Presentation on theme: "Darcy’s law in heterogeneous medium - Introduction Averages"— Presentation transcript:
1Darcy’s law in heterogeneous medium - Introduction Averages Susanne Rudolph
2Heterogeneity of porous media Homogenous: A medium is homogenous with respect to a property if the property is independent of position within the medium.Isotropic: A medium is isotropic with respect to a property if the property is independent of the direction within the medium.Anisotropic: If at one point in the medium a property such as permeability varies with the direction.
3Anisotropy Reservoirs are commonly anisotropic with respect to the permeability.Anisotropy of permeability is due to evolution of formations;e.g. in carbonate rocks formation of channels within theformation rock due to dissolution of carbonates in water.Sedimentary porous media (e.g. sandstone) have layeredstructure. Permeability parallel to layers is mostly greater thanperpendicular.
4Anisotropy Stratified formation are defined as anisotropic homogeneous medium when the thickness of the individuallayer is smaller than the length of interest. In such cases thepermeability cannot be determined from core samples becauseit would not display the real permeability.So far, only isotropic media have been considered wherein thepermeability as constant factor (scalar) in Darcy’s law.Due to anisotropy is the direction of the pressure gradientvector different than the direction of the Darcy velocity vectorat a point in the medium.
5Anisotropy Assume a porous medium with an arbitrary orientation with respect to the coordinate system and the pressure gradientpoints in the x-direction.Due to anisotropy the flow rates in the different directionsare not the same.Darcy’s law for anisotropic media is (Ferrandon 1948):
6Anisotropy Herein kij are the elements of the permeability tensor with Permeability values depend on the orientation of the medium with respect to the coordinate system:With this, Darcy’s law can be written in vector notation as:
7AnisotropyIf it is assumed that the anisotropic medium is ‘orthotropic’(they have 3 mutually orthogonal principal axes) and if thecoordinate axes are aligned with the principal axes of themedium permeability tensor is diagonalAn orthotropic material has two or three mutually orthogonal two-fold axes of rotational symmetry so that its mechanical properties are, in general, different along the directions of each of the axes. Orthotropic materials are thus anisotropic; their properties depend on the direction in which they are measured. An isotropic material, in contrast, has the same properties in every direction.One common example of an orthotropic material with two axes of symmetry would be a polymer reinforced by parallel glass or graphite fibers. The strength and stiffness of such a composite material will usually be greater in a direction parallel to the fibers than in the transverse direction. Another example would be a biological membrane, in which the properties in the plane of the membrane will be different from those in the perpendicular direction. Such materials are sometimes called transverse isotropic.
8Heterogeneous MediaReservoirs are commonly heterogeneous. A reservoir consists of ‘patches’ with different properties.Often reservoir simulations are performed applying a cylindrical homogeneous structure.Main interest of reservoir engineers is the flow through porous medium and its understanding.Only in recent years heterogeneity is taken into account in order to analyze reservoir behavior.Geological or geostatistical models provide are detailed description of the heterogeneity of the reservoir.
9Heterogeneous MediaDetails can often not be incorporated in reservoir simulation models.Permeability values have to be averaged.Averaging procedure has to be conducted with care to avoid erroneous averaged values.Mathematics provides computation of means such as arithmetic, harmonic and geometric mean.
10Arithmetic mean- Sum of all the values of which the arithmetic mean has to be determined divided by the number of summed values.- For a set of data X = (x1, x2, x3,…,xn) the arithmetic mean is:Note:- If the arithmetic mean is determined of values varying strongly in value/order of magnitude, it might give an erroneous high average value.- Arithmetic mean can only be taken from values with the same reference.
11Harmonic mean- The number of values divided by the sum of the reciprocal values of the property.For a set of data X = (x1, x2, x3,…, xn) the harmonic mean is:Derived from electrotechnique computing the avarage resistance of a electrical circuit with two resistors in parallel
12Geometric meanIndicates the central tendency or a typical value of a set of parameters.For a set of data X = (x1, x2, x3,…, xn) the geometric mean is:Can only applied to possitive values.Is smaller or equal to the arithmetic mean of the same data setIs closely related to arithmetic mean
13Relation between arithmetic and geometric mean For a set of data X = (x1, x2, x3,…,xn) the geometric mean can be written as an arithmetic mean by taking the natural logarithm:For positive values of xi: Hn(x) ≤ Gn(x) ≤ An(x)
14Average permeability for heterogeneous media After recalling the meaning of the different ways to take mean values, the averaging of the permeability is considered for heterogeneous reservoirsUse of average permeability only for simple flow cases.Rock system composed of distinct layers with different properties.Only flow of a homgeneous fluid (, = constant); therefore use of hydraulic conductivity analogous to use of permeability.
