1. Imbibition Assisted Recovery
Orkhan H Pashayev
Petroleum Engineering Department
Texas A&M University
Masters Division
February 2004

2. Presentation Outline Introduction Problem Statement
Background and Literature Review
Objectives
Numerical Modeling
Grid Sensitivity
Matching Experimental Results
Numerical Analyses of Spontaneous Imbibition
Imbibition Upscaling
Conclusions
Imbibition Assisted Recovery
February 2004

3. Problem Statement
An understanding the role of imbibition in Naturally Fractured Reservoirs in order to achieve maximum recovery
Lack of knowledge in upscaling laboratory imbibition experiments to field dimensions
Imbibition Assisted Recovery
February 2004

4. Background and Literature Review
Two methods of modeling Naturally Fractured Reservoirs
Numerical model with sufficiently refined grid to adequately represent matrix/fracture geometry
Dual Porosity Model (Warren and Root, 1963)
The disadvantage of the first method is need for extensive computing power.
Warren and Root (1963) introduced the concept of dual porosity medium. Their idealized dual model is widely used in today’s commercial reservoir simulators to simulate fluid flow in naturally fractured reservoirs. In their model, the isolated cubes represent matrix blocks and the gaps between cubes represent well-connected fractures. The fracture system is further assumed to be the primary flow paths but have negligible storage capacity. Also the matrix is assumed to be the storage medium of the system with negligible flow capacity.
Primary porosity is homogeneous and isotropic, and is contained within a systematic array of identical, rectangular parallelepipeds.
All of the secondary porosity is contained within an orthogonal system of continuous, uniform fractures which are oriented parallel to the principal axes of permeability
Flow, can occur between the primary and secondary porosities, but flow through the primary-porosity elements can not occur
The primary and secondary porosities are coupled by a factor called the transfer function or the inter-porosity flow.
In dual porosity model, the fluid flow between the matrix blocks and the surrounding fractures are characterized by transfer functions
Transfer functions assume the transfer or inter-porosity flow can be attributed to imbibition phenomenon.
Imbibition Assisted Recovery
February 2004

5. Background and Literature Review
Expulsion of oil from matrix block to the surrounding fractures by capillary imbibition of water is the most important oil recovery in Naturally Fractured Reservoirs
Schematic representation of the displacement process in fractured porous medium
Capillary imbibition is described as a spontaneous penetration of a wetting phase into a porous media while displacing non-wetting phase by means of capillary pressure, e.g. water imbibing into an oil-saturated rock. This is the primary component of transfer of fluids from the matrix to the fracture.
Imbibition Assisted Recovery
February 2004

6. Background and Literature Review
Transfer Functions:
Transfer functions that use Darcy’s Law
Diffusivity transfer functions
Empirical transfer functions
Scaling transfer functions
1.The shape of the model must be identical to that of the matrix block.
2. The reservoir water-to-oil viscosity ratio must be duplicated in the laboratory tests.
3. Initial fluid distributions in the reservoir matrix block and the pattern of water movement in the surrounding fractures must be duplicated in the laboratory tests.
4. The relative permeability functions must be the same for the matrix block and the laboratory model.
5. The capillary pressure functions for the matrix block and the laboratory model must be related by direct proportionality.
When these conditions are satisfied, saturations in the laboratory model will be the same as those in the reservoir matrix block at “corresponding” times. Corresponding times for the model and the matrix block are related as follows (see Rapoport’s Eq. 37).
Imbibition Assisted Recovery
February 2004

7. Background and Literature Review
Scaling transfer functions:
Rapoport (1952)
Graham and Richardson (1959), Mattax and Kyte (1962)
Hamon and Vidal (1986), Bourblaux and Kalaidjian (1995), Akin and Kovsek (1998), etc
Du Prey (1978), Kazemi (1992), Ma et.al (1996), etc
Gravity effects are negligible
The shape of a laboratory model is identical to that of the reservoir matrix block
The laboratory model has the same water-to-oil viscosity ratio as that of the reservoir
The laboratory model duplicates the initial fluid distributions in the reservoir matrix block and the pattern of water movement in the surrounding fractures
Relative permeabilities as functions of fluid saturation are the same for the reservoir matrix block and the laboratory model
The capillary pressure of the reservoir matrix block and the laboratory model are related by direct proportionality, such as Leverett’s dimensionless J-function
Scaling transfer functions are used to predict recovery in field size cases with the results from lab experiments. Rapoport proposed the “scaling laws” applicable in case of water-oil flow.
Imbibition Assisted Recovery
February 2004

