# Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

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1 Module Te Science of Surface and Ground Water Version CE IIT, Karagpur

2 Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur

3 Instructional Objectives On completion of te lesson, te student sall be learn 1. Te description of steady state of ground water flow in te form of Laplace equation derived from continuity equation and Darcy s law for ground water movement.. Te quantitative description of unsteady state ground water flow. 3. Te definition of te terms Specific Yield and Specific Storage and teir relationsip wit Storativity of a confined aquifer. 4. Te expressions for ground water flow in unconfined and confined aquifers, written in terms of Transmissivity. 5. Expression for two dimensional flow in unconfined and confined aquifers; Boussinesq equation. 6. Expression for two dimensional seepage flow below dams. 7. Analytical solution of steady one dimensional flow in simple cases of confined and unconfined aquifers..6.0 Introduction In te earlier lesson, qualitative assessment of subsurface water weter in te unsaturated or in te saturated ground was made. Movement of water stored in te saturated soil or fractured bed rock, also called aquifer, was seen to depend upon te ydraulic gradient. Oter relationsips between te water storage and te portion of tat wic can be witdrawn from an aquifer were also discussed. In tis lesson, we derive te matematical description of saturated ground water flow and its exact and approximate relations to te ydraulic gradient..6.1 Continuity equation and Darcy s law under steady state conditions Consider te flow of ground water taking place witin a small cube (of lengts x, y and z respectively te direction of te tree areas wic may also be called te elementary control volume) of a saturated aquifer as sown in Figure 1. Version CE IIT, Karagpur

4 It is assumed tat te density of water (ρ) does not cange in space along te tree directions wic implies tat water is considered incompressible. Te velocity components in te x, y and z directions ave been denoted as ν x, ν y, ν z respectively. Since water as been considered incompressible, te total incoming water in te cuboidal volume sould be equal to tat going out. Defining inflows and outflows as: Inflows: In x-direction: ρ ν x ( y. x) In y-direction: ρ ν y ( x. z) In z-direction: ρ ν z ( x. y) Outflows: ν x In X-direction: ρ (ν x Δ x x x) ( y. z) Version CE IIT, Karagpur

5 ν y In Y-direction: ρ (ν y Δ y ) ( x. z) y ν z In Z-direction: ρ (ν z. z) ( y. x) z Te net mass flow per unit time troug te cube works out to: Or ν x ν x x x ν ν y y y y ν z ν z z z ( x. y. z) = 0 () Tis is continuity equation for flow. But tis water flow, as we learnt in te previous lesson, is due to a difference in potentiometric ead per unit lengt in te direction of flow. A relation between te velocity and potentiometric gradient was first suggested by Henry Darcy, a Frenc Engineer, in te mid nineteent century. He found experimentally (see figure below) tat te discarge Q passing troug a tube of cross sectional area A filled wit a porous material is proportional to te difference of te ydraulic ead between te two end points and inversely proportional to te flow lengt L. It may be noted tat te total energy (also called ead, ) at any point in te ground water flow per unit weigt is given as = Z γ p v g (3) Were Z is te elevation of te point above a cosen datum; p is te pressure ead, and γ v is te velocity ead g Since te ground water flow velocities are usually very small, and v is neglected g Version CE IIT, Karagpur

6 = Z γ p is termed as te potentiometric ead (or piezometric ead in some texts) Tus Q α A P L Q (4) Introducing proportionality constant K, te expression becomes Q = K.A. P L Q (5) Since te ydraulic ead decreases in te direction of flow, a corresponding differential equation would be written as Q = -K.A. d dl (6) Were (d/dl) is known as ydraulic gradient. Te coefficient K as dimensions of L/T, or velocity, and as seen in te last lesson tis is termed as te ydraulic conductivity. Version CE IIT, Karagpur

7 Tus te velocity of fluid flow would be: Q d ν = = -K ( ) (7) A dl It may be noted tat tis velocity is not quite te same as te velocity of water flowing troug an open pipe. In an open pipe, te entire cross section of te pipe conveys water. On te oter and, if te pipe is filed wit a porous material, say sand, ten te water can only flow troug te pores of te sand particles. Hence, te velocity obtained by te above expression is only an apparent velocity, wit te actual velocity of te fluid particles troug te voids of te porous material is many time more. But for our analysis of substituting te expression for velocity in te tree directions x, y and z in te continuity relation, equation () and considering eac velocity term to be proportional to te ydraulic gradient in te corresponding direction, one obtains te following relation K x K x x K y z = 0 z x y z (8) Here, te ydraulic conductivities in te tree directions (K x, K y and K z ) ave been assumed to be different as for a general anisotropic medium. Considering isotropic medium wit a constant ydraulic conductivity in all directions, te continuity equation simplifies to te following expression: = 0 (9) x y z In te above equation, it is assumed tat te ydraulic ead is not canging wit time, tat is, a steady state is prevailing. If now it is assumed tat te potentiometric ead canges wit time at te location of te control volume, ten tere would be a corresponding cange in te porosity of te aquifer even if te fluid density is assumed to be uncanged. Wat appens to te continuity relation is discussed in te next section. Important term: Porosity: It is ratio of volume of voids to te total volume of te soil and is generally expressed as percentage. Version CE IIT, Karagpur

