These models have been used to history match heavy oil production, but their common weakness is that the dynamic processes which are important features of foamy oil ?ow were not included properly. Although it may be possible to get an acceptable history match using these models, the predictive ability is likely to be limited . . . A more successful approach to modeling was recently advanced by Sheng et al. (1999b). This model also requires the modeling of nucleation, bubble growth and disengagement of gas bubbles from the oil. The model ultimately leaves undetermined two adjustable parameters which ?t the theory to experimental data better than previous models. To predict foamy oil behavior, Mastmann et al. (2001) use ‘‘Eclipse’’ which is a commercial ‘‘black oil’’ simulator. The model equations at the foundation of this simulator are the conventional multiphase ones based on the concept of relative permeability (see Section 2) which is not well suited to the description of a homogeneous mixture in which the oil and gas move in lock step. Obviously, in such a state the molecular viscosity of the gas is not important. The black oil model allows for gas evolution through solubility tables and not through an equation of state like our (2.14). The present theory could be called a continuum mixture theory which is appropriate for foamy oil ?ow with dispersed gas of low mobility relative to the liquid and leads to three coupled nonlinear partial di?erential equations for u, p and the gas fraction /, ?ve scalar equations in ?ve unknowns. Our model has a few features in common with the excellent early work of Leibenson (1941) on the motion of gas saturated ?uid in a porous media. The model proposed here does not require information about nucleation, bubble growth, compressibility or forces which produce relative velocity. We put up a one-phase or mixture theory in which the dispersed gas is described by a gas fraction ?eld in a single ?uid in which the viscosity, density and mobility in Darcy?s law all depend on the gas fraction. This ?uid satis?es the

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usual Darcy law, and the continuity equation together with a kinetic (constitutive) equation required by the condensation and outgassing of methane (or other gases) in heavy crude. The theory depends only on parameters which can be measured in a PVT cell and sandpack. The virtue of the model is simplicity, but it can work only for relatively immobile dispersed gas bubbles in which divergence-free velocities are excluded (see the discussion following (4.4)). Certainly such a theory could not be expected to give rise to a percolation threshold or even to a critical gas fraction. We shall show that it can describe many features of solution gas drive of foamy oils in the regimes when the bubbles in the mixture are dispersed and even when they are trapped in foam. The assumption that bubbles move in lock step with the oil could not hold when the diameter of a large number of the nucleated bubbles grow larger than a typical pore diameter. Nor would our theory work if the diameter of many of the bubbles were so small as to produce a sensible e?ect on the mixture pressure. Under the high pressure conditions of a typical reservoir one atmosphere is very small and even a 1 lm bubble with a tension of 30 dynes/cm would produce a capillary pressure of six atmospheres. Our theory then assumes a large fraction of small bubbles but is not restricted to small gas fractions. It is our idea that the increased recovery and production are generated by the pumping of nucleating and growing gas bubbles embodied in (1.1). However recovery factors and production rates are not the same and we might test some ideas: if two foamy oils have the same viscosity, the one with higher solubility having more gas to release will have higher primary recovery and production rate; if two foamy oils have the same solubility, the one with lower viscosity will have a higher rate of production because the mixture is more mobile, but a lower primary recovery because the percolation threshold is lower. If the oil foams in situ, oil and gas move in lock step and the primary recovery factor increases while due to the increased viscosity of foam the rate of production could decrease.

2. Equation of state for dispersed gas Solubility isotherms are an equation of state for foamy oil in which all of the gas not in solution is dispersed in small bubbles. Free connected gas is not allowed. When the pressure is dropped, the mass of dissolved gas released goes entirely into dispersed gas bubbles. We have obtained solubility isotherms from measurement in PVT cells and applied the results to characterize the outgassing and absorbing of gasses in foamy oil in porous media. We have adopted the view that foamy oil will have the same equation of state in a porous media as a PVT cell; the thermodynamics of the ?uid are independent of the wall of the containing solid provided that these walls are not so closely spaced as to e?ect the thermodynamic properties of the bulk ?uid. In the experiments of Svrcek and Mehrotra (1982) the pressure is dropped from p and T to pref and Tref where in the experiments pref is atmospheric and Tref ? 373:2 K. It is assumed that all the gas in the live oil at p; T comes out. De?ning now: VG ?p; T ? is the volume of dispersed gas, VL ?p; T ? is the volume of live oil, V ? ?p; T ? is the volume of dispersed gas which evolves from the condensed gas when p; T are dropped to pref , Tref .

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In this model we avoid all constitutive equations regarding nucleation rates and bubble growth. In our model we have only a mixture of liquid and dispersed gas, and the dispersed gas enters only through the volume fraction VG : ?2:1? /? VL ? VG Eq. (2.1) may be solved for VG ? /VL : 1?/ ?2:2?

