Vienna 2011 Modeling Gas Flooding in the Presence of Mixing Dr. Russell T. Johns Energy and Minerals Engineering The Pennsylvania State University
Why is Mixing Important? A) Mixing of gas and oil alters the compositions and decreases the strength of the miscible solvent. B) Mixing causes a miscible gas process to develop two-phase flow, decreasing recovery. C) Both A and B
Recovery in 1-D Depends on Mixing Oil recovery, fraction 1.00 0.90 0.80 Lab Scale Mixing Increases Field Scale 0.70 From Solano et al. (2001) 0.00 0.02 0.04 0.06 0.08 0.10 1/(Peclet Number)
Key Questions/Objectives What level of mixing is present at field scale? What grid-block sizes should be used in compositional simulation? What value of miscible residual oil saturation (S orm ) should be used?
Outline Definition of mixing Estimation of dispersion at field scale Upscaling with reservoir mixing Miscible residual oil saturation (S orm ) Conclusions
What is Mixing? An Illustration (pictures from Jacques Vanneste, University of Edinburgh) Red: High concentration (gas) White: Intermediate concentration (mixing zone) Blue: Low concentration (oil) After Stirring Diffusion is small and there is little surface area for mixing After stirring, there is a significant increase in surface area, but no time for diffusion to mix. Contrasts are large.
What is Mixing? An Illustration Red: High concentration (gas) White: Intermediate concentration (mixing zone) Blue: Low concentration (oil) Time Increases With time and enhanced surface area, diffusion significantly reduces the highs and lows (mixing is enhanced)
Key Points Thus Far Mixing occurs only by diffusion. Mixing is considerably enhanced or accelerated by stirring or in porous media by variations in velocity. Mixing is not spreading of the gas (say through high permeability pathways.)
Dispersion at the well-patternscale
How is Mixing Measured? Local Inj. C Prod. Echo Time Transmission C C C Time Time Is the transmission Transmission dispersion is apparent mixing and is mixing representative unrealistically large owing to convective spreading of true diffusive mixing? Local and echo dispersion are more representative of actual mixing caused by diffusion Time
Evidence for Scale-Dependent Dispersion Measured dispersivities from echo tests are several orders of magnitude larger than those measured in the lab Dispersion (m 2 / ) s = Dispersivity (m) Velocity (m/s) Arya et al., 1988; Lake, 1989; Mahadevan et al. 2000;
Estimate Level of Mixing from Simulation Calculate local dispersivity for each small grid-block of firstcontact miscible (FCM) simulation Solvent 1.0 Normalized Concentration, C D 0.8 0.6 0.4 0.2 Simulation data Solution of 1-D CDE 0.0 0.4 0.8 1.2 1.6 2 PVI (t D ) Dispersion increases with distance travelled owing to greater contact area between gas and oil
Estimation of Dispersion in FCM Gas Flood Dispersion (normalized by the well spacing) is a function of 7 dimensionless groups (Garmeh et al. 2010): 1) Peclet number 2) Dispersion number 3) Effective aspect ratio 4) Mobility ratio 5) Dykstra-Parsons coefficient N N R M u L inj Pe = φdxx L D zz D = H Dxx L V L k z = H kx µ = µ 0 o DP s = 1 exp( σ ) ln k 6) Horizontal correlation length 7) Vertical correlation length λ λ Dx Dz n r 1 r r r γ ( h) = k( xi ) k( xi + h) 2n i= 1 2
Sensitivity of Scale-Dependent Dispersion 1.000 1.000 λ XD = 2.00 Normalized dispersivity (αd) 0.100 0.010 λ XD = 0.10 Normalized dispersivity (αd) 0.100 0.010 λ ZD = 0.02 λ ZD = 0.50 0.001 0.00 0.25 0.50 0.75 1.00 Normalized length traveled (x D ) 0.001 0.00 0.25 0.50 0.75 1.00 Normalized length traveled (x D ) 1.000 1.000 Normalized dispersivity (αd) 0.100 0.010 V DP = 0.80 Normalized dispersivity (αd) 0.100 0.010 M = 25.0 V DP = 0.40 0.001 0.00 0.25 0.50 0.75 1.00 Normalized length traveled (x D ) M = 1.0 0.001 0.00 0.25 0.50 0.75 1.00 Normalized length traveled (x D )
Importance of Mixing to Gas Floods All cases have same mean permeability, but different correlation lengths. Case 1 Case 2 md Case 3 Case 4 Which case will have the greatest amount of mixing?