15Average permeability for heterogeneous media Two situations are considered:Flow is parallel to layersFlow is normal/perpenticular to layerski: permeability;bi: thickness of layer i;Qi: flow rate through layer I;W: width of the layers; same for all layersμ: viscosity of fluid; assumed to be equal for all systems and constantΦ: fluid potentialA: cross-sectional area
17Flow parallel to layers 1-D & linear flow Driving force described by difference of fluid potentials (piezometric head) 1 and 2.
18Flow parallel to layers 1-D & linear flow Flow rate of each layer expressed by Darcy’s law:Total flow rate Q: sum of the flow rates through each layer:With the cross-sectional area:
19Flow parallel to layers 1-D & linear flow Transmissitivity: product of the thickness and the permeability over the visocsityThis gives then:Combining this result with the same flow rate Q through a porous medium of the thickness b described in terms of the equivalent permeability kparallel or transmittivity Tparallel:
20Flow parallel to layers 1-D & linear flow Gives:
21Flow parallel to layers 1-D & linear flow Rewriting the effective or equivalent permeability for the horizontal flow parallel to the layers gives:Which is the arithmetic average of the permeability.
23Flow parallel to layers 1-D &radial flow Driving force described by difference of fluid potentials (piezometric head) 1 and 2.hT
24Flow parallel to layers 1-D & radial flow Flow rate of each layer expressed by Darcy’s law:Total flow rate Q: sum of the flow rates through each layer:
25Flow parallel to layers 1-D & radial flow Transmissitivity: product of the thickness and the permeability over the visocsityThis gives then:Combining this result with the same flow rate Q through a porous medium of the thickness hT described in terms of the equivalent permeability kparallel or transmittivity Tparallel:
26Flow parallel to layers 1-D & radial flow Gives:
27Flow normal to layers 1D & linear flow Datumlevel
28Flow normal to layers 1D & linear flow Horizontal flow normal or perpendicular to layers:Total flow rate per unit width is unchanged. The total drop of the head Δ is now the sum of the drop of heads for each layer ΔiDatumlevelRock Fluid Interactions – Part 1 AES1310
29Flow normal to layers 1D & linear flow The drop of the piezometric head of each section of the layer is described by Darcy’s law:The total piezometric head is then:
30Flow normal to layers 1D & linear flow Combining the equation again with the result obtained regarding the porous medium described by an equivalent or effective permeability knormal for the flow with the same flow rate Q through a porous medium of the length L:Gives:
31Flow normal to layers 1D & linear flow Rewriting the permeability for the horizontal flow normal to the layers gives:Which is the harmonic average of the permeability.
33Flow normal to layers 1D & radial flow Now we consider the horizontal flow normal or perpendicular to layers occurs. For this case the total flow rate per unit width is unchanged. The total drop of the head Δ is now the sum of the drop of heads for each layer Δi. Horizontal flow is considered, thus change of fluid potential is equal to pressure drop.
34Flow normal to layers 1D & radial flow The drop of the piezometric head of each section of the layer is described by Darcy’s law:The total piezometric head is then:And the flow rate:
35Flow normal to layers 1D & radial flow Combining the equation again with the result obtained regarding the porous medium described by an equivalent or effective permeability knormal for the flow with the same flow rate Q through a radial porous medium with an outer radius re and a inner radius rw:Gives:
36Averaging of permeability From the equations to describe the flow parallel and normal to layers follows that the equivalent permeabilities of parallel flow are larger than of the normal flow:This can be prooved by considering frist two layers and then increasing the number of layers while computing the knormal and kparallel.
37Averaging of permeability Note: The geometric mean is commonly used for the description of the average permeability in a chessboard reservoir (= area is subdivided in blocks of equal size)Cardwell and Parsons showed for chessboard arrangement that the equivalent permeability lays between the one of parallel flow and the one of normal flow.This is in agreement what we saw before:Hn(x) ≤ Gn(x) ≤ An(x)
38Averaging of permeability Determine the average permeability of the situation described in the tables for linear flow and radial flow.What are the ratios of the separate flows in these beds?What are the ratios of the separate piezometric heads in these beds?Perpendicular flowParallel flowBedLength orradiusL or R [ft]Perm[mD]125025250350010041000200BedThicknessH [ft]Perm[mD]12010021520031030045400