1
B015 SIMULATION OF AIR INJECTION IN LIGHT-OIL
FRACTURED RESERVOIRS – FROM THE
MATRIX BLOCK SCALE TO THE CROSSSECTION SCALE BY USING A DUALPOROSITY MODEL
Philippe DELAPLACE, Sébastien LACROIX, Bernard BOURBIAUX (IFP) and Yann LAGALAYE (TOTAL)
Institut Français du Pétrole, 1&4 avenue de Bois Préau, 92852 RUEIL-MALMAISON Cedex, FRANCE
Abstract
Air injection can be an economical alternative for pressure maintenance of fractured reservoirs as it avoids
re-injecting a valuable associated gas and/or generating or importing a make-up gas. In addition, the oil
recovery can be enhanced thanks to the thermal effects associated with oil oxidation. However, such an
improved recovery method requires a careful assessment of the involved reservoir displacement mechanisms,
in particular the magnitude and kinetics of matrix-fracture transfers. Considering the situation of a light-oil
fractured reservoir, compositional thermal simulations of matrix-fracture transfers were carried out on a finegrid single-porosity model of a matrix block or a stack of blocks surrounded by air-invaded fractures then on
the equivalent (up-scaled) dual-porosity model, respectively (a) to identify the main physical mechanisms
controlling matrix-fracture transfers during air injection (b) to dispose of a reliable simulation tool usable for
field-scale predictions.
Firstly, the reference fine-grid simulations show that gas diffusion and thermodynamic transfers are the
major physical mechanisms controlling the global kinetics of matrix-fracture transfers and the resulting
oxidation of oil. The chronology of extraction of oil components from the matrix blocks is clearly interpreted
in relation with phase transfers. Then, the predictions of the dual-porosity model are shown to be in very
good agreement with those of the reference model, thanks to a specific numerical formulation which ensures
a proper up-scaling of diffusion and inter-phase transfers at the overall scale of matrix blocks.
1 - Physical Data
Petrophysical Data
The petrophysical and thermodynamic properties used in our simulations are largely inspired from the
Ekofisk field (Thomas et al., 1983, 1991, Agarwal et al., 1999, Jensen et al. 2000). The matrix medium
has a permeability K of 1mD, a porosity Φ equal to 30%. The calorific capacity of the unsaturated rock is
equal to 2.35 Jg-1°C-1 and the overall thermal conductivity of the fluid-saturated rock equals 1.8 Wm-1°C-1.
The working pressure is 5600 Psi, very close to the bubble point pressure and temperature is 266°F. The
irreducible water saturation, Swi, is 0.15, the critical gas saturation, Sgc, is 0.01 and the residual oil
saturations, Sorw and Sorg, are both equal to 0.25. Capillary pressures and relative permeability curves are
shown figure 1.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Kr WATER-OIL
Pc WATER-OIL
1
2
kr WATER
kr OIL
0.9
1.5
0.8
0.7
1
Capillary Pressure (Bar)
0.6
0.5
0.4
0.3
0.2
0.5
0
-0.5
-1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Water Saturation
0.7
0.8
0.9
1
-1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
Water Saturation
0.7
0.8
1
Kr OIL-GAS
Pc GAS-OIL
1
0.03
kr GAS
kr OIL
0.9
0.025
Capillary Pressure (Bar)
0.9
0.8
0.7
0.02
0.6
0.5
0.015
0.4
0.01
0.3
0.2
0.005
0.1
0
0
0
0.1
0.2
0.3
0.4
Gas Saturation
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
Gas Saturation
0.7
0.8
0.9
1
Figure 1: Relative Permeabilities and Capillary Pressure Curves
Multi-component Oil
A light oil representative of Ekofisk field was adopted as the hydrocarbon phase for most simulations.
In order to take into account the specific volatility of the light fractions and the possibilities of oxidation
of the heavy fractions of reservoir oil, the following optimal set of pseudo-components (molar
concentrations in parenthesis) is considered to model accurately the oil phase :
- the pure component methane (C1 = 60%),
- two intermediate pseudo-components (C2C3 = 12% and C4C9 = 14%)
- and two heavy pseudo-components (C10-C17 = 8% and C18-C30 = 6%).
To complete this thermodynamic system, we need extra components namely nitrogen (N2 = 80%) and
oxygen (O2 = 20%) for air and carbon dioxide resulting from the oxidation of heavy components. The
Peng-Robinson Equation Of State (EOS) is used to model the fluid behaviour: gas-liquid equilibrium,
density of each component for each phase in the presence of air and/or CO2.
