Cours Hydrogeologie [PDF]

Zitiervorschau

COURSE OUTLINE COURSE DETAIL Sl. No.

Topic

No. of Hours

1

INTRODUCTION 05 Characteristic of ground water, Global distribution of water, Role of groundwater in water resources system and their management, groundwater column, aquifers, classification of aquifers. Hydrogeological cycle, water level fluctuations, Groundwater balance.

2

MOVEMENT OF GROUNDWATER 12 Darcy's Law, Hydraulic conductivity, Aquifer transmissivity and storativity, Dupuit assumptions Storage coefficient - Specific yield Heterogeneity and Anisotropy, Direct and indirect methods for estimation of aquifer parameters. Governing equation for flow through porous medium - Steady and unsteady state flow - Initial and boundary conditions, solution of flow equations.

3

WELL HYDRAULICS 10 Steady and unsteady flow to a well in a confined and unconfined aquifer - Partially penetrating wells - Wells in a leaky confined aquifer - Multiple well systems - Wells near aquifer boundaries Hydraulics of recharge wells.

Module 1 : INTRODUCTION Lecture 1 : Aquifer classification The residence time can be calculated as, (1.2) Where, tr is the residence time for groundwater, Vgr is the volume of groundwater and q av is the inflow or outflow at steady rate. Approximation of Groundwate r Table The water table is a boundary between saturated zone and unsaturated zone. The soil matrix is fully saturated below the water table. At the same time, the soil just above the water table is also saturated due to the capillary effect. The depth of capillary rise may be from few centimeters to few meters.

Fig. 1.3 Moisture distribution in a soil column Capillary rise may be around 2-5 cm in case of course sand, may be around 12-35 cm in case of sand, around 35-70 cm in case of fine sand, around 70-150 cm in case of silt and around 24 m and more in case of clay soil. The actual distribution can be approximate by a step function which is necessary to approximate the elevation of the groundwater table. The step defines the depth of the capillary rise, hc.

It can be assumed that up to the distance of hc, above the phreatic surface, the aquifer is fully saturated. The aquifer above hc line is completely dry, i.e. no moisture is present. The upper end of the capillary fringe may be taken as the groundwater table. However, when depth of capillary fringe, hc is much smaller than the thickness of the aquifer below the water table, the capillary fringe may be neglected in solving real world problems. The depth of the capillary fringe can be approximated as (Mavis and Tsui 1939) (1.3) where dm is the mean diameter of the soil grain, n is the porosity.

Polubarinova - Kochina (1952, 1962) approximated the capillary fringe as

(1.4) where d10 is the partical size at which 10% of the total partical is finer than that size.

Fig. 1.4: Approximation of groundwater table Geological formations and their classification Aquifer An underground geological formation which contains water and sufficient amount of water can be extracted economically using water wells. Aquifers comprise generally layers of sand and gravel and fracture bedrock. When water table serves as the upper boundary of the aquifer, the aquifer is known as unconfined aquifer (Fig. 1.5 (a)). In most of the analysis, the capillary zone is neglected and water table is considered as the upper boundary of the aquifer. The unconfined aquifer is also known as water table aquifer and phreatic aquifer. An impervious layer is generally served as the bottom boundary of an unconfined aquifer. Sometime, the bottom of an unconfined aquifer may be semipervious and water may gain and lose through the semipervious bottom layer. The aquifer is then known as leaky unconfined aquifer (Fig. 1.5 (b)).

Fig. 1.5 (a) Unconfined aquifer, (b) Leaky unconfined aquifer As such, the confined aquifer is also known as pressure aquifer. Top and bottom layer of a confined aquifer is generally impervious. However, sometimes these layers may be semipervious in nature. In such a situation, the water may gain or lose through these semipervious layers. The aquifer is then called leaky confined aquifer (Fig. 1.6 (b)).

Fig. 1.6 (a) Confined aquifer (b) Leaky confined aquifer When piezometric surface of a confined aquifer is above the ground level, the confined aquifer is then called an artesian aquifer. For artesian aquifer, if you put a well, the water will come out of the well automatically. Called homogeneous when aquifer parameters are constant throughout the medium, i.e. the properties of the medium are independent of space (Fig. 1.7(a)). The medium will be called non-homogeneous when aquifer properties are varying with space (Fig. 1.7(b)).

Fig. 1.7 (a) Homogeneous aquifer (b) Non-homogeneous aquifer A porous medium will be called isotropic when medium parameters are constant in all the directions, i.e. the parameters are independent of direction (Fig. 1.8(a)). The medium will be called an anisotropic when the parameters are different in different directions (Fig. 1.8(b)).

