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CHARLES G. DODDAND 0. GERALD KIEL
Vol. 63
EVALUATION OF MONTE CARLO METHODS I N STUDYING FLUID-FLUID DISPLACEMENT AND WETTABILITY I N POROUS ROCKS1 BY CHARLES G. DODDAND 0. GERALD KIEL The University of Oklahoma, Norman, Oklahoma Received March 6, 1060
Network models have been used to determine the roles of various parameters on the quasi-static capillary pressure desaturation of a wetting liquid from a porous solid matrix. We have modified the network model suggested by Fatt to permit wetting fluid to be trap ed during a desaturation rocess; and we have studied the effects of pore size distribution, network distribution, number o?connections per pore, anif network size on calculated capillary pressure desaturation curves. The results indicate that network models involving random distributions of selected properties are a valuable tool for investigating these parameters. Of more significance, however, is the indication that a still more valuable application of the network models may be made in the characterization of the wettability of orous media with respect to specified pairs of immiscible fluids saturating the solid. In the initial work reported herein tEe wettability concept was equated to the probability of entrance of displacing fluid into a pore filled with the fluid being displaced. Further considerations, and a review of recent experimental work, indicate a need for a more sophisticated ap roach to the problem which would include an assigned distribution of wettability to internal pores and the application o?Monte Carlo methods to the study.
Introduction The description of single and multiphase fluid flow through capillary pores in solid porous media has been unsatisfactory because the pore geometry generally is too complex t o be described adequately by analytical expressions. Physical models, such as bundles of parallel capillary tubes, heretofore used to develop analytical expressions, have been overly simplified. A new approach to this problem mas suggested by Fatt2 who set up a network model incorporating random assignments of parameters such as pore size distribution and number of connections between pores, within which fluid flow could be followed as one pore after another, or one set of pores after another, was desaturated according to the rules set up to describe a quasi-static or a dynamic desaturation process. A network model such as that devised by Fatt is but one of many discussed by Scheidegger3in his comprehensive review of the problem. Rose4 has criticized Fatt’s model because it did not describe rock pore geometry and orientation more exactly, but we have found a modification of Fatt’s model accurate enough to simulate actual capillary pressure curves of sandstone rocks. We have chosen to explore the potentialities of the network model further and have found it a promising means of applying stochastic processes such as Monte Carlo methods to n study of the complex and subtle problem of wettability in porous media. Fatt devised a network model in which he replaced each pore in a sphere-packed model with a cylinder. To facilitate calculations, the model was reduced to two dimensions. Each cylinder was viewed on end in the network where it was connected with other cylinders in a regular geometric pattern. The effects of pore-size distribution, network distribution and number of connections per pore on the model were studied. Four regular geo(1) This paper was based in part on work done by 0. G. Kiel and reported in a ill. Petroleum Engr. Thesis submitted in partial fulfillment of the requirements for the M.P.E. Degree to the Graduate College of the University of Oklahonia, 1957. (2) I. Fatt, Trans. Am. I n s t . Mining Met. Engra., 20T, 144, 160, 164 (1956).
(3) A. E. Scheidegger, “The Physics of Flow Through Porous Media,” Toronto Univ. Press, Toronto, Canada, 1957. (4) W. Roae, “Studies of Waterflood Performance. 111. Use of Network Models,” Illinois State Geol. Survey, Circ. No. 237, Urbana, Ill.. 1957.
