Experimental and numerical investigation of one-dimensional waterflood in porous reservoir

al and numerical investigation of one-dimensional waterflood in porous reservoir N. Hadia a, L. Chaudhari a, A. Aggarwal b, Sushanta K. Mitra a, *, M. Vinjamur b, R. Singh c a IITB ONGC Joint Research Centre, and Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai 4 76, India b IITB ONGC Joint Research Centre, and Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 4 76, India c IITB ONGC Joint Research Centre, and Institute of Reservoir Studies, ONGC, Ahmedabad, India Abstract al and numerical investigation of relative permeability and oil recovery from the porous reservoir are described for short and long core samples. The relative permeability ratios, which are function of water saturation, obtained from laboratory core flooding experiments have been used for prediction of oil recovery through numerical simulation of non-dimensional Buckley Leverett equation. The simulation results for oil recovery compared well with recovery results obtained from core flooding experiments. Keywords: Relative permeability; Oil recovery; Saturation; Core flooding 1. Introduction The oil reservoirs are porous in nature. The oil recovery processes involve the simultaneous flow of two or more fluids in the reservoir. In waterflooding process, water injected in the reservoir displaces the oil from the pores of reservoir bed. The property of the porous reservoir that plays an important role in determining the performance of the waterflooding process is relative permeability. The prediction of the relative permeability characteristics of oil reservoir is the primary task of any laboratory waterflood experiments. For this purpose, laboratory experiments are conducted on a small representative core samples obtained form the reservoir. The oil recovery characteristics can be predicted by using Buckley Leverett equation [2]. For such prediction, individual phase relative permeabilities are required. Geffen et al. [5] presented a comprehensive study on different methods available for determining relative permeabilities of reservoir rocks and suggested the direct measurement of relative permeabilities in the laboratory using representative core sample. There are two basic methods for obtaining relative permeabilities from laboratory experiments: steady state method, and unsteady state method. The calculation of relative permeabilities from steady state experiment requires very long experimentation time. Unsteady state method needs less time; this method estimates relative permeabilities by interpreting the data collected from laboratory experiments in which one fluid is injected into a core sample that is saturated with another fluid. The relative permeability can be calculated by application of Welge [16] equation and method developed by Johnson et al. [7] or by using graphical technique developed by Jones and Roszelle [8]. The basic limitation of this approach is the necessity to differentiate the experimental data which may yield large errors in estimation of relative permeability [15]. Another approach for determining the relative permeability is history matching [1]. In this method, the relative permeability curves are adjusted until calculated oil recovery curves match those obtained from the laboratory displacement experiments. The interpretation of relative

356 Nomenclature English letters d a mass coefficient F source term K absolute permeability k r relative permeability L length of core sample q flow rate per unit area S saturation t time x distance Greek symbols C flux vector l absolute viscosity / porosity Subscripts D non-dimensional o oil phase or residual oil t total w water phase wi irreducible water w2 terminal water Abbreviations PV pore volume OOIP original oil inplace permeability based on history matching technique is found to be less precise for the injected fluid than the displaced fluid because the displacement history is weakly dependent on the injected phase relative permeability [12]. Macary and Walid [1] used an X-plot technique for oil recovery prediction and for obtaining relative permeability ratio and fractional flow curves. However, the technique can only be applied to reservoirs producing with water fractions higher than 5% and requires linearization of the relationship between recovery and water cut. al studies by Richardson [13] indicated that the relative permeability ratio is independent of the viscosity of fluids and the rate of water advance. Moreover, Buckley Leverett theory can be used to predict the waterflood performance neglecting the capillary pressure effects. Numerical