Air, water and carbon dioxide
Air consists of 20% oxygen and 80% nitrogen which are declared as components of the EOS and may
be partially dissolved in the oil phase. In the same way, carbon dioxide is also declared as a component of
the EOS but the water component is considered separately because we don’t yet dispose of a complete
three-phase thermodynamical flash calculation in our dual porosity simulator. In consequence, water
vaporisation and dissolution of carbon dioxide in the water phase are neglected in this work.
Oxidation process
To formulate oxidation reactions, each pseudo-component k is modelled as the equivalent alkane,
CnkH2nk+2, with nk the number of carbons determined from the molecular weight of the pseudo-component:
< Mw > k = Mw [Cnk H2nk+2] = (12 nk + 2 nk + 2) = 14 nk +2
3
The reaction of oxidation of pseudo-component k is then approximated as:
[CH 2 ]nk + 3nk O2 → nk [CO2 ] + nkH 2 O
2
(+
∆Hk of Reaction ) . . . . . . . . . . . . . (1)
This reaction of complete oxidation (with CO2 and H2O productions) is applied to each of the three
heaviest pseudo-components with the following reaction rate:
Qr = VK r e − Ea / RT (φρo So xk )α ( PyO2 ) β . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . (2)
where:
- the reaction rate, Qr, in moles per time unit, is given for a reference component (the oil pseudocomponent in our case) in the reaction formula,
- V is the cell volume (if Qr is expressed at cell scale),
- Kr is the exponential pre-factor, i.e. a constant characteristic of the reaction considered, with a
dimension consistent with α and β values,
- ρo the molar density of the oil phase,
- xk the molecular concentration of the hydrocarbon pseudo-component,
- yO2 the partial pressure of oxygen,
- Ea, the energy of activation,
- α and β are the reaction orders with respect to the hydrocarbon component concentration and to the
oxygen partial pressure (α = 2 and β = 1 in this study).
Actually, only the heaviest pseudo-component, C18-C30, is oxidised which provides the necessary heat
enhancing the diffusion/vaporisation of the intermediate pseudo-components. Whereas reaction enthalpies
are rather well known (from the reaction enthalpy of a methyl radical for instance), the pre-exponential
factor and energy of activation require to be calibrated from representative experimental data. In this
study, we adopted estimated values. The energy of activation, Ea, closely controls the kinetics of reaction,
as the parameter of an exponential function. The same value (61250) was assumed for all three reactions,
which is close to existing values found in Burger (Burger et al., 1985).
Diffusion process
The molecular diffusion flux in the “p” phase for the component “k” is expressed as:
( p)
k
(F
⎡ φ
⎤
) = ⎢Tk ∆(Ck )⎥
⎣ τ
⎦
( p)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
where:
-
Tk = Dk.(A/∆X)eff is the diffusion transmissivity with (A/∆X)eff the effective surface to length
ratio and Dk the effective diffusion coefficient described below;
τ the matrix tortuosity;
and ∆(Ck) the component concentration difference.
The diffusion of components within the liquid phase can be neglected. For the vapor phase, effective
diffusion coefficients Dk are computed at each time step as a function of pressure, temperature and the
gas composition (using the Wilke’s Method described in Da Silva SPE 19672) because temperature and
composition may vary within a large range of values during air injection.
2 - Up-Scaling Strategy
This work is focused on the modelling of physical processes at different scales, from the 1D-X matrix
block to the 2D-XZ stack of blocks and from constant boundary conditions to complex boundary
conditions (variable in space and time). At each scale, single porosity simulations and equivalent dual
porosity simulations were performed. The single porosity simulations performed on fine grids give
insight of the physics of matrix-fracture transfers and provides reference solutions for the dual porosity
model. We will use four physical models corresponding respectively to four levels of increasing
complexity :
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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The “1D-X Block” model will be used for studying the pure matrix-fracture transfers, without any
gravity effect. Without limitation of meshing, this model allows us to accurately follow saturation, oil or
gas composition profiles along the X-axis, orthogonal to the matrix-fracture exchange surface.
The “2D-XZ Block” model will be used for studying at the matrix block scale, a complete and
realistic physical case including gas-oil gravity drainage.
