Osmotic Repulsive Pressure
Ladd 1960 explains that: "The exchangeable cations are attracted to the clay particles by the negative electric field arising from the negative charge on the particles. Hence, the electric field acts as a semi-permeable membrane in that it will allow water to enter the double layer but will not allow the exchangeable cations to leave the double layer." Thus, when the total ion cations plus anions concentration in the double-layer between the clay particles is higher than that in the aqueous pore fluid, the water in the pore fluid diffuses into the double-layer to dilute its ion concentration. This phenomenon creates an osmotic repulsive pressure between the clay particles. As a result, the interparticle distance increases causing the clay to expand and swell. Therefore, the driving force for osmotic pressure is the difference of the total ion concentrations between the clay double-layer, cc, and the surrounding pore fluid For only very dilute aqueous solutions, the van't Hoff equation given below can be used to estimate the osmotic pressure Ladd, 1960
Non-ideal models are required for concentrate solutions.
Contents
Water Absorption Rate
Civan et al. 1989 assumed that water diffuses through the solid matrix according to Pick's second law over a short distance near the surface of the solid exposed to aqueous solution, because the coefficient of water diffusion in solid is small. Thus, the water absorption in the solid can be predicted by the one-dimensional transientstate diffusion equation:
where c0 and c are the initial and instantaneous water concentrations, respectively, in the solid, Cj is the water concentration of the aqueous solution, z is the distance from the pore surface, t is the actual contact time, k is the film mass transfer coefficient, and D is the diffusivity coefficient in the solid matrix. Eq. 2-4 expresses that the water diffusion into clay is hindered by the stagnant fluid film over the clay surface. Thus, similar to Civan (1997), the analytical solution of Eqs. 2-2 through 2-5 can be used to express the cumulative amount of water diffusing into the solid surface as given by Crank (1956):
where h = kID. Civan et al. (1989) have resorted to a simplified approach by assuming that the film mass transfer coefficient k in Eq. 2-4 is sufficiently largeso that Eq. 2-4 becomes:
and, therefore, an analytical solution of Eqs. 2-2, 3, 8, and 5 according to Crank (1956) yields the expression for the cumulative and rate of water absorption, respectively, as:
The rate of formation damage by clay swelling also depends on the variation of the water concentration in the aqueous solution flowing through porous rock. Whereas, the analytical expressions given above assume constant water concentrations in the aqueous pore fluid. However, they can be corrected for variable water concentrations by an application of Duhamel's theorem. For example, if the time-dependent water concentration at the pore surface is given by:
where F(t) is a prescribed time-dependent function, the analytic solution can be obtained as illustrated, by Carslaw and Jaeger (1959). Then, using Eq. 2-10, the rate of water absorption can be expressed by:
However, in the applications presented here the water concentrations involved in the laboratory experiments are essentially constant. The preceding derivations assume a plane surface as supposed to a curved pore surface. From the practical point of view, it appears reasonable because of the very short depth of penetration of the water from the solid-fluid contact surface.
Clay Swelling Coefficient
The rate of clayey formation swelling is derived from the definition of the isothermal swelling coefficient given by Collins, 1961:
V and Vw are the volumes of the solid and the water absorbed, respectively. Ohen and Civan 1991 used the expression given by Nayak and Christensen 1970 for the swelling coefficient:
in which c is the water concentration in the solid and CI is the plasticity index. <;, and qt are some empirical coefficients, m is an exponent. Chang and Civan 1997 used the expression given by Seed et al. 1962:
where Cc is the clay content of porous rock as weight percent, PI is the plasticity index, and k' is an empirical constant.