Svrcek and Mehrotra (1982) give volumetric solubility curves (CO2 and methane, Figs. 8 and 12, p. 34 and 36). Their results show that b V ? V ? =VL ?p; T ? ?2:3?

is a linear function of pressure, with a larger slope at lower temperatures. We can assume that this tells you how much dispersed gas can come out of a solution of dissolved gas in oil which is at a b saturation value at any pressure and temperature. We are going to assume that this V determines the dispersed gas fraction / following an argument put forward in what is to follow. The data of Svrcek and Mehrotra (1982) show that ^ b ?2:4? p ? pref ? c?T ? V ; b ^ where c ? dp=d V is the slope of the solubility isotherms shown in Figs. 8 and 12. This slope is approximately constant. Here we have chosen pref as a small pressure at which a negligible amount of gas is dissolved in the oil. For practical purposes this could be standard (atmospheric) pressure. In the present approach we have no way to predict the size distribution of gas bubbles. This means b that we are free to choose the size and distribution to measure V and the most convenient choice is when all the released gas is collected at the top of a PVT such as in the experiment of Svrcek and Mehrotra. We ?rst suppose that all the gas which comes out of a solution is dispersed and does not percolate or foam. We must suppose that the nucleation, growth and compressibility of gas bubbles are working, but these microstructural features are not monitored in these experiments b which give only the solubility V . This is also what we do in the mathematical model. To convert (2.4) into a relation between p and / at equilibrium we note that the total mass M of gas in the live oil is invariant, independent of p and T and M ? MG ?p; T ? ? Mc ?p; T ?; ?2:5?

where MG ?p; T ? is the mass of dispersed gas; Mc ?p; T ?, the mass of condensed gas. Since the mass of condensed gas does not change when it is vaporized and assuming that this vapor is a perfect gas, we have Mc ? pref V ? =RTref ; where R is the gas constant. From the same gas law MG ? pVG =RT : ?2:7? ?2:6?

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Hence, from (2.5)–(2.7) we have M? pVG pref V ? ? RT RTref ?2:8?

and, using (2.3) and (2.4) pVG pref p ? pref M? ? VL : ^ RT RTref c

?2:9?

~ We next introduce the bubble point pressure p as the pressure at which there is no dispersed gas, e def b all the gas is dissolved in the live oil so that VG ? 0; V ? V when VG ? 0 and from (2.4) ^ e ~ p ? pref ? c?T ? V ; where e p V ?~; T ? ? V ? ?~; T ?=VL ?~; T ?: p p Since M is invariant, we may evaluate (2.8) at the bubble point pref e p M? VL ?~; T ? V ?~; T ?: p RTref e Using (2.10) to eliminate V in (2.12) and equating (2.12) and (2.9) we get ! ~ Tref p p ? pref p ? pref VL ?~; T ?: VG ? p VL ?p; T ? ? ^ ^ T pref c c ?2:11? ?2:10?

?2:12?

?2:13?

In most depressurization experiments the change of liquid volume due to outgassing and compressibility is small and VL ?p; T ? % VL ?~; T ?. In this case the terms proportional to pref in (2.13) p subtract out and after replacing VG with /VL =?1 ? /? we get b where b? ^ Tref c T pref ?2:15? ~ / p?p ? ; 1?/ p ?2:14?

is completely determined by the solubility isotherm. Since the variation of T is small on an absolute scale, the values T =Tref are just slightly larger than one. b e The variable V is called gas–oil ratio and V is the gas–oil ratio at saturation pressure. From (2.10): ^ c?T ? ? ~ p ? pref : e V ?2:16?

^ Substituting this value of c in (2.15) b? ~ Tref p ? pref : e T pref V ?2:17?

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~ In most practical situations p ) pref so that from (2.17) b? ~ Tref p : e T pref V ?2:18?

~ It is also customary in the oil industry to characterize live oil by its saturation pressure p and its e gas–oil ratio V at saturation pressure. By virtue of (2.18) one can calculate the solubility parameter b. Note that in most experiments pref is chosen as atmospheric pressure and Tref as 60 °F ? 15:6 °C. Values for b for two Canadian heavy oils, Lloydminster and Lindbergh (Maini and Sarma, 1994) and for a Venezuelan heavy oil, Cerro Negro, are given in Table 1. Graphs of the isotherm (2.14) for various values of b are shown in Fig. 1. It can be noted that the solubility value for heavy crude oils from very di?erent regions are very close, which indicated that they contain similar amounts of dissolved gases at the same pressures. The oil industry calls oil with dissolved gas ‘‘live oil’’.

Table 1 Solubility coe?cients for some heavy oils Oil Lloydminster Lindbergh Cerro Negro b 3.40 3.17 3.53 T (K) 293 293 327

Fig. 1. Graphs of the solubility isotherm (2.14) for various values of b. The limits of validity of the dispersed gas model can be roughly set at a close packing value, say /c ? 0:68. When / > /c some bubbles must touch and form foam ?lm or to coalesce. This implies that results for drawdown greater than those for which / ? 0:68 must take into account foaming and ?ngering of free gas.

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When / and p satisfying (2.14) vary from point to point rp ? ?bp2

2 ~ p?1 ? /?

r/:

?2:19?

According to Darcy?s law, u ? ?krp where k is the mobility of the foamy mixture in the porous media; hence the ?uid ?ows up the bubble gradient toward regions in which there are more bubbles where the pressure is smaller. Departures from the equilibrium solubility relation (2.14) are indicated as supersaturation or subsaturation. De?ning the function ~ f ?p; /? ? p ? p ? bp/=?1 ? /?;

def

?2:20?

