Darcy s law in 3-D. K * xx K * yy K * zz

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1 PART 7 Equations of flow Darcy s law in 3-D Specific discarge (vector) is calculated by multiplying te ydraulic conductivity (second-order tensor) by te ydraulic gradient (vector). We obtain a general form of Darcy s law. q q x î K xx K xy K xz x q y ĵ K yx K yy K yz y q z kˆ K zx K zy K zz z K xx x K xy y K + + î xz z K yx x K yy y K + + ĵ yz z K zx x K zy y K + + kˆ zz z In vector notation, tis general form of Darcy s law is q K If te system is rotated suc tat te principal axes are aligned wit t x, y and z, ten we ave only te tree diagonal components in te ydraulic conductivity tensor K K * xx K * yy K * zz and te specific discarge becomes q K * xx î K * yy x y ĵ K * zz kˆ z Note tat K * xx, K* yy and K* zz are different in te rotated system tan in te original tensor. But te flow (q) must be te same in bot cases (we must preserve te flow wen we rotate te system). Hydrogeology, 431/531 - University of Arizona - Fall 007

2 Equations of flow 5 General continuity equation Assume: (1) medium beaves as a perfect elastic solid; () fluid as nearly constant density (it is incompressible for some purposes); (3) Darcy s law is valid. z q y Look at conservation of mass in a cubic unit volume Statement of mass conservation in te unit volume: q x dy dz x volume in - volume out cange in storage dx Volume in: face x: q x A q x dydz face y: q y A q y dxdz face z: q z A q z dxdy Total volume in q x dydz + q y dxdz + q z dxdy Volume out: x direction inflow + extra volume y q z q x dydz + ( q x / x)dxa x q x dydz + ( q x / x)dxdydz [q x + ( q x / x)dx]dydz y direction [q y + ( q y / y)dy]dxdz z direction [q z + ( q z / z)dz]dxdy Total volume out [q x + ( q x / x)dx]dydz + [q z + ( q z / z)dz]dxdy + [q z + ( q z / z)dz]dxdy Hydrogeology, 431/531 - University of Arizona - Fall 007

3 Equations of flow 53 Now look at te balance: volume in - volume out cange of storage ds Plug in volumes in and volumes out: q x dydz + q y dxdz + q z dxdy - [q x + ( q x / x)dx]dydz - [q y + ( q y / y)dy]dxdz - [q z + ( q z / z)dz]dxdy ds -( q x / x)dxdydz -( q y / y)dydxdz -( q z / z)dzdxdy ds (Equation 1) (- q x / x - q y / y - q z / z)dxdydz ds Look at ds. It is cange of storage (or volume) in time, i.e., ds -dv w /dt From te lecture on storage of water (part 8 of notes), we know tat dv w -S s V T d wic, on substitution into ds, gives (Equation ) ds -(-S s V T d)/dt S s dxdydz (d/dt) Finally, we put equations 1 and togeter: (Equation 3) q x q -- y q -- z -- S x y z S Continuity equation Recall te definition of divergence of vector a (divergence operator is, or del dot ; it takes a vector as te argument and it produces a scalar): a a -- x x a y a z y z Now, we can write our continuity equation in vector notation: q S S Te equation means: te divergence of flux q equals te cange of storage. Hydrogeology, 431/531 - University of Arizona - Fall 007

4 Equations of flow 54 Substitute Darcy s law (under assumption of principal directions) q x -K xx (/ x) q y -K yy (/ y) q z -K zz (/ z) into equation 3 to get x Kxx x Kyy Kzz + + S z z S Tis is te general continuity equation for eterogeneous and anisotropic medium. Wy is tis eterogeneous? Wy is it anisotropic? In vector notation, it is: K S S Diffusion equation Tis is te diffusion-type equation. It as noting to do wit te process of diffusion, but te equation is matematically te same as te equation tat describes te diffusion process. Moreover, te solutions are te same as tat of te diffusion equation; we can use tose solutions to solve groundwater flow equations! Boundary conditions Te flow equation is second order partial differential equation wit spatial (second-order spatial derivatives) and temporal (first order time derivative) terms in it. We need two boundary conditions to account for spatial variability, and an initial condition to account for te transient term. Possible boundary conditions are: (1) Prescribed ead (Diriclet) boundary condition; () Prescribed flux (Neumann) boundary condition; (3) Mixed (Caucy) boundary condition. An initial condition is specified ead at specified time. Hydrogeology, 431/531 - University of Arizona - Fall 007

5 Equations of flow 55 Boundary conditions Prescribed ead (Diriclet) boundary condition Head on te boundary (ere - contact between te river and te aquifer, i.e., yy te bottom of te river) is known and independent of te flow in te aquifer. ;; ; y Examples: open water (river, lake, ocean). Prescribed flux (Neumann) boundary condition ;;;;;; yyyyyy ;;;;;; yyyyyy aquifer river red contact area as ead defined by te river Flux on te boundary is known, i.e., - K(d/dn) known value (or -T(d/dn) known value) Note tat (d/dn) is in te direction normal to te boundary. Two types of prescribed flux boundaries: no-flow boundary: d/dn 0, e.g., impervious layer or fault non-zero flux boundary, e.g., examples A and B below Example A: outcrop were aquifer is recarged: prescribed flux infiltration rate Example B: pumping well wit discarge Q; flux on boundary F is: Q F K dσ n (A) ;; yy ;yrainfall (B) Q ;; yy F Hydrogeology, 431/531 - University of Arizona - Fall 007