8 .6. Ground water flow equations under unsteady state For an unsteady case, te rate of mass flow in te elementary control volume is given by: ν ν x y ν z M ρ Δx Δy Δz = (10) x y z t Tis is caused by a cange in te ydraulic ead wit time plus te porosity of te media increasing accommodating more water. Denoting porosity by te term n, a cange in mass M of water contained wit respect to time is given by M t = ( nδx Δy Δz t ) ρ (11) Considering no lateral strain, tat is, no cange in te plan area x. y of te control volume, te above expression may be written as: M t ρ = t t ( nδx Δy Δz ) ( n Δz ). ρ Δx Δy. (1) Were te density of water (ρ) is assumed to cange wit time. Its relation to a cange in volume of te water V w, contained witin te void is given as: d( V V w w ) = dρ ρ (13) Te negative sign indicates tat a reduction in volume would mean an increase in te density from te corresponding original values. Te compressibility of water, β, is defined as: d ( Vw ) [ ] β = Vw (14) dp Were dp is te cange in te ydraulic ead p Tus, β = dρ ρdp (15) Tat is, Version CE IIT, Karagpur

9 d ρ =ρ dp β (16) Te compressibility of te soil matrix, α, is defined as te inverse of E S, te elasticity of te soil matrix. Hence 1 d ( σ Z ) = E S = d ( Δ z ) Δ z Were σ Z is te stress in te grains of te soil matrix. α (17) Now, te pressure of te fluid in te voids, p, and te stress on te solid particles, σ Z, must combine to support te total mass lying vertically above te elementary volume. Tus, Leading to Tus, pσ z = constant (18) dσ z = -dp (19) 1 α = dp d ( Δ z ) Δ z (0) Also since te potentiometric ead given by = γ p Z (1) Were Z is te elevation of te cube considered above a datum. We may terefore rewrite te above as d 1 dp = 1 dz γ dz () First term for te cange in mass M of te water contained in te elementary volume, Equation 1, is Version CE IIT, Karagpur

10 p. n. Δx Δy Δz (3) t Tis may be written, based on te derivations sown earlier, as equal to p n. ρ. β.. Δx Δy Δz (4) t Also te volume of soil grains, V S, is given as V S = (1-n) Δ x Δy Δz (5) Tus, dv S = [ d ( z) d ( n z) ] x y (6) Considering te compressibility of te soil grains to be nominal compared to tat of te water or te cange in te porosity, we may assume dv S to be equal to zero. Hence, Or, [ d ( z) d ( n z) ] x y = 0 (7) d ( z) = d ( n z) (8) Wic may substituted in second term of te expression for cange in mass, M, of te elementary volume, canging it to ( n Δz) ρ Δx Δy t ( Δz) = ρ Δx Δy t ( Δz) = ρ Δz ΔxΔyΔz t p = ρ α Δx Δy Δz (9) t Tus, te equation for cange of mass, M, of te elementary cubic volume becomes Version CE IIT, Karagpur

11 z y x t p n t M Δ Δ Δ = ρ β α. ) ( (30) Combining Equation (30) wit te continuity expression for mass witin te volume, equation (10), gives te following relation: t p n z y x z y x = ρ β α ν ν ν ρ ) ( (31) Assuming isotropic media, tat is, K X =K= Y K Z =K and applying Darcy s law for te velocities in te tree directions, te above equation simplifies to t p n z y x K = ) ( β α (3) Now, since te potentiometric (or ydraulic) ead is given as = γ p z (33) Te flow equation can be expressed as t K n z y x = γ β (α ) (34) Te above equation is te general expression for te flow in tree dimensions for an isotropic omogeneous porous medium. Te expression was derived on te basis of an elementary control volume wic may be a part of an unconfined or a confined aquifer. Te next section looks into te simplifications for eac type of aquifer..6.3 Ground water flow expressions for ground water flow unconfined and confined aquifers Unsteady flow takes place in an unconfined and confined aquifer would be eiter due to: Cange in ydraulic ead (for unconfined aquifer) or potentiometric ead (for confined aquifer) wit time. And, or compressibility of te mineral grains of te soil matrix forming te aquifer And, or compressibility of te water stored in te voids witin te soil matrix Version CE IIT, Karagpur