supersaturation (f > 0) corresponds to having more gas dissolved than there should be under thermodynamic equilibrium; subsaturation (f < 0) corresponds to having less gas dissolved than there should be under equilibrium. Supersaturation occurs when the pressure in the reservoir is drawn down, but the oil cannot evolve gas fast enough to keep up with the depressurization. Subsaturation, on the other hand, occurs when there is not enough gas available to dissolve in order to satisfy thermodynamic equilibrium at prevailing reservoir pressure and temperature. The function f in our theory is thus an indicator for departure from equilibrium solubility. The equilibrium isotherm f ? 0 allows for dispersed gas fraction / from zero to one; from all liquid to superdry foam. Our theory makes no distinction between bubbly mixtures, wet or dry foams. Consider a drawdown to atmospheric pressure pa ? 106 dynes/cm2 from a saturation pressure ~ p ? 4:83 ? 107 dynes/cm2 as in the experiment of Maini and Sarma (1994). Using b ? 3.4 for Lloydminster (from Table 1) and evaluating the gas fraction of / ? /a at atmospheric pressure we ?nd /a ? 0:93: ?2:21? This is too much gas to exist as a bubbly dispersion; either the bubbly mixture passed into foam near to some critical value, say / ? /c ? 0:680 ?2:22? or close packing or else some of the bubbles coalesced as free gas which ?ngers out of the sandpack. Probably foam and free gas both evolve at the outlet of a closed sandpack after a sudden drawdown to atmospheric from high saturation. We also note that the physics of supersaturation is complicated by the fact that bubbles cannot nucleate continuously and that the mechanism by which they nucleate is still not well understood. A bubble which might form in crude oil by the vaporization of dissolved gas at supersaturated conditions can be expected to satisfy Laplace?s law pvapor ? p ? 2c=R ?2:23? where c is surface tension (say 30 dynes/cm) and R is the bubble radius. Under mildly supersaturated conditions p is slightly smaller than the vapor pressure pvapor ; hence, R satisfying (2.23) cannot be very small. In deriving (2.14) we have assumed the continuity of pressure across the bubble surface, ignoring the small pressure drop implied by (2.23).

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It is argued that the vaporization of dissolved gas under supercritical conditions requires the simultaneous presence of undissolved gas hidden in crevices of impurities which are wet by gas preferentially. The curvature of the gas–oil interface in such a crevice is opposite to a bubble and it is controlled by capillarity rather than interfacial tension. The supersaturated dissolved gas vaporizes at the undissolved gas hidden in the crevice and the volume of the gas grows there until a bubble breaks away restoring the nucleation site to its original condition. The train of gas bubbles which emanate usually from a single site on a glass of beer as the gas comes out of the solution is a convenient example of outgassing at a nucleation site. The pore walls in a porous media are nucleation sites for outgassing of foamy oil. The possibility that asphaltenes in the oil are nucleation sites for dissolved gas is a current but unresolved question.

3. Live oil and dead oil Oil without dissolved gas is called dead oil. Oil saturated with dissolved gas is called live oil. The dissolved gas can be considered condensed and live oil is a mixture of miscible liquids, dead crude oil and condensed gas. The mixture of miscible liquids is like glycerin and water with the caveat that the oil and dissolved gas do not mix in all proportions; the fraction of dissolved gas at saturation is a function of temperature and pressure. The weight fraction of dissolved gases range from 0.3% to 6% (see the tables corresponding to Figs. 8 and 12 in Svrcek and Mehrotra (1982)). The viscosity of live oil can be orders of magnitude smaller than the viscosity of dead oil; the viscosity of live oil is a strongly decreasing function of the amount of dissolved gas in solution just as the viscosity of glycerol strongly decreases with the water fraction. In a pressure decline the viscosity of the live oil will increase because less gas is dissolved and because the presence of dispersed gas should increase the viscosity of the composite ?uid. We may seek to answer the question ‘‘what is the density of the dissolved gas in solution’’. We are not able to measure the density of the dissolved gas in oil, but the density q of the mixture is available in data presented by Svrcek and Mehrotra (1982, see Figs. 6 and 11, p. 34 and 36). Let us note that this data shows that the density of the CO2 in bitumen is a weak function of the pressure; hence the density of the non-saturated bitumen is independent of the volume ratio W of soluble gas in bitumen. Dissolved methane has a density only slightly di?erent than bitumen (see their Figs. 8 and 12). A theory of miscible mixtures which applies to live oil can be found in Chapter X of Joseph and Renardy (1992).

4. Model description In this model we avoid all constitutive equations regarding nucleation rates and bubble growth. In our model we have only foamy oil and dispersed gas and the dispersed gas enters only through its volume ratio /. The model combines Darcy?s law, with a / dependent mobility, a mass conservation law for ideal mixtures together with a constitutive equation governing the evolution of departures from equilibrium solubility. For ?ow not in equilibrium gradients and time derivatives are crucial. The time derivatives which are used here have a material derivative

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D o ? a ? u ? r; Dt ot where a is the porosity. The continuity equation is given by dq D/ ? q?/? div u ? 0; d/ Dt where q?/? ? qG / ? qL ?1 ? /? % qL ?1 ? /?

?4:1?

?4:2?

?4:3?

because qG ( qL where qL is the density of live oil which depends only weakly on the volume ratio of dissolved gas. Combining (4.3) and (4.2) we ?nd that D log?1 ? /? ? div u ? 0: ?4:4? Dt Eq. (4.4) restricts the theory to dispersions of low mobility relative to the suspending liquid. In any motion u?x; t? of the composite which is divergence free div u ? 0, the dispersed gas fraction does not change following the motion of the ?uid D/ ow ? ? u ? Dw ? 0: Dt ot Bubbles rising under gravity would lead to divergence free motions as would any motion of the bubbly mixture in which dispersed bubbles do not nucleate, di?use or compress. Motions with non-zero divergence satisfy (1.1); the ?ux out of any closed volume, over which the div u does not sum to zero, must be non-zero. This is the simpli?ed way that our theory accounts for nucleation and di?usion. Turning next to Darcy?s law, we let x increase in the direction of gravity. Then u ? ?kfrp ? qgex g % ?kfrp ? qL g?1 ? /?ex g; where k?/? ? k=l?/? ?4:6? ?4:5?

is the mobility, l?/? is the viscosity of live oil with dispersed gas of volume ratio / and k?/? is the permeability. The viscosity l?/? increases with / for two reasons: dispersed gas bubbles and the reduction of live oil viscosity due to outgassing of light components. 4.1. Relative permeability Eqs. (4.2), (4.5) and (4.6) are implied by the usual theory for two-phase ?ow through porous media based on relative permeabilities under the assumption that the average liquid and gas velocities are equal. The demonstration of this fact leads also to a formula (4.3) for ratio of the relative permeabilities under the same assumption. The composite volume average velocity is given by u ? UL ? UG ; ?4:7?