metrical patterns were used to show the effects of number of connections per pore. Different random number tables were used to define network distribution. Three pore-size distributions were used to study that effect upon the desaturation process. I n order to determine the effect of varying the size of the individual cylinders, Fatt assumed the length of each cylinder was proportional to its radius raised to the power, alpha. He tried different values for alpha and selected a value of - 1. All of Fatt’s calculations were performed assuming a continuous wetting phase. As a consequence, no wetting phase fluid could be trapped in a desaturation process. We have extended the work of Fatt by employing desaturation steps such that wetting’ fluid can be trapped, and we have studied parameters similar to those he studied in the desaturation process. More important, however, we have attempted to consider the elusive factor of wettability by treating the probability of entrance of displacing fluid into a wetting fluid-filled pore as a stochastic process. Experimental The Network Model.-We have defined a probabilistic model representing the pore geometry of porous media and have carried through an operation on the model analogous to the capillarypressure desaturation process employed in the “restored state” laboratory testing of petroleum reservoir rock core sample^.^ The system employed consisted of two fluids, analogous to a n oil-water system, in which the contact angle measured through the fluid initially found in the network was less than 90 degrees. The cylinders representing the pores were assumed to contain only one fluid a t a time; this allowed the displaced fluid to be trapped if no continuous path to the effluent end was available. Variations in the parameters of pore-size distribution, network distribution and number of connections per pore were studied to indicate their effects on the desaturation process. A modification of Fatt’s network model was employed in order to make calculations more convenient when studying the trapping of displaced fluid. Rather than showing a series of basic geometrical designs with an end view of the cylinders connected (5) W. Rose and W. A. Bruce, Trans. Am. Inat. Mining Met. Engrs., 886. 137 (1949),
Oct., 1959
FLUID-FLUID DISPLACEMENT AND WETTABILITY IN POROUS ROCKS
by lines to common junction points, the lines in the geometrical designs were replaced with side views of the cylinders. In this manner a certain number of cylinders were shown to terminate a t a given point according to the number of connections per pore. This alteration facilitated keeping track of the possible paths of fluid movement in the desaturation process. Three such regular networks, shown in Fig. 1, mere used as models, one having four connections per pore, one having six connections per pore, and one having ten connections per pore. An effort to distribute the number of connections per pore in a random manner was tried, but the manner of allocating the various connections proved inadequate for the size of model studied. The graphical representations of pores, such as those in Fig. 1, show a distinct orientation for each pore with respect to the other pores, but this orientation was not effective in the desaturation process. For any pressure gradient applied across the network, displacement was equally likely in all pores contiguous to the displacing phase having radius size corresponding to the imposed pressure gradient. Additional assumptions required for this model were that both fluids in the system were incompressible, the viscosity of each phase was constant and played no role in the quasi-static capillary pressure desaturation process under study, the displacement was piston-like with only one fluid occupying a pore a t a given time, and there was no interaction between fluid and solid matrix. In order that the wetting fluid might be trapped during the displacement process, it was necessary to devise a suitable method of entry for the displacing fluid and exit for the displaced fluid. The model used represented a vertical section of a core sample in a restored state capillary pressure desaturation apparatus. The displacing fluid was allowed to enter on three sides and the displaced fluid to exit on the fourth side. Further, it was assumed that a membrane containing pores, the largest size of which was less than the smallest pore in the network model, was placed a t the effluent end. This membrane was assumed to be preferentially wet by the wetting fluid which was displaced from the network model. The effects of gravity in the systems were neglected to simplify the problem. The interfacial tensionat fluid-fluid interfaces within the network was assumed constant. The wettability of the solid matrix, as measured by the fluid-fluid interfacial contact angle, was considered constant except in those modifications of the network where this parameter was specifically varied or otherwise investigated. Using the assumptions and simplifications discussed above, it was possible to express capillary pressure in the classical manner as a function of the inverse value of the cylindrical pore radius, and, further, to define a scaled capillary pressure equal to the inverse of pore radius, using Fatt's alpha of -1. I n carrying out a desaturation operation with the network model, a stepwise process was employed. The first increment of pressure to be selected for application to a model was infinitesimally greater than that necessary to displace wetting
1647
1~"
0
Ih NN
1mnnnnnr 6 CONNECTIONS-PPR-PORE.MODE L
8 0
DISPLACED
PHASE
D l l r L A C l N O PHASE
Fig. 1.-Network models having four, six and ten connections per pore. The distribution of displaced fluid that was trapped upon complete desaturation is shown for the model having ten connections per pore, and using pore size distribution one and raiidom number table two.
1.50
1.25 i2
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a e] 0.75
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20 40 60 80 100 Wetting phase saturation, %. Fig. 2.-Capillary pressure curves showing the effect of pore size distribution for a model having six connections per pore and using random number table one.