Multicontact Miscible Flow at 0.6 PVI Initial oil saturation is at S orw = 0.18 (already water flooded) 0.33 PV solvent slug followed by 1.66 PV water S o Case 1 Case 2 Case 3 Case 4
Incremental Oil Recovery for 0.33 PV Gas Slug 20 Incremental Recovery % 15 10 5 Case 1 Case 2 Case 3 Case 4 0 0.50 1.00 1.50 2.00 2.50 3.00 HCPVI How can we improve recovery?
Incremental Oil Recovery with 0.5 PV Gas Slug 20 Incremental Recovery % 15 10 5 Case 1 Case 2 0 0.50 1.00 1.50 2.00 2.50 3.00 HCPVI Optimum slug size required for tertiary oil recovery increases as reservoir heterogeneity (channeling) increases.
Grid-Block Upscaling Considering Reservoir Mixing
Upscaling Methodology α L tot = α L res + α L num + 0 α Linput Scaling groups Response functions Grid-block size x + vx t 2 Key: We want to match α α tot T tot α ( fine-scale) tot ( upscaled) vz = f ( α L, z, ) tot v x Example: Model Fine-Scale Upscaled α res num 14 ft 4.5 ft α α α L T tot tot 0.5 ft x = 1 ft 10 ft x = 20 ft 14.5 ft 0.5 ft 14.5 ft 0.5 ft
Example: Maximum Grid-Block Size for Accurate Simulation Uncorrelated
FCM displacement Uncorrelated Uncorrelated
Oil recovery from FCM displacement 1 1 Oil recovery, fraction 0.8 0.6 0.4 0.2 Upscaled model (32x32) Fine-scale model Oil recovery, fraction 0.8 0.6 0.4 0.2 Upscaled model (32x32) Fine-scale model 0 0.50 0.75 1.00 1.25 1.50 Pore volume injected (PVI) 0 0.50 0.75 1.00 1.25 1.50 Pore volume injected (PVI) Correlated Uncorrelated Maximum grid-block size is proportional to the level of physical mixing
Impact of Mixing on Miscible Residual Oil Saturation (S orm )
Why do we need S orm? Compositional simulation for MCM flow predicts excessive vaporization of oil (100% recovery) especially near the well grid blocks. Lab experiment and field observations indicate that recovery is not 100% owing to bypassed oil (channeling) and in rare cases water shielding. S orm also accounts for incomplete vaporization when FCM is not achieved. From Teletzke et al. (2005)
Effect of Mixing on S orm Gas channels through high permeability layers. Inj. Large grid blocks create artificially large mixing, which is often more than true mixing. Large mixing causes enhanced sweep, but lower displacement efficiency. S orm depends on grid block sizes used, physical mixing, and volume injected.
How is S orm currently used in simulators? S orm is typically the same value for every grid block. Key methods to include S orm : Alpha factors (Barker et al. 2005) Excluded oil (Eclipse) Dual-porosity (Coats et al. 2004) S orm likely depends on the FCM key scaling groups, volume injected, and grid-block sizes used. It should not be constant for every grid block. After T. Hirawa et. al
Conclusions Mixing of gas and oil is considerably accelerated by velocity fluctuations. The spreading caused by the velocity fluctuations leads to the formation of fingerlike structures with sharp concentration contrasts, and results in fast mixing (homogenization) by diffusion. Mixing is scale dependent. The scale dependence of dispersion is related to the increase in finger-like structures (and contact area) with distance travelled. Upscaling and the miscible residual oil saturation should account for mixing at the pattern scale.
