1. Two-Stage Upscaling of Two-Phase Flow: From Core to Simulation Scale
Arild Lohne*
George Virnovsky**
*Stavanger University College **Rogaland Research
Lou Durlofsky
Stanford University
SPE 89422, Tulsa, April, 2004

2. 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

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

4. 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 m
Core scale heterogeneities accounted for through upscaled effective absolute and relative permeabilities, capillary pressure

5. 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 heterogeneities Size and geometric distribution
Spatial location High rate near wells, low rate away from wells

6. 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

7. 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
Periodic boundary conditions Qy Qx
Classical boundary conditions Qx
Pressure field with macro Dp in x-direction

8. 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

9. 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

10. 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
Core
Geostatistical
Simulation
1 cm
10 cm
1 m
10 m
100 m
1000 m CL
Rate sensitive VL

11. 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 CL VL
NC > NG > NS

12. Layered example Idealized layered model
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 steps to coarse homogeneous model: 25  20 blocks

13. 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 CL+VL. Along (xx) and across (yy) layers.

14. Oil production: Fine scale versus Simulation scale
Simulation at the coarse scale with upscaled curves (CL+VL) matches the fine scale oil production in both wells.
Single step upscaling using either CL or VL gives much poorer match

15. 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

16. 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

17. Stochastic 2D example Scale Fine Medium Coarse Gridblocks 5002 502 102
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)
lx=0.2, ly = 0.1
G-block 10·10
Permeability distribution at Geostatistical scale (Lx·Ly=100·100 m2)

18. 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

19. Upscaled relative permeability
1st stage
Low permeable facies
1st stage
High permeable facies
2nd stage
block i=7,j=(1,10)

20. Quarter five-spot simulation
Fine grid simulation vs. 1st step Geostatistical model
Oil production rate
Fine
0.2 bar/G VL CL

21. Water saturation for CL and VL, 1st stage

22. 2nd stage - Simulation scale
Coarse model: 10·10 grid blocks and upscaled properties
Fine scale model: 500·500 grid blocks
0.2 bar/G+VL
Fine
VL+VL
CL+CL
CL+VL

23. Water saturation distribution
Upper row: Fine scale, Lower row: Simulation scale

24. Water saturation: Fine scale vs. refined simulation model
Upper row: Fine scale, Lower row: Coarse properties

25. Left to right flow Oil production rates
1st step, Fine vs. Geostatistical scale
2nd step, Fine vs. Simulation scale
Fine
0.2
Fine
0.2
0.1 VL
VL+VL CL
CL+VL

26. Water saturation, Fine vs. Simulation scale
Upper row: Fine, Lower row: Coarse (0.2 bar/G-block + VL)

27. 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

28. 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