methods have also been developed to predict the waterflood behavior including capillary effects [4]. Sheldon and Cardwell [14] solved the Buckley Leverett equation using the method of characteristics and used the treatment analogous to that used in the theory of supersonic compressible flow. Solution of Buckley Leverett equation needs the individual relative permeability values, which are obtained either from JBN method [7] or from empirical correlations [3]. However, their solutions are not validated through experiments. McEven [11] presented the results of numerical solution to one dimensional waterflooding in which outlet end effects are neglected. Gottfried et al. [6] presented two numerical schemes for solving the equations for one dimensional, multiphase flow in porous media with assumed correlations for relative permeabilities and negligible capillary pressure. However, the results obtained numerically have not been validated through laboratory waterflood experiments. In this paper, the experimental investigation of one dimensional waterflood performance of short and long Berea core samples are presented. The waterflood performance has been predicted by solving the Buckley Leverett equation using FEMLAB solver. In general, it is observed that for oil recovery estimation in one-dimensional core flood, individual phase relative permeabilities are used in the simulation. In this study, however, instead of using individual relative permeability values, a novel method is used where the relative permeability ratios obtained from the saturation experiments are used. This reduces the computational time significantly. The results obtained from the numerical simulation are validated by the laboratory one dimensional waterflood experiments and a good agreement has been found between them. 2. s The schematic of the experimental apparatus used for permeability measurement and waterflooding studies is shown in Fig. 1. The dual cylinder syringe pump whose inlet is connected to the brine reservoir maintained the desired flow rate through the core. The outlet of the pump is connected to the three way valve which in turn is connected to brine and oil accumulators at a common junction. Hence, by operating the three way valve appropriately, either oil or brine can be injected into the core. The effluents from the core outlet are collected in a fraction collector. A differential pressure transmitter has been connected between the inlet and outlet of the core to measure the pressure difference across the core length. A digital pressure gage is provided at the inlet of the core holder to record the system pressure. s have been performed at room temperature of 24 C and atmospheric pressure on cylindrical Berea sandstone core sample, 3.8 cm in diameter and 7 cm in length, as well as on rectangular Berea sandstone core sample, 2.5 2.4 54 cm. The properties of the core samples are provided in Table 1. Brine (water with 1% KCl by volume) has been used as a saturating and a displacing fluid while heavy liquid paraffin oil is the displaced fluid and its viscosities at room temperature of 24 C are.97 and 13 cp, respectively.

357 Brine accumulator 3 Way valve Oil accumulator Dual cylinder syringe pump Brine reservoir Pressure gage Core holder Differential pressure transmitter To Vacuum pump Fraction collector Fig. 1. Schematic of experimental set-up for 1-D waterflood experiments. Table 1 Properties of core samples Core sample Porosity (%) Cylindrical Berea core Absolute permeability (md) S wi %PV S or %PV 29.4 1131 37.5 23.2 5 29.4 1131 48. 5.1 8 29.4 1131 45. 14.8 1 38.5 1584 3 34.7 1 Long Berea core 38.5 1584 32.5 33.4 5 38.5 1584 32.2 31.7 1 S wi = irreducible water saturation. S or = residual oil saturation. md = mili Darcy (1 md = 1 15 m 2 ). Injection Rate ml/h The cylindrical core sample is first coated with resin and kept in a stainless steel cylindrical core holder of 5.4 cm internal diameter. The annular space between the core and the core holder is filled with an alloy called Cerro Metal (42% tin and 58% Bi by weight) to prevent leakage of fluids from the surface of the core sample. Fig. 2 shows schematic of apparatus for long core with rectangular cross-section. For experiments on this apparatus, the core is first coated with a thick layer of resin. After drying the resin, leakage test is carried out using soap bubble test with nitrogen(up to 2 bar pressure). The leakage prone areas are then sealed and the core is made leakproof. Pressures have been measured at three different locations along the length of the core with pressure transmitters. At the outset of experiments, air is removed from the cores by a vacuum pump. When sufficient vacuum level is achieved, the core sample is disconnected from the vacuum INLET P 1 P 2 P 3 5 25 19 5 RESIN COATING 54 P 1 ;P 2 ;P 3 Pressure Transmitters All dimensions are in cm OUTLET Fig. 2. Schematic of long core for waterflood experiments. 