THANK YOU Acknowledgments: gas flooding JIP members at Penn State University
Review of dispersion theory in literature Taylor (1953) suggested a Fickian representation for dispersion in porous media Dispersion is scale dependent (Schwartz, 1977; Peaudecerf and Sauty, 1978; Sudicky and Cherry, 1979; Pickens and Grisak, 1981; Wheatcraft and Scott 1988; Sternberg and Greenkorn 1994) Mathematical dispersivity models Continuous time random walk model (Berkowitz and Scher, 1995; Bijeljic and Blunt, 2006; Rhodes et al., 2007) Fractional derivatives, time and scale dependent models (Benson et al., 2000; Berkowitz et al., 2002; Cortis et al., 2004; Dentz et al., 2004; Su et al. 2005)
Spreading versus mixing Flow reversal to distinguish between convective spreading and mixing (Hulin and Plano (1989); Rigord et al., 1990; Leroy et al., 1992; Berentsen et al. 2005 ; John et al., 2008) Drift may impact echo tests (Coats et al., 2009) Local measurement to distinguish between convective spreading and mixing. Kitanidis (1994) defined dilution index Cirpka and Kitanidis (2000), Jose et al. (2004) used point breakthrough curve to get information about mixing and spreading Fiori (2001) explained anomalous transport Dentz and Carrera (2007) estimated effective mixing from local mixing information Jha et al. (2008) experimentally and at the pore-scale showed the occurrence of local mixing
Hydrodynamic dispersion in porous media Hydrodynamic dispersion Molecular diffusion Mechanical dispersion Velocity fluctuations Flow through different pore geometries Parabolic distribution of velocity Variations in flow direction Flow through different pore Parabolic distribution of v Variation in flow direction Molecular diffusion Hydrodynamic dispersion in porous media (Bear, 1972)
Convective spreading versus mixing 0.30 X D =0.3 Normalized concentration (C D ) 0.20 0.10 Overal concentration Local concentration 0.00 0.00 0.15 0.30 0.45 0.60 Pore Volume Injected (td)
Flow reversal (echo) eliminates convective spreading 0.16 Local Overall X D =0.0 Normalized concentration (C D ) 0.12 0.08 0.04 0.00 0.40 0.50 0.60 0.70 0.80 Pore Volume Injected (t D )
Objectives Understand the origin of scale dependent dispersivity from pore-scale to field-scale. Determine dispersivity (reservoir mixing) as function of reservoir parameters and stochastic heterogeneity of the reservoir. Determine appropriate grid-block sizes that should be used in both directions in compositional simulations to accurately predict oil recovery from miscible displacements.
Pore-scale model Pore model length: 30 mm Grain size: 5-40 µm Injection fluid velocity of 1E-5 m/s 0.86 m/day Diffusion coefficient: 1E-9 m 2 /s D o water
Pore-scale model Multi physics simulator COMSOL Flow and transport around grains V(x,y) C(x,y,t) u o Do Step 1: Solve Navier-Stokes equation: Incompressible, steady-state, no gravity 2 µ v + ρ v v + P = ( ) 0 Step 2: Solve convection-diffusion equation: C + C ( D 0 C + Cv) = 0 t + ( D 0 C + Cv) = 0 t
Longitudinal dispersion and pore-peclet number Max: 1.0 Min: 0.0 10000 1000 D L /D o = 0.015*N Pe 1.89 D L /D o 100 10 1 0.1 0.1 1 10 100 1000 Pore peclet number (N Pe ) Porosity 0.61 0.51 0.39 Experimenta l data exponent of the convection dominant region 1.89 1.68 1.55 1.1-1.3
Echo dispersion in parallel layer model 0.08 Longitudinal dispersivity (mm) 0.06 0.04 0.02 Echo with crossflow Echo without crossflow 0 0 4 8 12 16 20 24 Traveled length (mm) Echo dispersivities versus mean distance traveled by the solute
Transverse dispersion decreases with distance traveled Max: 1.0 u o Min: 0.0 1.E-06 Dispersion coefficient (m 2/ S) 1.E-07 1.E-08 1.E-09 Longitudinal dispersion Transverse dispersion D T /D L decreases as porosity decreases at the same traveled distances 1.E-10 0 2 4 6 8 Length traveled (mm)
Conclusions The pore-scale simulations produce similar features that are observed in experimental displacements. We confirmed that echo dispersivities are more accurate than transmission dispersivities. Both echo and transmission dispersivities are scale dependent. Convective spreading is not mixing, but can enhance mixing by diffusion across increased contact surface area. Transverse dispersivities are also scale dependent, but decrease with distance traveled.
