Romenski GV Reshetova Institute of Geology amp Mineralogy SB RAS Institute of Mathematics SB RAS Institute of Computational Mathematics amp Mathematical Geophysics ID: 809449
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Slide1
Yu.V. Perepechko*, E.I. Romenski†, G.V. Reshetova‡*Institute of Geology & Mineralogy SB RAS†Institute of Mathematics SB RAS‡Institute of Computational Mathematics & Mathematical GeophysicsSB RAS
Modeling
of
Compressible Multiphase
Flow through Porous Elastic Medium
Slide2IntroductionMotivation: Modelling of multiphase flow in elastic porous media is a topical area of research in the general study of fluid flow regimes in petroleum reservoirs. The deep understanding of the coupled problem for multiphase mixture flow and finite elastoplastic deformation of a skeleton is very important for the development of new technologies of intensifying oil extracting.Despite the intensive research efforts there is still no a common approach and conventional mathematical model for the description of above mentioned processes.
Goal
of the present research is to propose
a new computational model for the multiphase flow in elastically deforming porous media in case of finite deformation of skeleton. The derivation of the model is based on the thermodynamically compatible hyperbolic systems of conservation laws theory (Godunov and Romenskii 2003) and irreversible thermodynamics. Results include derivation of new governing equations for the flow of compressible liquids mixture in elastic porous medium and high accuracy Runge-Kutta-WENO numerical method for solving differential equations of the model. The equations of the model comprise balance laws for phase masses, total momentum and total energy conservation laws supplemented by the equations for relative velocities and for deformation gradient in divergent form. The Runge-Kutta-WENO method up to 4th order in time and 5th order in space is developed for two-dimensional equations.
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk
, September 22-25 2014
Slide3Theory I. Parameters of state of the medium is the volume fraction of elastic skeleton (phase 1) is the volume fractions of the fluids (n=2,…,N) is the mass density of the phase is the total mass density of the medium is the mass fraction of phase
is
the velocity vector of
phase i (i=1,2,3) is the relative velocity of phase with respect to the elastic porous phase is the mixture velocity of the medium is the deformation gradient of the whole medium is the entropy for the whole medium saturation constraints
porosity
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk
, September 22-25 2014
Slide4Theory I. The mixture energyThe mixture internal energyThe mixture full energyFirst and second principle of thermodynamics
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk , September 22-25 2014
Slide5Theory I. The generating potential25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014The partial derivations of the generating potential
Slide6Theory II. The master equations25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014The set of the master equations
Godunov
, S.K. and
Romenskii, E.I. Elements of continuum mechanics and conservation laws. Kluwer Academic/Plenum Publ., NY. 2003.
Slide7Theory II. Governing Equations - total mass conservation - phase mass fraction balance - phase mass conservation - relative velocities
equation
- mixture momentum conservation - deformation gradient conservation
dissipative free
thermodynamically compatible model
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk
, September 22-25 2014
Slide8Theory II. Governing Equations - total mass conservation - phase mass fraction balance - phase mass conservation - relative velocities
equation
- mixture momentum conservation - deformation gradient conservation
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk
, September 22-25 2014
dissipative
thermodynamically compatible model
Slide9Theory II. Governing EquationsAn artificial variable is introduced in order to keep the equation for the relative velocities in a divergent form.The term can be considered as the true source term because the following compatibility condition holds
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk , September 22-25 2014dissipative thermodynamically compatible model
Slide10Theory II. Governing Equationsdissipative productionenergy conservationThe system of the thermodynamically compatible equations is hyperbolic and all equations are in a divergent form.dissipative production & energy conservation
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk , September 22-25 2014
Slide11Numerical methodHigh accuracy Runge-Kutta-WENO method 3rd order in time and 5th order in space (Shu, C.W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA/CR-97-206253, ICASE. report No.97-65. 1997)Differential equations of the model in a two-dimensional coordinate system may be write out in the following vector formthe Runge-