The “2D-XZ Continuous Column” model will be used for studying a realistic air injection in a
fractured reservoir where gas contact moves down slowly in the fracture network inducing a complex
boundary condition in terms of available oxygen for matrix oil oxidation.
The “2D-XZ Stack of Blocks” model is a composite model consisting in a stack of continuous
matrix blocks without capillary continuity to validate the new dual-porosity formulation at a scale close
to a reservoir cell.
3 – 1D-X Block Results
For this first study, the boundary conditions are controlled by a high vertical flux of air in the fracture
that ensures a constant gas composition of 80% nitrogen and 20% oxygen and drains most of the heat
generated within the matrix block, holding a quasi constant temperature two or three degrees above the
initial temperature(131°C to 134°C). Only the single porosity model is gridded along the x-axis and
enables us to see the evolution of the gas saturation or oil composition during diffusion/vaporisation
exchange between matrix and fracture.
Figs. 2 gives, for different times, the saturation and oil composition profiles along a large matrix block
(half block length = 50 cm): the pure diffusion case (equivalent to a nitrogen injection) is shown on the
left whereas Low Temperature Oxidation effects (about 270°F) are included on the right. The diffusion
process seems to be the dominant process at the short time scale and it is determinant in the control of
the oxidation process. We can observe two different profiles of gas invasion : a single “piston”
displacement in the purely-diffusive case and a “double piston” when the oxidation process occurs.
These observations are at the starting point of a new dynamic formulation of matrix-fracture gas
diffusion transfers in the “dual medium” version. This new formulation takes into account variable
diffusion transmissivity values instead of constant ones in the conventional formulation.
4 - 2D-XZ Block Results with Ideal Boundary Conditions
This model was used to study, at the matrix block scale, a complete and realistic physical case including
gravity drainage. The boundary conditions are controlled by a significant gas diffusion exchange
between the fracture and a large external air tank that also maintains a constant gas composition of 80%
nitrogen and 20% oxygen. This new boundary condition radically differs from the previous one from
the point of view of the thermal behaviour. In this case, the heat is drained off the matrix only by
thermal conduction to the surrounding infinite medium having the same thermal properties as the matrix
block. This boundary condition revealed itself to be close to an adiabatic condition for the matrix block
where the temperature strongly increased, up to 400°C. It is nevertheless closer to realistic conditions
than the previous isothermal conditions because the latter required a fracture displacement rate much
higher than practical reservoir values. Figure 3 shows the evolution with time of oil components (mass
in place and mass produced).
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OIL COMPOSITION PROFILES at time = 160 days
OIL COMPOSITION PROFILES at time = 160 days
1
1
Water Phase
0.9
0.6
Gas Phase
0.5
0.4
0.3
C1
0.2
O2
0.1
and N2 dissolved
in oil phase
0
0
5
10
15
20
25
30
X-Depth (cm)
35
40
45
0.7
0.6
C1
0.3
0.2
CO2
0.1
and N2 dissolved
in oil phase
5
0.9
0.8
0.8
Oil Sat. Norm. Composition
Oil Sat. Norm. Composition
0.9
0.7
0.6
0.5
0.4
0.3
45
0
0
50
5
0.9
0.9
0.8
0.8
0.7
0.6
0.5
0.4
C18-C30
C10-C17
10
15
20
25
30
X-Depth (cm)
35
40
45
50
45
50
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
OIL COMPOSITION PROFILES at time = 1800 days
1
Oil Sat. Norm. Composition
Oil Sat. Norm. Composition
OIL COMPOSITION PROFILES at time = 1800 days
0.2
45
0.3
1
0.3
40
0.4
0.1
40
35
0.5
0.1
35
20
25
30
X-Depth (cm)
0.6
0.2
20
25
30
X-Depth (cm)
15
0.7
0.2
15
10
OIL COMPOSITION PROFILES at time = 400 days
1
10
C4-C9
C2-C3
OIL COMPOSITION PROFILES at time = 400 days
5
C10-C17
0.4
0
0
50
Gas Phase
0.5
1
0
0
C18-C30
0.8
Oil Sat. Norm. Composition
Oil Sat. Norm. Composition
0.8
0.7
Water Phase
0.9
0.1
5
10
15
20
25
30
X-Depth (cm)
35
40
45
50
0
0
5
10
15
20
25
30
X-Depth (cm)
35
40
Figure 2: Saturation and Oil Composition profiles versus time with only diffusion (on the left) and
with Low Temperature Oxidation (on the right)
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Figure 3: Oil components history for Drainage (….. ),Diffusion, (
) and Reaction (______) .