2.5 2.4 pump. Pore volume (PV) is found by determining the volume of brine needed to saturate the evacuated core. The absolute permeability to brine is then determined by measuring the pressure drop across the core length for a known flow rate. To mimic the reservoir condition of irreducible water saturation, heavy paraffin oil is injected till no more brine at outlet is observed. After the core is prepared for the recovery experiments, the brine is injected at a known flow rate through the core using dual cylinder syringe pump. Three experiments have been performed on each core at different injection rates. In actual reservoirs, the flow velocity is about 1 ft/day. The injection rate in the reservoir is porosity times the average velocity times cross-sectional area available to the flow. However, for relative permeability measurement by laboratory experiments, the maximum injection rate can be up to 1 PV/h for which the fluid velocity may exceed 1 ft/day. Accordingly, injection rates are chosen such that they do not exceed the injection rates of 1 PV/h. For cylindrical core, the flooding experiments have been performed at injection rates of 5, 8, and 1 ml/h, respectively whereas for long core, injection rates of 1, 5, and 1 ml/h are used. The injection rates are chosen as per the flood front stability criteria [17] such that for cylindrical short core, the front is unstable whereas for long square cross-section core it is stable. The experiments are terminated after 2 PV of brine injection; after 2 PV, the recovery reached a plateau. During the displacement test, the fractions of oil and brine are collected at the outlet in the fraction collector. The relative permeability ratios, krw k ro, are then calculated from the measured oil fractions and known viscosities of oil and brine. After completion of the experiment at a particular injection rate, the same core is flushed with oil and brought to the irreducible water saturation and again used for experiments at different injection rates. 2.1. Errors in s The measurement errors in the oil recovery curve are attributed to the errors involved in the measurement of volume fractions of oil. The fraction measurements are made in graduated glass tubes which have a least count of.2 ml.

358 The errors involved in the relative permeability ratios, based on the least count, are ±2%. Based on the least count and the difference in original oil inplace (OOIP) values, errors in the recovery curve for short and long Berea core are ±3% and ±1%, respectively. 3. Numerical simulation In numerical simulation, the capillary pressures and gravity effects are neglected. Under such conditions, the non-dimensional form of Buckley Leverett equation for the displacing phase (water) is [9] os w þ of wðs w Þ ¼ ; ð1þ ot D ox D where S w is the water saturation, f w is the fractional flow of water, t D is the non-dimensional time, and x D ¼ x is the L non-dimensional distance from the inlet. The fractional flow of water neglecting the capillary pressure, f w is given as f w ¼ 1 1 þ l w l o k ro ; k rw where l w and l o are the viscosities of displacing phase and displaced phase (oil), respectively, and k rw and k ro are the relative permeabilities of displacing phase and displaced phase, respectively. The non-dimensional time can be expressed as, t D ¼ tq t, /L where, q t is the flow flux at the inlet, L is the length of the core, / is the porosity of the porous medium, and t is the time for injection. (Here t D represents the cumulative water injection as a fraction of pore volume (PV).) Eq. (1) is solved using general PDE model of FEMLAB given as os w d a þrc ¼ F ; ð3þ ot where d a is mass coefficient, F is the source term, and C is flux vector. For modeling Eq. (1) using Eq. (3), following parameters are used d a ¼ 1; C ¼ ; and F ¼ of wðs w Þ ; ð4þ ox D where the source term F can be rewritten as F ¼ of wðs w Þ osw : ð5þ os w ox D It is to be noted that the experimental relative permeability ratio curves for short cylindrical and long square cross-section Berea core have been used for the numerical simulations. 