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Reservoir Simulation and
Potentials for Fractional Calculus Ozgur U. Kirlangic
Saudi Aramco /Expec ARC
Computational Modeling Technology (CMT) Team
Email: [email protected]
Saudi Aramco: Company General Use
Darcy’s Law: From integer ordered to fractional ordered calculus
𝐮 = 𝜆𝜕𝑃
𝜕𝑥 𝐮 = 𝜆𝜕𝛼𝑃
𝜕|𝑥|𝛼
Saudi Aramco: Company General Use
Why would it not be a trivial journey?
• Because, Darcy relationship is not the whole story…
• Reservoir simulation is an essential tool for reservoir management
• It is a highly complex interdisciplinary subject based on various scientific fields
Engineering: geology,
petrophysics, geophysics,
…
Physics: PVT, Rock physics,
…
Computer science: Software
engineering, Algorithms,
HPC,…
Mathematics: ODE, PDE, Numerical
techniques, Schemes, linear
algebra, …
Chemistry: EOR, …
further known/unknown
branches, high end advances in
science and technology
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Basic definitions for non-reservoir engineers
Hydrocarbon Reservoir • Porous• Permeable• Confined
gas
water
oil
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Basic definitions for non-reservoir engineers
Permeability (𝕜)
Porosity (𝝓)
Saturation (𝑺𝒍)
Saudi Aramco: Company General Use
Basic definitions for non-reservoir engineers
Compressibility (𝒄𝒇, 𝒄𝑹)
and Density (𝝆𝒍)Viscosity (𝝁𝒍)
Saudi Aramco: Company General Use
Capillary pressure (𝑷𝒄𝒐𝒘, 𝑷𝒄𝒐𝒈)
Basic definitions for non-reservoir engineers
Relative Permeability (𝒌𝒓𝒍)
Water-wet Oil-wet
Wettability
Saudi Aramco: Company General Use
FLU
X
Mass conservation: Single Phase
𝑑
𝑑𝑡𝜌∅ + 𝛻 ∙ 𝜌𝐮 = 𝑞𝑤𝑒𝑙𝑙
𝐮 = −𝕜
𝜇∙ 𝛻𝑃 + 𝜌𝐠
Darcy:𝕜 =
𝑘𝑥 0 00 𝑘𝑦 0
0 0 𝑘𝑧
,
if isotropic: 𝑘𝑥 = 𝑘𝑦 = 𝑘𝑧Fractional Derivatives?
FLUX
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Mass conservation: Multi Phase - Black Oil Model Equations
𝑑
𝑑𝑡
1
𝐵0∅𝑆𝑜 + 𝛻 ∙
1
𝐵0𝐮𝒐 = 𝑞𝑜
𝑑
𝑑𝑡
1
𝐵𝑤∅𝑆𝑤 + 𝛻 ∙
1
𝐵𝑤𝐮𝒘 = 𝑞𝑤
𝑑
𝑑𝑡∅𝑅𝑠𝐵𝑜𝑆𝑜 +1
𝐵𝑔𝑆𝑔 + 𝛻 ∙
𝑅𝑠𝐵0𝐮𝒐 +1
𝐵𝑔𝐮𝒈 = 𝑞𝑓𝑔 + 𝑅𝑠𝑞𝑜
constant temperature
thermodynamic equilibrium
𝐮𝒍 = −𝕜𝑘𝑟𝑙𝜇𝑙∙ 𝛻𝑃𝑙 + 𝜌𝑙𝐠Darcy:
Highly Nonlinear! Variable Coeff PDE’s
Fractional Derivatives?
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Mass conservation: General Equations
𝑑
𝑑𝑡∅𝑆𝑜𝜌𝑜𝜛𝑜𝑖 + ∅𝑆𝑔𝜌𝑔𝜛𝑔𝑖 + 𝑞𝑖
= −𝛻 ∙ 𝜌𝑜𝜛𝑜𝑖𝐮𝒐 + 𝜌𝑔𝜛𝑔𝑖𝐮𝒈 + 𝛻 ∙ ∅𝑆𝑜𝜌𝑜𝕂𝑜𝑖𝛻𝜛𝑜𝑖 + ∅𝑆𝑔𝜌𝑔𝕂𝑔𝑖𝛻𝜛𝑔𝑖
𝒊 = 𝟏,… ,𝑵
𝝕𝒐𝒊 and 𝝕𝒈𝒊 : mass fractions of hydrocarbon
components in oil and gas phases
𝑖=1
𝑁
𝜔𝑜𝑖 = 1 𝑖=1
𝑁
𝜔𝑔𝑖 = 1
𝜔𝑔𝑖
𝜔𝑜𝑖= 𝑲𝑖 = 𝑓(𝑃𝑙 , 𝜔𝑖) 𝒊 = 𝟏,… ,𝑵
Driven by Darcy’s Law! (Not exact) Analogy: If Black Oil Model is a black/white photo, Compositional Model is a colorful photoFractional Derivatives?
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Enhanced Oil Recovery (EOR) Processes (sources of nonlinearity)
Miscible displacement,Convection Diffusion Equations
• approaching in critical point within 2-phase region oil and gas become miscible
• Kro=Krg
New physics means : additional Non-linearies additional equations + more terms additional components additional complications additional dependencies
Driven by Darcy’s Law!
Salinity Modeling
Polymer Injection
Surfactant Injection
CO2 (or other gas) injection
Foam Injection
Alkaline Injection
Nano-particles
Chemical reactions
…
Fractional Derivatives?
