32 Reservoir Formation Damage (2-13) V and V w 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: (2-14) in which c is the water concentration in the solid and CI is the plasticity index. <;, and q t are some empirical coefficients, m is an exponent. Chang and Civan (1997) used the expression given by Seed et al. (1962): c - 10) 244 (2-15) where C c 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: = k w S(w t -w) subject to the initial condition (2-16) (2-17) where k w is a water retainment rate constant, w denotes the weight per- cent 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: = w t -(w t -w 0 ) exp (-k w S) (2-1 8) Osisanya and Chenevert (1996) measured the variation of the water content of the Wellington shale exposed to deionized water. Figure 2-20 Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 33 10 15 1 1 « (hr • Osisanya and Chenevert Gage 1 data Correlation of the Gage 1 data % Osisanya and Chenevert Gage 2 data Correlation of the Gage 2 data X Osisanya and Chenevert Gage 3 data Correlation of the Gage 3 data Figure 2-20. Correlation of water pickup during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers). shows the correlation of their data with Eq. 2-18 using Eq. 2-6. The best fits were obtained using w 0 = 2.7 wt.%, w t = 3.27 wt.%, A = k w (c { - c 0 )/ h = 0.26 and h-Jo = \ for their Gage 1 data, w 0 = 2.77 wt.%, w t = 3.28 wt.%, A = 0.06 and h^D = 0.8 for their Gage 2 data, and w 0 = 2.77 wt.%, w t = 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. Figure 2-21 shows a correlation of the data with Eq. 2-18 using Eq. 2-6. The best fit was obtained using w 0 = 0%, w t = 14.2 wt.%, A = 0.2 and Time-Dependent Clay Expansion Coefficient By contact with water the swelling clay particles absorb water and ex- pand. 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, (V t - V). Therefore, the rate equation is written as: 34 Reservoir Formation Damage e \3 o 1 S n t\ . ft . • Brown Brown x" ^ * en data tion of the eO data , / ; X lx » 468 t 1/2 (hr in ) 10 12 Figure 2-21. Correlation of water pickup during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers). = k a s(v t -v) subject to (2-19) (2-20) A: is a rate constant. Thus, solving Eqs. 2-19 and 20 yields: V = V t -(V t -V 0 )exp(-k a S) (2-21) from which the expansion coefficient of a unit clay volume is determined as: (2-22) where a t is the terminal expansion coefficient at saturation. k a 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-stabi- lized kaolinite cylinders containing gypsum and ground granulated blast furnace slag, respectively. Figures 2-22 and 2-23 show the correlation of their data with Eq. 2-22 using Eq. 2-6. The best matches of the data Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 35 25 2 . 1 *i •J . 0 5 n > U I C : ! .,& f t • I 1 ;*/ \/S+ £# F i i f ^ 4 \S^ /- \s^ S 5 10 15 20 2f t ira (day 1 «) + A X Seed et al data Correlation of Seed et al data Bfomqutet and Fbrtigo data Correlation of Btomquist and Portigo data Chenevert data Correlation of Chenevert data Figure 2-22. Correlation of volume change during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers). •\ 0 , 0 i. 1 1 1 1 k ^ ^< 0.5 1 • Wil CYi .>, Wil Co X W8 fV, X y / •*> y / ^» / 1.5 2 2.5 3 t 1/2 (day in ) d et al Fig.5 data rrelation of Wild et al Fig.5 data d et al Fig.6 data -relation of Wad et al Rg.6 data d et al Rg.8 data •relation of Wild et al Fig.8 data Figure 2-23. Correlation of volume change during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers). 36 Reservoir Formation Damage shown in Figure 2-22 were obtained using A = k(c { - c^lh = 0.085, h-\TJ5 = 0.67, and a t = 100(V, - V 0 }IV 0 = 3.7 vol.