8. Objectives
Conduct numerical studies with matrix block surrounded by fractures to better understand the characteristic of spontaneous imbibition
Evaluate dimensionless time tD and investigate the limitations of the upscaling laboratory imbibition experiments to field dimensions
Imbibition Assisted Recovery
February 2004

9. Presentation Outline Introduction Problem Statement
Background and Literature Review
Objectives
Numerical Modeling
Grid Sensitivity
Matching Experimental Results
Numerical Analyses of Spontaneous Imbibition
Imbibition Upscaling
Conclusions
Imbibition Assisted Recovery
February 2004

10. Simulation Parameters
Two phase black-oil commercial simulator, CMG™
Core = 3.2cm x 3.2cm x 4.9cm
K = 74.7
SWi = 41.61%
Φ = 15.91%
μOIL = 3.52 cp
μWATER = 0.68 cp
APIOIL = 31°
Berea core was selected cause it is widely used.
Imbibition Assisted Recovery
February 2004

11. Grid Sensitivity Analyses No. of gridblocks in I, J and K directions
Cartesian grid system
Simulation
Run
No. of gridblocks in I, J and K directions
Total No. of gridblocks
I - Direction
J - Direction
K - Direction 1 7
343 2 12
1,728 3 16
4,096 4 20
8,000 5 25
10,000
In order to numerically simulate the experiment the core was discretized into a grid model.
Imbibition Assisted Recovery
February 2004

12. Grid Sensitivity Analyses
The main yardstick in grid sensitivity analyses was Recovery vs. Time
Imbibition Assisted Recovery
February 2004

13. Grid Sensitivity Analyses
Water saturation profiles in the core
Middle gridblock at last timestep
Imbibition Assisted Recovery
February 2004

14. Grid Sensitivity Analyses
10000
8000
According to theory at some certain number of gridblocks, computer time required to solve pressure equation in gridblock simulation will increase exponentially.
4096
1728
Imbibition Assisted Recovery
February 2004

15. Reservoir Grid I = 20, J = 20, K = 20 No. of gridblocks = 8000
Grid dimensions
I: 1x0.01cm 18x0.178cm 1x0.01cm
J:1x0.01cm 18x0.178cm 1x0.01cm
K:1x0.01cm 19x0.259cm
An extra grid block of very small dimensions was added at the bottom to account for the boundary condition.
This grid block was assigned a water saturation value of 1.0 at all times.
This represents the contact of water with the core.
Imbibition Assisted Recovery
February 2004

16. Matching Experimental Results
Imbibition Assisted Recovery
February 2004

17. Matching Experimental Results
The following logarithmic capillary pressure relationship was used
PC° - threshold capillary pressure
SW – water saturation
Capillary Pressure is also modeled using a non-linear function.
The capillary pressure is traditionally known to be a logarithmic function of water saturation.
In this model the initial value capillary pressure or Pc0 is experimental fit parameter, is varied to obtain a match of the recovery.
With low capillary pressure the waterfront takes longer to reach the other end of the core.
By trial and error solution the value of capillary pressure “end point value” was found to be 2
Imbibition Assisted Recovery
February 2004

18. Matching Experimental Results
Imbibition Assisted Recovery
February 2004

19. ρWATER = 1 g/cc ρOIL = 0.8635 g/cc ρWATER = ρOIL = 0.8635 g/cc
Gravity Effect
Bond number
ρWATER = 1 g/cc
ρOIL = g/cc
ρWATER = ρOIL = g/cc
The ratio of gravity to capillary forces is called the Bond number
Bo>1 – capillarity is negligible
Bo<1 – gravity is negligible
Imbibition Assisted Recovery
February 2004

20. Different Boundary Conditions
All Faces Open
Two Ends Closed
Two Ends Open
One End Open
No Flow Surfaces
Imbibition Assisted Recovery
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21. Different Boundary Conditions
Imbibition Assisted Recovery
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22. Different Boundary Conditions
This figure is a plot of absolute time for imbibition as a function of the number of faces available for imbibition.
It is observed from the figure that the time required to saturate core 60% with water increases exponentially as the number of faces available for imbibition decreases.
A comparison of all cases shows that non-wetting recovery in TEC and AFO cases are most efficient and fast as compared to other cases.
Imbibition Assisted Recovery
February 2004

23. Case 1: K1 > K2 > K3 > K4 Case 2: K1 < K2 < K3 < K4
Heterogeneities K1 K2 K3 K4
One End Open
Case 1: K1 > K2 > K3 > K4
Case 2: K1 < K2 < K3 < K4
Case 1 is when oil was being displaced by water and permeability was decreasing along the way
Case 2 when oil was being displaced by water but in this case permeability was increasing along the way
water
Imbibition Assisted Recovery
February 2004

