Calculation of Initial Fluid Distribution in Oil Reservoirs

Transcription

1 AMERICAN INSTITUTE OF MINING AND METALLURGICAL ENGINEERS Technical Publication No Class G, Petroleum Technology, July 1948 DISCUSSION OF THIS PAPER IS INVITED. Discussion in writing (I co ies) may be sent to the Secre- tary. American Institute of,mining and MeJallurgic?l Engineers. 29 West 39tg Street, New.York.18. N. Y. Unless special arrangement IS rnade, dlscuss~on of th~s paper will close Sept. 30, Any discuss~on offered thereafter should preferably be m the form of a new paper. Calculation of Initial Fluid Distribution in Oil Reservoirs BY MORRIS MUSKAT,* MEMBER AIME IT is pointed out that the application of capillary pressure curves obtained by drainage or desaturation processes to the calculation of the fluid distribution in interphase transition zones involves a number of difficulties; namely: (I) the development of very low nonwetting phase saturations appears - - to be in contradiction with the lack of mobility of such distributions indicated by permeability-saturation curves, (2) dispersed nonwetting phases are thermodynamically unstable, and (3) discontinuous phases should not be subject to hydrostatic equilibrium requirements. While these difficulties could be obviated by assuming that the capillary pressure drainage curve has an initial horizontal segment, they are automatically circumvented by application of the imbibition capillary pressure curve to the lower part of the water-oil transition zone. These generally show zero displacement pressure at only partial wetting-phase saturations. The countercurrent upward flow of oil into the main oil-saturated pay and downward drainage of water also suggests that wetting-phase imbibition processes will control the saturation distribution immediately above the watersaturated section. Similar considerations, with a generalized interpretation of the apparent wetting-phase behavior of the oil and gas phases, provides a basis for constructing the curve for fluid distribution in the oil-gas transition zone. In the transition zones so derived, the oil begins with a nonvanishing saturation at the water-oil contact and terminates with a similar saturation at the top of the gas-oil contact. The gas-oil transition zone begins with an equivalent nonvanishing gas saturation. It has been a common assumption for some years that the nature of the fluid dis- Manuscript received at the office of the Institute July Gulf Research and Development Co., Pittsburgh, Pa. tribution in virgin reservoirs, and in particular that in the transition zones between the oil and water and between the oil and gas sections, can be computed by a simple application of capillary pressure data. The results of such calculations appear to have been first reported by Leverett.' The latter, however, indicated only the numerical values of the parameters used in the computations, without explicitly describing the procedure. Leverett's illustrative calculated transition zones are reproduced in Fig I. Although no critical study of the apparently obvious method of calculation has been published, one cannot proceed very far in the computation without encountering some rather fundamental questions not yet answered in the literature. This paper is not written under the pretense that the whole transition-zone prob- lem has been fully and satisfactorily solved. Its purpose, rather, is to discuss suggestions for its treatment, much of it admittedly hypothetical, and emphasize the nature of the uncertainties arising therein; for the literature, as now available, gives little indication that there are any problems still outstanding. The basic equation, which presumably gives directly the fluid distribution without recourse to any further consideration, is: where yl, yt are the densities of the two contiguous fluids, h is the height above the "capillary-free " zone of complete saturation with fluid I, g the acceleration of gravity, and A# is the capillary pressure across the average interface at the height References are at the end of the paper. Copyright, 1948, by the American Institute of Mining and Metallurgical Engineers. Inc. Printed in USA

2 2 CALCULATION OF INITIAL FLUID DISTRIBUTIONS.IN OIL RESERVOIRS h. Eq I is inherently correct. It is its appli- C(p) represents a universal relation,. for cation which gives rise to questions, as will the particular porous medium of interest, be seen presently. and can be used to transform Eq I as: Unless the capillary pressure curve is initially obtained using the actual fluids I YW - YY C@w) = a,,ghw [3I and 2 to which Eq I is applied, which apparently is seldom done in practice, the capillary pressures are inferred from their when applied specifically to the water-oil FIG I-WATER-OIL AND GAS-OIL Rr Cent Soturotion TRANSITION-ZONE FLUID DISTRIBUTION CALCULATED BY LEVERETT. relationship to the interfacial curvatures, interfaces in a sand in the water-oil transias: tion zone. Choosing the values of 7, - y. AP = i2cb) [2] and ow, pertaining to the particular oilwater system of interest, Eq 3 presumably whereo12 is the interfacial tension between pmvides an immediately applicable forthe two fluids' and C(p) 's the mean mula for computing the relation between curvature of the interfaces at the liquid hw and p,. saturation p. Thus from the capillarypressure data for the specific fluids used in CHOICE OF THE CAPILLARY PRESSURE the measqrements the function Cb) is CURVE computed, also by applying Eq 2. From The essence of the discussion to be prethe capillary pressures as usually reported, sented here concerns the type of capillary this function will have a graphical representation as shown by the solid curve in a While such is plausible. in clean sands, evidence that it is not always valid has been reported Fig 2. It is now assumed that the function by G. L. Hassler, E. Brunner, and T. J. Deahl.1 0