Water Content During Clay Swelling
The rate of water retainment of clay minerals is assumed proportional with the water absorption rate, 5, and the deviation of the instantaneous water content from the saturation water content as:
where kw is a water retainment rate constant, w denotes the weight percent of water in clay and the subscripts o and t refer to the initial (t = 0) and terminal (t -» °o) conditions, respectively. An analytical solution of Eqs. 2-16 and 17 yields. Osisanya and Chenevert (1996) measured the variation of the water content of the Wellington shale exposed to deionized water.
shows the correlation of their data with Eq. 2-18 using Eq. 2-6. The best fits were obtained using w0= 2.7 wt.%, wt= 3.27 wt.%, A = kw(c{ - c0)/ h = 0.26 and h-Jo = \ for their Gage 1 data, w0 = 2.77 wt.%, wt = 3.28 wt.%, A = 0.06 and h^D = 0.8 for their Gage 2 data, and w0= 2.77 wt.%, wt = 3.28 wt.%, A = 0.035 and h-^j~D = 0.8 for their Gage 3 data. Brownell (1976) reports the data of the moisture content of a dried clay piece containing montmorillonite soaked in water.
Time-Dependent Clay Expansion Coefficient
By contact with water the swelling clay particles absorb water and expand.The rate of volume increase is assumed proportional to the water absorption rate, 5, and the deviation of the instantaneous volume from the terminal swollen volume that will be achieved at saturation, (Vt - V). Therefore, the rate equation is written as:
where at is the terminal expansion coefficient at saturation. ka is the rate coefficient of expansion. Seed et al. (1962), Blomquist and Portigo (1962), Chenevert (1970), and Wild et al. (1996) measured the rates of expansion of the samples of compacted sandy clay, hydrogen soil, typical shales, and lime-stabilized kaolinite cylinders containing gypsum and ground granulated blast furnace slag, respectively.
Wild et al. (1996) tested lime-stabilized compacted kaolinite cylinders containing gypsum and ground granulated blast furnace slag. After oistcuring for certain periods, they soaked these samples in water and measured the linear expansion of the samples. The second set of data is for a 28-day moist-cured kaolinite containing 6% lime and 4% gypsum. The third set of data is for a 28-day moistcured kaolinite containing 2% lime, 4% gypsum and 8% ground granulated blast furnace slug. The best fits of Eq. 2-22 using Eq. 2-6 to the first, second, and third data sets were obtained with A = k(cl - c0}lh =1.1, W# = 1.0 and a, = 10.8 vol.%, A = 20, h^D = 0.2 and cc, = 1.48 vol.%, and A = 2.4, H-jD = 0.7 and oc, = 0.655 vol.%, respectively. Ladd (1960) measured the volume change and water content of the compacted Vicksburg Buckshot clay samples during swelling. For a linear plot of Ladd's data first, the S term is eliminated between Eqs. 2-18 and 22 to yield.
References
[1] Amaefule, J. O., Kersey, D. G., Norman, D. L., & Shannon, P. M., "Advances in Formation Damage Assessment and Control Strategies," CIM Paper No. 88-39-65, Proceedings of the 39th Annual Technical Meeting of Petroleum Society of CIM and Canadian Gas Processors Association, Calgary, Alberta, June 12-16, 1988, 16 p.
[2] Amaefule, J. O., Altunbay, M., Tiab, D., Kersey, D. G., & Keelan, D. K., "Enhanced Reservoir Description: Using Core and Log Data to Identify Hydraulic (Flow) Units and Predict Permeability in Uncored Intervals/Wells," SPE 26436 paper, Proceedings of the 68th Annual Technical Conference and Exhibition of the SPE held in Houston, Texas, October 3-6, 1993, pp. 205-220.
[3] Ballard, T. J., Beare, S. P., & Lawless, T. A., "Fundamentals of Shale Stabilization: Water Transport through Shales," SPE Formation Evaluation, June 1994, pp. 129-134.
[4] Blomquist, G. C., & Portigo, J. M., "Moisture, Density and Volume Change Relationships of Clay Soils Expressed as Constants of Proportionality," Bull. Highway Res. Board, No. 313, 1962, pp. 78-91.
