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Single phase flow in porous media: Darcy’s law

Single phase flow in porous media: Darcy’s law. Oil or gas reservoir. Sandstone reservoir. Limestone reservoir. Porosity. Rock matrix. Pore space. Rock Matrix and Pore Space. Typical Pore Structure. Pore Structure. Porosity in Sandstone. Pore Throat. Pores Provide the

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Single phase flow in porous media: Darcy’s law

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  1. Single phase flow in porous media: Darcy’s law

  2. Oil or gas reservoir Sandstone reservoir Limestone reservoir

  3. Porosity

  4. Rock matrix Pore space Rock Matrix and Pore Space

  5. Typical Pore Structure Pore Structure

  6. Porosity in Sandstone Pore Throat Pores Provide the Volume to Contain Hydrocarbon Fluids Pore Throats Restrict Fluid Flow Scanning Electron Micrograph Norphlet Formation, Offshore Alabama, USA

  7. Porosity of common rock types

  8. Permeability Permeability • Permeability k [D, mD] • ‘capacity of rock to transmit fluid’ • function of open space and its interconnection • depends on properties of rock formation

  9. Permeability (Darcy’s Law) Darcy’s experiment was performed to design a filter large enough to ensure the daily requirement of water for the city of Dijon (1856). • q: volumetric rate [m3/s] • K: hydraulic conductivity [m/s] • A: Cross-sectional area of sand pack [m2] • h: piezometric head [m] • L: length of sand pack [m]

  10. Permeability (Darcy’s Law) • In Petroleum Engineering we use phase potentials • k is permeability and property of a rock • Usually expressed in D or mD (D stands for Darcy) • 1 Darcy = 10-12 m2 = 1 µm2

  11. Darcy’s Law and dip angle • u is Darcy’s velocity

  12. Definition of parameters Total flow rate = total discharge [m3 s-1]: Q Darcy velocity u = specific discharge q [m s-1]: Interstitial velocity = linear velocity or pore velocity v:

  13. Definition of parameters Hydraulic gradient: In empirical Darcy’s law; ratio of difference in piezometric head (P/ρg + z) and length of the sand pack; sand pack arranged in vertical position Potential gradient: Analogous to hydraulic gradient; sand pack position not restricted to vertical position; in more generalized Darcy’s law; ratio of difference in fluid potential and length of the sand pack

  14. Bundle of capillary tubes • Many pore space models are based on capillary tube bundle • Darcy’s law: • Hagen-Poiseuille law for laminar flow:

  15. Restrictions and assumptions ofDarcy’s law • Laminar flow • For Reynolds numbers between 1 and 10 • No inertial forces • Viscous forces predominant • No slip (zero velocity of fluid at wall) • Incompressible fluid (=constant density) • Viscosity of water •  Application to gas flow through porous medium not appropriate • Note: Transition to turbulent flow for Re between 60 and 150.

  16. Permeability • Permeability is influenced by: • Pore size and pore-size distribution • Grain size • Grain-distribution • Compaction (which is function of pressure) • Grain shape • Klinkenberg effect

  17. Permeability measurement methods • core scale • Inject a fluid with defined properties • Use Darcy’s law to calculate permeability • Well test • Measure flow and pressure • Calculate permeability

  18. Representative Elementary Volume (REV) • To use equations we need average values for permeability, saturations, porosity, … over a volume • The volume must be small with respect to our problem of interest and large enough such that the averaged quantity does not change significantly if we increase the averaging volume by a factor of say two.

  19. Darcy’s law and Navier-Stokes equations • NS is continuum form of Newton’s second law • We neglect inertia forces:

  20. Limitations of applicability of Darcy’s law • Due to skin formation permeability in vicinity of wellbore is changing (decreasing) • At higher flow rates inertial force, acting due to convective acceleration of fluid particles through porous medium, have to be taken into account

  21. Forchheimer equation • Modification of Darcy’s law taking the inertial forces into account • Inertial forces need to be attributed for Re numbers higher than 10. • In porous media inertial forces need to be taken into account because of acceleration and decelerations of fluid particles through pore spaces NOT because of turbulence flow. • - Originally derived for flow of fluids through pipes where at high velocity distinct transition from laminar to turbulent flow • - Additional pressure drop due to skin formation can be determined applying Forchheimer equation.

  22. Forchheimer equation Forchheimer equation is a phenomenological approach. It was recognized that Darcy’s law deviates for Re numbers of around 10. To correct for this the inertial forces were also taken into account. These forces describe the fact that in porous media the fluid flow is accelerating or decelerating due to the tortuosity. The Forchheimer equation was stated on this picture empirically” Herein is βtheinertial parameter.

  23. Forchheimer equation As common for phenomenological approaches, the phenomenological parameters are related to measurable parameters by correlations. αandβ can be related to pore structure parameters by: With the constants A=180 and B=1.8 In general, 1/kand the inertia parameter β can be deduced from experimental data on the drop of the piezometric head as function of the Darcy velocity.

  24. Permeability-porosity correlations • General form of correlations: k = Shape factor * Porosity factor * square of grain size diameter Carman-Kozeny correlation: Tortuosity is a variable that defines the “straightness” of the flow paths

  25. Other Data Used in Well Testing

  26. Formation Volume Factor

  27. v + dv dy v Viscosity

  28. Fluid Compressibility

  29. Pore Compressibility

  30. Shale h1 Sand h2 h3 Net Pay Thickness h = h1 + h2 + h3

  31. Net Pay Thickness Case 1 Case 3 Case 2 Case 4

  32. rw Wellbore Radius

  33. Total Compressibility

  34. Radial (steady state) Darcy’s Law Radial system in steady state qB rw h Note: B assumed constant qB pe pe pr p qB Permeability can be derived: pw rw r re

  35. Can be negative too ! Stimulated Undamaged Damaged Less Pressure drop due to hydraulic frac “Expected” flowing pressure undamaged Extra Pressure drop due to damage skin Actual flowing pressure pskin Skin Skin is any near wellbore phenomenon that causes an additional pressure drop extra to that expected from Darcy inflow (Delta P-skin), e.g. damaged rock: Positive Skin: drilling mud filtrates, clay swelling, mechanically destroyed rock, gravel pack Negative skin: acid jobs, extra deep perforations, hydraulically fractured

  36. Skin and Productivity Index van Everdingen equation: Productivity Index: Ways to improve PI: • Skin removal • Increasing effective permeability • Viscosity reduction • Reduction of Bo • Increasing well penetration h

  37. Geometric Skin Partially perforated Fully perforated Pressure drop due to geometric skin

  38. Flow in parallel QT re QT h1 k1 h1 Q1 Q1 h2 k2 h2 Q2 Q2 P2 hT Pw Pe k3 h3 Q3 Q3 h3 P1 L Linear Radial

  39. Q L re P2 P1 k2 k1 k3 r2 Q Q h r1 Pe Pw L1 L2 L3 Linear Often used for vertical permeability Radial Flow in series

  40. Permeability averaging Parallel Flow Arithmetic Average Series Flow Harmonic Average Random Flow Geometric Average

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