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CHARLES G. DODD AND 0. GERALD KIEL
Vol. 63
20 40 60 80 100 Wetting phase saturation, yo. Fig. 4.-Capillary pressure curves showing the effect of number of connections per pore for pore size distribution three and random number table one.
pressure gradient at the second step was selected t o be just greater than the capillary pressure required to desaturate the next smaller group of pores contiguous to those pores now filled with displacing fluid. This stepwise process was continued for the number of steps required to complete the desaturation process. The various pore-size distributions assumed were distributed in a random manner throughout the model. Sequences of unique random numbers were obtained from two sourceslBby copying a sequence selected at a random place in one of the tables and deleting any duplicates. After obtaining the desired sequence of random numbers, a one-to-one correspondence between these numbers and the ordered sequence of network position numbers was constructed. This gave each position a unique identification. The one-to-one correspondence was then re-ordered with respect to the random numbers. The pore-size distributions used were those cited by Fatt as being representative of a narrow distribution, an intermediate distribution, and a wide distribution, referred to herein as pore-size distributions one, two and three, respectively. The number of pores to be associated with each radius size was determined by multiplying the percentage times the total number of pores in the network and adjusting to the nearest whole number. A final check was made to assure that the sum of the number of pores equalled the total number of pores. If nl was the number of pores having a radius T I , then the first nl ordered random numbers were assigned the relative radius size rl. The next n2 pores were assigned rzl and the process was repeated until the entire group had been assigned. Having obtained a one-to-one relation between network location, its corresponding random number and its assigned pore radius, the groups were re-ordered so the position numbers formed a sequence from smallest to largest. The stepwise desaturation process was carried out in the manner indicated above by repeatedly imposing a capillary pressure gradient required to displace wetting fluid from the next smaller set of pores, checking to determine whether any pores containing displaced fluid were isolated so there was no continuous path for movement of displaced fluid, considering wetting fluid in such pores to be trapped and to remain in that condition for the entire desaturation process, and, finally, checking pores contiguous to those displaced by the displacing phase to ascertain if any additional pores would be displaced a t the existing pressure gradient. This sequence of steps of displacing, checking for trapping and checking for further displacement was repeated for each pore size in the network The number of displaced pores resulting from each successive sequence of steps was then tallied and assembled in an appropriate table outlining the results of the process. Allowance was made for those cases when pores having radii corresponding to pressure gradients smaller than that imposed a t the step under (6) (a) G . W. Snecdor, “Statietical Methods,” Iowa State College
fluid from the largest pores in the network contiguous to the upper three sides of the network. The
Press. Arnes. Is., 1946, p. 11-12. (b) The Rand Corporation, “A Million Random Digits with 100,000 Normal Deviates,” The Free Press, Glencoe, Ill., 1956, p. 191-192.