[5] Brownell, W. E., Structural Clay Products, Springer-Verlag, New York, 1976, 231 p.
[6] Bucke, D. P., & Mankin, C. J., "Clay-Mineral Diagenesis Within Interlaminated Shales and Sandstones," /. Sedimentary Petrology, Vol. 41, No. 4, December 1971, pp. 971-981.
[7] Carslaw, H. S., & Jaeger, J. C., Conduction of Heat in Solids, Second ed., Oxford University Press, 1959, 510 p.
[8] Chang, F. F., & Civan, F., "Predictability of Formation Damage by Modeling Chemical and Mechanical Processes," SPE 23793 paper, Proceedings of the SPE International Symposium on Formation Damage Control, February 26-27, 1992, Lafayette, Louisiana, pp. 293-312.
[9] Chang, F. F., & Civan, F., "Practical Model for Chemically Induced Formation Damage," /. of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 123-137.
[10] Chenevert, M. E., "Shale Alteration by Water Adsorption," JPT, September 1970, pp. 1141-1148.
[11] Chilingarian, G. V., & Vorabutr, P., Drilling and Drilling Fluids, Developments in Petroleum Science, 11, Elsevier Scientific Publishing Co., New York, 1981.
[12] Civan, R, "Predictability of Formation Damage: An Assessment Study and Generalized Models," Final Report, U.S. DOE Contract No. DE-AC22- 90BC14658, April 1994.
[13] Civan, F., "Modeling and Simulation of Formation Damage by Organic Deposition," Proceedings of the First International Symposium on Colloid Chemistry in Oil Production: Asphaltenes and Wax Deposition, ISCOP'95, Rio de Janeiro, Brazil, November 26-29, 1995, pp. 102-107.
[14] Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31101, Proceedings of the SPE Formation Damage Symposium, Lafayette, LA, February 14-15, 1996, pp. 311-326.
[15] Civan, F, "Interactions of the Horizontal Wellbore Hydraulics and Formation Damage," SPE 35213, Proceedings of the SPE Permian Basin Oil & Gas Recovery Conf., Midland, TX, March 27-29, 1996, pp. 561-569.
[16] Civan, F., "Model for Interpretation and Correlation of Contact Angle Measurements," Jour. Colloid and Interface Science, Vol. 192, 1997, pp. 500-502.
[17] Civan, F., "Interpretation and Correlations of Clay Swelling Measurements," Paper SPE 52134, Proceedings of the 1999 SPE Mid-Continent Operations Symposium, 28-31 March 1999, Oklahoma City, OK.
[18] Civan, F., & Knapp, R. M., "Effect of Clay Swelling and Fines Migration on Formation Permeability," SPE Paper No. 16235, Proceedings of the SPE Production Operations Symposium, OKC, 1987, pp. 475-483.
[19] Civan, F., Knapp, R. M., & Ohen, H. A., "Alteration of Permeability by Fine Particle Processes," J. Petroleum Science and Engineering, Vol. 3, Nos. 1/2, October 1989, pp. 65-79.
[20] Collins, E. R., Flow of Fluids Through Porous Materials, Penn Well Publishing Co., Tulsa, Oklahoma, 1961, 270 p.
[21] Crank, J., The Mathematics of Diffusion, Oxford University Press, London, 1956, 347 p.
[22] Degens, E. T., Geochemistry of Sediments (A Brief Survey), Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965, 342 p.
[23] Ezzat, A. M., "Completion Fluids Design Criteria and Current Technology Weaknesses, SPE 19434 paper, the SPE Formation Damage Control Symposium held in Lafayette, Louisiana, February 22-23, 1990, pp. 255-266.
[24] Grim, R. E., "Modern Concepts of Clay Minerals," Jour. Geology, Vol. 50, No. 3, April-May 1942, pp. 225-275.
[25] Grim, R. E., Clay Mineralogy, McGraw-Hill, New York, New York, 1953, 384 p.
