TwoStage Upscaling of TwoPhase Flow: From Core to Simulation Scale

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Two-Stage Upscaling of Two-Phase FlowFrom Core
to Simulation Scale
Lou Durlofsky Stanford University
Arild Lohne George Virnovsky Stavanger
University College Rogaland Research
SPE 89422, Tulsa, 17-21 April, 2004
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Acknowledgement

• The reported results are obtained within a
  co-operative project performed by Statoil,
  Rogaland University College and RF-Rogaland
  Research on Two-phase upscaling
• The work is funded by Statoil

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Reservoir scale and upscaling steps
Geocellular scale ( 20 m)
Pore scale ( 100 mm)
Core scale ( 10 cm)
Reservoir scale ( 100 m)
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Introduction

• The problem studied
• 2 phase flow (e.g., oil and water) in
  heterogeneous reservoirs
• Small to medium scale heterogeneities in both
  absolute permeability and in capillarity (cm to
  meters scale)
• Typical gridblock sizes in reservoir models are
  100 m in horizontal direction and 1 m in
  vertical direction
• Field simulation models are typically constructed
  from geostatistical models with horizontal grid
  size 20-50 m
• Core scale heterogeneities accounted for through
  upscaled effective absolute and relative
  permeabilities, capillary pressure

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Introduction cont.

• Effective two-phase properties
• relative permeability and capillary pressure
• Depend on balance between
• viscous, capillary and gravitational forces
• This balance of forces depends on
• Subgrid heterogeneitiesSize and geometric
  distribution
• Spatial locationHigh rate near wells, low rate
  away from wells

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Numerical experiments

• 2D horizontal models
• capillary and viscous forces
• Heterogeneities at two scales
• Core scale
• Geostatistical scale
• Numerical experiments Compare displacement of
  oil by water in
• Fine scale models
• Coarse simulation models with upscaled properties

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Method Steady state two-phase upscaling

• Numerical solution of steady state flow through a
  heterogeneous flow unit corresponding to one
  coarse block
• Upscaled properties are calculated from total
  flow and pressure drop

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Uspcaling of capillary heterogeneities

• Two limiting cases
• CL - capillary limit conditions (low rate)
• Saturation distribution is given by the
• capillary-gravity equilibrium
• VL viscous limit conditions (high rate)
• Fractional flow is constant
• In both limits the saturation distribution is
  known a priori, and the upscaling problem is
  reduced to a sequence of one-phase problems
• At intermediate rates, saturation distribution
  must be solved implicitly

Two adjacent rock types with same kr and
different pc-curves
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Evaluation of rate dependency

• Capillary number
• ?pg is the global scale pressure gradient
• lpc is a characteristic length for the capillary
  heterogeneity
• Dpc is the capillary contrast (at VL-conditions)
• Note Dpc Dpc (Sw, x, wettability)
• At the intermediate saturation range, the pc
  level can be estimated

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Rate dependency at different scales

• Typical ?pg in a field 0.1 bar/m
• Assume the capillary contrast is of same order as
  the estimated pc-level
• Then
• Upscaling from Geostatistical to Simulation scale
  should use VL-conditions
• Capillary heterogeneities should be accounted for
  through upscaling from Core to Geostatistical
  scale

Geostatistical
Simulation
Core
1 cm
1000 m
100 m
10 m
1 m
10 cm
CL
Rate sensitive
VL
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2-stage upscaling

• Assumption
• Capillary forces are most important on the small
  scale
• Method
• 2-phase upscaling from Core to Geostatistical
  scale accounting for capillary heterogeneities
  (CL or rate dependent)
• Upscaling from Geostatistical to Simulation scale
  assuming viscous forces are dominant (VL)

Core Geostatistical Simulation
NC Ci
NG Gi
NS Si
VL
CL
NC gt NG gt NS
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Layered example
Model (schematic)

• Idealized layered model
• Production along and across layers
• Isotropic absolute permeability
• Contains capillary heterogeneities at two scales
• Simulation scale Coarse model 20 equal layers
• Geostatistical scale Each coarse layer contains
  two facies, F1 and F2
• Core scale Each facies containes 25 periods of
  two sublayers
• Sublayers are represented with 4 grid blocks in
  the fine scale model
• Fine scale grid 25 ? 8000 blocks will be
  upscaled in two stepsto coarse homogeneous
  model 25 ? 20 blocks