Fig. 1.8 (a) Isotropic aquifer (b) Anisotropic aquifer

Module 2: MOVEMENT OF GROUNDWATER Darcy’s Experiment Based on his experiments, he concluded that the rate flow through the porous media is proportional to the head loss and is inversely proportional to the length of the flow path. Figure 3.1 shows the setup of Darcy's experiment. As shown in the figure, the length of the vertical sand filter is L, the cross sectional area of the filter is A, the piezometric heads at top and bottom of the filter are h1 and h2 . Thus the head loss is (h1 - h2 ). The piezometric heads are measured with respect to an arbitrary datum. As per the conclusions made by Darcy, the flow rate Q is

Fig. 3.1 Darcy’s Experiment in vertical sand filter   

proportional to the cross sectional area (A) of the filter proportional to the difference in piezometric heads inversely proportional to the length (L) of the filter After combining these conclusions, we have (3.1) Where, Q is the flow rate, i.e. the volume of water flows through the sand filter per unit time. K is the coefficient of proportionality and is termed as hydraulic conductivity of the medium. It is a measure of the permeability of the porous medium. It is also known as coefficient of permeability. h1 and h2 are the piezometric heads.

Now, defining

and

Where J is the hydraulic gradient and q is the specific discharge, i.e. the discharge per unit area. The equation 3.1 can also be written as, (3.2) q = KJ Now consider an inclined homogeneous sand filter as shown in Fig. 3.2 In this case, the Darcy’s formula can be written as,

Fig. 3.2: Darcy’s Experiments in inclined sand filter (3.3) or, (3.4) or, q = KJ

(3.5)

Where,

and

and

z1 and z2 are the datum head or elevation head and

are the pressure head

It should be noted here that q and K have the same dimension with the velocity. The value of q will be equal to K for unit hydraulic gradient. As such for the case of isotropic medium, the hydraulic conductivity (K) may be defined as the specific discharge (q) occurs under unit hydraulic gradient (J= 1).The hydraulic conductivity is dependent on both porous matrix properties and fluid properties and can be expressed as (3.6) Where, ρ is the density of the fluid, µ is the viscosity of the fluid, v is the kinematic viscosity, k is the intrinsic permeability of the soil which depends on the properties of the porous matrix. Considering (3.6), the Darcy’s Law can be written as (3.7) It may be noted that in Darcy's law, we have neglected the kinetic energy of water. The velocity of water in case of porous medium is very low and along the flow path, the change in piezometric head is much smaller than the change in kinetic energy. Hence, k inetic energy can be neglected. Further, it may be noted that the flow takes place from higher piezometric head to lower piezometric head and not from higher pressure to lower pressure. Only in case of horizontal flow (z1 = z2 ), the flow takes place from higher pressure to lower pressure. Thus incase of horizontal flow, the Darcy’s formula can be written as,

(3.8) Moreover, In case of flow through porous medium, the flow takes place only through the pores of the medium. Therefore, the cross sectional area through which the flow actually takes place is ηA. Where η is the porosity of the porous medium. As such, the average velocity of the flow can be expressed as (3.9) Validity of Darcy’s Law Darcy's law stated that the relation between specific discharge (q) and hydraulic gradient (J) is linear. However in real world situation, the relation becomes non- linear for higher values of specific discharge. As such, the Darcy’s law is not valid for higher specific discharge. The relation is generally linear as long as the Reynolds number does not exceed some value between 1 and 10. The Reynolds number, which is expressed as the ratio between inertial force and viscous force acting on the fluid, can be written as,

Fig. 2.3 Variation of specific discharge with hydraulic gradient (3.10) Where d is some representative length of the porous medium. It is often taken as the mean particle size of the porous medium. Sometime d 10 is also taken as the representative length. d10 denotes the diameter at which 10 percent (by weight) of the soil grain size are smaller than that diameter. ϑ is the kinematic viscosity of the fluid. q is the specific discharge. Darcy's law also stated that the flow is proportional to the hydraulic gradient. As per the law, if there is a small hydraulic gradient, there should have a very small specific discharge. However, in real situation there is a threshold value of hydraulic gradient. The flow will take place only when the hydraulic gradient is more than a threshold gradient. The threshold value of hydraulic gradient depends on the properties of porous matrix and properties of the fluid. For example, threshold value of the gradient may be high in case of clayey soil as the resistive force will be higher. The value of threshold gradient will also be high for the contaminated water which has higher viscosity than the uncontaminated water.