[26] Hart, R. T., Fekete, T., & Flock, D. L., "The Plugging Effect of Bacteria in Sandstone Systems," Canadian Mining and Metallurgical Bulletin, Vol. 53, 1960, pp. 495-501.
[27] Hayatdavoudi, A., "Changing Chemophysical Properties of Formation and Drilling Fluid Enhances Penetration Rate and Bit Life," Paper No. SPE 50729, Proceedings of the SPE International Symposium on Oil Field Chemistry held in Houston, Texas, 16-19 February 1999, pp. 273-285.
[28] Hayatdavoudi, A., private communication, 1999.
[29] Hughes, R. V., "The Application of Modern Clay Concepts to Oil Field Development," pp. 151-167, in Drilling and Production Practice 1950, American Petroleum Institute, New York, NY, 1951, 344 p.
[30] Khilar, K. C., Fogler, H. S., "Water Sensitivity of Sandstones," SPEJ, February 1983, pp. 55-64.
[31] Kia, S. F., Fogler, H. S., & Reed, M. G., "Effect of Salt Composition on Clay Release in Berea Sandstones," SPE 16254, February 1987.
[32] Ladd, C. C., "Mechanisms of Swelling by Compacted Clay," in Highway Research Board Bulletin 245, High Research Board, National Research Council, Publication 731, Washington, D.C., 1960, pp. 10-26
[33] Liu, X., Civan, F, & Evans, R. D., "Correlation of the Non-Darcy Flow Coefficient, J. of Canadian Petroleum Technology, Vol. 34, No. 10, 1995, pp. 50-54.
[34] Lynn, J. D., & Nasr-El-Din, H. A., "Evaluation of Formation Damage due to Frac Stimulation of Saudi Arabian Clastic Reservoir," J. of Petroleum Science and Engineering, Vol. 21, Nos. 3-4, 1998, pp. 179-201.
[35] Mancini, E. A., "Characterization of Sandstone Heterogeneity in Carboniferous Reservoirs for Increased Recovery of Oil and Gas from Foreland Basins," Contract No. FG07-90BC14448, in EOR-DOE/BC- 90/4 Progress Review, No. 64, pp. 79-83, U.S. Department of Energy, Bartlesville, Oklahoma, May 1991, 129 p.
[36] Mohan, K. K., & Fogler, H. S., "Colloidally Induced Smecticic Fines Migration: Existance of Microquakes," AIChE J., Vol. 43, No. 3, March 1997, pp. 565-576.
[37] Mondshine, T. C., "A New Potassium Based Mud System," Paper No. SPE 4516, 48th Annual Fall Meeting of SPE of AIME, October 1973.
[38] Mondshine, T. C., "New Technique Determines Oil Mud Salinity Needs in Shale Drilling," Oil and Gas J., July 14, 1969.
[39] Mungan, N., "Discussion of An Overview of Formation Damage," JPT, Vol. 41, No. 11, November 1989, p. 1224.
[40] Nayak, N. V, & Christensen, R. W., "Swelling Characteristics of Compacted Expansive Soils," Clay and Clay Mineral, Vol. 19, No. 4, December 1970, pp. 251-261.
[41] Neasham, J. W., "The Morphology of Dispersed Clay in Sandstone Reservoirs and Its Effect on Sandstone Shaliness, Pore Space and Fluid Flow Properties," SPE 6858, Proceedings of the 52nd Annual Fall Technical Conference and Exhibition of the SPE of AIME held in Denver, Colorado, October 9-12, 1977, pp. 184-191.
[42] Ngwenya, B. T., Elphick, S. C, & Shimmield, G. B., "Reservoir Sensitivity to Water Flooding: An Experimental Study of Seawater Injection in a North Sea Reservoir Analog," AAPG Bulletin, Vol. 79, No. 2, February 1995, pp. 285-304.
[43] Norrish, K., "The Swelling of Montmorillonite," Discussions Faraday Soc., Vol. 18, 1954, p. 120.