3 MORRIS MUSKAT-TP pressure or curvature curve to be used in Eq I or Eq 3. This is a matter which has been given virtually no detailed consideration. In fact, recent work on the application ward pull of gravity. While these two usually merge in the range of low wettingphase saturation, they differ markedly at high liquid saturations. Typical of some FIG 2-HYPOTHETICAL CURVATURE-SATURATION Solid curve, drainage; dashed lower segment, imbibition. of capillary pressure data to the determination of connate water saturations seems to have given rise to the impression that the curves for capillary pressure vs. saturation obtained in such experiments also control all aspects of the fluid distribution in interphase transition zones. As originally emphasized in the soil science literat~re,~ and also observed in early investigationsls2 pertaining to oil production, there are two basic types of relationships for capillary pressure vs. saturation. The one refers to a desaturation or drainage of the wetting phase from a rock under pressure or gravity. The other describes the process of wetting-phase absorption or imbibition against an applied pressure or the down- RELATION IN A POROUS MEDIUM. of the imbibition curves shown by consolidated porous materials is the lower dashed segment in Fig 2. In his original study of capillary phenomena Leverettl suggests that "the conditions under which hydrocarbons accumulate in and are produced from the earth will lead to distribution of the fluids corresponding more closely to the imbibition equilibrium than to the drainage equilibrium.'' He indicates that where "it is necessary to choose between the two sets of data we shall use the lower (imbibition)." In view of the qualitative similarity between the imbibition and drainage capillary pressure curves found by Leverett, his calculated transition distributions (cf. Fig I) may well have been

4 4 CALCULATION OF INITIAL FLUID DISTRIBUTIONS IN OIL RESERVOIRS the result of applying the imbibition curve, although otherwise it suggests the use of the drainage curve. On the other hand, the remarks quoted do not indicate why imbibition processes may be expected to control the fluid distribution in transition zones, nor do they suggest any inherent difficulties associated with the application of the drainage curves. Aside from the question of its applicability to the problem of the fluid distribution in the transition zone, there is inherent interest in an examination of the implications of the capillary pressure drainage curve. The solid curve of Fig 2, if reconverted to capillary pressure ordinates, is typical of those observed with consolidated rocks2 This curve, it will be noted, begins at a nonvanishing value at IOO pct liquid saturation. The corresponding value of the capillary pressure is the "displacement" pressure, which represents the maximum that can be applied to a liquid-saturated rock before entry of nonwetting phase will develop. The existence of a displacement pressure or "displacement curvature'' immediately appears to invalidate Eq I and Eq 3 with respect to values of h lower than given by these equations for the displacement values of Ap or Up,). This, in itself, is not serious, and can be interpreted as indicating simply that the rock will be fully saturated until this minimum value of h is reached. More disturbing, however, are thd implications of the drainage curves immediately above the displacement pressure or curvature. In accordance with most published data, the solid curve of Fig 2 has been drawn to show a nonvanishing slope at IOO pct liquid saturation. This implies the existence and development of stable distributions of arbitrarily low nonwetting-phase saturations as the pressure is increased slightly above the displacement value. If uniformly distributed, the nonwetting phase would certainly be in a dispersed and discontinuous distribution at such low saturations. But a bubble or globular distribution of gas or oil will have zero permeability, and it is difficult to understand how it could be created throughout a rock sample when only its surface is exposed to the nonwetting phase. Moreover, small bubbles, especially, and any dispersed nonwetting phase are inherently unstable thermodynamically, and ultimately would tend to disappear by solution and diffusion through the wetting phase. Finally, if the nonwetting phase is discontinuous, it should not be subject to the requirement of hydrostatic equilibrium of Eq I. An individual globule of oil trapped in a pore will remain so without appreciable change in curvature at any hydrostatic pressure, and the latter alone will not directly influence its stability, except for the effects of solution and diffusion. It is possible to explain these difficulties associated with arbitrarily low nonwettingphase saturations by assuming that instead of representing a discontinuous distribution of separated bubbles or globules the nonwetting phase is localized in continuous iilamentary channels comprising only a small part of the total pore volume. While such a situation is conceivable, it would suggest a surprising degree of nonuniformity in the typical small rock sample. Moreover, it would be expected that if such continuous channels could be set up in a capillary-pressure desaturation process they would also be observed in multiphase flow experiments so as to give vanishing equilibrium gas saturations. Yet even the permeability-saturation curves obtained by liquid desaturation processes show in virtually all cases that continuous flow channels of the nonwetting phase do not develop until its saturation builds up to at least 5 to 10 pct. Although experimentation with multiphase flow in this region is very difficult and subject to relatively large experimental errors, the frequency of occurrence of vanishing equilibrium saturations for the nonwetting phase is certainly

