Permeability

Permeability

1. PERMEABILITY

2. Definition (ABW, Ref: API 27) … permeability is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids … permeability is the fluid conductance capacity of a rock, or it’s the a measure of the ease with which the rock will permit the passage of fluids. Reservoir Rocks and Fluid properties Permeability

3. Permeability theory Permeability is an INTENSIVE property of a porous medium (e.g. reservoir rock) Reservoir Rocks and Fluid properties Permeability

4. Reservoir Rocks and Fluid properties Permeability Permeability which will permit flow of one centipoise fluid to flow at linear velocity of one cm per second under a pressure gradient of one atmosphere per centimetre.

5. Reservoir Rocks and Fluid properties Permeability Three types of permeability Absolute permeability - the permeability of a porous medium with only one fluid present (single-phase flow). When two or more fluids are present permeability of the rock to a flowing fluid is called effective permeability (k o , k g , k w ). Relative permeability is the ratio of absolute and effective permeabilities k ro =k o /k, k rg =k g /k, k rw =k w /k.

6. A French hydrologist named Darcy did the first work on permeability. He was concerned about flow of water through filters. He found that flow rate Q, is proportional to area of flow A,h, and 1/L. This is expressed mathematically as flows: 1 Darcy’s “K” was determined to be a combination of k, permeability of the sand pack (porous medium, e.g. reservoir rock) K is a constant of proportionality , viscosity of the liquid K constant may be written as; 2 Reservoir Rocks and Fluid properties Permeability

7. Alternatively, eqn.1; can also be expressed in terms of the pressure gradient dp over a section dL as; 3 Where dp = ∆h ρ g 4 dp is the diff. b/w the upstream & downstream pressures ∆h the diff. b/w the upstream and downstream hydraulic gradients ρ fluid density g is the acceleration due to gravity (9.81 m/sec 2 ) Reservoir Rocks and Fluid properties permeability

8. Darcy’s Apparatus for Determining Permeability Reservoir Rocks and Fluid properties Permeability theory h 1 -h 2 h 1 h 2 q A WATER WATER q ( Sand Pack Length) L A

9. Reservoir Rocks and Fluid properties Extension of Darcy’s Equation Darcy did not consider the fluid viscosity. Restricted to a medium with 100% saturated by water Other researchers discovered that Q is inversely proportional to viscosity.

10. Assumptions Used in Darcy Equation: Steady state flow, under laminar regime i.e. Q in = Q out Viscous flow - rate of flow directly proportional to pressure gradient The flowing fluid is incompressible. Porous media 100% saturated with fluid which flowed Fluid and porous media not reacting Reservoir Rocks and Fluid properties Extension of Darcy’s Equation

11. Darcy’s Law can be extended to other fluids, if K is expressed as a ratio of / μ , where μ is the viscosity of a given fluid and  the permeability of the porous medium. Eqn. 3 can be written to account for viscosity μ as; Q = - A 5 Eqn.5 can be integrated b/w the limits of length from 0 to L and pressure from P 1 (upstream) to P 2 (downstream), for a fluid flow case. Reservoir Rocks and Fluid properties Extension of Darcy’s Equation

12. Therefore, 6 7 or 8 Hence, the original equation of Darcy was modified to account for viscosity as follows: 9 Reservoir Rocks and Fluid properties Extension of Darcy’s Equation

13. Permeability is determined by: Core analysis Well test analysis (flow testing) RFT (repeat formation tester) provides small well tests Production data production logging measures fluid flow into well Log data MRI (magnetic resonance imaging) logs calibrated via core analysis Reservoir Rocks and Fluid properties Sources for Permeability Determination

14. Shape and size of pore system Sorting Cementation Fracturing and solution Lithology or rock type Reservoir Rocks and Fluid properties Factors affecting the magnitude of permeability