4. Initial and boundary conditions Initially, the core will be at the irreducible water saturation, S wi which is determined from the laboratory experiments. At the inlet (x D = ), water saturation, S w is 1. ð2þ 5. Prediction of oil recovery The average water saturation inside the core is estimated by integrating water saturation through the core length and can be expressed as ðs w Þ av ¼ Z 1 S w dx D : The percent original oil in place (OOIP) recovered is calculated from ð6þ %OOIP ¼ ðs wþ av S wi 1 S wi 1: ð7þ 6. Results and discussion The experimental and simulation results of one dimensional waterflood on two different cores are presented here. The effects of injection rates on water oil relative permeability ratios and total oil recovery for short cylindrical core are shown in Fig. 3 and 4, respectively. It can be observed from Fig. 3 that as the terminal water saturation, S w2, increases, the water to oil relative permeability ratio increases. Also, as the injection rate increases from 5 to 8 ml/h, the relative permeability ratio decreases which in turn indicates the increase in the oil relative permeability. However, as injection rate increases from 8 to 1 ml/h, the relative permeability ratio increases which indicates the decrease in oil relative permeability for the same terminal water saturation. This concludes that for the injection rate of 8 ml/h, the recovery should be maximum as the oil relative permeability is maximum for the same terminal water saturation. From Fig. 4, it can be observed that as injection rate increases from 5 to 8 ml/h, the total recovery increases. However, for an injection rate of 1 ml/h, the total recovery decreases which indicates that there is an optimum injection rate for which the recovery is maximum. This is Relative permeability ratio (K rw /K ro ) 1 1 1 1 1 5 ml/hr 8 ml/hr 1 ml/hr 1 2.4.5.6.7.8.9 1 Terminal water saturation (S w2 ), % PV Fig. 3. al water to oil relative permeability ratio curve for short Berea core sample for different injection rates. PV = 23.3 ml, K = 1131 md. Lines joining the points show the trend of curves.

359 1 8 6 4 2 5 ml/hr 8 ml/hr 1 ml/hr.5 1 1.5 2 2.5 Fig. 4. Effect of injection rate on recovery of short Berea core sample. PV = 23.3 ml, K = 1131 md. Lines joining the points show the trend of curves. also supported by the relative permeability ratio curves (Fig. 3). At an injection rates higher than the optimum, the recovery reduces, which can be attributed to the phenomena of viscous fingering [18]. Moreover, the flood front in cylindrical core is unstable and for such flood fronts, the recovery depends on injection rates. Hence, there is an optimum injection rate for such unstable fronts and above this injection rate, due to viscous fingering, recovery decreases. The relative permeability ratios obtained from the experimental data have been directly implemented in the numerical simulation. The experimental relative permeability ratio curves for long Berea core at different injection rates are shown in Fig. 5. It can be observed from Fig. 5 that the relative permeability ratio curves exhibit an exponential relationship with water saturation. Moreover, there is marginal effects of injection rates on relative permeability ratios for the injection rates considered in flooding experiments. The effect of injection rates on recovery performance of long Berea core is shown in Fig. 6. It can be observed from Fig. 6 that as injection rate increases from 1 to 1 ml/h, the recovery increases. Unlike the recovery performance of 6 5 4 3 2 1 1 ml/hr 5 ml/hr 1 ml/hr.5 1 1.5 2 2.5 Fig. 6. Effect of injection rate on recovery of long Berea core. PV = 126 ml, K = 1584 md. Lines joining the points show the trend of curves. short core, optimum injection rate is not observed for the injection rates considered here. Moreover, as the injection rate is increased from 5 to 1 ml/h, only 3% increase in the ultimate recovery is observed after 2 PV of injection. This can be attributed to the marginal decrease in the water oil relative permeability ratios as can be observed Pressure (psi) 2 15 1 5 P 1 P 2 P 3.25.5.75 1. Fig. 7. Measured pressure variations with PV injection for pressure transmitters P 1, P 2, and P 3 in long Berea core. 1 psi = 6987.9 N/m 2. Relative permeability ratio (K rw /K ro ) 1 1 1 1 1 1 ml/hr 5 ml/hr 1 ml/hr 1 2 3 35 4 45 5 55 6 Terminal water saturation (S w2 ), % PV Fig. 5. Effect of injection rate on water/oil relative permeability ratio for long Berea core. PV = 126 ml, K = 1584 md. Lines joining the points show the trend of curves. Water saturation (S w ) 1.8.6.4 Brine Oil S wi.2.2.4.6.8 1 Non dimensional distance (x D ) Fig. 8. Numerical simulation flood front position in a short Berea core sample after.1 PV of injection at an injection rate of 5 ml/h. PV = 23.3 ml, K = 1131 md, S wi = 37.5% PV, S or = 23.2% PV. The black region is thick because of more data points.

