Reservoir Simulation and

Potentials for Fractional Calculus Ozgur U. Kirlangic

Saudi Aramco /Expec ARC

Computational Modeling Technology (CMT) Team

Email: ozgur.kirlangic@aramco.com

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Darcy’s Law: From integer ordered to fractional ordered calculus

𝐮 = 𝜆𝜕𝑃

𝜕𝑥 𝐮 = 𝜆𝜕𝛼𝑃

𝜕|𝑥|𝛼

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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)

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Basic definitions for non-reservoir engineers

Compressibility (𝒄𝒇, 𝒄𝑹)

and Density (𝝆𝒍)Viscosity (𝝁𝒍)

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Capillary pressure (𝑷𝒄𝒐𝒘, 𝑷𝒄𝒐𝒈)

Basic definitions for non-reservoir engineers

Relative Permeability (𝒌𝒓𝒍)

Water-wet Oil-wet

Wettability

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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?

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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

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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?

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Investment in Modeling

• # of runs/day

• # of slots

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Darcy’s Law: From integer ordered to fractional ordered calculus

𝐮 = 𝜆𝜕𝑃

𝜕𝑥𝐮 = 𝜆

𝜕𝛼𝑃

𝜕|𝑥|𝛼

(Semi empirical)

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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.