Nuevos Enfoques sobre Medición y Escalamientos de Desplazamientos Inmiscibles en Medios Porosos

1. Nuevos Enfoques sobre Medición y Escalamientos de Desplazamientos Inmiscibles en Medios Porosos W W W Darcy rong hat´s ith A journey through the fantasy of Relative Permeability Concept Marcelo Crotti - INLAB S.A.

2. Contents • A brief analysis of the origin and uses of Darcy´s law in Reservoir Engineering. • Differences between: • Conduction of fluids. • Injection of fluids. • Production of fluids. • Limitations of the Relative Permeability concept. • Different Sceneries. • Conclusions.

3. Darcy´s Law (I) • Darcy´s law describes homogeneous fluid flow through linear porous materials. This law, which relates flow velocity with pressure gradient, has three components. • A geometric factor given by length and area of porous medium. • A fluid related factor (viscosity). • A property of the porous material (Permeability). • Q = K . A . DP / (µ . L) 

4. Darcy´s Law (II) • Permeability is usually defined as • “The ability of a porous material to conductfluids".

5. Darcy´s Law (III) • Only injection rate or production rate can be measured experimentally. • But... during the flow of an uncompressible homogeneous fluid: • The rate of conduction is equal to: • Injection rate and • Production rate. • So, measuring one of these rates is enough to obtain the value of the other rates.

6. Darcy´s Law (IV) • For multiphase fluid flow, Darcy´s law was “corrected” using a different factor for each flowing phase. This factor is known as relative permeability curve. • Qw = K . Krw. A . DPw / (µw . L)  .......... [2] • Qo = K . Kro. A . DPo / (µo . L)  .......... [3] • The value of relative permeability is different for every fluid saturation.

7. Darcy´s Law (V) • However ..... • A new problem arises during unsteady multiphase fluid flow. • The rate of conduction of each phase losses its equivalence with • Injection rate, and • Production rate. • When more than one phase is flowing, it is not possible to measure production rate or injection rate to determine the conduction rate of each phase. • And Darcy´s law applied to each phase is based on conduction rate exclusively.

8. Darcy´s Law (VI) • In order to solve this "inconsistency" two ways were found. One of them during measurement, and the other during calculation. • Experimentally a method was developed in order to re-create Darcy´s requirements (Injection = Conduction = Production). This methodology was known as “steady state”. • Simultaneous Injection rate is maintained until production rate equals injection rate of each phase. • Through calculation, equations were solved during unsteady state displacement in order to obtain values in a dimensionless section. This is known as “unsteady state” or Welge methodology.

9. Darcy´s Law (VII) • Briefly, to allow the use of Darcy´s law during multiphase fluid flow, it was necessary to generate a unique saturation at the position of calculation. In this way, once again: • Injection = Conduction = Production • During “steady-state” measurements the whole sample has the same saturation. • Using “unsteady-state” methodology, all calculations are made in a single point (the outlet face) were the saturation is unique (point saturation).

10. A Question (I) • Both methodologies (“steady-state” and “unsteady-state”) gives the same result when applied to homogeneous media. • However ..... • Is this a good answer for Reservoir Calculations?

11. A Question (II) • In other words:The solution for the particular case whereInjection = Conduction = Productionis useful forunsteady reservoir conditions?.

12. Answer (I) • A very simple example will give a satisfactory answer for this question. • Let us suppose that we have a thin, horizontal porous system. • If the system is filled with gas it is obvious that its ability to conduct water is null. Using Darcy law we can say that: • Kw = 0 when Sw = 0 • However..... • Although water cannot be “conducted” there is no special impediment to inject water in the porous system. An empty rock allows water injection. • Although water can be injected, water cannot be produced until it reaches de outlet end.

13. Answer (II) • Using this model we may ask: • Which is the system ability to conduct water whenSw = 50%? • Half of the system with Sw = 100% and the other half with Sw = 0% • There are two “well defined” capacities to conduct water at this moment: • The capacity "X" (obtained through Darcy´s calculations) where Sw = 100% • A null capacity where Sw = 0 • So.... Which is the overall value for the water conduction ability?