NUMBER OF CONNECTIONS PER PORE 0 FOUR QSlX
A S I X TO TEN
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.TEN
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.--9 9 3
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u
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20 40 60 80 100 Wetting phase saturation, yo. Fig. 3.-Capillary pressure curves showing the effect of number of connections per ore for pore size distribution two and random number t a b g two. 0
OF CONNECTIONS PER PORE FOUR
NUMBER
OSlX
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FLUID-FLUID DISPLACEMENT AND WETTABILITY IN POROUS ROCKS
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study were found contiguous to the displacing phase. RANDOM NUMBER TABLE 0 , . ONE The over-all process was repeated until desaturation 0,. Two was complete. The results then were tabulated. PORE SIZE DISTRIBUTION Finally, the capillary pressure-fluid saturation curve 1.50 0,O ONE was plotted using the functional relationship de0,. THREE veloped by Fatt. Representative capillary pressure NUMBER OF CONNECTIONS desaturation curves obtained in this manner are PER PORE shown in Fig. 2 for the three pore-size distributions. 0 , O FOUR 1.25 I o,m TEN Results Obtained with the Model.-The effect of the number of connections per pore was not $P separated easily from those of the other parameters. I I C 0 1.00 This was due to the slight variation necessary in the size and arrangement required to obtain a geometric configuration corresponding to a given number of a connections. The predominant effect shown in a d 0.75 Figs. 3 and 4 is believed to result from changes in % the number of connections. The effects of network distribution were deter- m mined by the use of two ordered random number 0.50 tables to define the distribution of pore sizes. Examples of the resulting capillary pressure curves are shown in Fig. 5. 0.25 The selection of the number of cylinders to be included in the various network models, i.e., the network size, was a compromise between the large number of pores required to represent adequately 01 I I the physical prototype and justify the use of a sto0 20 40 60 80 100 chastic process, and the limitations imposed by the Wetting phase saturation, yo. task of accounting manually for each step in the Fig. 5.-Capillary pressure curves showing the effect of desaturation process. This first study was done network distribution in two models. without the help of a digital computer, and it was limited to two-dimensional models containing from place observations as the jerky, irregular move310 to 480 pores in order to permit several sets of ments of liquid droplets moving over heterogeneous parameters to be evaluated in the time available. solid surfaces, the intermittent advance of fluidThe capillary pressure curves obtained from this fluid interfaces in glass tubes not uniformly clean, work, however, were sufficiently like those obtained and the displacements of microscopic interfaces in from rock core samples to give confidence in the porous bead packs’ support the plausibility of the significance of the results. Extensions of the work proposition that fluid-fluid interfacial movement now are being programmed for digital computers and displacement in a porous solid is a random procemploying larger models of varying geometry. ess. The unsolved problem of contact angle hysComparisons of three of the curves obtained with teresis supports the stochastic concept, but it also leads to the thought that the statistics involved in 310- and 480-pore models are shown in Fig. 6. Characterization of Wettabi1ity.-During the the process should be biased in terms of surface enlast few steps of the desaturation process where ergies, or some related physical factor, when assignsimultaneous movement in pores of equal size was ing probabilities of interfacial movement in a given assumed, the possible continuous paths for desatu- situation. This line of reasoning has led to an exration were observed to be few in number. As few amination of the possibility of applying stochastic as one continuous path to the effluent end was processes such as Monte Carlo methods* to the defound to exist for several of the desaturation calcu- termination or characterization of the effective wetlations. This made it necessary for numerous tability of a given sample of porous solid with repores having the same radius to empty through the spect to the displacement of one fluid by another existing path. As a result, the movement was not within its pores. The balance of the work with the first simple netsimultaneous but rather one in which the more remote pores were desaturated first followed by work model described above was based on the assuccessively less remote pores. This problem made sumption of a zero pore-size distribution network, i t necessary to consider the necessity of modifying i.e., a network containing equal-sized pores. Since one of the basic premises upon which the network the problem of direction between pores did not exist, study had been based up to this time, i.e., the con- this problem was approached in a manner analogous cept of simultaneous desaturation of all pores con- to that of a random walk problem in one dimension tiguous to the displacing fluid and having a pore having no reflection barrier^.^ For exemplary purposes, the network model having four connections size such that displacement could occur. Further reflection upon this development led to per pore was used for these calculations. (7) A. Chatenever and J. C. Calhoun, Jr., Trans. Am. Inst. Mining the conclusion that the problem presented was very Engrs., 196, 149 (1952). similar to that of characterizing the “wettability” Met. (8) H. A. Meyer, Ed., “Symposium on Monte Caylo Methods.” of the internal surfaces of a porous rock sample. John Wiley and Sons. New York, N. Y., 1956. Intuitive considerations based on such common(9) S. Chandrasekhar, Rev. Mod. P h y s . , 16, 1 (1948).