5 MORRIS MUSE: not comparable with that in which the capillary pressure apparently rises gradually from its displacement value in desaturation experiments. While it may seem unjustified to question so much of the evidence reported on capillary pressure measurements, it is felt, nevertheless that this apparent discrepancy largely reflects experimental errors in the experiments on capillary pressure drainage. It is anticipated that if such experiments were performed with special care and so as to ensure equilibrium conditions when the capillary pressure just exceeds the displacement value, the curve would have a strictly flat initial segment, with the capillary pressure beginning to rise only after the wetting-phase saturation had been reduced to a value approximating the equilibrium saturation. These difficulties can be circumvented with respect to the transition zone, even if it should be established definitely that the curves of capillary pressure drainage do not begin with a horizontal segment. For if the capillary pressure curves obtained by imbibition are applied near the zone of complete water saturation, the questions of nonwetting-phase mobility, its stability, and control by hydrostatic forces simply do not arise. In the lower dashed segment of Fig 2, no displacement pressure is shown. Moreover, a zero capillary pressure is exhibited at less than IOO pct wettingphase saturation. The resulting nonwettingphase saturation is presumably at least the limiting value for a continuous and mobile distribution, although hysteretic and metastable inclusions of dispersed elements may be part of the whole. While these features automatically avoid the difficulties arising from the application of the drainage curve to the lower regions of the transition zone, it is instructive to consider in detail the circumstances that give inherent plausibility, at least, to the introduction of the imbibition curve. The intuitive feeling that the transition zones are established entirely by drainage processes raises doubt as to the applicabil- ' ity of curves for capillary pressure imbibition. Because of the uncertainty of the detailed mechanism of oil accumulation in reservoir traps, the nature of the fluid redistribution following the initial entry of the petroleum fluids is likewise not definitely established. Nevertheless, the physical processes determining the fluid distribution in the transition zone can be analyzed without defining in detail the basis mechanism of oil migration and accumulation. There is general agreement that prior to the accumulation of the oil the reservoir rock was saturated with water. There can also be little doubt that the initial replacement of the water by the invading oil left a water saturation higher than the "irreducible" water content subsequently found on discovery at appreciable elevations above the water contact. In the upper parts of the pay the establishment of the connate water saturation therefore must have involved a de- saturation process of gravity drainage opposed by capillary forces, as commonly assumed. Associated with this drainage, however, there must have occurred an upward movement of oil from the lower parts of the oil zone to replace the downward draining water. And the latter, in turn, must have involved an inkrease in water saturation-an imbibition-in the region near the oil-free and water-saturated section. It is therefore to be expected that the final fluid distribution developed by the countercurrent flow of water and oil will be a Strictly speaking, this is an assumption, though it would seem to be surprisingly accidental if the rate of accumulation were everywhere and always exactly such that gravity forces could suffice to establish complete drainage equilibrium throughout the formation during the initial accumulation process, and even before the position of the ultimate capillary-free water level were definitely established.