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where UL ? ?1 ? /?uL ; UG ? /uG ?4:8? are super?cial velocities (volume ?ow rate over total area) and UL is the average oil velocity and UG is the average gas velocity. The mixture density q?/? is given by (4.3) and the composite mass average velocity is given by ?q ?1 ? /?uL ? qG /uG ? : ?4:9? um ? L q?/? The equations expressing the conservation of mass of each of the two phases are oqL ?1 ? /? ? r ? qL ?1 ? /?uL ? 0; ot oq / a G ? r ? qG /uG ? 0: ot When uL ? uG , the average gas and liquid velocities are equal and from (4.7)–(4.9) a um ? u ? uL ? uG : ?4:10? ?4:11?

?4:12?

After adding (4.10) and (4.11), using (4.12) we get Eq. (4.2). We now introduce relative permeability and show that we may replace the concept of relative permeability with an equivalent concept of e?ective viscosity, which is a more familiar concept in the general theory of mixtures. Neglecting gravity and capillarity we have k krL k rp; ?4:13? rp ? ? ?1 ? /?uL ? UL ? ? lL ?/? gL k krG k /uG ? UG ? ? rp ? ? rp; ?4:14? lG ?/? gG where k is the permeability which we will identify as the k in (4.6), krL and krG are relative permeabilities, gL and gG are molecular viscosities and def g def g ?4:15? lL ? L ; lG ? G krL krG are e?ective viscosities. The introduction of e?ective viscosities is nothing more than a de?nition; however, the de?nition implies that the di?erential ?ow resistance can be associated with two ?uid phases and not with the porous media. Noting now that u in (4.5) is the composite velocity, we may write (4.5) and (4.6) neglecting gravity as k rp ?4:16? u?? l?/? and using (4.7) together with (4.13) and (4.14), we get 1 1 1 ? ? : l?/? lL ?/? lG ?/?

?4:17?

Eq. (4.17) and (4.15) relate the e?ective viscosity of the gas–liquid mixture to the relative permeabilities.

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The consequences of our basic assumption that the dispersed gas moves with the liquid u ? uG ? uL may be extracted from the above equations. We ?nd that 1 1?/ ? lL ?/? l?/? and 1 / : ? lG ?/? l?/? ?4:19? ?4:18?

The e?ective gas and liquid viscosities are completely determined by the e?ective viscosity of the mixture when u ? uG ? uL . Moreover, the ratio of relative permeabilities is krL gL lG ?/? 1 ? / gL ? ? / gG krG gG lL ?/? is determined by / and the ratio of molecular viscosities. When uG 6? uL , we may introduce v ? uL =uG and then krL gL lG ?/? ?1 ? /?gL v ? ? : /gG krG gG lL ?/? The ratio v of local velocities is related to the ratio of super?cial velocities by v? UL / UG ?1 ? /? ?4:23? ?4:22? ?4:21? ?4:20?

v is a useful and easily computed parameter for ?ow diagnosis. When v ? 1, the ?ow is homogeneous. When v ? 0, the ?ow is all gas and the gas has percolated. When v 6? 1, the homogeneous theory leading to the continuity equation (4.2) does not apply. Instead, the conservation equations (4.10) and (4.11) for the gas and liquid phase must be used. 4.2. Constitutive equations relating the dispersed gas fraction to the pressure We are proposing models in which the basic variables are the pressure and dispersed gas fraction and are such that disturbed systems which are not forced will relax to equilibrium with pressure and temperature on the solubility isotherm. f ?p; /? ? 0 given by (2.14). There are many possible ways to build models with the above properties. The simplest conceptual model with the desired properties is a ?rst evolution model Df ? ?f ; Dt which guarantees that a disturbed stationary system will relax exponentially. The implied assumption that both factors in the equilibrium isotherm relax at exactly the same rate is probably overly restrictive. s

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The constitutive equation ! ~ ~ D / D p?p b/ p?p ? ? s2 s1 ? Dt 1 ? / Dt 1?/ p p

?4:24?

allows for separate rates of relaxation. Obviously this equation is very closely tied to the equilibrium isotherm. To evaluate the possibility that a simple model based on (4.24), involving only the pressure and dispersed gas fraction, can describe solution gas drive of foamy oil, we must integrate the equations and apply the results to experiments and ?eld operations. In this work, we apply our model to the experiments of Maini and Sarma (1994). The model requires that we identify a viscosity function g?/? of the dispersed gas fraction and two relaxation times s1 and s2 . In our ?rst attempt at evaluation we put s1 ? 0; in this case (4.24) reduces to a Maxwell-like constitutive equation with pressure relaxation leading a hyperbolic system of PDEs that support wave propagation of pressure and bubble fronts. This system could not be made to simultaneously match experiments on steady ?ow, the rate of production of oil in the same steady ?ow, and the cumulative production in a blowdown experiment. We get a better match when s2 ? 0 with s1 6? 0 and we are able to select a viscosity function g?/? ? ?1 ? /??11 and s1 to match all the sandpack experiments, but the match of model to drawdown experiments (Figs. 10 and 11) are less good. We speculate that the mismatch is due to percolation of free gas from the sandpack due to the massive pressure decline in blowdown that must, at best, lead to the copious formation of relatively dry foam. Obviously we may get a better agreement with the model by ?tting two relaxation parameters (s1 and s2 ) rather than one; in this case we might expect to model short time phenomenon associated with nucleation. We prefer to proceed in the future by keeping ?tting to a minimum and to look carefully at ?eld data, not only the experiment of Maini and Sarma. In this work we con?ne our attention to the cases in which all the gas that comes out of a solution is dispersed in gas bubbles. In a pressure drawdown the gas bubbles nucleate and grow until the gas fraction / is in equilibrium with the pressure, f ?p; /? ? 0. We shall describe relaxation to equilibrium with a ?rst order rate equation of the form (4.24) with s2 ? 0; s1 ? s written as sp D/ ? f ?p; /?: ?4:25? 2 ?1 ? /? Dt D=Dt is the usual expression (4.1) for the substantial derivative in a porous media. It is easy to verify that (4.24) and (4.25) give rise to relaxation. If the pressure is dropped from saturation, at ~ which p ? p and / to p1 , the initial response is given by s ~ D?/=1 ? /? p ? pi >0 ? Dt pi ?4:26?

to that the dispersed gas will begin to increase toward the value of / on the isotherm f ?pi ; /? ? 0. After eliminating u in (4.4) and (4.24) using (4.5), we ?nd that ~ o o / / p?p ? ; ?4:27? ?b s a ? krp ? r ? kqL g?1 ? /? ot ox 1 ? / 1?/ p 1 o/ o/ o/ a ? krp ? r/ ? ?div?krp? ? 2kqL g ? k0 qL g?1 ? /? : ?4:28? 1?/ ot ox ox

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A mathematical theory for the system (4.27) and (4.28) is not yet available. We note that except for div krp the system is of ?rst order, so that we might look at the problem as initial value problem for a second order PDE in p for which end conditions on the pressure and initial conditions on the pressure and gas fraction are appropriate. We will solve these equations for sandpack problems in the sequel. The system (4.24) and (4.5) may be regarded as describing the ?ow of a relaxing compressible ?uid through a porous media. To see this, we replace / with q ? qL ?1 ? /?; using (4.3). Then (4.24), (4.4) and (4.5) may be written as o q ?q q ?q ~ s a ?u?r L ? p ? p ? bp L ; ot q q ?4:29? oq a ? u ? rq ? q div u ? 0; ot u ? ?krp ? kgqex :

5. Constant state solutions 5.1. Constant state solutions, drainage and stability The constant state solution are uniform solutions p0 , /0 ; u0 of (4.27), p0 and /0 satisfying ~ f ?p0 ; /0 ? ? p ? p0 ? bp0 /0 =?1 ? /0 ? ? 0 and u0 ? ex u0 ; u0 ? k?/0 ?qL g?1 ? /0 ?: ?5:2? The constant state solution is a drainage ?ow; this ?ow may be used to determine the mobility k?/?. This solution is stable to small perturbations using normal modes (see Joseph et al., in press). ?5:1?

6. Sandpack experiments Sandpack experiments are used as laboratory surrogates for the ?ow of oil and gas in porous reservoirs. In Fig. 2 a cartoon of a typical sandpack experiment copied from a paper by Sheng et al. (1996) is displayed. The pack may be loaded with sand from reservoirs. Sandpacks provide an excellent way to get precise data under controlled conditions simulating ?ow in reservoirs; they are convenient for mathematical modeling because they lend themselves to one-dimensional treatments. It is useful to look at these one-dimensional models for horizontal sandpacks in which gravity may be neglected and for vertical ?ow in which gravity may be important. In both these cases we have our governing equation (4.28) with r ? ex o=ox. Terms proportional to gravity are put to zero in horizontal sandpacks. Di?erent experiments can be carried out in a sandpack corresponding to di?erent conditions listed below.

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Fig. 2. Cartoon of a sandpack experiment. Live oil can be injected at the inlet and a pressure depletion rate controller is at the outlet. Di?erent experiments are described by prescribed conditions at the inlet and outlet (Sheng et al., 1996). Table 2 Properties of the porous medium Parameter Length (m) Cross-sectional area (m2 ) Sand size (lm) Porosity (fraction) Pore volume (ml) Permeability (lm2 ) Con?ning pressure used (MPa) Value 2.0 16:1 ? 10?4 74–105 0.33 1062 3.35 14.0

Maini and Sarma (1994) reported results of experiments in a horizontal sandpack like that shown in Fig. 2 where pressure at inlet and outlet are controlled. A table of properties of the sandpack is given in Table 2. ‘‘Prior to the start of the ?ow experiments, each oil sample was cleaned of its suspended materials. The oil was then recombined with methane gas in the recombination equipment at a pressure of 4.83 MPa’’. They did steady ?ow experiments using Lloydminster and Lindbergh crude oil in which the inlet pressure was at saturation ~ p ? p ? 4:83 MPa ? 4:83 ? 107 dynes=cm

2

varying the ‘‘drawdown’’ pressure at the outlet. Table 3 gives the properties of the two ‘‘live’’ oils at saturation. The pressure distribution along the core sample which was measured in steady ?ow by Maini and Sarma (1994) is shown in Figs. 3 and 4. Six pressure transducers were placed at intervals along the pack. Each transducer measures the pressure drop between two taps equally spaced along the pack; the pressure drop across two taps is called a ‘‘di?erential pressure’’. The plots given in Figs. 3 and 4 are of straight

D.D. Joseph et al. / International Journal of Multiphase Flow 28 (2002) 1659–1686 Table 3 Properties of live oil at saturation Oil Lloydminster Lindbergh Density (g/cm2 ) 0.968 0.978 Viscosity (Poise) 30.07 39.70 Core average / at maximum drawdown 0.138 0.148 b 3.40 3.17

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Fig. 3. Pressure distributions in steady ?ows of Lindbergh oils at various pressure drawdowns (Maini and Sarma, 1994).

Fig. 4. Pressure distributions in steady ?ow of Lloydminster oils at various pressure drawdowns (Maini and Sarma, 1994).

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line segments between pressure taps. Maini and Sarma (1994) also did ‘‘blowdown’’ experiments in which the inlet was closed and pressure at the outlet suddenly dropped to 0.1 MPa (see Fig. 9). 7. Dimensionless equations, rate constant and viscosity function We are going to choose model parameters to get a best ?t to the experiments of Maini and Sarma (1994). To get the maximum generality from our study and to identify the ?tting parameters it is useful to express the governing equations (4.27) and (4.28) in dimensionless form. We introduce dimensionless variables X ? x=L; T ? t=h; P ? p=~; p K?/? ? k?/?=k0 ? g?0?=g?/? and ?nd that if h ? aL2 =k?0?~ p then o/ o oP ? ?1 ? /?K?/? oT oX oX and J ?1 ? /?

2

?7:1?

?7:2?

?7:3? ?

o/ oP o/ ? K?/? oT oX oX

1?P b/ ? ; P 1?/

?7:4?

where L is the pack length and J ? s~k?0?=L2 p ?7:5?

is the dimensionless relaxation parameter. The outlet is at X ? 0 and the inlet at X ? 1. The system (7.3) and (7.4) is completely determined by a rate constant J and a permeability function K?/? or equivalent viscosity function g?/?. We are dealing with dispersions of small gas bubbles that are spherical when not crowded. It is usual to ?t the viscosity function of equivalent dispersions of solid spheres with ?m g?/? / ; ?7:6? ? 1? g?0? A where A is a maximum packing fraction, / 6 A, and m > 0 is selected from data; m % 2 works well. It is altogether di?erent for gas bubbles at large / where stabilized bubbles would change phase, ?rst to wet and then to dry foams. To be consistent with the solubility expression (2.14) we should seek a viscosity that increases continually with / from dilute dispersions with / % 0 to dry foam with / % 1; hence A ? 1 and m is to be determined. The viscosity function

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g?/? ?

g?0? ; ?1 ? /?m

m>0

says that dry foam has an in?nite resistance to ?ow. Of course, the foam would break under pressure for dispersed fraction / near one; the gas bubbles would collapse with free gas channels percolating through the pack. We have a one parameter family of viscosity functions depending on the exponent m. After ?tting our model to the experimental data of Maini and Sarma (1994) we found that m ? 11, which at small / gives g?/? ? g?0??1 ? 11/ ? ? ? ??: ?7:7?

The value 11 is greater than the Einstein value m ? 2:5 for dilute dispersions of solid spheres possibly because of the additional increases in the viscosity of the liquid phase caused by the vaporization of dissolved gases; in other words, the viscosity of mixture increases because there is less dissolved and more dispersed gas. It follows now that the system (7.3) and (7.4) is well characterized when the parameter J and m K?/? ? ?1 ? /?m ?7:8?

are known. We have tried to ?t the model predictions to the experiments with a nearly best, but imperfect ?t, when m ? 11; J ? 5: ?7:9? In the next section we describe the experiments and compare the data with the model prediction. Using the data given in Section 6 we found that J ? 1:345 ? 10?6 s: Our data ?tting gives J ? 5 or s ? 3:71 ? 106 s. ?7:10?

8. Steady ?ow in a horizontal sandpack Foamy oil is driven from a reservoir at saturation P ? 1 at X ? 0 to another reservoir at the outlet pressure pL ; PL ? pL =~ < 1. Since o/=oT ? 0 for steady ?ow, Eq. (7.3) implies that p ?K?/??1 ? /? dP ? C ? constant: dX ?8:1?

Eq. (8.1) says that the oil velocity ?k dP ?1 ? /? dX ?8:2?

is a constant in steady ?ow, independent of X. Combining (8.1) and (7.4) we get o/ 1 ? P b/ ? ? : oX P 1?/ ?1 ? /? JC

3

?8:3?

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Eqs. (8.1) and (8.3) are to be solved subject to the conditions that P ?1 P ? PL when J ? 0 P? and /?0 1?PL / ? /L ? 1??b?1?PL at X ? 0; at X ? 1: ?8:6? 1?/ ; ?b ? 1?/ ? 1 dP ?b d/=dX ? dX ??b ? 1?/ ? 1?2 ?8:5? at X ? 0; at X ? 1; ?8:4?

Elimination of dP =dX in (8.1) then gives rise to the following quadrature: Z / bK?g??1 ? g? dg : CX ? ?1 ? ?b ? 1?g?2 0

?8:7?

The constant C is then determined by the outlet pressure given by (8.6). The integral in (8.7) can be done when K?g? is given by (7.8). In general, when J ? 0, the ?rst order system (8.1) and (8.3) may be solved numerically. The steady distributions of pressures along the sandpack are compared with data from experiments in Figs. 5–7. Maini and Sarma (1994) also reported the rate of oil production in steady state as a function of the drawdown pressure. The oil production is the mass ?ow rate, the volume ?ow rate of oil times the oil density q % 0:97 g/cm3 . The super?cial mixture velocity u is given by Darcy?s law (4.16), Au, where A is the cross-sectional area, is the volume ?ow rate of the mixture of gas and oil, which is larger than the volume ?ow rate u?1 ? /? of oil alone. Hence the mass ?ow rate of oil alone is

Fig. 5. Comparison of theoretical and experimental pressure distributions at drawdown pressure pL ? 3 MPa.

