Geol_127_Lecture_11_Specific discharge_ equations of flow by keralaguest


									Geology 127                                                                        Lecture #11
Hydrogeology                                 Reading assignment: Fetter, pp. 122-132, 138- 146

- Last time: we talked about hydraulic head and refraction of flow lines
- Today: We will discuss specific discharge (Darcy flux), average linear flow velocity,
groundwater flow equations, and equations of flow in confined and unconfined aquifers

I) Darcy's law and moving water

       A) Specific discharge and Darcy velocity

               - Darcy’s law could be rearranged to predict ground water flow velocity, BUT it
                       does not account for all variables.
               - The problem: in the next section we will find we need to add a correction to
                       account for actual particle path. Particles take a longer, more tortuous path
                       than Darcy assumed.
               -If we used Darcy’s law to predict particle velocity:

                      - First: convert from Q to v:
                              Q = vA         OR: v = Q/A

                      - Substituting into Darcy's equation:
                              Q = -KA dH                    or   v = Q = - K dh
                                         L                           A       L
                      - This gives the equation for SPECIFIC DISCHARGE (also called Darcy
                              Flux/ Darcy Velocity

               - What is the problem here?

We now realize that calling this a “Darcy Velocity” isn’t correct, because it implies that
water is moving at this velocity. It isn’t!! Water moves faster than predicted by the
specific discharge equation when flowing through a porous medium.

               - Specific discharge would predict accurate velocity for flow through a pipe, but
                       doesn’t account for the extra travel path length that water molecules
                       take in a porous medium
               - The solution: To find the actual velocity of the moving water: we must include a
                       porosity term (see section C below)

       B) Laminar and turbulent flow

               - Another factor that must be considered
               - Some of Darcy's assumptions begin to break down with faster-flowing water
               - Darcy didn't account for turbulence (he assumed laminar flow)
               - This approach gives an unrealistic view of each particle path
               - Also gives an unrealistic view of particle velocity

See Figure 4.6 from Fetter, p. 123

               - At many “natural” flow velocities: turbulence is created around grains
               - You can calculate the flow velocity (based on temp, density, grain size,
                       viscosity) that occurs when Darcy’s law begins to break down: see p. 123-
                       124 in Fetter.
               - Geologists refer to a Reynolds number (Re) as the boundary between
                       laminar and turbulent flow.
               - In natural conditions (aquifers): experiments have shown the boundary between
                       laminar and turbulent flow occurs at Re ~10.
               - This is MUCH lower than the boundary of Re = 500-2000 used in open flow
               - The good news here: Water usually travels at slow enough velocity for
                       Darcy’s Law to be valid (true laminar flow)
               - Most GW flow equations assume true laminar flow
               - So: Darcy’s law produces an empirical result for K, but can’t accurately
                       predict flow velocity for a given particle

       C) Seepage velocity (average linear flow velocity)

               - Represents the average rate at which water moves between 2 points
               - Includes the effective porosity:

                      Vx = Q = - K dh
                          ne A   ne dl

       where           Vx = average linear flow velocity
                       ne = effective porosity: the volume of the void spaces through which water
                               or other fluids can travel in a rock or sediment divided by the total
                               volume of the rock or sediment. Units = dimensionless porosity;
                               this value is a decimal (NOT a %). Ex: 18% porosity = 0.18 ne
               - Note: this equation does not account for diffusion or dispersion
               - Therefore: cannot be used for calculations of contaminant plume velocity.

II) Equations of groundwater flow

       A) Introduction

               - Use law of conservation of mass, law of conservation of energy
               - The theory: a fluid contained in a small area can neither gain nor lose energy
               - Energy of the fluid is described by looking at fluid flux in x,y,z directions
               - Note: Equations of flow are different for confined and unconfined aquifers
               - Also: there are 2 basic aquifer conditions:
                       - Some aquifers are steady state (flow conditions do not vary)
                       - Some aquifers are variable or transient (often from pumping)
              - In class: I will give final flow equation, but won’t go through the derivation; you
                       can read this in the book.
              - The basic approach: take a controlled look at the aquifer, describe changes in
                       head (h) in terms of changes in x,y,z, t
              - Assumptions:
                       - Aquifer is homogeneous, isotropic
                       - All water is derived from the aquifer
              - Solution uses partial differential equations: Solves Darcy=s Law for Qx, Qy, Qz

