DISCRETIZATION AND GRID BLOCKS NTNU Author: Professor Jon Kleppe Assistant producers: Farrokh Shoaei Khayyam Farzullayev

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Discrete system:  Continuous system: Contineous and discrete systems

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  For i=2 to N-1: Summation of forward and backward equations give us: Constant grid block sizes i-1ii+1 xx xx i-1ii+1i-1ii+1  Backward expansions of pressure:  Forward expansion of pressure:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Solving backward equation and Eq. II give us:  For i=N:  For i=1: Eq. I  Solving forward equation and Eq. I give us: 21 xx PLPL N-1 N xx PRPR Eq. II

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling Variable grid block sizes  Finer description of geometry  More realistic grid system  Better accuracy in areas of rapid changes in pressures and saturations  Specially useful in the neighborhood of production and injection wells i-1ii+1 xx  x i+1  x i-1  The Taylor expansions:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  The flow term : Where :  For i=2 to N-1 :  Due to the different block sizes, the error terms are of first order only.  Flow equation for i=1 and i=N depends to boundary conditions.

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling x1x1 x2x2 1 2 PLPL  Boundary conditions  Pressure condition at the sides of slab:  Same for pressure at the right hand side:  For i=1:  For i=N:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Flow rate specified at the sides of slab: QLQL 12 x1x1 x2x2  Same for flow rate at the right hand side:  For flow rate at the left hand side:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling Time discretization  Expansion forward:  Expansion backward:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  All parameters except p i at (t+ Δt) are in time t and are known, so simply by solving equation you can find p i at (t+ Δt).  Use the forward approximation of the time derivative at time level t.  The left hand side is also at time level t.  Solve for pressures explicitly. Numerical formulations xx i-1ii+1 t This formulation has limited stability, and is therefore seldom used. i-1ii+1 t + Δt  Explicit method:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Use the backward approximation of the time derivative at time level t+ Δt.  The left hand side is also at time level t+ Δt.  Solve for pressures implicitly. xx i-1ii+1 t i-1ii+1 t + Δt  A set of N equations with N unknowns, which must be solved simultaneously.  For instance using the Gaussian elimination method. This formulation is unconditionally stable.  Implicit method:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Use the central approximation of the time derivative at time level t+ Δt / 2.  The left hand side is also at time level t+ Δt / 2.  The resulting set of linear equations may be solved simultaneously just as in the implicit case.  The formulation is unconditionally stable, but may exhibit oscillatory behavior, and is seldom used.  Crank-Nicholson method:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling S Wir  100 grid blocks:  40 grid blocks:  20 grid blocks:  10 grid blocks: Sensitivity to number of grid blocks 1 i 10 1 i 20 1 i 40 1 i SWSW X / L  The more grid blocks we have, the smaller are the blocks sizes (Δx), smaller is the numerical dispersion because the discretization error is proportional to Δx 2. 1-S or

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling Sensitivity to time step Δt = 20 sec Δt = 10 sec Δt = 1 sec  The smaller are the time steps (Δt), the smaller is the numerical dispersion due to the discretization where the error is proportional to Δt. S Wir SWSW X / L 1-S or

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling Capillary and viscous forces  pressure difference across a grid block will be directly proportional to the size of this ( in the flow direction ).  The direct effect of capillary pressure will therefore often be dominating in a core-sized grid block while it is negligible in a full field simulation formation scale grid block.  Core plug:  Simulation grid block: Capillary forces (capillary endpoint pressure) Viscous forces (viscous pressure drop) 10 cm 0.01 bar productioninjection 0.7 bar 150 m production injection 21 bar 0.7 bar

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Geological models may contain millions of grid blocks representing geologically interpolated data (geostatistical realizations).  Numerical simulators cannot handle this level of detail due to cost limitations (applicable with less than millions of grid blocks).  The magnitude of the difference between fine and coarse scales is very significant.  The key problem is how to obtain effective input for the numerical flow simulator from data on finer scales.  This process is called upscaling. Upscaling

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Laminar sclae grid blocks: 1.5 * 0.5 m core plug  Formation scale grid blocks: 60 * 5 m  Formation scale grid blocks: 12 * 2.5 m  Formation scale grid blocks: 1.5 * 0.5 m High permiability Low permiability

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Fine grids and coarse grids:  Fine grids:  Coarse grids:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling C ij F ij  The permeability tensor of a porous medium is specified on each fine-scale cell F ij, and must be upscaled or homogenized over each coarse-scale or computational cell C ij  Permeability tensor:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling Recovery Time Fine grid model with original relative permeability Coarse grid model with upscaled relative permeability  Every single upscaling step is quality controlled during the upscaling by comparing recovery from the fine model with the recovery from the upscaled coarse gridded model incorporating the pseudo curves.  Quality control:

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling Questions 1.Use Taylor series to derive the following approximations (include error terms): 4.Write the discretized equation on implicit and explicit forms. a)Forward approximation of d)Central approximation of b)Backward approximation of c)Central approximation of (constant  x) (variable  x) 2.Modify the approximation for grid block 1, if the left side of the grid block is maintained at a constant pressure, P L. 3.Modify the approximation for grid block 1, if grid block is subjected to a constant flow rate, Q L.

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling References  Kleppe J.: Reservoir Simulation, Lecture note 3 Kleppe J.  EPS EPS

Discretization and Grid Blocks Contineous and discrete systems Constant grid block sizes Variable grid block sizes Time discretization Numerical formulations Sensitivity to number of grid blocks Sensitivity to Time step Capillary and viscous forces Upscaling  Title: DISCRETIZATION AND GRID BLOCKS (PDF) (PDF)  Author:  Name: Prof. Jon Kleppe  Address: NTNU S.P. Andersensvei 15A 7491 Trondheim  Website Website   Size: 660 KB About this module