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Fig. 6. Comparison of theoretical and experimental pressure distributions at drawdown pressure pL ? 1 MPa.

Fig. 7. Comparison of theoretical and experimental pressure distributions at drawdown pressure pL ? 0:75 MPa.

Q qL ? ?qL A?1 ? /?k?/?

dP ; dX

?8:8?

which according to (8.2) is a constant, independent of X. A comparison of the measured and theoretical results for m ? 11, J ? 5, b ? 3:4, A ? 16:1 ? 10?4 m2 is given in Fig. 8. The modest discrepancy between measured and theoretical values for large drawdowns may be due to the formation of foam. This foaming might be expected to reduce the rate of production because of ?ow resistance, and to increase the primary recovery because foam e?ectively traps the gas; all the oil must come out with trapped gas.

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Fig. 8. Comparisons of oil production rates in steady ?ows at various pressure drawdowns.

9. Blowdown experiment We turn next to the blowdown experiment of Maini and Sarma (1994) (their Fig. 9, our Fig. 9). They describe their experiment as follows (the emphasis indicated is ours). A di?erent type of experiment was needed to estimate the total recovery potential of solution gas drive. This experiment started with the sandpack at maximum live oil saturation. The pack was allowed to blowdown to atmospheric pressure through the outlet end, and the inlet end remained closed. Fig. 9 shows the recovery and pressure-drop behavior. More than 20% of

Fig. 9. Change in the pressure drop across di?erent core segments and cumulative oil production with time during the blowdown experiment with the Lloydminster system (Maini and Sarma, 1994).

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the original oil was recovered in this primary depletion experiment. The value is surprisingly high for the viscous oil system and suggests that the critical gas saturation was much higher than what would be measured by an external gas drive experiment. Typically, the external drive experiments in such systems show the critical gas saturation to be <5%. Therefore, this experiment also suggests that a mechanism is present in heavy-oil systems to increase the critical gas saturation. We suggest that this mechanism is the formation of an oil-continuous foam. We have already remarked that the solubility isotherm with b ? 3:4 or 3.17 leads to dispersed gas fractions of the order 0.93; foaming is inevitable. The blowdown experiment is unsteady. The experiments shown in Fig. 9 show that the pack is still producing oil after 200 h; the terminal steady and uniform state has not been achieved. The initial value problem corresponding to the blowdown problem is to solve (7.3) and (7.4) subject to the initial conditions P ? 1; / ? 0 dP ?0 dX and P ? PL ?1 ? H ?t?? at X ? 0; ?9:3? for 0 6 X 6 1 when T 6 0 ?9:1?

and boundary conditions at X ? 1 ?9:2?

~ where H ?t? is unit step function pL ? 0:1 MPa, p ? 4:83 MPa, PL ? pL =~ ? 0:0205: The intep gration of these equations was carried by numerical methods. The cumulative production may be obtained as the integral over time of the rate of production Z t ?t? dt; Q ?9:4? q0 Q?t? ? q0

0

where ? Ak?1 ? /? dP =dX Q ?9:5?

is evaluated at X ? 0, the open end of the sandpack. A comparison of the model prediction with m ? 11 and J ? 5 and the experiment are shown in Fig. 10. The values m ? 11, J ? 5 were selected for a best ?t for the steady pressure distributions given in Figs. 5–7, the rate of production in steady ?ow given in Fig. 8 and the comparison of theory and experiment in drawdown given in Fig. 10. The di?erential pressure in the six segments de?ned by pressure tabs is given in Fig. 11. The theory greatly overpredicts the drop in pressure in drawdown. The blowdown experiment produces large amounts of gas because the pressure drop is very severe. The appearace of foam was observed, consistent with rapid outgassing. Blowdown also favors the formation of connected gas, due to bubble rupture, which percolates out of the sandpack without a trace reducing both the pressure and the emulative production. These e?ects of free gas are not allowed in our model and may account for the discrepancies between theory and experiment exhibited in Figs. 10 and 11.

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Fig. 10. Comparison of theory and drawdown of pressure in experiment for Lloydminster oil.

Fig. 11. Model prediction of the evolution of the di?erential pressure in six segments for the blowdown experiment in Fig. 9. b ? 3:4; m ? 11; J ? 5.

10. Discussion and conclusion The theory which we have developed could be called a continuum mixture theory for foamy oils ?ow with dispersed gas of low mobility relative to the liquid which leads to three coupled nonlinear partial di?erential equations for u, p and the gas fraction /, ?ve scalar equations in ?ve unknowns. We derived an equation of state (called an equilibrium isotherm) for foamy oil. This equation gives the volume fraction of dispersed gas at a given temperature and pressure at equilibrium for