Kutta
-WENO method for solving the system
in a rectangular area with cells , where ).The difference approximation of the equation may be written out in the formIf the values of fluxes on lateral faces of a cell are known, one can obtain the differential equation for solution values in a cell which is solved by the Runge-Kutta method of the required order of accuracy. (Kennedy, C.A. and Carpenter, M.H. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. NASA Langley Technical Report Server, 2001)25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk
, September 22-25 2014
Slide12Numerical methodHigh accuracy Runge-Kutta-WENO method 3rd order in time and 5th order in spaceThe formulae for the TVD Runge-Kutta method of the third order of accuracy are of the form
(
Drikakis
, D. and Rider, W.J. High-resolution methods for incompressible and low-speed flows. Springer-Verlag, 2004)25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014
Slide13Numerical methodHigh accuracy Runge-Kutta-WENO method 3rd order in time and 5th order in spaceTo specify the values of fluxes on the faces of the computational cells it is sufficient to know the values of on the cell faces. To calculate fluxes on cell faces the WENO approximation of the solution by polynomials (reconstruction) is implemented using the solution values in neighbor cells. A decomposition with respect to space variables is employed, i.e. for calculating and one-dimensional systems are used which describe the propagation of waves along the axes The
Lax–
Friedrichs
method is employed to calculate fluxes in the equations. For the equation in the direction of the axis, assuming that in the initial data the values and on the left and right of point are known, calculate the flux in this point by the formulaAnalogically for axis,25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014
Slide14Numerical methodAn example of WENO reconstruction providing for the fifth order of accuracyInside the computational cell we define “right” and “left” values, which should be calculated with the use of reconstruction Nonlinear weight coefficients are defined by the relationswith a smoothness indicator The constant is introduced to avoid division by zero
when
and is usually set to
. Optimal weight coefficients for “right” values for left values .We get two reconstructed functions and correspondingly on the left and right on each face of a cell in each internal cell of the computation area. Performing reconstruction for all variables we obtain the value and on the left and right of the cell borders which are then used to calculate flows by the Lax–Friedrichs formula.25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014
Slide15Wave propagation in the porous formationthe saturated porous formation with porosity 0.2 pressure x-component of stress deviator
Ricker
source
o
il-water
interface
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk
, September 22-25 2014
Slide16Filtration of water-oil mixtureThe filtration of water-oil mixture in the space beyond the well when the mixture is injected into the well.Flow velocity fields of water (horizontal (left) and vertical (right) components) and oil (horizontal component). A pressure source is absent.
Flow velocity fields of water (horizontal (left) and vertical (right) components
)
and oil (horizontal component). A pressure source of the Ricker type is located in the center of the lower part of the well.25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014
Slide17Physical parameters 25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk , September 22-25 2014The values of the phases’ physical parameters were set in the model to the following:
Slide18in addition …effectivity of parallel code (Siberian Supercomputer Center, NCS-30Т cluster )Kalgin K.V., Kireev S.E., Malishkin V.E., 2013(Contract of the Ministry of Education and Science of Russian Federation #07.514.11.4156)
25.09.2014
2nd Russian-French Workshop "Computational Geophysics",
Berdsk , September 22-25 2014
Slide19ConclusionsThe new computational model of multiphase compressible flow in saturated elastic porous media is proposed. The governing equations of the model form the hyperbolic system of differential equations in a divergent form. The above properties allow one to apply high accuracy numerical methods to solve the variety of problems.The proposed model allows one to study the propagation of seismoacoustic waves accompanied by dissipative effects and temperature variations in well systems, the influence of stress waves on the flow of fluid (water, oil, their mixture) in deforming porous media, and processes of finite elastic deformations of porous medium saturated by the compressible multiphase fluid.25.09.20142nd Russian-French Workshop "Computational Geophysics", Berdsk
, September 22-25 2014