% for the Figure 2 data of Seed et al. (1962), A = 2.2, h^D = 1.1, and a, = (V t - V 0 }IV 0 = 95/V 0 volume fraction for the Figure 9 data of Blomquist and Portigo (1962), and A = 0.4, /zVZ) = 1.37, and a, = 0.55% for the Figure 4 (Curve F) data of Chenevert (1970). Note that the initial sample volume V 0 is not given in the original data. However, this value is not required for the plots of (1 - oc/oc,) because the V 0 value cancels out in the ratio of oc/a,. Note that the data points shown in Figure 2-21 are the tick-mark readings of the plots of the original reported data. Wild et al. (1996) tested lime-stabilized compacted kaolinite cylinders containing gypsum and ground granulated blast furnace slag. After moist- curing for certain periods, they soaked these samples in water and mea- sured the linear expansion of the samples. Figure 2-23 shows the representation of the three typical data sets selected from their Figures 5, 6, and 8 by Eq. 2-22 using Eq. 2-6. The first set of data was obtained using a 7-day moist-cured kaolinite containing 6% lime and 4% gypsum. 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 moist- cured kaolinite containing 2% lime, 4% gypsum and 8% ground granu- lated 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(c l - c 0 }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 lin- ear plot of Ladd's data first, the S term is eliminated between Eqs. 2-18 and 22 to yield: 1 ^L = \^L- W w, -w. (2-23) Then, inferred by Eq. 2-23, Ladd's data can be correlated on a log- log scale by a straightline as shown in Figure 2-24. The best linear fit of Eq. 2-23 was obtained using w 0 = 0.8g, w t = 32g, a, = 13.2/V 0 and k/k w = 1.907. Note that the value of V 0 is not given and not required be- cause Eq. 2-23 employs the ratio of a/a r Porosity Reduction by Swelling Based on the definition of the swelling coefficient, Civan and Knapp (1987) expressed the rate of porosity change by swelling of porous matrix as: Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 37 5 Figure 2-24. Correlation of volume change vs.water pickup during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petro- leum Engineers). (2-24) where § is porosity, t is time, S is the rate of water absorbed per unit bulk volume of porous medium. Civan (1996) developed an improved equation assuming that the rate of porosity variation by swelling is proportional to the rate of water ab- sorption and the difference between the instantaneous and the terminal or saturation porosities: (2-25) (2-26) subject to Integrating Eqs. 2-25 and 26 yields: In (2-27) from which the porosity variation by swelling can be expressed by: 38 Reservoir Formation Damage = 4> - 4> 0 = 4>, - 1 - (2-28) where k sw is the formation swelling rate constant, t is the actual time of contact with water. Therefore, the swelling rate constant can be deter- mined by fitting Eq. 2-27. However, due to the lack of experimental data, the application of Eq. 2-28 could not be demonstrated. It is difficult to measure porosity during swelling. Permeability can be measured more conveniently. Ohen and Civan (1993) used a permeability-porosity rela- tionship to express porosity reduction in terms of permeability reduction. Permeability Reduction by Swelling Civan and Knapp (1987) assumed that the rate of permeability reduc- tion by swelling of formation depends on the rate of the water absorp- tion and the difference between the instantaneous permeability and terminal permeability that will be attained at saturation as: = a sw S(K-K t ) subject to the initial condition K = K 0 ,t = 0 where a sw is a rate constant. Thus, solving Eqs. 2-29 and 30 yield: (2-29) (2-30) (2-31) where a sw is the rate constant for permeability reduction by swelling, from which the permeability variation by swelling is obtained as: - exp(-a w 5)] (2-32) Civan and Knapp (1987) and Civan et al. (1989) have confirmed the validity of Eq. 2-31 using the Hart et al. (1960) data for permeability reduction in the outlet region of a core subjected to the injection of a suspension of bacteria. Because bacteria is essentially retained in the inlet side of the core, the permeability reduction in the near-effluent port of the core can be attributed to formation swelling by water absorption. The Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 39 best linear, least-squares fit of Eq. 2-31 to Hart et al. (1960) data using Eq. 2-9 for S yields (Civan et al., 1989): (2-33) with (K t /K 0 ) = 0.57 and B = 2a sw (c l -c 0 )Jo/n = QMhr~ {/2 with a corre- lation coefficient of R 2 = 0.93 as shown in Figure 2-25. However, as