IMMISCIBLE DISPLACEMENT IN TWO DIMENSIONS- AREAL If permeability varies, If gravity segregation occurs, If capillary forces are large compared to viscous forces. If the reservoir homogeneous and isotropic, the flood front may be vertical: displacement. 2-D Otherwise, the displacement is 3D. PREDICTION OF DISPLACEMENT PERFORMANCE: Earlier methods: quick estimate. Today, more computer skills. All models have one common factor: 2D fluid flow equations: where ko o Po ko o Po o So x o x y o y t k w w Pw k w w Pw w S w x w x y w y t Tayfun Babadagli, PhD, PEng Short Course - EOR kw kw ( S w ) and ko ko ( S w ) Po Pw Pc ( S w ) File-5 1 DISPLACEMENT OF 5-SPOT PATTERN INJECTOR PRODUCER ASSUMPTIONS and SIMPLIFICATIONS: Large reservoirs: injector / producer 1. Injection rate = production rate Five spot in a reservoir can be simplified by examining the behavior of a single five-spot. 2 Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 CORRELATIONS DEVELOPED FROM SCALED LAB MODELS: The Craig-Geffen-Morse (CGM) Model: Experimental data for a variety of oil and aqueous systems were correlated empirically. E A E Abt 0.633 log Wi Wibt E A E Abt 0.274 ln Wi Wibt EAbt : areal sweep efficiency at BT : fraction of area swept to an average water saturation EA was correlated to mobility ratio ( S wf MS) Mobility ratios can be calculated from relative permeability data when the water permeability was evaluated at the average water saturation behind the front as determined from the frontal advance theory. E Abt 0.546 0.0317 0.3022 M 0.0051M S MS e S Tayfun Babadagli, PhD, PEng Short Course - EOR 3 File-5 Prediction of displacement performance: Oil displaced by water in a five-spot pattern, no initial gas. At BT, oil recovery is given by: N pbt E Abt S wf S wi V p S wf is the average displacing-phase saturation at BT in a linear flood as computed from frontal advance solution. Production after BT: N pbt E Abt S w5 S wi V p S w5 : average water saturation in a region swept by the injected fluid. The key to this model is the assumption made to evaluate this parameter ( S w5 ). A new variable is defined as the volume of water injected divided by the volume of the 5-spot contacted by the * injected water: Qi Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 At BT: * Qibt For 5-spot After BT: pattern: Qi dQi* Wi Vp Wibt S wf S wi E Abt Thus, Qibt E Abt ( S fw S wi ) dWi E AV p Integrating this equation from BT (Wi/Wibt=1.0) to (Wi/Wibt) : Qi* 1 E Abt * Qibt Wi / Wibt 1 W d i Wibt 1 a e a1 Ei( a ) Ei( a ) 1 2 1 EA Tayfun Babadagli, PhD, PEng Short Course - EOR 5 File-5 where a1 = 3.65EAbt a2 =a1+ln(Wi/Wibt) and Wibt ≤ Wi ≤Wi100 Wi100 : value of Wi required to sweep the pattern completely. Ei( x ) 0.577 ln( x ) x n /( nn! ) n 1 It is possible to compute EAbt and Wi.. Tayfun Babadagli, PhD, PEng Qi* * Qibt for any Short Course - EOR File-5 6 PRACTICAL ESTIMATION OF WATERFLOOD PERFORMANCE Material Balance: From core data and estimates of sweep efficiency. Divide reservoir into two part: (1) swept volume, and (2) unswept volume. The volume of oil displaced: Ev N d ( So )V p Bo Ev : fraction of reservoir volume that is swept by the injected water when the economic limit is reached. So : change in average oil saturation within swept volume. N d : oil displaced from the volume swept by the waterflood. Assume piston like displacement: Oil saturation in the swept region is the average residual saturation determined from flood tests. Thus, S o So1 Sor , where So1is the volumetric average oil saturation in the reservoir at the beginning of waterflood: 7 Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 Oil remaining in the reservoir when the unswept pore space is resaturated to the initial water saturation: Nr Ev SorV p Bo ( 1 Ev )SoiV p Bo Oil recovered by waterflooding,N pw , V N pw So1 Ev Sor ( 1 Ev )Soi p Bo Oil potentially recoverable by waterflooding, as STB: N pw ( N N p ) N Boi Bo Sor 1 E 1 v S oi N : Initial oil in place, STB. Np : Oil produced during primary operations, STB. Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 8 Waterflooding performance prediction requires estimation of the residual oil sauration (core data) and volumetric sweep efficiency (VSE, EV). VSE can be estimated by three methods: 1) Using Vp N pw So1 Ev Sor ( 1 Ev )Soi Bo 2) VSE and ASE : Form interpretations of well logs fluid-rock properties, and simplified displacement models. Ev EI E A EA EI : Fraction of the reservoir area within the vertical portion of the reservoir that has been swept to residual oil saturation. : Fraction of the reservoir cross section that has been displaced by the injected water. 3) Using correlations based on simulation of waterflood performances (Hirasaki et al, SPE 13415, see two figures on Page 10). Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 9 Engineering approaches: Scaled-Model Correlations: CGM (for 5-spot). Kimbler et al (JPT, Feb. 1964) developed correlations for 9-spot. They used the following graphs and parameters: krw w S or M kro o S oi Vd : number Qi Wi : PV’s of Vd Wi V p ( 1 S wi Sor ) of displaceable HC pore volumes: water injected, dimensionless. : Cumulative volume of water injected, bbl. Tayfun Babadagli, PhD, PEng Vd Short Course - EOR Qi ( 1 S wi Sor ) File-5 10 EXERCISE #8 (G.P. Willhite, Waterflooding Ex. 6.3): Waterflood in a 9-spot pattern. Vp =38,790 bbl (pore volume). M is based on end points of the relative permeability curves is 1.0. Soi =0.7 and Sor = 0.25. Plot Qi and WOR/Recovery. Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 11 After (Hirasaki et al, SPE 13415) Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 12 NUMERICAL MODELING OF EOR APPLICATION Pressure Saturation for each grid. By solving the continuity equations for each phase numerically. TWO PARAMETERS : Numerical Grids Injection Well Production Well 13 Saturation contour lines Tayfun Babadagli, PhD, PEng Short Course - EOR File-5 CHALLENGE : Data Preparation Permeability – porosity distribution (2 or 3 Dimensional) Relative permeability and capillary pressure curves Fluid properties (reservoir conditions) 1-D and 2-Phase Flow ( o u o ) q o (S o o ) t ( wu w ) q w (OIL) (S w w ) t (WATER) uo and uw are darcy velocities. Neglecting gravity: S uo o t x S u w w t x Tayfun Babadagli, PhD, PEng (OIL) (WATER ) Short Course - EOR 14 File-5 uin1 uin11 Sin1 Sin x t (Solve for P and S) DATA NEEDED: k (distribution), kr and Pc (uncertainty and up-scaling involved!) PVT (reliable lab measurement) Other numerical models for EOR: IMMISCIBLE FLOW (DISPLACEMENT) Darcy’s law (Black-oil) MISCIBLE FLOW (DISPLACEMENT) Fick’s law THERMAL INJECTION Fourier’s law Tayfun Babadagli, PhD, PEng Short Course - EOR 15 File-5

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