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CHARLES G. DODDAND 0. GERALD KIEL
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Vol. 63
The second case considered was that the probability of movement or no movement was equal, Le., M = S = 1/2. Again, by substituting these values along with the value for n, each of the terms was evaluated. Since M = S, it was necessary to evaluate only the coefficients of the terms. The general rth coefficient is n(n
1
2 I *a 1.00 -
- l ) ( n - 2 ) . . .(n - r ( r - 1)1
+ 2)
If this coefficient has the maximum numerical value, the most probable number of simultaneous occurrences would be n - T 1. I n terms of n, this R 2 value is 4 2 or '/z(n f l),according to whether n is r;l even or odd. 20.75 After determining the number of pores into which the displacing phase would enter simultaneously, a z means had to be devised to differentiate among the d u3 n possible locations to select the network locations 0.50 9 where displacement would occur. The uniqueness was defined by the random numbers associated with each network position number. The method devised to determine the positions was to extract from 0.25 a table of one-to-one correspondence between the random numbers and network locations the n network locations contiguous to the displacing fluid. This set of n random numbers was ordered sequen0 20 40 60 80 100 tially with the first pair having the smallest random number and the last pair the largest. I n the first Wetting phase saturation, %. Fig. 6.-Capillary pressure curves showing the effect of step of the desaturation process the first n/2 or model size for pore size distribution one in a model having 'lz(n 1) pairs from the ordered group were sesix connections per pore. lected starting with the pair having the smallest If one considers a zero pore-size distribution net- random number having zero as the hundreds digit. work model in its initial state, totally saturated with The displacing fluid was allowed to desaturate these the fluid to be displaced, and with n pores contigu- pores. A check was made to determine whether any ous to the displacing phase, one would expect all n pores containing the displaced phase were cut off pores to be desaturated when the (scaled) capillary- from a continuous path for desaturation. Again, the pressure gradient exceeded the displacement pres- process of ordering the set of n pairs of numbers corresure of the individual pores. Under the conditions sponding to all network positions contiguous to the set up for these models, displacement would occur displacing phase was carried out. Then the n/2 over an infinite period of time, and each set of pores or ' l z ( n 1) pairs were taken in sequential order contiguous to the displacing phase would be de- starting with the smallest random number having a saturated before desaturation of the next set would three in the hundreds position. The starting point begin. To express the probability of displacement for the next step was the smallest number having a in a pore, it was assumed that, of the n possible six in the hundreds position. Each succeeding positions of entry for the displacing fluid, a certain starting point was advanced by adding three to the number would occur. This number of identical hundreds position, modulus ten, until the desaturaevents which was most probable was obtained from tion process was completed. The results indicated a larger residual than when the probability of movethe binomial expansion ment was one. ( M + 8)" = M" + nM"-'S ... For the final case the displacement process was n(n - 1). . , ( n - r + 2)Mndr+1P - 1 considered as a series of displacements where one ( r - l)! pore was displaced before another displacement be. . . + nMS"-' + S" gan. This was equivalent to considering the term where nM8n-l as the term having the maximum value. M = probability of success for instantaneous movement, The reason for not considering the case where S = S = probability of failure for instantaneous movement, 1was that no displacement would occur under these M + S = l conditions. The procedure of ordering the random The first case considered was that for which M = numbers and selecting the starting positions was the 1 and S = 0, ie., it was certain that the displacing same as that described above. The resulting refluid would enter the pore and displace the fluid on sidual saturation was the largest. If oil were the displacing fluid and water the discontact. Under this assumption the most probable number of events occurring simultaneously would be placed fluid, it seems reasonable to interpret these n. The calculations for this case were made ac- results as indicating a greater probability of diucording to the procedure used for the desaturation placement for the least water-wet system. The case process described above. Under this most extreme where the probability of movement is in the neighborhood of one would correspond to an oil-wet syscondition there was not total desaturation.