Generalized equations for first contact miscible process (FCM) Pressure Equation : (incompressible fluid and no gravity effects ( z=0 )) Axkx P Ayky P (( ) ) x + (( ) ) y + qsc = 0 x µ B x y µ B y Darcy s Law u u x y kx = µ m ky = µ m P ( ) x P ( ) y
T Material balance equation for components 2 2 Ci Ci Ci Ci Ci Φ ( ) + u ( x ) + uy( ) = Φ KL( ) + ΦKT( ) 2 2 t x y x y Dispersion equation K K Li Ti u D α = + ΦF Φ D α = + ΦF Φ i Li xi u i Ti yi where α i = dispersivity of component i F = formation resistivity factor D i = molecular diffusion of component i Φ = porosity
Continuity equation Mixing rule u x x uy + = 0 y µ m = [ µ + β ( µ µ )] 1/4 1/4 1/4 4 o s o β = C C C C I 1 1 J I 1 1 C 1 =injection component Solution technique: Pressure is calculated first since steady-state is assumed. Concentrations are then calculated based on the velocity field from the pressure solution.
Scaling groups from inspectional analysis There are four scaling groups for 2-D FCM flow: (Gharbi et al 1998) Peclet number (u inj L/ΦK L ) Mobility ratio (µ o /µ s ) Effective aspect ratio (L/H(k y /k x )) Dispersion number (L/H(K T /K L )) Variables in scaling groups: L, H, k x, k y, µ s, µ o, u inj, Φ, K L, K T We will examine the effect of scaling groups by changing the parameters that go into them
Most accepted current methods Alpha factors: (Barker and Fayers 1994, Barker et al. 2005) Transport coefficients applied to each component in mass balance equation, e.g. (α oi x i, α gi y i ) becomes (x i, y i ) Analogous to pseudo relative permeabilities, but for a component. Hence, similar problems are found, e.g. rate dependence, Representative fine-grid needed to determine coefficients (Barker et al. 2005 shows how to calculate them exactly under FCM conditions) Excluded volume: Easy to implement as specified constant S orm excluded from flash calculations Keeps oil at initial composition in grid block, hence no blow down
Most accepted methods (Cont d) Dual-porosity: (Coats et al. 2004) Can handle both MCM and blowdown. Uses a second continuum (low permeable zone) for unswept oil Determining input parameters (k, Swc, ) for second continuum uncertain, as well as exchange model between the continuum. Not demonstrated to work with WAG Is it correct to use a constant value of S orm?
Estimation of the dispersion number group Dispersion number including numerical dispersion (Fanchi, 1983) N D L H 2 2 α v / v + 2 α + v / v z L z x T z x 2α + x L Assumptions: Time step sizes are small (N CO <0.05) vinj v v x A response function is generated to estimate velocity ratio
Estimation of the velocity anisotropy ratio 3.0 Velocity ratio is a function of five scaling groups Box-Behnken experimental design is used to estimate velocity ratio The prediction matches well with simulations Predicted v ZD / v XD 2.0 1.0 0.0 0.0 1.0 2.0 3.0 Simulation values of v ZD / v XD R L Scaling groups λ Dz M V DP λ Dx 0 1 2 3 4 Absolute t- values
Also valid for MCM floods Saturation profile accurate only when maximum gridblock size not exceeded Uncorrelated Correlated Uncorrelated
Upscaling in x- and z- directions First step: Determine scaling group values V = 0.60 λ = 0.25 λ = 0.10 DP xd zd N = 2.28 M = 1.0 R = 6.0 D L