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Upscaling layered example

• 1st step
• Core to Geostatistical scale using CL
• Produces two sets of curves, one for each facies
  F1 and F2
• Diagonal tensors (different curves along and
  across layers)
• 2nd step
• Geostatistical to Simulation scale using VL
• Produces a single set of curves, i.e., the model
  is homogeneous
• The upscaled relative permeabilities are diagonal
  tensors

Effective relative permeability at the simulation
scale. Upscaled in two steps using CLVL. Along
(xx) and across (yy) layers.
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Oil production Fine scale versus Simulation scale

• Simulation at the coarse scale with upscaled
  curves (CLVL) matches the fine scale oil
  production in both wells.

• Single step upscaling using either CL or VL gives
  much poorer match

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Heterogeneities at two scales

• Upscaling from Core to Simulation scale
• In a single step rate dependent upscaling (R)
• Two steps (R or CL) (R or VL)

Average saturation in a simulation block at
different rates Selection of appropriate
2-step method
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Upscaling in two steps

• To use CL VL in two steps
• The large scale flow must be insensitive to small
  rate variations
• Sufficient difference between heterogeneity
  scales to give a rate window where we have
• capillary equilibrium between small scale
  heterogeneities
• viscous dominance between large scale
  heterogeneities
• Outside this window
• Include rate dependency in one of the steps

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Stochastic 2D example

• Two facies types at the Geostatistical
• scale
• 1) Low K, water wet
• 2) High K, mixed wet
• At the Core scale (fine) each facies is
  represented by two subtypes in a checker board
  pattern
• The subtypes have
• -Different pc-curves and
• k1/k2 5 (facies 1)
• k1/k2 2 (facies 2)

lx0.2, ly 0.1
G-block 1010
Permeability distribution at Geostatistical scale
(LxLy100100 m2)
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1st step Core to Geostatistical scale

• Using rate constraints
• Start with CL and VL upscaling from Core scale to
  Geostatistical scale
• Simulate on Geostatistical level to obtain an
  average pressure gradient for the field (between
  the VL and CL solutions)
• Repeat the upscaling with the estimated pressure
  gradient and simulate again with the new curves
• Proceed with second step upscaling when model is
  self-consistent
• Alternatively peform both upscaling steps and
  estimate pressure gradient at the Simulation scale

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Upscaled relative permeability
1st stage Low permeable facies
2nd stage block i7,j(1,10)
1st stage High permeable facies
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Quarter five-spot simulation

• Fine grid simulation vs. 1st step Geostatistical
  model
• Oil production rate

Fine
0.2 bar/G
VL
CL
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Water saturation for CL and VL, 1st stage
CL
VL
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2nd stage - Simulation scale

• Coarse model 1010 grid blocks and upscaled
  properties
• Fine scale model 500500 grid blocks

0.2 bar/GVL
Fine
VLVL
CLCL
CLVL
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Water saturation distribution
Upper row Fine scale, Lower row Simulation
scale
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Water saturation Fine scale vs. refined
simulation model
Upper row Fine scale, Lower row Coarse
properties
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Left to right flow
Oil production rates
1st step, Fine vs. Geostatistical scale
2nd step, Fine vs. Simulation scale
Fine
0.2
0.2
Fine
0.1
VLVL
VL
CL
CLVL
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Water saturation, Fine vs. Simulation scale

• Upper row Fine, Lower row Coarse (0.2
  bar/G-block VL)

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Computation time

• CPU for stochastic 2D case (1.4 MHz PC-platform)
• Fine scale
• Diagonal flow 558 CPU hrs
• Left to right flow 759 CPU hrs
• Coarse scale 3-5 CPU seconds
• Upscaling 5 CPU hours, includes
• 1st stage 3 runs at Geostatistical scale (1.5
  hrs each)
• 2nd stage 90 seconds for calculation of 100
  curves
• CPU speedups of a factor 100 or more

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Summary

• Developed and implemented a 2-stage upscaling
  procedure
• Method accounts for capillary heterogeneities at
  very fine scales
• 1st stage CL was valid in some cases. In other
  cases, a self-consistent iterative upscaling
  approach is required.
• 2nd stage VL was applicable in all cases.
• Demonstrated accurate simulation with upscaled
  properties for different flow scenarios