6 determined by the characteristics of the capillary pressure drainage curve in the main body of the oil pay and the upper layers of the water-oil transition zone, and detailed mechanism of initial accumulation and distribution of the free gas phase is even more conjectural than when the oil phase alone enters a reservoir trap. Again, ~oturbtion in Per Cent FIG 3-CALCU~ATED SATURATION DISTRIBUTIONS USING THE CURVATURE-SATURATION FUNCTIONS OF FIG 2. Water-oil and oil-gas density differences assumed = 0.3 and'o.5 g/cc; water-oil and oil-gas interfacial tensions assumed = 30 and 20 dynes/cm. by those of the imbibition capillary pressure curve in the region immediately adjacent to the water contact. Since the imbibition and drainage curves merge at moderate saturations, the exact point of change-over between the two will be of little importance. In any case, the oil saturation in the transition zone will begin at a nonvanishing value in a manner quite similar to that which would follow from a drainage curve with initial horizontal segment. The transition water-oil zone distribution. as calculated from the imbibition curve of Fig 2, assuming y, - yo = 0.3 g/cc, u,, = 30 dynes/cm, is plotted in Fig 3. While the general problem of the gas-oil transition zone when a free gas phase is present is basically similar to that of the water-oil zone, the former gives rise to additional complications. In fact, here the IN WATER-OIL AND OIL-GAS TRANSITION ZONES, however, regardless of the manner in which the gas and oil entered the trap, the application of the curve for capillary pressure drainage at high liquid saturations encounters the difficulties of gas-phase mobility, stability, and the validity of the hydrostatic pressure balance requirement, unless an initial horizontal segment be assumed. And here, too, although the latter condition may be considered as a probable characteristic of the drainage curve, independently of the transition-zone problem, the use of the curve for imbibition capillary pressure near the gas-oil contact appears to be inherently the more reasonable procedure. For if the initial gas-oil distribution be assumed as substantially uniform, as the result of the dynamic phase displacement process, the redistribution to the state of increasing oil saturations on approaching the gas-oil contact will again require

7 MORRIS MUSWT-TP countercurrent gas and oil movements in which the lower parts of the transition zone "imbibe" the oil draining down from above. Thus the gas-phase saturation will also start at the nonvanishing value as a limiting continuous phase left by the imbibition process. Although in the theory outlined above only the lower parts of the oil-water transition zone are considered as controlled by imbibition processes, the application of the desaturation curve raises questions of physical interpretation when applied to the top of the gas-oil transition zone. For, as indicated by Fig I, which apparently is the result of such application, the oil saturation will decline continuously to zero, as the total liquid saturation approaches the irreducible minimum already occupied by the connate water. Evidently the previously mentioned difficulties of mobility, stability, and hydrostatic pressure control, associated with the necessarily discontinuous distribution at the very low oil saturations-unless accidental filamentary distributions are assumed-must be faced here, too. However, in this region the drainage curve does not permit a simple modification for resolving the dilemma. In the light of the almost complete absence of basic information on capillary equilibrium in three-phase sy~tems,~ the difficulty just mentioned might well be considered simply as an unsolved problem. However, it is possible to construct a plausible tentative solution by generalizing the concepts pertaining to two-phase distributions. First, it is to be noted that both the oil and gas are basically nonwetting phases in a water-wet rock. The conventional two-phase capillary pressure experiments are commonly interpreted as applying to three phases with the oil adding to the water as a composite liquid. But this is only an assumption. It is reasonable a Since this paper was written some interesting three-phase capillary pressure studies have been reported by J. H. Welge. when the gas-phase saturation is low and does not appreciably exceed the minimum needed for phase continuity. Certainly, however, when the oil phase approaches the discontinuous distribution, the assumption that it forms a continuous extension of the water phase with curvatures the same as for an equivalent total water saturation becomes highly questionable. Indeed, it is just this which leads to the difficulties mentioned. More plausible is the supposition that the gas and oil change places, as it were, when the gas-phase saturation appreciably exceeds that of the oil phase. While intuitively it would appear that the oil phase will always lie adjacent to the water and the gas phase will occupy only the central pore channels, probably this reflects mainly a dynamic behavior largely controlled by their relative viscosities. In fact, the only reported three-phase permeability studies4 show that at moderate water contents the relative permeabilities, in which the direct viscosity effects are removed, for the gas and oil vary with their own saturations in a very similar manner. It does not seem unreasonable, therefore, to consider the high saturation gas phase in the upper parts of the gas-oil transition zone as the supplementary wetting phase superposed on the water phase, analogous to the oil near the oil-saturated zone. In a strict sense, of course, neither the gas nor the oil will actually be true wetting phases, and if the irreducible water saturation should leave completely dry and exposed parts of the grain surface between the pendular rings, the oil probably would cover them in preference to the gas. It is only suggested that the gross interfacial geometry between the gas and oil be determined by considering the gas and water as the wetting-phase equivalent rather than the oil and water. In any case, the previous difficulties are automatically resolved by this supposition, for now the countercurrent upward migration of gas and