See Figure 4.7 from Fetter, p. 127

              - Solve Darcy's law for Qx, Qy, Qz

       B) Confined aquifer

              - Remember: storativityconfined = b Ss

See Equation 4.42 from Fetter, p. 128 or simplified form: Equation 4.44

              - Pick out components that we know:
                      - flux terms for x,y,z axis
                      - Compressibility term
                      - d/t term = velocity
                      - K term
              - The result = total flow through the tube
              - This is a steady state condition (no change in the Z dimension with time)
              - The result: this equation gives total flow through the tube/layer/box (aquifer)
              - Can be modified to account for leakage (book gives equation, but we
                      won’t discuss it in class)

       C) Unconfined aquifers

              - Are a different situation
              - Water levels change with time: this produces a non-linear equation that cannot
                      be solved with Calculus (except in some specific instances!)
              - Pumped water comes from specific yield
                      Remember: storativity = Sy + hSs
              - Equation includes a Sy term

See overhead, equation 4.46, 4.47 (simplified version)
              - The equation can be made linear by assuming that drawdown is very small
                     compared to average aquifer thickness.

III) Solving flow problems
               - Remember: flow is different in confined and unconfined aquifers

       A) Steady flow in a confined aquifer
              - Flow implies that the aquifer has a gradient (Remember: water flows downhill)
              - Gradient of the aquifer is linear
See Figure 4.16 from Fetter, p. 139
              - Note: the potentiometric surface is above the confining layer
              - (Due to storativity, compressibility of the aquifer skeleton and water)
              - We discussed this before: In a confined aquifer

                                S = b Ss
              - and:
                                Ss = wg(+ n)

                     where:  = compressibility of the mineral skeleton
                              = compressibility of water
                             w = density of water
              - Now: how do you calculate the head for this confined aquifer between known
              - Formula:

                       h = h1 -    q1 x
                                x = some intermediate distance between h1 and h2
                                b = aquifer thickness
                                h1 = height of the water table in highest well

              - Hint: put all length terms in meters or feet

              - There are two ways to find q1:

                       1) q1 = discharge per unit width (a variation on "big Q")
                               - calculate q1 by finding "Q" first, dividing "Q" by the total
                                        width of the aquifer
                               - Width of aquifer must be given in problem
                                        ex: 7000 ft wide gravel lense
                               - gives "discharge per unit area"
                               - units: L2/t (ex: m2/day)

              Or:    2) q1 = Kb dh
              - Now: the other possibility is that the aquifer is unconfined:

       B) Steady flow in an unconfined aquifer

              - This is different!!!
              - The big difference:
                      The aquifer isn't always fully saturated,
                      Another consequence:
                               The gradient is not constant now:
                               Gradient increases in the direction of flow
              - From a mathematical standpoint: this complicates the picture

See Figure 4.17 from Fetter, p. 141

              - 2 problems:
                      - Equation for this sloping surface is much more complicated
                      - Also: recharge from the surface affects the gradient
              - The solution: make some assumptions about boundary conditions, calculate
                      discharge through a section of the aquifer

See Figure 4.18 from Fetter, p. 142

              - Assume: no flow in the "z" direction
              - The equation for discharge through a face of this cube (Dupuit equation):

                     q1 = 1 K (h12 - h22)
                          2       L

                     where:           L = flow length
                                      q1 = flow per unit width; units= L2/T (ft2/day, m2/day)

                     - The value of this equation: lets you calculate discharge per foot
                             of aquifer
                     - This determines discharge; what we really started out to calculate was
                             hydraulic head at some distance "x"
                     - With infiltration/evaporation included: Equation is fairly complex

See equation 4.70 from Fetter, p. 143
Write equation on board:

                      - For our purposes: realize that it is possible to calculate hydraulic
                              head at a point that is distant from a known hydraulic head
                      - "w" term here: is a recharge factor (necessary in unconfined

              2) Neglecting recharge:
                      - Equation is simpler (since w = 0)
See equation 4.71, p. 143
Write equation on board

                      - We just need to know:
                             x = distance of unknown point from origin

Draw picture on board

                             L = total distance between known wells
                             K = hydraulic conductivity
                             h1 = water level in first known well
                             h2 = water level in second known well
                      - Can also calculate the position of a groundwater divide, using some
                             different assumptions:

See Figure 4.19 from Fetter, p. 143
See eqtn. 4.73, p. 144

                      - "d" = distance from origin to water divide
                      - Be aware that it IS possible to make some of these predictions with
                                      unconfined aquifers

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