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AES1310: Rock Fluid Interactions - Part 1 1 Susanne Rudolph Darcys law in heterogeneous medium - Introduction - Averages

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AES1310: Rock Fluid Interactions - Part 1 2 Heterogeneity 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.

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AES1310: Rock Fluid Interactions - Part 1 3 Anisotropy 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 the formation rock due to dissolution of carbonates in water. Sedimentary porous media (e.g. sandstone) have layered structure. Permeability parallel to layers is mostly greater than perpendicular.

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AES1310: Rock Fluid Interactions - Part 1 4 Anisotropy Stratified formation are defined as anisotropic homogeneous medium when the thickness of the individual layer is smaller than the length of interest. In such cases the permeability cannot be determined from core samples because it would not display the real permeability. So far, only isotropic media have been considered wherein the permeability as constant factor (scalar) in Darcys law. Due to anisotropy is the direction of the pressure gradient vector different than the direction of the Darcy velocity vector at a point in the medium.

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AES1310: Rock Fluid Interactions - Part 1 5 Anisotropy Assume a porous medium with an arbitrary orientation with respect to the coordinate system and the pressure gradient points in the x-direction. Due to anisotropy the flow rates in the different directions are not the same. Darcys law for anisotropic media is (Ferrandon 1948):

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AES1310: Rock Fluid Interactions - Part 1 6 Anisotropy Herein k ij 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, Darcys law can be written in vector notation as:

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AES1310: Rock Fluid Interactions - Part 1 7 Anisotropy If it is assumed that the anisotropic medium is orthotropic (they have 3 mutually orthogonal principal axes) and if the coordinate axes are aligned with the principal axes of the medium permeability tensor is diagonal

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AES1310: Rock Fluid Interactions - Part 1 8 Heterogeneous Media Reservoirs 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.

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AES1310: Rock Fluid Interactions - Part 1 9 Heterogeneous Media Details 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.

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AES1310: Rock Fluid Interactions - Part 1 10 Arithmetic 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 = (x 1, x 2, x 3,…,x n ) 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.

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AES1310: Rock Fluid Interactions - Part 1 11 Harmonic mean - The number of values divided by the sum of the reciprocal values of the property. -For a set of data X = (x 1, x 2, x 3,…, x n ) the harmonic mean is: -Derived from electrotechnique computing the avarage resistance of a electrical circuit with two resistors in parallel

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AES1310: Rock Fluid Interactions - Part 1 12 Geometric mean -Indicates the central tendency or a typical value of a set of parameters. -For a set of data X = (x 1, x 2, x 3,…, x n ) the geometric mean is: -Can only applied to possitive values. -Is smaller or equal to the arithmetic mean of the same data set -Is closely related to arithmetic mean

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AES1310: Rock Fluid Interactions - Part 1 13 Relation between arithmetic and geometric mean -For a set of data X = (x 1, x 2, x 3,…,x n ) the geometric mean can be written as an arithmetic mean by taking the natural logarithm: -For positive values of x i : H n (x) G n (x) A n (x)

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AES1310: Rock Fluid Interactions - Part 1 14 Average 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 reservoirs Use 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.

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AES1310: Rock Fluid Interactions - Part 1 15 Average permeability for heterogeneous media Two situations are considered: 1)Flow is parallel to layers 2)Flow is normal/perpenticular to layers k i : permeability; b i : thickness of layer i; Q i : flow rate through layer I; W: width of the layers; same for all layers μ: viscosity of fluid; assumed to be equal for all systems and constant Φ: fluid potential A: cross-sectional area

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AES1310: Rock Fluid Interactions - Part 1 16 Flow parallel to layers 1-D & linear flow

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AES1310: Rock Fluid Interactions - Part 1 17 Flow parallel to layers 1-D & linear flow Driving force described by difference of fluid potentials (piezometric head) 1 and 2.

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AES1310: Rock Fluid Interactions - Part 1 18 Flow rate of each layer expressed by Darcys law: Total flow rate Q: sum of the flow rates through each layer: With the cross-sectional area: Flow parallel to layers 1-D & linear flow

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AES1310: Rock Fluid Interactions - Part 1 19 Transmissitivity: product of the thickness and the permeability over the visocsity This 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 k parallel or transmittivity T parallel : Flow parallel to layers 1-D & linear flow

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AES1310: Rock Fluid Interactions - Part 1 20 Gives: Flow parallel to layers 1-D & linear flow

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AES1310: Rock Fluid Interactions - Part 1 21 Rewriting the effective or equivalent permeability for the horizontal flow parallel to the layers gives: Which is the arithmetic average of the permeability. Flow parallel to layers 1-D & linear flow

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AES1310: Rock Fluid Interactions - Part 1 22 Flow parallel to layers 1-D &radial flow hThT

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AES1310: Rock Fluid Interactions - Part 1 23 Flow parallel to layers 1-D &radial flow Driving force described by difference of fluid potentials (piezometric head) 1 and 2. hThT

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AES1310: Rock Fluid Interactions - Part 1 24 Flow rate of each layer expressed by Darcys law: Total flow rate Q: sum of the flow rates through each layer: Flow parallel to layers 1-D & radial flow

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AES1310: Rock Fluid Interactions - Part 1 25 Transmissitivity: product of the thickness and the permeability over the visocsity This gives then: Combining this result with the same flow rate Q through a porous medium of the thickness h T described in terms of the equivalent permeability k parallel or transmittivity T parallel : Flow parallel to layers 1-D & radial flow

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AES1310: Rock Fluid Interactions - Part 1 26 Gives: Flow parallel to layers 1-D & radial flow

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AES1310: Rock Fluid Interactions - Part 1 27 Flow normal to layers 1D & linear flow Datum level

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AES1310: Rock Fluid Interactions - Part 1 28 Flow 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 Δ i Rock Fluid Interactions – Part 1 AES1310 Datum level

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AES1310: Rock Fluid Interactions - Part 1 29 Flow normal to layers 1D & linear flow The drop of the piezometric head of each section of the layer is described by Darcys law: The total piezometric head is then:

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AES1310: Rock Fluid Interactions - Part 1 30 Flow 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 k normal for the flow with the same flow rate Q through a porous medium of the length L: Gives:

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AES1310: Rock Fluid Interactions - Part 1 31 Flow 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.

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AES1310: Rock Fluid Interactions - Part 1 32 Flow normal to layers 1D & radial flow

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AES1310: Rock Fluid Interactions - Part 1 33 Flow 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 Δ i. Horizontal flow is considered, thus change of fluid potential is equal to pressure drop.

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AES1310: Rock Fluid Interactions - Part 1 34 Flow normal to layers 1D & radial flow The drop of the piezometric head of each section of the layer is described by Darcys law: The total piezometric head is then: And the flow rate:

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AES1310: Rock Fluid Interactions - Part 1 35 Flow 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 k normal for the flow with the same flow rate Q through a radial porous medium with an outer radius r e and a inner radius r w : Gives:

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AES1310: Rock Fluid Interactions - Part 1 36 Averaging 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 k normal and k parallel.

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AES1310: Rock Fluid Interactions - Part 1 37 Averaging 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: H n (x) G n (x) A n (x)

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AES1310: Rock Fluid Interactions - Part 1 38 Averaging 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? BedThickness H [ft] Perm [mD] BedLength or radius L or R [ft] Perm [mD] Parallel flow Perpendicular flow

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