12 We may visually express te above conditions as sown in Figure 3, assuming an increase in te ydraulic (or potentiometric ead) and a compression of soil matrix and pore water to accommodate more water Since storability S of a confined aquifer was defined as S = bγ ( α n β ) (35) Te flow equation for a confined aquifer would simplify to te following: = x y z S K b t (36) Defining te transmissivity T of a confined aquifer as a product of te ydraulic conductivity K and te saturated tickness of te aquifer, b, wic is: T = K b (37) Version CE IIT, Karagpur

13 Te flow equation furter reduces to te following for a confined aquifer = x y z S T t (38) For unconfined aquifers, te storability S is given by te following expression S = S y S s (39) Were S y is te specific yield and S s is te specific storage equal to γ ( α n β ) Usually, S s is muc smaller in magnitude tan S y and may be neglected. Hence S under water table conditions for all practical purposes may be taken equal to S y..6.4 Two dimensional flow in aquifers Under many situations, te water table variation (for unconfined flow) in areal extent is not muc, wic means tat tere te ground water flow does not ave muc of a vertical velocity component. Hence, a two dimensional flow situation may be approximated for tese cases. On te oter and, were tere is a large variation in te water table under certain situation, a tree dimensional velocity field would be te correct representation as tere would be significant component of flow in te vertical direction apart from tat in te orizontal directions. Tis difference is sown in te illustrations given in Figure 4. Version CE IIT, Karagpur

14 In case of two dimensional flow, te equation flow for bot unconfined and confined aquifers may be written as, x y = S T t (40) Tere is one point to be noted for unconfined aquifers for ydraulic ead ( or water table) variations wit time. It is tat te cange in te saturated tickness of te aquifer wit time also canges te transmissivity, T, wic is a product of ydraulic conductivity K and te saturated tickness. Te general form of te flow equation for two dimensional unconfined flow is known as te Boussinesq equation and is given as x x y y S y = K t (41) Were S y is te specific yield. If te drawdown in te unconfined aquifer is very small compared to te saturated tickness, te variable tickness of te saturated zone,, can be replaced by an average tickness, b, wic is assumed to be constant over te aquifer. For confined aquifer under an unsteady condition toug te potentiometric surface declines, te saturated tickness of te aquifer remains constant wit time and is equal to an average value b. Solving te ground water flow equations for flow in aquifers require te elp of numerical tecniques, except for very simple cases..6.5 Two dimensional seepage flow In te last section, examples of two dimensional flow were given for aquifers, considering te flow to be occurring, in general, in a orizontal plane. Anoter example of two dimensional flow would tat be wen te flow can be approximated. to be taking place in te vertical plane. Suc situations migt occur as for te seepage taking place below a dam as sown in Figure 5. Version CE IIT, Karagpur

15 Under steady state conditions, te general equation of flow, considering an isotropic porous medium would be = 0 (4) y z However, solving te above Equation (4) for would require advanced analytical metods or numerical tecniques. More about seepage flow would be discussed in te later session..6.6 Steady one dimensional flow in aquifers Some simplified cases of ground water flow, usually in te vertical plane, can be approximated by one dimensional equation wic can ten be solved analytically. We consider te confined and unconfined aquifers separately, in te following sections Confined aquifers If tere is a steady movement of ground water in a confined aquifer, tere will be a gradient or slope to te potentiometric surface of te aquifer. Te gradient, again, would be decreasing in te direction of flow. For flow of tis type, Darcy s law may be used directly. Version CE IIT, Karagpur

16 Aquifer wit constant tickness Tis situation may be sown as in Figure 6. Assuming unit tickness in te direction perpendicular to te plane of te paper, te flow rate q (per unit widt) would be expressed for an aquifer of tickness b According to Darcy s law, te velocity v is given by q = b *1 * v (43) v = -K x (44) Were, te potentiometric ead, is measured above a convenient datum. Note tat te actual value of is not required, but only its gradient in te direction x of flow, x, is wat matters. Here is K is te ydraulic conductivity Hence, q = b K x (45) Version CE IIT, Karagpur

17 Te partial derivative of wit respect to x may be written as normal derivative since we are assuming no variation of in te direction normal to te paper. Tus q = - b K d d x (46) For steady flow, q sould not vary wit time, t, or spatial coordinate, x. ence, d q = b K d x d d x = 0 (47) Since te widt, b, and ydraulic conductivity, K, of te aquifer are assumed to be constants, te above equation simplifies to: d = 0 (48) d x Wic may be analytically solved as = C 1 x C (49) Selecting te origin of coordinate x at te location of well A (as sown in Figure 6), and aving a ydraulic ead, A and also assuming a ydraulic ead of well B, located at a distance L from well A in te x-direction and aving a ydraulic ead B, we ave: Giving A = C 1.0C and B = C 1.LC C 1 = - A /L and C = A (50) Tus te analytical solution for te ydraulic ead becomes: H = B L A x A (51) Aquifer wit variable tickness Consider a situation of one- dimensional flow in a confined aquifer wose tickness, b, varies in te direction of flow, x, in a linear fasion as sown in Figure 7. Version CE IIT, Karagpur