8 8 CALCULATION OF INITIAL FLUID DISTRIBUTIONS IN OIL RESERVOIRS downward drainage of oil during the establishment of equilibrium in the transition zone again becomes an imbibition process in the upper layers with respect to the composite water and gas equivalent wetting phase. Eq I is then applied using the curve for capillary pressure imbibition with h representing the depth below the top of the transition zone. The exact point of change-over between the gas and oil as the apparent single nonwetting phase is not clear at present. Probably it will occur when theoil and gas phases have approximately equal saturations. Moreover, since the capillary pressure curves are inherently subject to hysteretic effects and depend on the past history and initial conditions of the system, no single curve will necessarily apply to the oil-water transition zone and both the lower and upper parts of the gas-oil transition zone. However, in principle these are subject to experimental determination, and indeed the investigation of three-phase capillary pressure curves merits much further study. From a practical standpoint, these uncertainties will affect only the quantitative details of the transition-zone distributions. Their gross characteristics and thickness will be largely fixed by the sharpness of the "bend " in the capillary pressure curve and the density and interfacial tension constants. A gas-oil transition-zone distribution calculated by the procedure outlined, using the imbibition curvature relation of Fig 2 and IT. = 20 dynes/cm, yo - y, = o..~ g/cc, is plotted in the upper part of Fig 3, assuming a change-over nonwetting phase saturation of 39 pct. Both the gas and oil saturations of the transition-zone boundaries begin with the nonvanishing values presumably left by the imbibition processes postulated above. Moreover, as is evident immediately from Eq 3, when transformed to the gas-oil interface equilibrium, the gas-oil transition zone is thinner, by several fold, than the water-oil transition zone, for the same curvature function. The calculation of the distribution of the transition zone fluid does not of itself determine the thickness of the oil zone. The latter is fixed by the total oil con- tent of the reservoir. The transition zones merely represent boundary layers adjoining and superposed on the oilsaturated section. Of course, if the latier is very thin the transition zones may span the whole of the oil pay. While such extreme cases apparently have been observed, they represent exceptional situations. On the other hand, if the capillary pressure drainage curve is considered as approaching the "irreducible " saturation asymptotically, the transition zone with respect to the water phase would, in principle, extend to the very top of the reservoir structure. From a practical point of view this is of little importance, since the saturation varies but slowly in the region of high capillary pressures. However, if the irreducible saturation is visualized as developing suddenly by a breaking of the continuous funicular films when a critical capillary pressure is exceeded, the transition zone will be correspondingly limited. As the latter situation appears to be more plausible, it has been so assumed in Fig.2. Under this assumption, generally there will be no need to take into account the capillary pressure balance in the gas zone with the water phase. It should be emphasized that the suggestion for calculating the nature of the gas-oil transition zone by an application of a modified interpretation of the conventional type of two-phase capillary pressure curves is not proposed as a final solution to the problem. It represents only a working hypothesis and convenient artifice when only the normal two-phase curves are available. Except for thefact that the her pore interstices are actually occupied by the connate water, the behavior of the gas and oil phases in the gas-oil transition zone

9 MORRIS MUSKAT-TP would simulate that of a two-phase system in which neither the gas nor the oil completely wets the solid internal surface. For quantitatively accurate calculations, it will be necessary to apply empirically determined curves for three-phase capillary pressure obtained by desaturation or inhibition processes corresponding to those actually occurring in the reservoir. The illustrative calculation underlying the gasoil transition zone plotted in Fig 3 has been presented only to show the qualitative characteristics of this transition zone and the gross features of the physical processes involved. It should be noted, finally, that the considerations and calculations set forth here refer only to strictly uniform media of identically the same capillary functions throughout. Unfortunately, in practice the reservoir formation usually has such vertical variations, directly reflected in its permeability, as may well mask completely any regular trend in the saturation distribution in the transition zone. The ideal situations offering opportunities for quantitatively checking calculated saturation distributions will therefore be limited. It is felt, nevertheless, that it would be highly desirable to attempt such comparisons, where possible, in order to help to clarify the physical principles of capillary phenomena. The writer is indebted to Dr. Paul D. Foote, Executive Vice-President of the Gulf Research and Development Co., for permission to publish this paper. REFERENCES I. M. C. Leverett: Capillary Behavior in Porous Solids. Trans. AIME (1941) 142, G. L. Hassler. E. Brunner and T. J. Deahl: The Role of Cavillaritv in Oil Production. Trans. AIME (19~4) i55, Cf. for example B. A. Keen: Physical Properties of the Soil M. C. Leverett and W. B. Lewis: Steady Flow of Gas-oil-water Mixtures through Unconsolidated Sands. Trans. AIME