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Enhanced Oil Recovery (EOR) Processes (sources of nonlinearity) E.g. Polymer Equation
𝑉∗ (𝑚𝑝,𝑗𝑡+Δ𝑡 −𝑚𝑝,𝑗
𝑡 ) + 𝑚𝑝,𝑎𝑑𝑡+∆𝑡 −𝑚𝑝,𝑎𝑑
𝑡 + Δ𝑡
𝑒𝜖𝑁𝑗
𝑇𝑒𝜆𝑤,𝑝,𝑒∗,𝑡+Δ𝑡ΔΦ𝑤,𝑒
𝑛+1 + Δ𝑡𝑄𝑝,𝑗𝑡+Δ𝑡 + Δ𝑡𝐻𝑝,𝑗
𝑡+Δ𝑡 = 0
𝑗 = 1…𝑁𝑐𝑒𝑙𝑙
Accumulation term modified to account for dead pore space from
adsorption
Adsorption term added
Water Mobility changed by water
viscosity
Water Mobility changed by
resistance factor from adsorption
Well model can inject and
produce polymer
DEAD_PORE_SPACE ADSORPTION_PL_TABLE
ROCK_MASS_DENSITY
VISCOSITY_PL_TABLE
Todd_Mixing_Parameter
RESIDUAL_RESIST_F
MAXIMUM_ADSORPTION
Inj_WTR_PL_Conc
Half life degradation
term
HALF_LIFE
DELAY_TIME
Polymer
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Boundary Conditions: Well modeling
𝑞𝑝,𝑙 = 𝑊𝐼𝑙𝜆𝑝(𝑃𝑖 − 𝑃𝑤,𝑙)
𝑞𝑝,𝑙 : The well inflow rate of phase p of completion lDriven by Darcy’s Law !Fractional Derivatives?
Saudi Aramco: Company General Use
High flow rates (Non-Darcy Effects)
−𝛻Φ𝑙 =𝜇𝑙𝑘 ⋅ 𝑘𝑟𝑙
𝑢𝑙 + 𝛽𝜌𝑙𝑢𝑙 ⋅ 𝑢𝑙
Unconventional: Hydraulic fracturing
Driven by Darcy’s Law !Fractional Derivatives?
Φ = 𝑃0
𝑃 𝑑𝑃
𝛾− 𝑧
𝛽: non-Darcy-flow beta factor
Saudi Aramco: Company General Use
Gridding: Structured/Unstructured Grids, LGR
Figure: Structured (left) vs Unstructured (right) Grid
Figure: Horizontal wells, faults, and the unstructured grid
• accuracy issues• difficulties in handling complex geometries
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Naturally Fractured Reservoirs
• Dual-Porosity Dual-Permeability (DPDP) Model:
1. Matrix Blocks: • high porosity• small permeability
2. Fractures: • low porosity• high permeability
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Multi-Porosity Multi-Permeability (MPMP)Figures: show a triple porosity system
Intercontinuum Mass transfer by Darcy’s Law ! Fractional Derivatives?
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Numerical Models
Methods such as:• finite differences • finite volumes• finite elements• …
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Linear Solver
Figure: From Notes on Reservoir Simulation,Khalid Aziz, Lou Durlofsky, Hamdi Tchelepi
• Our matrices are super-sparse (~97% zeros)
• Challenge:• fractional calculus works with dense matrices
• HPC, Machine Learning, 4IR technologies?
Saudi Aramco: Company General Use
Darcy’s Law: From integer ordered to fractional ordered calculus
𝐮 = 𝜆𝜕𝑃
𝜕𝑥𝐮 = 𝜆
𝜕𝛼𝑃
𝜕|𝑥|𝛼
(Semi empirical)
Saudi Aramco: Company General Use
Fractional Derivatives in Reservoir Simulation? What we want to learn
1. Mathematical Derivation of a simple Fractional Derivative?
2. Graph of a simple fractional derivative and discussion?
3. Physical Meaning of Fractional Derivatives?
4. The problem of "Transient testing in fractured porous media"?
5. What is Memory Formalism? What is Fluid memory?
6. What is the problem of "longtailed non-Fickiansolute transport in fractured porous rock" ?
7. What is anomalous transport?
8. How can "Fractal Models" and "fractional diffusion models" improve the limitations such as "lateral non-heteogeneity" assumption?
9. How is "non-local flux constitutive law" is employed to relate "volumetric flux" and "pressure gradient"?
10. What is "nonlocal flux relationship"?
11. What are the other factors that influence the observed/predicted flux is other than the "pressure gradient" at the desired location at any instant in time?
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Fractional Derivatives in Reservoir Simulation? What we want to learn
15. What is the difference between Normal vs Anomalous Diffusion Models? the "mean square displacement (MSD)" of the diffusing particle as a "linear" function of time. MSD = Diffusion Coeff * Elapsed Time the "mean square displacement (MSD)" or "mean square variance" of the diffusing particle is a "non-linear" function of time.
16. What is "random walks"? What does the classical "random walk" problems have a mean waiting time that is finite and solutions are based on the knowledge of the current state of the system mean?
17. Why would Anomalous diffusion models be better for describing flow in naturally fractured and disordered nano-porous media? They require more information other than the current state (i.e. history of the process)..
18. Fractional Derivatives and HPC, Machine Learning and 4IR?
12. The "non-local flux constitutive equations“ compared with the "classical diffusion approach“
13. How does the classical diffusion approach: based on "random Brownian motion of particles“ ?
14. How can "non-local flux constitutive equations" improve below problems against the classical diffusion approach? fluid flow in porous media of fractal geometry, naturally fractured unconventional shale reservoir, and nano-porous and porous materials.