36 from Fig. 5. Moreover, the flood front in long core is stabilized and hence oil recovery is independent of injection rates. The pressure variations with respect to the volume of brine injection, along the length of the long core, is shown in Fig. 7 for three pressure transmitters P 1, P 2, and P 3, respectively. It can be observed from the pressure plot that as PV injection increases, the pressure inside the core decreases. Eventually, the pressure at each location becomes constant when no further oil is recovered. The numerical simulation of the flood front position in a short cylindrical core for an injection of.1 PV at an injection rate of 5 ml/h is shown in Fig. 8. The area below the curve represents the total oil displaced by the brine above the irreducible water saturation, S wi, whereas area above the curve represents the amount of oil left in the core after.1 PV of brine injection. The comparison of numerical simulation results with the experimental results at different injection rates for short cylindrical core are shown in Fig. 9. In the numerical simulation, experimentally obtained relative permeability ratios are used instead of individual phase relative permeability curves. It can be observed from Fig. 9a that for injection rate of 5 ml/h, the simulation and experimental results are in good agreement. The initial slope, however, of the experimental recovery curve is not well predicted by the simulation. For injection rates of 8 and 1 ml/h (Fig. 9b and c, respectively), it can be observed that the simulation results are underpredicting the experimental 6 4 2.5 1 1.5 2 2.5 1 8 6 4 2.5 1 1.5 2 2.5 8 6 4 2.5 1 1.5 2 2.5 Fig. 9. Comparison of numerical and experimental recovery performance of 1-D waterflood of short Berea core sample for injection rates of (a) 5 ml/h (S wi = 37.5% PV, OOIP = 14.5 ml, S or = 23.2% PV) (b) 8 ml/h (S wi = 48% PV, OOIP=12.1 ml, S or = 5.1% PV), and (c) 1 ml/h (S wi = 45% PV, OOIP=12.8 ml, S or = 14.8% PV). PV = 23.3 ml, K = 1131 md for all the cases. 6 5 4 3 2 1.5 1 1.5 2 2.5 6 5 4 3 2 1.5 1 1.5 2 2.5 6 4 2.5 1 1.5 2 2.5 Fig. 1. Comparison of numerical and experimental recovery performance of 1-D waterflood of long Berea core sample for injection rates of (a) 1 ml/h (S wi = 3% PV, OOIP = 88.8 ml, S or = 34.7% PV) (b) 5 ml/h (S wi = 32.5% PV, OOIP = 85 ml, S or = 33.4% PV), and (c) 1 ml/h (S wi = 32.2% PV, OOIP = 85.4 ml, S or = 31.7% PV) PV = 126 ml for all the cases.