15. Two sets of units are generally used in reservoir engineering: Darcy units and Oil-field units For the purpose of the mathematical derivations, a system of units commonly referred to as Darcy units are used. These units are: V s = cm/sec; μ = cP; ρ = gm/cc K = Darcy; g = cm/sec 2 ; dp/ ds = atm/cm For application to field data of the various mathematical expressions, the second system of units called the practical oil-field units are used. Reservoir Rocks and Fluid properties Units of permeability

16. From dimensional analysis, the dimension for K is [L] 2 . We could use ft 2 , ins 2 , or cm 2 for measure of permeability, but these units are all too large to be applied in porous media. So Darcy unit is used and recommended , but in many cases the millidarcy ( mD ) 1 Darcy = 1000 miliDarcy 1 Darcy ~ 10 -8 sm 2 (10 -6 mm 2 ) 1 miliDarcy ~ 10 -11 sm 2 (10 -9 mm 2 ) Reservoir Rocks and Fluid properties Units of Permeability

17. Darcy’s Law - Darcy Units Linear (1-D) flow of an incompressible fluid where, q cm 3 /s k darcies A cm 2  p atm  cp L cm The Darcy a derived unit of permeability, defined to make this equation coherent (in Darcy units)

18. Darcy’s Law - Oilfield Units Linear (1-D) flow of an incompressible fluid where, q bbl/D k millidarcies A ft 2  p psia  cp L ft

19. Reservoir Rocks and Fluid properties Common Oil Field Units Quantity Symbol Dimension Oilfield Units SI Units Mass m m Ibm Kg Moles n n Ibmol Kmol Force F ML/ t 2 Ibf N Length L L ft m Area A L 2 acres m 2 Volume-liquids v L 3 bbl m 3 Volume-gases v L 3 ft 3 m 3 Pressure p m/L t 2 psi kPa Temperature T T R K Flow rate-liquids q L 3 /t bbl/d m 3 /d Flow rate -gases q L 3 /t Ft 3 /d m 3 /d Viscosity μ m/Lt cP mpa.s Permeability k L 2 md m 2

20. Permeability is an important parameter that controls the reservoir performance. Its importance is reflected by the number of available techniques typically used to estimate it. These different techniques provide formation permeability that represents different averaging volumes. The quality of the reservoir, as it relates to permeability can be classified as follows < 10 md fair 10 – 100 High 100 – 1000 Very High >1000 Exceptional This scale changes with time, for example 30 years ago k< 50 was considered poor. Reservoir Rocks and Fluid properties Uses of Permeability

21. What is 1 Darcy? A porous medium has a permeability of one Darcy, when a single-phase fluid of one centipoises viscosity, that completely fills the voids of the medium will flow through it, under conditions of viscous flow (also known as laminar flow), at a rate of one cubic centimeter cross sectional area, and under a pressure or equivalent hydraulic gradient of one atmosphere per centimeter. Reservoir Rocks and Fluid properties Definition of a Darcy

22. Important Characteristics that must be considered in the reservoir: Types of fluids in the reservoir Flow regimes Reservoir geometry Number of fluid flowing in the reservoir Reservoir Rocks and Fluid properties Permeability

23. Types of Fluids Incompressible fluids Slightly compressible fluids Compressible fluids Flow Regimes Steady state flow Unsteady state flow Pseudosteady state flow Reservoir Rocks and Fluid properties Permeability

24. Reservoir Geometry Linear flow Radial flow spherical and hemispherical flow Number of fluid flowing Single phase flow Two phase flow Three phase or multiple phase flow Reservoir Rocks and Fluid properties Permeability

25. Reservoir Rocks and Fluid properties Types of Fluids Incompressible fluids Slightly compressible fluids Compressible fluids

26. Flow Regimes Steady state flow Unsteady state flow Pseudosteady state flow Reservoir Rocks and Fluid properties Permeability