18 Te unit discarge, q, is now given as d q = - b (x) K dx (5) Were K is te ydraulic conductivity and d/dx is te gradient of te potentiometric surface in te direction of flow,x. For steady flow, we ave, dq db = K dx dx d dx d b = 0 d x (53) db Wic may be simplified, denoting as b dx d d b b = 0 (54) d x dx A solution of te above differential equation may be found out wic may be substituted for known values of te potentiometric eads A and B in te two observation wells A and B respectively in order to find out te constants of integration. Version CE IIT, Karagpur

19 .6.6. Unconfined aquifers In an unconfined aquifer, te saturated flow tickness, is te same as te ydraulic ead at any location, as seen from Figure 8: Considering no recarge of water from top, te flow takes place in te direction of fall of te ydraulic ead,, wic is a function of te coordinate, x taken in te flow direction. Te flow velocity, v, would be lesser at location A and iger at B since te saturated flow tickness decreases. Hence v is also a function of x and increases in te direction of flow. Since, v, according to Darcy s law is sown to be d ν = K (55) dx te gradient of potentiometric surface, d/dx, would (in proportion to te velocities) be smaller at location A and steeper at location B. Hence te gradient of water table in unconfined flow is not constant, it increases in te direction of flow. Tis problem was solved by J.Dupuit, a Frenc ydraulician, and publised in 1863 and is assumptions for a flow in an unconfined aquifer is used to approximate te flow situation called Dupuit flow. Te assumptions made by Dupuit are: Te ydraulic gradient is equal to te slope of te water table, and Version CE IIT, Karagpur

20 For small water table gradients, te flow-lines are orizontal and te equipotential lines are vertical. Te second assumption is illustrated in Figure 9. Version CE IIT, Karagpur

21 Solutions based on te Dupuit s assumptions ave proved to be very useful in many practical purposes. However, te Dupuit assumption do not allow for a seepage face above an outflow side. An analytical solution to te flow would be obtained by using te Darcy equation to express te velocity, v, at any point, x, wit a corresponding ydraulic d gradient, as dx Tus, te unit discarge, q, is calculated to be d ν = K (56) dx d q = K (57) dx Considering te origin of te coordinate x at location A were te ydraulic ead us A and knowing te ydraulic ead B at a location B, situated at a distance L from A, we may integrate te above differential equation as: Wic, on integration, leads to L 0 q dx = K B A d (58) q x L 0 K. B = (59) A Or, q. L = K B A (60) Rearrangement of above terms leads to, wat is known as te Dupuit equation: q = 1 K B L A (61) Version CE IIT, Karagpur

22 An example of te application of te above equation may be for te ground water flow in a strip of land located between two water bodies wit different water surface elevations, as in Figure 10. Te equation for te water table, also called te preatic surface may be derived from Equation (61) as follows: x ( ) L 1 1 = (6) In case of recarge due to a constant infiltration of water from above te water table rises to a many as sown in Figure 11: Version CE IIT, Karagpur

23 Tere is a difference wit te earlier cases, as te flow per unit widt, q, would be increasing in te direction of flow due to addition of water from above. Te flow may be analysed by considering a small portion of flow domain as sown in Figure 1. Considering te infiltration of water from above at a rate i per unit lengt in te direction of ground water flow, te cange in unit discarge d q is seen to be Version CE IIT, Karagpur

24 d q = i.dx (63) Or, dq = i (64) dx From Darcy s law, d q = K.. = dx dq 1 = dx d K dx 1 ( ) d( ) k dx (65) (66) dq Substituting te expression for, we ave, dx Or, 1 d ( ) K = i dx (67) d ( ). i = dx k (68) Te solution for tis equation is of te form K x = C1x C If, now, te boundary condition is applied as, At x = 0, = 1, and At x = L, = Te equation for te water table would be: (69) ( ) 1 1 i = 1 x ( L x) x L K (70) And, Version CE IIT, Karagpur

25 Were to be q 0 q = x q x (71) 0 is te unit discarge at te left boundary, x = 0, and may be found out q 0 ( ) 1 1 il = (7) L Wic gives an expression for unit discarge ( ) 1 1 L q x = K i x L q x at any point x from te origin as (73) For no recarge due to infiltration, i = 0 and te expression for become independent of x, ence constant, wic is expected. q x is ten seen to References Ragunat, H M (00) Ground Water (Second Edition), New Age International Pvt. Ltd. Version CE IIT, Karagpur

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