Upscaling in x- and z- directions Example: Fine-scale (480x60) Procedure: 1) Determine total dispersivity in both directions for fine-scale V First: Determine scaling group values = 0.60 λ = 0.25 λ = 0.10 DP xd zd N = 2.28 M = 1.0 R = 6.0 D L 2) Fix the number of grid blocks in transverse direction and decrease the number of grid blocks in longitudinal direction 3) Repeat step 2 using fewer number of grid blocks in transverse direction No. of Grids λ xd λ zd V DP N D α LDres x z 480x60 0.25 0.10 0.600 2.28 0.045 1.56 1.56 0.081 34.74 0.294 120x20 120x30 0.25 0.10 0.587 0.574 1.56 1.86 0.034 0.036 6.25 4.69 3.12 0.072 0.076 29.86 28.30 0.324 0.285 60x20 60x30 0.25 0.10 0.586 0.573 1.10 1.31 0.026 0.028 12.50 4.69 3.12 0.072 0.076 27.00 26.07 0.307 0.270 30x20 30x30 0.25 0.10 0.583 0.571 0.77 0.92 0.021 25.00 4.69 3.12 0.071 0.075 28.39 27.90 0.274 0.31 15x20 15x30 0.25 0.10 0.577 0.566 0.47 0.65 0.013 0.016 50.00 4.69 3.12 0.070 0.073 36.78 36.64 0.341 0.310 v Z v X α L to t α T α T to t
Upscaling in x- and z- directions Fine-scale (480x60): α Ltot =34.74 ft, α Ttot =0.294 ft α L Total longitudinal reservoir mixing contour map (ft) 53
Upscaling Results (cont d) Fine-scale (480x60): α Ltot =34.74 ft, α Ttot =0.294 ft α L Best upscaled match Total longitudinal and transverse reservoir mixing contour map
First-Contact Miscible Flow at 0.6 PVI Case 1 S g Case 2 Case 3 Case 4 Channeling in reservoirs 1 and 2 causes early breakthrough Mixing of gas and oil is increased in reservoirs 1 and 2 owing to increased surface area contact between gas and oil
Multicontact Miscible Flow at 1.2 HCPVI S o Case 1 Case 2 Case 3 Case 4 Large reservoir spreading causes greater remaining oil saturation in the reservoir. Oil redistributes for case 1. Inject more solvent for case 1 reservoirs! This has been a major problem for past field projects.
Impact of Heterogeneity and Dispersion Compare four different permeability distributions based on various single and bimodal correlation lengths Reservoir dimensions are 512x64x1 m Grid blocks are 1.0 m so that numerical dispersion is small Correlation lengths vary from 1 m to 250 m, with a mean permeability of 1700 md
Miscible residual oil saturation (S orm ) Gas channels through highly permeable zones in heterogeneous reservoirs Oil is bypassed in less permeable zones Bypassed oil is not vaporized and this oil saturation is quantified as miscible residual oil saturation (S orm ) Typical permeability map for heterogeneous reservoirs
How grid block size affects S orm (importance of mixing) Grid blocks create artificial mixing 1.00 Oil recovery strongly depends on mixing (Baker 1977; Fayers and Lee, 1994; Johns et al., 1994, 2002) Oil recovery, fraction 0.90 0.80 In 1-D S orm should decrease as mixing increases 0.70 0.00 0.02 0.04 0.06 0.08 0.10 1/N Pe In 2-D, however, S orm could increase as mixing decreases owing to more bypassed oil
S orm as a function of pore volume injected Near the injection well, more pore volumes of injected gas are flowing through the grid block and more oil should be vaporized than in areas farther away from the injector which makes S orm a function of thruput (pore volume injected) S orm S orm is also a function of the gridblock size since sweep is not correctly predicted with large grid blocks Distance Current techniques use constant S orm, which is likely not physical
Estimate Level of Mixing in a Given Pattern Procedure: Estimated local dispersivity for each grid-block of FCM simulation (normalized by well spacing) Validated the scaling groups Did sensitivity analysis Used experimental design and generated response functions Normalized dispersivity (ΠD) Normalized dispersivity (αd) ( D) 1.000 1.000 R - 1 R - 2 0.100 R - 3 V DP = 0.80 0.100 R - 4 R - 5 0.010 0.010 V DP = 0.40 0.001 0.00 0.20.25 0.4 0.50 0.6 0.75 0.8 1.00 1.0 0.001 Normalized length traveled (x D )(x D ) 0.00 0.25 0.50 0.75 1.00 0.24 Normalized length traveled (x D ) Predicted αd Scaling groups 0.18 M 0.12 0.06 x D =1 R L N D V DP Dz Dx 0.00 0 0 0.06 4 0.12 8 0.18 0.24 12 Simulation Absolute t- values of α D
Sensitivity of Scale-Dependent Dispersion