27. Reservoir Rocks and Fluid properties Reservoir Geometry Linear flow Radial flow spherical and hemispherical flow

28. Steady state flow, i.e. Q in = Q out Viscous or Laminar flow: the particles flow in parallel paths Rock is 100% saturated with one fluid Fluid does not react with the; this a problem with shaly-sand (interstitial particles) The formation is homogeneous and isotropic: same porosity, same permeability and same fluid properties Reservoir Rocks and Fluid properties Assumptions Used in Darcy Equation

29. Generalized form of Darcy’s Law The generalized Darcy’s law is given by the following equation: V s = Q/A and dp/ ds = dp/ dx , hence Reservoir Rocks and Fluid properties Horizontal, Linear, Liquid System Separating the variables and integrating gives the following eqn.: Note that P1>P2

30. Fluid not compressed (no density change with pressure) Steady flow (mass in = mass out) In horizontal flow (Linear) :

31. Reservoir Rocks and Fluid properties Flow Potential The generalized form of Darcy’s Law includes pressure and gravity terms to account for horizontal or non-horizontal flow The gravity term has dimension of pressure / length Flow potential includes both pressure and gravity terms, simplifying Darcy’s Law  = p -  gZ/c; Z+ ; Z is elevation measured from a datum  has dimension of pressure

32. The gravitational conversion constant, c, is needed to convert the gravity term to units of pressure gradient [pressure/length] A clear way to determine the sign of the gravity term (+ or -) is to consider the static case. If no fluid is flowing, then there is no potential gradient (potential is constant), while pressure changes with elevation, due to fluid density (dp=(  g/c) dZ ). Reservoir Rocks and Fluid properties Flow Potential

33. Reservoir Rocks and Fluid properties

34. Reservoir Rocks and Fluid properties

35. Fluid not compressed Steady flow (stream mass was constant) Radial Flow System r w r e P wf P e h r

36. Q = flow rate, m 3 /sec K = permeability, m 2 h = thickness, m P e = pressure drainage radius, N/m 2 P wf = flowing pressure, N/m 2 = fluid viscosity, N/m 2 re = drainage radius, m r w = the well-bore radius, m Reservoir Rocks and Fluid properties Horizontal, Radial, and Liquid System

37. Linear, Parallel Flow Discrete changes in permeability Same pressure drop for each layer Total flow rate is summation of flow rate for all layers Average permeability results in correct total flow rate Permeability varies across several horizontal layers (k 1 ,k 2 ,k 3 )

38. Substituting, Rearranging , Average permeability reflects flow capacity of all layers

39. Serial Flow Discrete changes in permeability Same flow rate passes through each layer Total pressure drop is summation of pressure drop across layers Average permeability results in correct total pressure drop Permeability varies across several vertical layers (k 1 ,k 2 ,k 3 )

40. Linear, Serial Flow Substituting, Rearranging, If k 1 >k 2 >k 3 , then Linear pressure profile in each layer x  L p  p 1 p 2

41. Radial, Parallel Flow Discrete changes in permeability Same pressure drop for each layer Total flow rate is summation of flow rate for all layers Average permeability results in correct total flow rate Permeability varies across several (3) horizontal layers (k 1 ,k 2 ,k 3 )

42. Radial, Serial Flow Substituting (r w =r 1 , r 2 ,r e =r 3 ), Rearranging,

43. Reservoir Rocks and Fluid properties Permeability It is necessary to determine an average value of permeability. Three common types of computed averages are as follows:   i ) arithmetic average ii) harmonic average ii) geometric average Selection of the averaging technique should be based primarily on the geometry of the flow system.

44. Reservoir Rocks and Fluid properties Averaging Permeability k 1 k 2 k 3 h 1 h 2 h 3 Q Arithmetic Average Parallel Flow

45. Reservoir Rocks and Fluid properties Averaging Permeability L 1 L 2 L 3 k 1 k 2 k 3 Harmonic Average Series Flow

46. Reservoir Rocks and Fluid properties Averaging Permeability Geometric Average Random Flow

47. Reservoir Rocks and Fluid properties Interaction between porosity & permeability