[44] Ohen, H. A., & Civan, E, "Simulation of Formation Damage in Petroleum Reservoirs," SPE 19420 paper, Proceedings of the 1990 SPE Symposium on Formation Damage Control, Lafayette, Louisiana, February 22-23, 1990, pp. 185-200.
[45] Ohen, H. A., & Civan, F., "Simulation of Formation Damage in Petroleum Reservoirs," SPE Advanced Technology Series, Vol. 1, No. 1, pp. 27-35, April 1993.
[46] Osisanya, S. O., & Chenevert, M. E., "Physico-Chemical Modelling of Wellbore Stability in Shale Formations," J. of Canadian Petroleum Technology, Vol. 35, No. 2, February 1996, pp. 53-63.
[47] Pittman, E. D., & Thomas, J. B., "Some Applications of Scanning Electron Microscopy to the Study of Reservoir Rock," SPE 7550 paper, November 1979, pp. 1375-1380.
[48] Reed, M. G., "Formation Permeability Damage by Mica Alteration and Carbonate Dissolution," JPT, September 1977, pp. 1056-1060.
[49] Rogers, 1963.
[50] Sahimi, M., Flow and Transport in Porous Media and Fractured Rock, VCR, Weinheim, 1995, 482 p.
[51] Schechter, R. S., Oil Well Stimulation, Prentice Hall, Englewood Cliffs, New Jersey, 1992, 602 p.
[52] Seed, H. B., Mitchell, J. K., & Chan, C. K., "Studies of Swell and Swell Pressure Characteristics of Compacted Clays, Bull. Highway Res. Board, No. 313, 1962, pp. 12-39.
[53] Sharma, M. M., & Yortsos, Y. C., "Fines Migration in Porous Media," AIChE J., Vol. 33, No. 10, 1987, pp. 1654-1662.
[54] Simpson, J. P., "Drilling Fluid Filtration Under Simulated Downhole Conditions," SPE Paper 4779, 1974, 14 p.
[55] Weaver, C. E., & Pollard, L. D., The Chemistry of Clay Minerals, Elsevier, New York, 1973, 213 p.
[56] Welton, J. E., "SEM Petrology Atlas," American Association of Petroleum Geologists, Tulsa, Oklahoma, 1984, 237 p.
[57] Wild, S., Kinuthia, J. M., Robinson, R. B., & Humphreys, I., "Effect of Ground Granulated Blast Furnace Slag (GGBS) on the Strength and Swelling Properties of Lime-Stabilized Kaolinite in the Presence of Sulphates," Clay Minerals, Vol. 31, 1996, pp. 423-433.
[58] Wojtanowicz, A. K., Krilov, Z., & Langlinais, J. P., "Experimental Determination of Formation Damage Pore Blocking Mechanisms," Trans, of the ASME, Journal of Energy Resources Technology, Vol. 110, 1988, pp. 34-42.
[59] Wojtanowicz, A. K., Krilov, Z., & Langlinais, J. P., "Study on the Effect of Pore Blocking Mechanisms on Formation Damage," Paper SPE 16233 presented at Society of Petroleum Engineers Symposium, Oklahoma City, Oklahoma, March 8-10, 1987, pp. 449-463.
[60] Zhou, Z., "Construction and Application of Clay-Swelling Diagrams by Use of XRD Methods," JPT, April 1995, p. 306.
[61] Zhou, Z., Gunter, W. D., Kadatz, B., & Cameron, S., "Effect of Clay Swelling on Reservoir Quality," Paper No. 94-54, Journal of Canadian Petroleum Technology, Vol. 35, 1996, pp. 18-23.
[62] Zhou, Z., Cameron, S., Kadatz, B., & Gunter, W. D., "Clay Swelling Diagrams: Their Applications in Formation Damage Control," SPE Journal, Vol. 2, 1997, pp. 99-106.
