1. Algorithm of the explicit type for porous medium flow simulation
International Conference
“Mathematical Modeling and Computational Physics” July 3-7, 2017
Dubna
Algorithm of the explicit type for porous medium flow simulation
Marina Trapeznikova, Natalia Churbanova and
Anastasiya Lyupa
Keldysh Institute of Applied Mathematics RAS
The work is supported by the Russian Foundation for Basic Research

2. Outline Introduction: applications and goals of the research
Mathematical model of multiphase fluid flow in a porous medium
Computational algorithm of the explicit type
Remarks on multicomponent fluid flow modeling
Verification of the approach by means of a drainage test problem
Isothermal and non-isothermal three-phase infiltration problems
Efficiency of parallelization on K-100
Conclusions

3. Applications Oil and gas recovery Land reclamation facilities
Hydraulic facilities
Solution of ecological problems

4. Goals of the Research
Development of an original computational technology including mathematical models, logically simple numerical algorithms and parallel codes to predict large-scale processes in the subsurface (contaminant infiltration into the soil, hydrocarbon recovery etc.)
Generalization of the kinetically-based model of porous medium flow to the case of multiphase multicomponent fluid in view of possible heat sources
Implementation of the model by explicit numerical methods
Some oil recovery problems (problems with combustion fronts, phase transitions, complicated functions of relative phase permeability) require calculations with a very small space step what constrains a time step strictly. Then explicit schemes can gain in terms of the total run time in comparison with implicit schemes. Besides explicit algorithms are preferable for adaptation to HPC systems.

5. Model of Multiphase Fluid Flow in a Porous Medium
α = w, n, g – water, NAPL or gas; r – rock (a porous skeleton)
S α the phase saturation
ρ α the phase density
P α the phase pressure
u α the Darcy velocity of α –phase
φ the porosity
K the absolute permeability
k α the relative phase permeability
µ α - the dynamic viscosity
β α - the coefficient of isothermal
compressibility
g the gravity vector
q α - the source term
l the minimal reference length
τ the minimal reference time
c α - the sound speed in α-phase
T the temperature
Eα - the internal energy
H α - the enthalpy
λeff - the effective coefficient
of heat conductivity
ηα - the coefficient of phase
thermal expansion
l is the minimal length of averaging at which the rock microstructure is negligible, l ~ 102 rock grains;
τ is the time interval for inner equilibrium establishing in the volume with size l

6. Generalization to the Nonisothermal Case
The temperature of all phases and the rock is considered to be identical therefore the model involves a single equation of the total energy conservation.
It is found by the analogy with the quasigasdynamic system.
It includes the next quantities:
S α the phase saturation
ρ α the phase density
P α the phase pressure
φ the porosity
T the temperature
Eα - the internal energy
H α - the enthalpy
CPα - the phase heat capacity
λeff - the effective coefficient
of heat conductivity
λα the coefficient of phase
heat conductivity
r denotes a hard rock
ρ r = const - the rock density
(including enthalpy of the rock)
µα (T), CPα(T) and λα (T) are described by empirical relations

7. Capillary Pressure and Relative Phase Permeability
The model by Parker et al.
The model by Stone
Sαr – the residual saturation
γ and N are Van Genuchten parameters

8. Algorithm of Computations
Primary variables are Pw , Sw , Sn and T, initial and boundary conditions are set for them. Operations to be performed on each time level:
Calculation of capillary and phase pressures, phase densities, relative phase permeability and dynamic viscosities
Calculation of heat conductivity coefficients and heat capacities by empirical formulas as well as phase enthalpies
Calculation of Darcy velocities
Calculation of on the next time level via the three-level explicit scheme, convective terms are approximated by central differences
Calculation of the internal energy
on the next time level via the explicit scheme
Solving the system of nonlinear algebraic equations at each computational point locally by Newton’s method, as a result Pw , Sw , Sn and T are obtained
Data exchange at multiprocessor computations

9. Multicomponent Fluid Flow
Let the fluid consist of nα phases and nc components. One of primary variables is
the mass concentration of j-th component in α phase:
Cjα = mjα/mα , α = 1, ..., nα , j = 1, ..., nc
Darcy law (all components of the phase have one and the same Darcy velocity)
Single energy conservation equation (temperature of all components is identical)
- Capillary pressures and relative phase permeability are functions of saturations
- As additional closing relations one can use constants of the phase equilibrium:

10. Verification by a Drainage Test Problem (1. Pinder G. F. , Gray W. G
Verification by a Drainage Test Problem (1. Pinder G.F., Gray W.G. Essentials of Multiphase Flow and Transport in Porous Media, Wiley, 2008)
Two-phase (water/air) isothermal infiltration Initially the medium is fully saturated with water The top boundary is open to the atmosphere with a no-flow condition on water Water drains from the column, a no-flow condition on air exists at the bottom
Desired (the picture from [1])
Obtained

11. Problem of Phase Redistribution under the Gravity
Gas saturation
Oil saturation
At the initial moment water, oil and gas are distributed uniformly
Impermeable boundaries
Gravitation is taken into account
Capillary forces are negligible
T1>T0 in the non-isothermal case
Infiltration process is accelerated by the temperature gradient, fluid layering occurs more actively
Water saturation
Temperature

12. Efficiency of parallelization on K-100
The isothermal problem of phases redistribution under the gravity is solved.
The computational grid is 200×200×100 = 4 million points, the run time of 50 time steps is measured for each number of CPU cores.
The procedure of optimal domain partitioning is applied.

13. Infiltration Problem with a Water Source
Impervious box
Uniform initial distribution of water, oil and gas
Constant water source on the top
Gravitation, capillary forces
Temperature is constant
Water saturation
at some time moment
Weak scaling on K-100: GPUs versus one CPU core at simultaneous increase of the number of computational grid points and employed GPUs

14. Infiltration Problem with a Hot Source

15. Dynamics of Nonisothermal Infiltration

16. Dynamics of Nonisothermal Infiltration

17. Fields of Primary Variables at Some Time Moment
Water saturation
Oil saturation
Gas saturation
Water pressure
Temperature

18. Conclusions
At present the proposed kinetically-based model of multicomponent fluid flow in a porous medium is tested on oil recovery problems. The created approach can be used for simulation of perspective thermal methods of oil recovery (such as heat carrier pumping into the stratum) aimed at increasing the oil production rate of difficult-to-recover hydrocarbon reserves.