6 Equations of flow 56 Mixed (Caucy or Fourier) boundary condition Te relationsip between flux and ead is known (altoug neiter flux nor ead are known). Example: leaky river bed Leakage troug te aquitard: q K' H s ---- b' were K is te ydraulic conductivity of te aquitard and b is its tickness; Hs is te ead in te river (it is independent of te ead in te aquifer). Flux at te aquifer/aquitard boundary (point A): ;;;; yyyy Aquifer: K A H s river Aquitard: K', b' q K n Te two fluxes (across aquitard and at point A) must be equal, so we can write: K K' H s ---- n b' tereby establising a relationsip between te ead in te aquifer and te flux -K(/ n). Hydrogeology, 431/531 - University of Arizona - Fall 007

7 Equations of flow 57 Simplified flow equations Te flow equation (p. 60, diffusion equation) is valid for any type of porous media. We now consider media wit specific properties. Anisotropic and omogeneous K is not a function of space, but is a function of direction, i.e., K(x1) K(x), but Kx Ky. Because K is not spatially variable (i.e., K/ x 0, and similarly for directions y and z) Kx x x K --- x + K x x x K x x x and similarly for y and z. Tus, te continuity equation becomes: K x + K y + K z S x y z S or, in vector notation: K S S Isotropic and eterogeneous Kx Ky Kz, but tey vary in space (e.g., K(x1) K(x)). x K K K + + S x z z S or, in vector notation: ( K ) S S Hydrogeology, 431/531 - University of Arizona - Fall 007

8 Equations of flow 58 Isotropic and omogeneous Kx Ky Kz constant in space K + + y z x S S or, in vector notation: K S S or S S ---- K t were S S /K is called ydraulic diffusivity. Steady state constant in time (tus, / 0); no oter simplifying conditions. K 0 If also omogeneous and isotropic K 0 and K does not matter as we can divide bot sides by K 0 Laplace equation Hydrogeology, 431/531 - University of Arizona - Fall 007

9 Equations of flow 59 Horizontal flow and transmissivity PDE: S S K K + y x z were is a -D Laplace operator. Multiply bot sides by (constant?) tickness b S S b K Kb + x y b ;yflow ;y x In tis equation: y S S b S is te storativity [-], Kb T is te transmissivity [L T -1 ]. Tis equation describes transient orizontal flow in omogeneous and isotropic, confined aquifer. Oter forms: T -- S D were D T/S is te aquifer diffusivity (most important factor for flow). Or, in general form: S T were T double bar is te transmissivity tensor (equivalent to ydraulic conductivity tensor). Hydrogeology, 431/531 - University of Arizona - Fall 007

10 Equations of flow 60 Horizontal flow in preatic aquifer Tickness b is now variable and it depends on te position of te water table. ;;; Confined Preatic yyb Lake b datum ;;yyy Saturated tickness independent of Saturated tickness varies in time wit Simplifying assumptions (Dupuit assumptions): (1) Essentially orizontal flow () Hydraulic gradient equal to te slope of te water table Note: (1) and () assume vertical equipotentials. (3) No seepage face. (4) Neglect compressibility (S S 0). (5) Horizontal bottom of te aquifer. water table t t + dt Flow in x direction: in: -K x dy (/ x) out: - / x[-k x dy (/ x)] - K x dy (/ x) net: - / x[-k x dy (/ x)]dx y dx dy Similarly net in y direction. Cange in storage: ds S y (/) dx dy x Putting all togeter: x Kx dy dx x Ky dx + dy Sy dxdy Hydrogeology, 431/531 - University of Arizona - Fall 007

11 Equations of flow 61 x Kx Ky + S x y Tis is Bousinesq equation for orizontal flow in a preatic aquifer wit orizontal bottom. Because K x T x and K y T y, we can write Tx Ty + S x x y or ( K ) S y were (x, y, t) and K is a -D tensor defined as K K xx K xy K yx K yy T const; T canges in time because of canges in te ead (water table) Important!!! T T() T((x, y, t)) Confined aquifer: T const Kb S Tx + Ty x x linear equation Preatic aquifer: T T() S y x ( ) x ( ) Tx + Ty nonlinear equation Hydrogeology, 431/531 - University of Arizona - Fall 007

12 Equations of flow 6 To solve nonlinear equation, assume T x () T x const and T y () T y const. Tis is called linearization. K K + S x x y y y Look at (/ x) 0.5( / x). Now we can write 1 --K K -- + S x x y y y or in vector notation K --- S y Tis is Bousinesq equation (for omogeneous, isotropic, orizontal bottom aquifer). Te equation is linear in (it was made linear in troug te process of linearization of second type) For steady state S y (/) 0 and we ave K or 0 and K does not matter. Suppose d is muc smaller tan (d<<). Look at K ( ) S y K ( ) K( + ) K T were is te average ead. Tis simplification was possible because d is small, ence is close to zero, and ence te term 0. Now we ave T S y preatic, omogeneous, isotropic Tis equation is linear in. It was made linear in troug te process of linearization of first type. Hydrogeology, 431/531 - University of Arizona - Fall 007