Computational Continuum Mechanics in M&M SMR A. V. Myasnikov D.Sc., Senior Research Scientist, SMR Stavanger, 28 th April, 2006 From Seismics to Simulations and back

1.Streamline Reservoir Simulations; 2.Finite-difference/volume simulations of poroelastic waves propagation in partially saturated layers; 3.Rheological monitoring for pay-zones High Road Steps:

Theorem #1: If Reservoir simulations, then Streamline technologies. Theorem #2: If Streamline technologies, then FrontSim. From Seismics to Simulations:

FrontSim: I.Development of an effective 1D simulators for multi-component two-phase flows II.Development of 3 Phase Compressible Dual Porocity Models III.Extanding of Front Tracking Technology beyond two phase black oil model IV.Effective parallelizing of pressure solver V.… and so on ….

a) To increase the performance by development of high resolution modern techniques (TVD, ENO, AMR, Front Tracking) b) To take the PVT-flash procedure off the hydrodynamic “body” of the code c) To search for improved algorithms for representation of phase equilibrium in terms of alternative thermodynamic variables. Development of an effective 1D simulators for multi-component two-phase flows

Results: a) I order vs. II order high compressive TVD schemes I order scheme II order scheme “exact” solution The same C1-CO2-C4-C10 mixture 200 pts instead of 800!

Alternative Set of Independent Variables Bubble points Dew points Tie lines Plait point Two-phase domain Tie-line extension C 1, C 2 C 1, γ Oil Gas Tie-line equation:

C1-CO2-C4-C10 mixture Red – 1D ECLIPSE Black –  -parametrisation Results: b) PVT-flash procedure is taken off: 20 times CPU advantage!

One Remarkable Feature of Alternative Variables THEOREM: The image of a Riemann problem solution for the auxiliary system coincides with the projection (red circles) of the C- image (red line) of a Riemann problem solution for the general system (blue line) (not proved yet…)

“direct” I order scheme “projective” scheme Results: c) “projective” scheme – 3 effective components instead of 100 Some problems in the vicinity of the root corner points!

II. Implementation of DPSP 3-Phase Compressible Model into FrontSim Porous Matrix continuum Fracture continuum Fractured Porous medium

DPSP Results: 3 Phase compressible flow Oil production rate vs time FrontSim Eclipse Water saturation matrix fracture

1.Streamline Reservoir Simulations; 2.Finite-difference/volume simulations of poroelastic waves propagation in partially saturated layers; 3.Rheological monitoring for pay-zones High Road Steps:

Finite volume and finite difference schemes for acoustic/elasticity equations: 1. Godunov’ s finite volume scheme of 1 st order accuracy in space and time and its “2 nd order” TVD extension; 2. Two step finite different Virieux’s-like staggered grid scheme 3. >>> + non-reflecting boundary conditions

15Initials Wavelet Propagation by Virieux or Godunov (rough grid) Virieux SOG FOG

16Initials FLAC SOG FOG Wavelet Propagation by Virieux or Godunov (finer grid)

17Initials N-R conditions in 2D Elasticity (SOG) “exact” N-R BC

18Initials Artificial attenuation in buffer zones (PML) Top and bottom– rigid walls

1.Streamline Reservoir Simulations; 2.Finite-difference/volume simulations of poroelastic waves propagation in partially saturated layers; 3.Rheological monitoring for pay-zones High Road Steps:

20Initials Experimental (?) data, no fit by viscoelastic and classical Biot models Uniform elastic media Layered media, no pay-zone Layered media, pay-zone

21Initials 1D for visco-X-ticity: Bottom boundary u(x,t) p(x,t) Top boundary Top boundary: Bottom boundary: nonreflecting (Korneev, Goloshubin, et al. Geophysics, 2004, 69, 522)

One phase models: conservation laws 1. Continuity equation: 2. Equations of motion: 3. Dynamic momentum equations : 4. Energy equation:

Linear and nonlinear liquids - definition of “fluid” - Gibbs identity Generelized Onsager relations for nonlinear processes:

The simplified diagram of the process of displacement oil by gas. Their movement paths are shown with arrows. One phase liquids: conservation laws Some special cases:

Linear and nonlinear solids Definition of “solid” - Gibbs identity - Strain tensor

Elastic solids By definition: elastic is the medium where all isothermal process are reversible So that:

Non-elastic models Let: As before, - System of ODE

1.Comprehensive study of acoustic/seismic feature of rhelogicaly complex multiphase single-layered pay-zone 2. The same for multiple-layered zones 3. Through inverse problem to 4D seismic Rheological monitoring: what is that?

Biot M.A. Nonlinear and semilinear rheology of porous solids. J. Geophys. Res Biot M.A. Variational irreversible thermodynamics of heat and mass transfer in porous solids: new concepts and methods. Q. Appl. Math Biot M.A. New variational-lagrangian irreversible thermodynamics with applications to viscous flow, reaction-diffusion and solid mechanics. Advances in Applied Mechanics Backgreoung: