Two-dimensional flow in a porous medium with general anisotropy P.A. Tyvand & A.R.F. Storhaug Norwegian University of Life Sciences 143 Ås Norway peder.tyvand@umb.no 1
Darcy s law for flow in an isotropic porous medium µ 0= p v K p is pressure μ is dynamic viscosity K is permeability Henry Darcy (1956) formulated this linear law v is velocity where a driving force is balanced by a resistance. (vector) Analogous linear constitutive laws are: Ohm s law of electric conduction Fourier s law of heat conduction Fick s law of mass diffusion Essentially these are first-order Taylor expansions
Darcy s law for flow in an isotropic porous medium: Newton s. law µ 0= p v K mass x acceleration = pressure force + resistance force (taken per volume, with acceleration term neglected) Basic experiment for obtaining Darcy s law 3
Basic 1D experiment for obtaining Darcy s law Tube (a)-(b) filled with a porous medium. A p p A p Q = Kfall Δp=p = K Pressure µ L µ a x -pb. Tube length L. Cross-section area A. Volume flux Q. a b A pa pb A p Q= K = K µ L µ x 4 This is the observed relationship between pressure fall and volume flux. The permeability K is introduced as a proportionality constant.
5 Darcy s law for flow in an isotropic porous medium µ 0= p v K p is pressure μ is dynamic viscosity K is Follows by generalizing to 3Dpermeability v is velocity the empirical relationship (vector) A pa pb A p Q= K = K µ L µ x by introducing as velocity the specific flux vx=q/a. Alternatively one may work with the average pore velocity nq/a (n is porosity).
Darcy s law for 1D flow in an anisotropic porous medium The 1D version of Darcy s law A pa pb A p Q= K = K µ L µ x is rewritten for an anisotropic medium as Qx K xx p vx = = A µ x 6 where Kxx is the permeability for flow in the x direction as driven by a gradient in the x direction
7 Darcy s law for 1D flow in an anisotropic porous medium Darcy s law for an anisotropic porous medium with 1D pressure gradient p/ x=δp/lx Qx K xx p vx = = Ax µ x Qy K yx p vy = = Ay µ x K zx p Qz vz = = Az µ x vz vy x p/ x vx Kzx Kyx Kxx Kyx, Kzx are the permeabilities for flows in the y and z directions as driven by a gradient in the x direction. Given by Kyx/Kxx=vy/vx, Kzx/Kxx=vz/vx
Darcy s law for 3D flow in an anisotropic porous medium Darcy s law for an anisotropic porous medium with 3D pressure gradient ( p/ x, p/ y, p/ z) µ ( vx vy K11 v z ) = K 1 K 31 or, in index form 8 K1 K K 3 K13 p / x K 3 p / y K 33 p / z p µ vi = K ij xj Kij (i=1,,3) (j=1,,3) are the permeabilities for flow in the x1=x, x=y and x3=z directions as driven by gradients in the xj (j=1,,3) directions
3D anisotropic porous media We consider a homogeneous porous medium with general 3D anisotropy The anisotropy can be caused by fibres with preferred directions The anisotropy can be caused by periodic layering These cases may have transverse isotropy: One permeability along fibres or across layer. permeability in perpendicular direction the Another Our analysis will assume general anisotropy, with arbitrary 9 directions of principal axes
Flow in a 3D anisotropic porous medium Coordinate axes x,y,z. Principal axes of permeability X, Y, Z, with permeabilities KI, KII, KIII. The two coordinate systems are linked by the three Euler angles α, β, γ KIII KII KI 10
11 Flow in a 3D anisotropic porous medium Principal axes of permeability are X, Y, Z. The permeability matrix is KI 0 0 0 K II 0 KIII 0 0 K III KII KI in the X, Y, Z system. The matrix is transformed to the x,y,z system
1 D flow in a 3D anisotropic porous medium The permeability matrix in the x,y,z system K11 K 1 K 31 K1 K K 3 K13 K 3 K 33 KIII KII KI is given by the three principal permeabilites and the Euler angles (formulas omitted) Now we restrict our attention to general D flow ( / z=0)
D flow in a 3D anisotropic porous medium We introduce the pressure gradient that drives a D flow (Gx,Gy) = - ( p/ x, p/ y) Note that this flow has a passive component in the z direction, even though there is no pressure gradient in the z direction Darcy s law for the 3D porous medium is written as v = K G = G K Here K is the 3D permeability 13 tensor (matrix), which is symmetric. The velocity vector v has three components (vx,vy,vz) The idea is now to introduce an effective D description for the flow in the (x,y) plane. K1 and K are defined as effective D principal permeabilities, with axes rotated an angle φ compared with the coordinate axes x and y φ y K1 K φ x
D flow in a 3D anisotropic porous medium An effective D description for the flow in the (x,y) plane takes K1 and K as effective D principal permeabilities, with axes rotated an angle φ Darcy s law for D flow in a 3D porous medium with (Gx,Gy) = ( p/ x,- p/ y) v = K G = G K The velocity vector v has three components (vx,vy,vz), but we are only interested in its D projection (vx,vy), which is given by the D formula φ K1 cos ϕ + K sin ϕ = (1 / )( K1 K ) sin ϕ 14 K eff K1 K (v x, v y ) = G K eff y φ (1 / )( K1 K ) sin ϕ K1 sin ϕ + K cos ϕ x
D flow in a 3D anisotropic porous medium The equations governing K1, K and φ are shown here. They can easily be solved numerically. K1 cos ϕ + K sin ϕ = K I a11 + K II a1 + K III a31 K1 sin ϕ + K cos ϕ = K I a1 + K II a + K III a3 (1 / )( K1 K ) sin ϕ = K I a11a1 + K II a1a + K III a31a3 φ The coefficients consist of products of of the rotation matrix 15 a11 a1 a1 a a 31 a3 yelements K a13 a3 = a33 cos γ cos α cos β sin α sin γ sin γ cos α cos β sin α cos γ sin β sin α K1 φ cos γ sin α + cos β cos α sin γ sin γ sin α + cos β cos α cos γ sin β cos α sin γ sin β cos γ sin β cos β x
Numerical results for KI=1, KII=, KIII=3 α β γ ϕ K1 K π /6 0 π /6 π /3 1 π / 6 π / 6 π / 6.896 1.103.09 π / 6 π / 3 π / 6.670 1.18.719 KIII φ KII y K1 K φ 16 KI x
17 Numerical results for KI=3, KII=, KIII=1 α β π /6 0 π /6 π /8 π /6 π /4 π / 6 3π / 8 γ ϕ K1 K π / 4 5π / 1 3 π / 4 1.19.865 1.916 π / 4 0.90.640 1.610 π / 4 0.669.578 1.19 Increasing β from zero, we see how the effective permeabilities K1 and K are influenced more and more from KIII. The values of these effective permeabilities are between the extremal 3D values
Original motivation for this work We investigated drainage flow from a D anisotropic porous medium into a ditch. General D permeability matrix y K xx K xy K yx K yy x Our question was: Will our D drainage theory for a 18 uniformly draining ditch be valid for a class of porous media with 3D anisotropy? The answer is yes. It is valid for general 3D anisotropy, defining the effective D permeability matrix presented here
Summary and conclusions A D flow in a general 3D anisotropic porous medium does not involve all six independent components of the full permeability tensor (matrix). It involves five, and reduces them to three. Three independent components of an effective D permeability tensor are sufficient to describe the D flow in the plane. These three are expressed by the principal values K1, K and the tilt angle φ of the principal axes. One disadvantage with this approach is that it overlooks the passive flow perpendicular to the x,y plane of the D flow Another disadvantage is that we cannot reconstruct the full 19 permeability tensor. Even if we took the passive flow components into account, we only have 5 equations to determine the 6 independent tensor components. The single tensor component that we cannot find with D flow is Kzz
APPENDIX: Mathematica program for calculating the effective permeability matrix K eff K1 cos ϕ + K sin ϕ = (1 / )( K1 K ) sin ϕ 0 (1 / )( K1 K ) sin ϕ K1 sin ϕ + K cos ϕ