The simulation restricted to air-oil Drainage presents a trivial oil recovery without any
compositional effect, contrary to Diffusion and Reaction cases: Figure 3 shows that vaporisation and
mass transfer by diffusion control the recovery of the light components. Comparing the pure Diffusion
case (close to a nitrogen injection) to Reaction case, we can see no sensitive effect on C1 produced but
a sensitive effect on the C4-C9 produced and heavier fractions.
If we focus on the Diffusion case about the two heaviest oil fractions, we can see the C10-C17 is
produced very slowly whereas C18-C30 is not produced at all. Because the vaporisation/diffusion
process is faster than drainage and firstly concerns the light components, the remaining oil becomes
highly concentrated in heavy components and air-oil drainage slows down due to the oil viscosity
increase which finally leads to the selective trapping of the C18-C30 fraction by capillary retention.
In addition, comparing mass in place to mass produced for the Reaction case, we can see that
half of the C18-C30 fraction is produced whereas all the C10-C17 fraction is rapidly produced.
Actually, the thermal effect induced by the oxidation of part of the heaviest fraction strongly enhances
the diffusion/vaporisation process.
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5 - 2D-XZ Model of a Stack of Blocks with Realistic Boundary Conditions
S in g le M e d iu m
M a tr ix
B lo c k
D u a l M e d iu m
1
D u a l M e d iu m
2 m
2
0 .2 5 m
F ra c tu re s 1 m m
2m
10
10 cells
Dual
Medium
Dual Medium
Cells for
for
10 10
Blocks
blocks
3
4
20 m
1
Dual
1 cell
Medium
Dual Medium
Cell for
for
10 Blocks
10 blocks
Figure 4: Validation of a new Dual Porosity formulation in 3 upscaling steps
New realistic boundary conditions are taken into account for these last numerical experiments
simulating a slow descent of the air-oil contact in the fracture network.
The first upscaling step consists in comparing the reference solution provided by a single-porosity
model with the solution provided by a dual-porosity model gridded in the vertical direction in order to
reliably predict gravity drainage. This step allowed us to validate the upscaling of lateral diffusion
exchanges between fracture and matrix for a variable gas composition along the fracture. The oxygen is
consumed in the upper part of the matrix rock while nitrogen, carbon dioxide and vaporised oil fractions
are drained off by the fracture.
The second upscaling step consists in comparing the results provided by this gridded dualporosity model with the results provided by a single-cell dual-porosity model where all the physics,
including gravity, are up-scaled. We don’t show any result about these two first upscaling step since the
upscaled results perfectly matched the references results.
The third upscaling step consists in comparing a dual-porosity model made up of the
superposition of N single-block cells as modelled in the previous step with a single-cell dual-porosity
model of N blocks subjected to variable boundary conditions.
A reference air injection flow rate was defined corresponding to a frontal velocity of one foot/day
assuming a total displacement of the oil-saturated matrix pore volume (85% of pore volume). The results
shown in Figure 5 correspond to the third up-scaling step (for an average frontal velocity of 1/5 foot/day).
The single-cell model (broken line) and the ten-cell model (solid line) give very close results for all light
and intermediate oil fractions. A time delay is however observed for the heaviest oil fraction, because of
the illegitimate averaging of oxygen concentration along the stack for the single-cell model.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Figure 5: Mass of Oil Components recovered from
the matrix rock for a stack of 10 matrix blocks
(DZ = 2m , DX = DY = 50 cm)
Conclusions
Reference fine-grid simulations revealed the major physical mechanisms involved in matrix-fracture
transfers during air injection in a fractured light-oil reservoir. Gas diffusion and thermodynamic phase
equilibria control the kinetics of matrix-fracture transfers and the resulting oxidation of oil.
We now dispose of a reliable field simulation tool disposing of all the required capabilities to
predict air injection scenarios in fractured reservoirs, and other gas injection scenarios involving
multiphase compositional matrix-fracture transfers.
Acknowledgements
This work was supported by Total and the authors would like to thank D. Foulon for fruitful discussions.
References
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Recovery Mechanism. Paper SPE 19672 presented at the Ann. Tech. Conf. & Exh., San Antonio, TX, Oct.
8-11, 1989.
Jensen T.B., Harpole K.J. and Osthus A.. EOR Screening for Ekofisk. Paper SPE 65124
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presented at
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