+
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+
+
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FLUID-FLUID DISPLACEMENT AND WETTABILITY IN POROUS ROCKS
Oct., 1959
PROBABILITY OF MOVEMENT OM 1
om :1/2
O l
I
I
0
20 40 60 80 100 Wetting phase saturation, %. Fig. 7.-Capillary pressure curves showing the effect of change in probability of movement for a model having four connections per pore, using pore size distribution three and random number table one.
tem. The case where the probability of movement is in the neighborhood of l / n would correspond to a water-wet model. Values between these two extremes, such as the case where the probability of movement is one-half, would define the degree of wettability of the solid for the fluid-fluid system. The residual saturations, calculated for a model having four connections per pore, were 46, 16 and 2%, respectively, for M values of l/n, l / Z , and 1. The residual saturations determined by assuming a value of M = ' / z for network models having four, six, ten and an infinite number of connections per pore were 16, 12,9 and O%, respectively. Finally, the probability of displacement principle was applied to a simple network model having four connections per pore and a wide distribution of pore sizes. Capillary desaturatioii curves were calculated for values of M = 1 and M = l/z. These curves are shown in Fig. 7. It is seen that the assumed variations in wettability affected only the residual fluid saturation. Discussion The results of the work described in this paper confirm and extend the developments of Fatt.2 I n general, our results indicate the wider the pore size distribution of the model, the lower is the residual displaced fluid saturation. In comparing the two random number tables used in constructing the network models it was found that, for a model of the size studied, there was a greater sensitivity to network distribution in models having a smaller number of connections per pore. I n particular, for the
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models incorporating pore-size distribution one, a narrow distribution, there was a considerable difference in the curves for the models obtained with different random number tables. The residual saturations were most sensitive to changes in the number of connections per pore. The distributions of trapped residual fluid, such as that shown in Fig. 1, when correlated with pore size, indicated that the displaced fluid was trapped largely in the smaller pores within the network, and this was emphasized in the models having the larger number of connections per pore. In general, it appeared that the models having ten connections per pore appeared to be more representative of the properties of relatively homogeneous sandstone rocks. The network model, when constructed with due consideration of the properties of the rock prototype it represents, appears to be a valid means of studying quasi-static fluid-fluid displacements such as those involved in the determination of capillary pressure desaturation curves, particularly when the object of the investigation is to study the interrelationships between the parameters that control capillary pressure equilibrium. Applications of the model to dynamic systems have been made by Fatt2and Rose.4 The subject of greater interest to the writers is the study of the complex problem of fluid-fluid wettability within porous media. The ideas concerning wettability discussed in this paper are of an elementary nature compared to the apparent potentialities for applying stochastic processes, such as the Monte Carlo method, to the problem of defining and determining wettability. The concept of equating the probability of entrance of displacing fluid to the wettability a t each step in the desaturation process is a simple one, but it does involve a more or less random determination of the pores to be displaced. The fraction of the total number of contiguous pores a t any step is determined by the assigned probability, or wettability, but the selection of the actual pores on the model for displacement is done in a pseudo-random manner. A more sophisticated approach to the problem might involve an assignment of an assumed distribution of wettabilities to each pore in the model initially. If the wettability of a pore were considered to be determined by the more conventional concept of the cosine of the interfacial contact angle, it would be logical to determine which pores were displaced a t each step by the quotient of the cosine divided by the pore radius (or by the product of interfacial curvature times the cosine), rather than by the reciprocal of the pore radius alone. The assumed distribution of wettabilities might take the form of a normal Gaussian distribution or it might be selectively biased in accordance with some sampling scheme determined by the perceptive intuition or experience of the investigator, Such an approach may be described as a Monte Carlo method. Brown and Fatt'" have advocated the use of the concept of fractional wettability which they defined in terms of the fractional internal surface area in contact with a single fluid phase. To determine fractional water wettability they utilized nuclear (IO) R. J. S. Brown and I. Fatt, Trnne. Am. Inst. Mining M e ; . Engrs., 207, 262 (1956).
