pt)x+ x control volume

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1 Appendix I DERIVATION OF EQUATIONS OF STEADY-STATE GROUNDWATER FLOW Consider a unit volume of saturated porous media (Figure AI. 1). In fluid mechanics, such a volume is called a control volume. The boundaries of the element are called control surfaces. portion of control surface pt)x+ x z Y control volume p13y+ ' (p'oy) Figure AI.1 Control volume for groundwater flow through porous media. The law of conservation of mass for steady-state flow requires that the rate at which fluid is entering the control volume is equal to the rate at which fluid is leaving the control volume or net rate of inflow = inflow - outflow = 0 (AI. 1) For purposes of analysis, consider the rate at which groundwater enters the control volume per unit surface area to consist of three components P x, PUy, and px> z where p is the density of water and x, uy and z are the apparent velocities of groundwater flow entering the control volume through control surfaces perpendicular to the x, y, and z coordinate axes. The dimensions of pv x, pry, and pv z are M/L2T. Using a Taylor Series approximation, the rate at which groundwater leaves the control volume in the x direction can be written 458

2 Appendix I 459 If we make the size of the control volume small, we can neglect higher-order terms (i.e., those involving A 2, A 3 etc.) and, because we have chosen a unit control volume (Ax = Ay = 1) the rate which groundwater leaves the control volume is ph+ x(pox). The net rate of inflow in the x direction is then net rate of inflow = rate of inflow - rate of outflow in x direction in x direction in x directon = - (AI.3) ) and the net rate of inflow the y and z directions are - (O and - (put), respectively. Because the net rate of inflow for the entire control volume must equal zero if the law of conservation of mass is to be satisfied, we can write - x(pu,,) - o o (p )y) - (p ),) = o - (AI.4) If we assume that groundwater density, p is constant (i.e., the fluid is incompressible), we can use the product role of calculus to evaluate a typical term in equation AI.4 a - ' (PUx) = - P' ' ' + = -P x (AI.5) Similarly for the x and y directions. Because groundwater density appears outside the derivative it cancels from equation AI.4 and we have i x i y /}z = 0 (AI.6) Now the apparent groundwater velocities are given by Darcy's Law

3 460 Derivation of Equations of Steady-State Groundwater Flow Problems where K x, Ky and K z are the hydrauliconductivities in the x, y, and z directions, respectively and h is the hydraulic head. Substituting equation AI.7 into equation AI.6. We arrive at the steady-state, saturated flow equation. If flow is two. dimensional, equation AI.8 simplifies to + o (^I.9) and if the flow is one-dimensional, we have ' k, x ' ',y: 0 (AI. 10) If a component of hydraulic conductivity is independent of position for a particular direction (i.e., is the same at all points along a line oriented in that direction), we can further simplify equation AI.8 using the product rule. For example, if K x is independent of postion x = )f 0h' _ O2h 0h 2 h = K x x 2 Similar terms can be obtained for Ky and K z if a ' :' ' media is homogenous and equation AI.8 simplifies to K 2h _ )2h _ )2h --+Ky y2 + = 0 X }x2 0 (ALIi) akv ak'z =0. In this case we say the porous (AI. 12) Finally, if K x = Ky = K z = K, a constant we say the porous media is homogeneous and isotropic and equation AI.8 simplifies to )2h )2h )2h = 0 x 2 y2 z 2 (AI. 13) which is known to mathematicians as La Place's equation.

4 Appendix I 461 If the porous media is not saturated, the value of hydrauliconductivity at a point is a function of the pressure head of the water in the voids at that point K = K(W) (AI. 14) where F is the pressure head. Substituting equation AI. 14 into equation AI.8 yields + + (Iq(v) ) = 0 h h h for the case where the unsaturated hydrauliconductivity function is different in the x, y, and z dir fion$. Recalling the definition of hydraulic head h -- ß + z* (AI. 16) where z* is the elevation head ( i.e., the vertical distance from any point to an arbitrary datum ). If the z coordinate axis is assumed to be vertical = (v+z*) = + = (AI.?) s'nnilarly and ah a a' = (V+z*)= + + (AI. 19) Substituting equations AI. 17, 18, and 19 into equation AI. 15 gives (AI. 20) which is the steady-state, unsaturated flow equation.

5 462 Derivation of Equations of Steady-State Groundwater Flow Problems Problems 1. Appendix I has presented the derivation of the equations of steady-state groundwater flow for a rectangular coordinate ystem i.e., a coordinate system defined by the three orthogonal coordinate axes x, y, and z. In some situations, for example in the case of groundwater flow to a well, it is more convenient to work in a cylindrical coordinate system i.e., in a coordinate system der'reed by the two orthogonal coordinate axes r, 0, and z (Fig. AI.2). (r + Ar)fil3 Figure AI.2 Control volume for groundwater flow through porous media in cylindrical coordinates. a. Using the same approach presented in this chapter derive the steady-state, saturated flow equation in cyh'ndrical coorob'nates J/ }h X KrSh 1 J/ Jh J/ }h' (AI.20) b. Derive the steady-state, unsaturated flow equation in cylindrical coordinates

6 Appendix I We can often use symmetry to reduce the dimensionality of a flow problem in cylindrical cordinates. In the case of groundwater flow to a well, it is common to consider the well to be an axis of symmetry. This is only role however if the aquifer geometry (i.e., the position of the soil surface and soil and rock layers), the components of hydraulic conductivity, and the specified boundary conditions are all independent of angular coordinate 0. In this case the derivatives of head with respect to 0 vanish and we say the problem is axisymmetric. Show that the axisymmetric forms of the steady-state saturated and the steady-state unsaturated flow equations can be written (AI.22a) (AI.22b) (Kr(¾) ) + r- -- +,,.(,. + 1)) = 0