shown in Figure 2-25, the Hart et al. (1960) data can also be correlated using Eq. 2-6 for S. In this case, the best fit is obtained using the pa- rameter values of A = a sw (q - c 0 )/h = 0.95, h^D = 1, and K t /K 0 = 0.57. Ngwenya et al. (1995) conducted core flood experiments by injecting an artificial seawater into the Hopeman (Clashach) sandstone. They re- port that the core samples used in their experiments contained trace amounts of clays. Therefore, they concluded that the effect of clay swell- ing, and entrainment and deposition of clay particles to permeability impairment would be negligible. However, their Table 1 data plotted according to Eq. 2-31 in Figure 2-26 indicates a reasonably well linear trend. Consequently, it can be concluded that the swelling of some con- stituents of the sandstone formation should be contributing to permeability reduction. The best least-squares linear fit of Eq. 2-31 was obtained using Eq. 2-9 for 5 with the parameter values of B = a sw (c, - c 0 )/h = 0.038hr~ 1/2 and (K t IK 0 } = 0.087 with a correlation coefficient of R 2 = 0.89. The best 3 - • Hart etal data • Correlation of Hart et al data •Linear (Hart et al data) Figure 2-25. Correlation of permeability reduction during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petro- leum Engineers). 40 Reservoir Formation Damage 1.2 1 ? 0.8 £f 0.6 0.4 0.2 0 Ngwenyaetaldata Correlation of Ngwenyaetaldata Linear (Ngwenyaet al data) 10 20 t i*(hr 1/2 ) 30 Figure 2-26. Correlation of permeability reduction during swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petro- leum Engineers). fit of Eq. 2-31 using Eq. 2-6 for 5 was obtained using the parameter values of A = a sw (c l - c 0 )/h = 0.035, hjD = 1, and K t IK 0 = 0.087. It is apparent from Figures 2-25 and 26 that the quality of both the Hart et al. (1960) and the Ngwenya et al. (1995) experimental data does not permit determining whether Eqs. 2-6 or 9 with Eq. 2-31 better rep- resents the data. Because Eq. 2-6 led to successful representation of the other data correlated in the preceding sections, it is reasonable to assume that Eq. 2-6 should also represent the permeability reduction data equally well. Therefore, Eq. 2-6 may be preferred over Eq. 2-9. Discussion and Generalization The preceding analyses of the various data indicate that the variation of the moisture, volume, and permeability of clayey formations during swelling by exposure to water is governed by similar rate equations, which can be generalized as (Civan, 1999): -d(f-f t )/dt = k f S(f-f t ) subject to the initial condition (2-34) (2-35) Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 41 Although the validity of Eq. 2-34 for porosity variation could not be dem- onstrated because of the lack of experimental data, porosity variation is also expected to follow the same trend because it is a result of solid ex- pansion by water absorption, for which case the validity of the proposed mechanism was confirmed with experimental data. Let / denote the properties of clayey formations that vary by swell- ing, that is /e(w,a,(t),AT),/ 0 and/, denote the initial and final values of / over the swelling period, t is time, k f is the rate constant for the prop- erty /, and S is the rate of water absorption controlled by the hindered diffusion of water into the solid according to Eq. 2-6. The analytic solution of Eqs. 2-34 and 35 can be written in the fol- lowing form: In f-f t (2-36) As demonstrated by Eq. 2-23, it is also possible to relate a property of f£(w,a,§,K) to another property of ge(w,oc,(|>,AT) for f*g. This can be accomplished by first applying Eq. 2-36 for g as: (2-37) The quantity S can then be eliminated between Eqs. 2-36 and 37 to obtain: (2-38) Eq. 2-38 is particularly useful to correlate between w,a,<|),and K with- out the involvement of the time variable. For example, applying Eq. 2-38, porosity and permeability variations can be correlated by the power law equation: K-K (2-39) where k K and are the rate coefficients for permeability and porosity reduction by swelling, respectively. [...]