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CHARLES G. DODD AND 0. GERALD KIEL
magnetic resonance experiments to estimate the fraction of internal rock surface in contact with water. Holbrook and Bernard” used a simple dye adsorption test to measure the same quantity. Iwankow12 prepared unconsolidated sand packs containing known fractions of water-wet grains and silane-treated, oil-wet grains. The sands were mixed well before packing, the packs were saturated with water, and the water-saturated packs were flooded with four pore volumes of oil. Holding other parameters constant, he found that the residual water saturations varied systematically with the fraction of oil-wet sand in the pack. I n these experiments a normal distribution of oil-wet sand grains, and hence oil-wet surface area, was assumed throughout the sand packs. Fatt and Klikoffla packed similar mixtures of water-wet and oil-wet sand grains for use in capillary pressure drainage experiments in which water was displaced by oil. In their experiments with a sand pack composed of uniformly sized grains, the capillary pressure curves were lowered and the residual saturations were reduced as the fractional oil-wettability increased. The decrease in residual water saturation is in agreement with our calculations referred to above. The decrease in the “plateau” of the capillary pressure curve with increasing oil-wettability is normal, but it was not found in the only pertinent calculation made with our network model, shown on Fig 7, because the assumption of a wettability number did not result in trapping any additional fluid in this pore size region. We did not assume a distribution of wettability such as characterized the Fatt and Klikoff experiment. Their experiments and Iwankow’s work show the need for the use of some kind of wettability distribution in the network models. I n other experiments Fatt and Klikoff found that the shape of the capillary pressure curves was changed when a wide sand-grain-size distribution was used to make the sand packs and the fractional oil wettability was increased in the same experi(11) 0. C. Holbrook and G . G . Bernard, Paper No. 896-G,Society of Petroleum Engineers-A.I.M.E. Meeting, Dallas, Texas, Oat. 6-9, 1957. (12) E.N. Iwankow, “A Correlation of Interstitial Water Saturation and Heterogeneous Wettability,” M. Petroleum Engr. Thesis, University of Oklahoma, Norman, 1958. (13) I. Fatt and W. A . Klikoff. Jr.. “Effect of Fractional Wettability on Multiphase Flow Through Porous Media.” paper presented at Amer. Inst. Chem. Eng. Meeting, Kansas City, Mo.. May 17,
1959.
Vol. 63
ment. I n one case they treated only the finest grains with silane and found a much higher residual water saturation as well as a change in shape of the curve. This work shows the need for considering the quotient of wettability divided by pore size in determining displacement of a given set of pores when desaturating the network models. DISCUSSION I. FATT(University of California at Berkeley).-I am keenly aware of the amount of time put into this work. I believe that this type of approach is applicable to many important unsolved problems. Your model is a static one in which the presstire in the network is constant a t each instant. It would be useful to make a dynamic model which takes the ressure gradient into account and shows how the system beRaves while fluids are actually moving through it. c. G. DoDD.-This is certainly worthy of further effort but it would require a faster computer and a very large machine memory in comparison with the quasi-static problem. M. J. VOLD(University of Southern California).-You have used Fatt’s relationship between pore diameter and pore length. Might it not be more realistic to use a random distribution of pore lengths? Doing this would enable you to calculate the pressure drop through each pore and would facilitate converting the static model into a dynamic one. C. G. DoDD.-This is an int,eresting suggestion. There is no analytical or procedural problem involved in distributing tbe pore lengths randomly throughout the model or in making the saturation calculations. The difficulty is to find a machine big enough to permit all of the data to be included in memory locations corresponding t80each pore location. I. FATT.-I made use of a random combination of lengths and radii in one calculation and discussed it in my AIME paper in 1956. The results are surprisingly close to those obtained with the inverse relation Dr. Dodd has used. I have no explanation as to why this should be so. M. J. VOLD.-DO you have any guess as to how many pores will have to be included in a model network to get, a statistically valid result? I. FATT.-A sandstone cube 1/4‘‘ on a side has about 10,000 pores. We made capillary pressure curves on a cube 2” on a side and got reliable results. Then we began slicing it up into smaller and smaller bodies. We began to get serious fluctuations when the cube was only l/d‘’ on a side, but this probably was due to mechanical inhomogeneity in the block. The computer model would be a homogeneous network and it may be that many less than 10,000 pores are required. C. G. DoDD.--We may be helped in this problem of determining network size and assigning pore lengths and wettability distributions if we have insight concerning the most applicable statistics for the problem. I n other words, if we know beforehand how the sampling scheme should be blased we can select the appropriate distribution. This is the Monte Carlo A proach. We need essentially to know part of our answer atead of time to minimize the parameters that must be varied and, hence, the machine time required.