24. Heterogeneities
When permeability decreases in the direction of water movement, water reaches the boundary of different permeabilities, and passes through it without any obstacles.
The movement of water doesn’t stop cause of capillary pressure increase.
Case 2: Water imbibes into the zone of low permeability and moves cuz of high capillary forces.
When water front reaches the zone of high permeability, with lower capillary forces, water movement here is not so fast as in previous case
Here some time oil is being displaced into the zone of high permeability, and water saturation increases in the zone of low permeability.
At some Sw, water starts to imbibing into the zone of higher permeability
Imbibition Assisted Recovery
February 2004

25. Presentation Outline Introduction Problem Statement
Background and Literature Review
Objectives
Numerical Modeling
Grid Sensitivity
Matching Experimental Results
Numerical Analyses of Spontaneous Imbibition
Imbibition Upscaling
Conclusions
Imbibition Assisted Recovery
February 2004

26. Spontaneous Imbibition Upscaling
Theory
Recovery behavior for a large reservoir matrix block could be predicted from lab experiments
Mattax and Kyte
Ma et.al
Ma et al introduced the idea of characteristic length into dimensionless scaling parameter.
A characteristic length can be defined for systems with different shapes and boundary conditions
lAi - the distance traveled by the imbibition front from the open surface to the no-flow boundary.
Vb - bulk volume of the matrix,
Ai - the area open to imbibition at the ith direction,
Imbibition Assisted Recovery
February 2004

27. Spontaneous Imbibition Upscaling
All Faces Open, Two Ends Closed, Two Ends Open and One End Open
Semi-log plot:
Normalized Recovery vs. Dimensionless Time
R8 – is ultimate recovery
Imbibition Assisted Recovery
February 2004

28. Spontaneous Imbibition Upscaling
Comparison
Imbibition Assisted Recovery
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29. Spontaneous Imbibition Upscaling Varying Mobility Ratio
Imbibition Assisted Recovery
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30. Spontaneous Imbibition Upscaling Varying Mobility Ratio
Mobility Ratio - not included
Need to include mobility ratio into the formulation of dimensionless time
Imbibition Assisted Recovery
February 2004

31. Spontaneous Imbibition Upscaling Varying Mobility Ratio
The new scaling improves correlation significantly by taking into account end-point mobilities and mobility ratio
Imbibition Assisted Recovery
February 2004

32. Spontaneous Imbibition Upscaling
TEO
Imbibition Assisted Recovery
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33. Spontaneous Imbibition Upscaling
TEC
Imbibition Assisted Recovery
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34. Spontaneous Imbibition Upscaling
Heterogeneous Core K1 K2 K3 K4
One End Open
Case 1: K1 > K2 > K3 > K4
Case 2: K1 < K2 < K3 < K4
Case 1 is when oil was being displaced by water and permeability was decreasing along the way
Case 2 when oil was being displaced by water but in this case permeability was increasing along the way
water
Imbibition Assisted Recovery
February 2004

35. Spontaneous Imbibition Upscaling
Heterogeneous Core
Imbibition Assisted Recovery
February 2004

36. Presentation Outline Introduction Problem Statement
Background and Literature Review
Objectives
Numerical Modeling
Grid Sensitivity
Matching Experimental Results
Numerical Analyses of Spontaneous Imbibition
Imbibition Upscaling
Conclusions
Imbibition Assisted Recovery
February 2004

37. Conclusions
It was observed that time required to saturate core to Sw=60% increases exponentially as the number of faces available for imbibition decrease
Results proved that using characteristic length in the equation of dimensionless time, instead of length of the core improves upscaling of spontaneous imbibition
Imbibition Assisted Recovery
February 2004

38. Conclusions
Further investigation revealed that upscaling correlations could be significantly improved by taking into account end-point mobilities and mobility ratio
Spontaneous imbibition recovery is higher for a flow in the direction of decreasing permeability than in the case of a flow in the direction of increasing permeability
Imbibition Assisted Recovery
February 2004

39. Conclusions
Some discrepancy observed in correlations, while upscaling heterogeneous core, indicated that existing transfer functions can not precisely account for heterogeneities in the core
Imbibition Assisted Recovery
February 2004

40. Thank You! Acknowledgement
Finally I would like to express my sincere gratitude and appreciation to my advisor Dr. David Schechter and Dr. Erwin Putra.
Thank You!
Imbibition Assisted Recovery
February 2004

41. Imbibition Assisted Recovery
Orkhan H Pashayev
Petroleum Engineering Department
Texas A&M University
Masters Division
February 2004