361 recoveries. The careful observation from Fig. 9 shows that the recovery is maximum for the injection rate of 8 ml/h which is also supported by the experimental recovery curves (Fig. 4). The error bars in the experimental recovery curve corresponds to the error in the volume fraction measurement, as discussed earlier. The comparison of numerical simulation results with the experimental results for long Berea core at different injection rates are shown in Fig. 1. It is found that for all injection rates, the initial slope of the recovery curve is well predicted by the numerical simulation. It is also observed that the simulation result greatly underestimated the experimental result for an injection rate of 1 ml/h (Fig. 1a), slightly underestimated for an injection rate of 5 ml/h (Fig. 1b) and overestimated for an injection rate of 1 ml/h (Fig. 1c). This is because for an injection rate of 1 ml/h, the flood front is just stable and water/oil relative permeability ratio is higher compared to higher injection rates of 5 and 1 ml/h for which the flood front is fully stabilized. However, with increase in injection rate from 5 to 1 ml/h, due to marginal decrease in the relative permeability ratios, the oil recovery increases. However, the maximum oil recovery predicted by the simulation differs from the experimental recovery by 3%. 7. Conclusions In this paper, the experimental and numerical studies have been carried out to predict the one dimensional waterflood performance of short and long Berea cores at different injection rates. The relative permeability ratios obtained from experiments are used in the numerical simulation. al studies on short core concludes that there is an optimum injection rate at which recovery is maximum. Comparison of experimental and numerical simulation results show reasonable agreement for short and long cores. It is also observed that there is a good match of the initial slopes of the recovery curves. References [2] S.F. Buckley, M.C. Leverett, Mechanism of fluid displacement in sands, Petroleum Transactions AIME 146 (1942) 17 116. [3] G.L. Chierici, Novel relations for drainage and imbibition relative permeabilities, SPEJ (1984) 275 276. [4] J. Douglas Jr., P.M. Blair, R.J. Wagner, Calculation of linear waterflood behavior including the effects of capillary pressure, Petroleum Transaction AIME 213 (1958) 96 12. [5] T.M. Geffen, W.W. Owens, D.R. Parrish, R.A. Morse, al investigation of factors affecting laboratory relative permeability measurements, Petroleum Transaction AIME 192 (1951) 99 11. [6] B.S. Gottfried, W.H. Guilinger, R.W. Snyder, Numerical solution of the equations for one dimensional multi phase flow in porous media, SPEJ (1966) 62 72. [7] E.F. Johnson, D.P. Bossler, V.O. Naumann, Calculation of relative permeability from displacement experiment, Petroleum Transaction AIME 216 (1959) 37 372. [8] S.C. Jones, W.O. Roszelle, Graphical techniques for determining relative permeability from displacement experiments, Journal of Petroleum Technology 3 (1978) 87 817. [9] L.W. Lake, Enhanced Oil Recovery, Prentice Hall, Englewood Cliffs, USA, 1989. [1] S. Macary, A.A. Walid, Creation of fractional flow curve from purely production data. SPE Paper 5683, in: 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 1999 October 3 6. [11] C.R. McEven, A numerical solution of the linear displacement equation with capillary pressure, Transactions of AIME 216 (1959) 412 415. [12] V.S. Mitlin, B.D. Lawton, J.D. McLennan, L.B. Owen, Improved estimation of relative permeability from displacement experiments, SPE Paper 3983, in: 1998 SPE International Petroleum Conference and exhibition of Mexico held in Villahermosa, Mexico, 1998 March 3 5. [13] J.G. Richardson, The calculation of waterflood recovery from steady state relative permeability data, Transactions of AIME 21 (1957) 373 375. [14] J.W. Sheldon, W.T. Cardwell, One dimensional, incompressible, non capillary flow, two phase fluid flow in a porous medium, Transactions of AIME 216 (1959) 29 296. [15] T.M. Tao, A.T. Watson, Accuracy of JBN estimates of relative permeability: Part 1 Error analysis, SPEJ (1984) 29 214. [16] H.J. Welge, A simplified method for computing oil recovery by gas or water drive, Transactions of AIME 195 (1952) 91 98. [17] G.P. Willhite, Waterflooding, SPE Textbook Series, vol. 3, Richardson, TX, USA, 24, p. 89. [18] Z.F. Zhang, J.E. Smith, Visualization of DNAPL fingering processes and mechanisms in water saturated porous media, Transport in Porous Media 48 (22) 41 59. [1] J.S. Archer, S.W. Wong, Use of reservoir simulator to interpret laboratory waterflood data, SPEJ (1973) 343 347.