14. Answer (III) • Option 1: Kw = X ? • Option 2: Kw = 0 ? • Option 3: Kw = X/2 ? • Option 4: ...... • While there is not a well defined Conduction ability, always exists a well defined Injection ability and also a well defined Production ability.

15. Real Scenaries • In Reservoir Simulation through Darcy´s calculations only Conduction ability is used. • For this reason Darcy´s equation is unable to reproduce Injection or Production rates in unsteady systems. • And ..... • All reservoirs are Unsteay Systems during production.

16. Reference Frame in Natural Reservoirs (The actual world) • In Reservoir Engineering frequently: • Natural porous media are heterogeneous. • Multiphase flow is the result of an equilibrium between viscous, capillary and gravity forces. This equilibrium varies with time and with physical location. • Reservoir calculations are founded on average phase saturation. • In a single cell (Reservoir Simulation) • The whole reservoir (Material Balance). • The properties of interest are Production and Injection abilities.

17. Solutions (I) • Every Scenery has its own solution. • Typical situations are: • Homogeneous systems with dominant viscous forces. Very few reservoirs fall in this category, but it is the typical laboratory scenery.. • Heterogeneous systems under viscous forces predominance. Stratified reservoirs without connected layers. • Heterogeneous systems under viscous and capillary forces predominance. Stratified reservoirs with connected layers. A cross-flow happens as a consequence of imbibition phenomena.

18. Solutions (II) • Other typical situations. • Heterogeneous systems with heavy oil where imbibition phenomena could be dominant. • Gravity dominated systems. Mainly in high permeability rocks with remarkable thickness and density differences and low viscosities (expanding gas cap, basal aquifers, ...). • Reservoirs dominated by capillary forces. Mainly in "tight sands“ reservoirs.

19. Conclusions (I) • Laboratory measurements are adequate to describe the ability to conduct fluids in porous media with homogeneous fluids saturation. • In Reservoir Engineering, the ability to inject or to produce fluids in non-homogeneous fluids saturation is needed. In unsteady systems the ability to conduct fluids losses its meaning and becomes useless.

20. Conclusions (II) • In order to fill the gap, experimental measurements must be made honoring real mechanisms occurring at reservoir scale. • During scaling-up stage, the following points must be considered: • The dominant displacement mechanism. • Laboratory “end point” measurements. • System geometry. • Heterogeneity. • The scaling-up process must be made by a multidisciplinary team. • Relative permeability curves must be “constructed” for every case. This operation must be based on available data and accepted reservoir models.

21. W W W Darcy • Thank you for your attention

22. Examples • The Relationship between production and fluid saturation will be analyzed using two very schematic models. • A: Water-Oil Displacement in a rectangular horizontal geometry under gravity forces dominance (Segregated flow). • B: Water-Oil Displacement in a rectangular non- horizontal geometry under gravity forces dominance (Segregated flow).

23. Q Q Sw Swirr Swirr Sw Segregated flow (A).Production at different faces. Sw avg = Swirr = 25%

24. Q Q Sw Swirr Swirr Sw Segregated flow (A).Production at different faces. Sw avg = 35%

25. Q Q Sw Swirr Swirr Sw Segregated flow (A).Production at different faces. Sw avg = 45%

26. Q Q Sw Swirr Swirr Sw Segregated flow (A).Production at different faces. Sw avg = 55%

27. Q Q Sw Swirr Swirr Sw Segregated flow (A).Production at different faces. Sw avg = 75%

28. First Remark • The Relationship between production rate and phases saturation may be different for different points in the same cell.

29. Q Q Sw Swirr Swirr Sw Segregated flow (B).Production at different faces. Sw avg = Swirr = 25%

30. Q Q Sw Swirr Swirr Sw Segregated flow (B).Production at different faces. Sw avg = 35%

31. Q Q Sw Swirr Swirr Sw Segregated flow (B).Production at different faces. Sw avg = 50%

32. Q Q Sw Swirr Swirr Sw Segregated flow (B).Production at different faces. Sw avg = 75%

33. Second Remark • If capillary forces take part of displacement mechanisms, the relationship between flow rate and phase saturation may depend on: • The selected production point. • The overall geometry and orientation of the system.