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

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

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

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.

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

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

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

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

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:

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

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

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.

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

Infiltration Problem with a Hot Source

Dynamics of Nonisothermal Infiltration

Dynamics of Nonisothermal Infiltration

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

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.