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the situation in which the gas that comes from releasing dissolved gas remains dispersed. This equation depends on only one material coe?cient, a function of the temperature which can be obtained from measurement in a PVT cell. Such measurements were made and yielded only modestly di?erent values of the coe?cient for two Canadian and one Venezuelan crude oils. The empirically based equation of state avoids constitutive hypotheses about nucleation and bubble growth, which are implicit in the measured equilibrium values of the volume fraction of dispersed gas. The derivation and application of equilibrium isotherms is one of the principal contributions of this paper. Another contribution is the development of a ?ow description for foamy oil whose center pieces are a continuity equation for the bubbly mixture, Darcy?s law and a relaxation equation for the equilibrium isotherm. The relaxation equation depends on a to-be-determined rate constant which allows for delay in the growth of the gas fraction / to a change of pressure expressing the idea that it takes time for bubbles to nucleate and grow. It is possible that the pressure also relaxes, but the modeling of pressure relaxation would introduce another rate constant and is left for future work. The mobility function k?/? which appears in Darcy?s law must also be modeled. The mobility function for our model is the measured permeability of the porous media over the to-be-determined e?ective viscosity of the mixture. The modeling of the viscosity of the mixture as a function of the dispersed gas is very di?cult and we have no guidelines from the literature. Our model, which depends only on the dispersed gas fraction, makes no formal distinction between bubbly mixture and the foam that arises inevitably under the kind of severe depressurization seen in the Maini–Sarma blowdown experiments. The range of the dispersed gas fraction is from / ? 0, in which the oil is saturated with dissolved gases, to / ? 1 which can be interpreted as a super-dry foam. The viscosity of the foam ought to be much greater than the bubbly mixture but less great than concentrated dispersions of solids that cannot move at all when they are closely packed. An additional complication in the modeling of the viscosity is that the viscosity of the liquid increases rather sharply when gas is released; its like lowering the fraction of water in a glycerin and water mixture. So the viscosity of the mixture increases both because of the presence of more bubbles and because the continuous liquid-phase viscosity also increases. We related our ?ow description to the conventional description of two-phase ?ow in porous media using the concept of relative permeability. We write down all the usual equations with the caveat that the mobility function for each phase is expressed in terms of an e?ective viscosity rather than a relative permeability; for the gas we have kG ?/? ? kkG k ? ; lG ?/? gG ?10:1?

where k is permeability, kG is relative permeability, gG is gas viscosity, and lG ?/? is e?ective gas viscosity. Of course, we have done nothing but change the names of symbols, but our concept is closer to mixture theory in which we describe ?ow properties of ?uid phases in a porous media of ?xed properties independent of the ?uid and ?ow. Moreover, our e?ective viscosities can be directly related to the measured values of the mixture viscosity ?1 1 1 ; ?10:2? ? l?/? ? lG ?/? lL ?/?

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where lL ?/? is the e?ective viscosity of the liquid phase. Our mixture theory arises as a rigorous consequence of the conventional theory under the assumption that the average velocities of dispersed gas and liquid are identical and in this case we get explicit lG ?/? ? lL ?/? ? l?/? ; / l?/? ?1 ? /? ?10:3? ?10:4?

relating e?ective viscosity of the phases to the e?ective viscosity of the mixture. In addition we ?nd that krL 1 ? / gL ? / gG krG ?10:5?

relating the ratio of relative permeabilities to the ratio of molecular viscosities. We applied our theory to the sandpack experiments of Maini and Sarma (1994), backing out the rate constant and mixture viscosity function to ?t data. The viscosity function was selected in the frequently used form m / ; ?10:6? l?/? ? l?0? 1? /m where /m is the maximum packing fraction. /m is a perfectly clear concept for solid spheres and might be considered for bubbly mixtures. But if our bubbly mixtures foam we can get dispersed gas all the way to / ? 1, hence /m ? 1. Our equilibrium isotherm also allows for a full range of dispersed gas fractions 0 6 / 6 1 up to dry foams. We then look for a single continuous expression for l?/? in the form (10.6) with /m ? 1. We think that dry foams are rigid unless riding on a liquid layer, which is not easily formed in a porous media; hence l?/? ! 1 as / ! 1. In the dilute limit l?/? ! l?0??1 ? m/?: ?10:7?

In foamy oil m should be larger than the Einstein value 2.5 because of the added viscosity increase of the continuous oil phase due to the release of dissolved gas. We backed out a viscosity function by ?tting, ?nding m ? 11 together with a relaxation time corresponding to a dimensionless time J ? 5. With these values we achieved some good agreements between the sandpack model prediction and data for steady ?ow but with larger discrepancies in blowdown experiments in which dispersed gas probably percolated into free gas. We think that this comparison of model and experiment establishes a direction for model building using the average velocity of phases, the pressure and volume fraction of dispersed gas as primitive variables. It is possible for the dispersed gas to move relative to the oil without forming connected gas through coalescence and percolation. To describe such a condition we would need to consider momentum and mass balances of the two phases, but the description could be formed with same set of primitive variables, replacing constitutive equations for nucleation, bubble growth and possibly some coalescence, with relaxation of equilibrium isotherms from PVT data.

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Our model has limitations; it only applies for small bubbles, but not too small either. Since monitoring of bubble size in porous medium is hard to do, or even presently impossible, there is no way of knowing whether the assumptions that we made are valid. The results seem to indicate that they might be. Models, like the one proposed here, should be tested against more data particularly against data from producing ?elds. For production the conditions for the breakdown of models that work when all the gas is dispersed due to the formation of connected gas is even more important than the models themselves. Acknowledgements The work of D.D. Joseph was supported in part by the DOE (Engineering Research Program of the Department of Basic Energy Sciences), the NSF under Grant Opportunities for Academic Liasons with Industry and the Minnesota Supercomputer Institute. We are indebted to Prof. G.I. Barenblatt for providing us with a translation of the prescient work of Leibenson (1941) and for calling our and the reader?s attention to the possibility that gas release in reservoirs could be quite di?erent than in a PVT cell or even in a sandpack. References

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