... "Predictability of Formation Damage by Modeling Chemical and Mechanical Processes," SPE 237 93 paper, Proceedings of the SPE International Symposium on Formation Damage Control, February 26-27, 19 92, Lafayette, Louisiana, pp 2 93- 31 2 Chang, F F., & Civan, F., "Practical Model for Chemically Induced Formation Damage, " / of Petroleum Science and Engineering, Vol 17 , No 1/ 2, February 19 97, pp 12 3 - 13 7 Mineralogy... Media," AIChE J., Vol 33 , No 10 , 19 87, pp 16 54 -16 62 Simpson, J P., "Drilling Fluid Filtration Under Simulated Downhole Conditions," SPE Paper 4779, 19 74, 14 p Weaver, C E., & Pollard, L D., The Chemistry of Clay Minerals, Elsevier, New York, 19 73, 2 13 p Welton, J E., "SEM Petrology Atlas," American Association of Petroleum Geologists, Tulsa, Oklahoma, 19 84, 237 p 48 Reservoir Formation Damage Wild, S.,... Clay Minerals, Vol 31 , 19 96, pp 4 23- 433 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 11 0, 19 88, pp 34 -42 Wojtanowicz, A K., Krilov, Z., & Langlinais, J P., "Study on the Effect of Pore Blocking Mechanisms on Formation Damage, " Paper SPE 16 233 presented at Society... 19 96, pp 53- 63 Pittman, E D., & Thomas, J B., "Some Applications of Scanning Electron Microscopy to the Study of Reservoir Rock," SPE 7550 paper, November 19 79, pp 13 75 - 13 80 Reed, M G., "Formation Permeability Damage by Mica Alteration and Carbonate Dissolution," JPT, September 19 77, pp 10 56 -10 60 Rogers, 19 63 Sahimi, M., Flow and Transport in Porous Media and Fractured Rock, VCR, Weinheim, 19 95, 482... Lafayette, LA, February 14 -15 , 19 96, pp 31 1 -32 6 Civan, F, "Interactions of the Horizontal Wellbore Hydraulics and Formation Damage, " SPE 35 2 13 , Proceedings of the SPE Permian Basin Oil & Gas Recovery Conf., Midland, TX, March 27-29, 19 96, pp 5 61- 569 Civan, F., "Model for Interpretation and Correlation of Contact Angle Measurements," Jour Colloid and Interface Science, Vol 19 2, 19 97, pp 500-502 Civan,... on Formation Damage Control, Lafayette, Louisiana, February 22- 23, 19 90, pp 18 5-200 Ohen, H A., & Civan, F., "Simulation of Formation Damage in Petroleum Reservoirs," SPE Advanced Technology Series, Vol 1, No 1, pp 27 -35 , April 19 93 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 19 96,... Highway Res Board, No 31 3 , 19 62, pp 78- 91 Brownell, W E., Structural Clay Products, Springer-Verlag, New York, 19 76, 2 31 p Bucke, D P., & Mankin, C J., "Clay-Mineral Diagenesis Within Interlaminated Shales and Sandstones," / Sedimentary Petrology, Vol 41, No 4, December 19 71, pp 9 71- 9 81 Carslaw, H S., & Jaeger, J C., Conduction of Heat in Solids, Second ed., Oxford University Press, 19 59, 510 p Chang, F F.,... Publication 7 31 , Washington, D.C., 19 60, pp 10 -26 Liu, X., Civan, F, & Evans, R D., "Correlation of the Non-Darcy Flow Coefficient, J of Canadian Petroleum Technology, Vol 34 , No 10 , 19 95, pp 50-54 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, 19 98, pp 17 9-2 01 Mancini,... defined as the ration of the lengths, Lt and L, of the tortuous fluid pathways and the porous media: T = L,/L (3 - 13 ) Liu and Masliyah (19 96a, b) recommend the Bruggeman (19 35 ) equation T1 ^1/ 2 (3 -14 ) for random packs of grains of porosity § > 0.2 and the Humble equation (Winsauer et al., 19 52) 1. 15 (3 -15 ) for consolidated porous media of porosity (|) < 0.45 They point out that the latter may have a variable... DE-AC2290BC14658, April 19 94 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, 19 95, pp 10 2 -10 7 Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31 1 01, Proceedings of the SPE Formation Damage Symposium, . 19 95, pp. 10 2 -10 7. Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31 1 01, Pro- ceedings of the SPE Formation Damage Symposium, Lafayette, LA, February 14 -15 , 19 96, . paper, November 19 79, pp. 13 75 - 13 80. Reed, M. G., " ;Formation Permeability Damage by Mica Alteration and Carbonate Dissolution," JPT, September 19 77, pp. 10 56 -10 60. Rogers, 19 63. Sahimi, . written as: 34 Reservoir Formation Damage e 3 o 1 S n t . ft . • Brown Brown x" ^ * en data tion of the eO data , / ; X lx » 468 t 1/ 2 (hr in ) 10 12 Figure 2- 21. Correlation