Lecture 2 : Extension of Darcy’s Law

The Darcy’s Law was derived experimentally for one dimensional flow in a homogeneous porous medium. The generalized three-dimensional form of the equation can be expressed as, q = KJ (4.1)

Where,

and

It may be noted that where is any direction. The negative sign also indicates that water is flowing from higher hydraulic head to lower hydraulic head. Thus hydraulic gradient is negative along the direction of flow. The equation (4.1) can also be written as

(4.2) The flow in x, y and z direction can be written as (4.3) (4.4) (4.5) In case of flow through homogeneous isotropic medium, the coefficient K is a scalar constant, i.e., the value of K is independent of direction and space. In that case the equation (4.3) to (4.5) become (4.6) (4.7) (4.8) It may be noted that the equations (4.6) to (4.8) are also valid for flow through nonhomogeneous porous medium as long as the medium is isotropic. In case of flow through homogeneous isotropic medium, the equations (4.6) to (4.8) are also expressed as (4.9) (4.10) (4.11) In this case the term Φ = Kφ is called specific discharge potential. It should be noted that the equations (4.9) to (4.11) cannot be used for flow through non-homogeneous and non- isotropic medium. In case of anisotropic aquifer, the Darcy’s law can be written as,

(4.12) This can also be written as

(4.13) The coefficients appear in equation (4.12) and (4.13) are the component of the second rank tensor of hydraulic conductivity. Since K xy = Kyx , Kxz = K zx and K yz = Kzy there are six distinct components in a three dimensional flow. In case of two-dimensional flow there are only three distinct components to fully define the hydraulic conductivity, i.e. (4.14) It may be noted here that the magnitude of the hydraulic conductivity of a porous medium is independent of the coordinate system. However the component K ij depends on the chosen coordinate system. Further, it is always possible to choose three mutually orthogonal directions in space in such a way that the component K ij = 0 for i ≠ j and K ij ≠ 0 for i = j. These chosen directions in space are called principal directions of an anisotropic medium. As such, when principal directions are chosen, the equation (4.13) becomes

(4.15) And, in case of 2-dimensional flow

(4.16) Or, (4.17)

Lecture 3 : Equivalent Hydraulic Conductivity

In case of horizontally stratified soil, the hydraulic conductivity of different layer may be different. However, if we consider a particular layer, the layer is homogeneous and isotropic in nature. For this type of field problem, it is always possible to obtain an equivalent hydraulic conductivity for the whole aquifer which will produce the same discharge as that of the stratified aquifer with different hydraulic conductivities. Consider an aquifer consisting with n horizontal layers as shown in Fig. 5.1 Each layer is individually isotropic in nature. Let the thickness of the layers are dz1 , dz2 , dz3 ..............dzn and hydraulic conductivity of the layers are K 1 , K2 , K3 ..............K n

Fig. 5.1 Aquifer with horizontal strata of different hydraulic conductivities The discharge per unit width through the first layer may be written as (5.1) (5.2) Similarly (5.3) (5.4) And (5.5) The total horizontal flow through the aquifer is

(5.6) (5.7) If we consider K x as the equivalent hydraulic conductivity of the aquifer the total flow through the aquifer is

(4.9) Equating (5.7) and (5.8) (5.9) (5.10) (5.11) (5.12) Where, H = dz1 , dz2 , dz3 ..............dzn , total depth of the aquifer.

Fig. 5.2 Equivalent homogeneous aquifer The aquifer medium can now be replaced by a homogeneous medium with horizontal hydraulic conductivity of K x as shown in Fig. 5.2. In some cases, the strata may be perpendicular to the flow direction. Fig. 5.3 shows an aquifer, where strata are perpendicular to the flow direction.

Fig. 5.3 Aquifer with vertical strata of different hydraulic conductivities In this case, the flow per unit width of the aquifer can be written as (5.13) (5.14) Where dφ1 is the head loss within the first layer. For continuity, the flow layers are

must be same for the other layers also. As such, the head in other

(5.15) (5.16) And (5.17) The total head loss through the aquifer is

(5.18) If we consider the aquifer as homogeneous with equivalent hydraulic conductivity at K x , the total head loss would be, (5.19) Equating (5.18) and (5.19), we have (5.20) Or (5.21) Or (5.22) Or (5.21) Where, L is the length of the aquifer. The aquifer medium can now be replaced by a homogeneous medium with horizontal hydraulic conductivity of K x as shown in Fig. 5.4.

Fig. 5.4 Equivalent homogeneous aquifer

Lecture 4 : Aquifer Transmissivity

Consider the flow through a confined aquifer as shown in Fig. 6.1. The width of the aquifer is W. The depth of the aquifer is B. The total discharge in the x direction through the area WB can be written as,

Fig.6.1 A confined Aquifer The discharge per unit width through the first Layer may be written as (6.1) (6.2) The discharge per unit width of the aquifer can be written as, (6.3) Putting Txx = BK xx and Txy = BK xy ', the equation (6.3) becomes (6.4) Similarly in the direction of y the discharge per unit width of the aquifer can be written as (6.5) Putting Tyy = BK yy and Tyx = BK yx ', the equation (6.5) becomes (6.6) In matrix form, it may be written as, (6.7) And in vector form, we can write as (6.8) Where T is the transmissivity of the aquifer which represents the discharge through the entire thickness of the aquifer under unit hydraulic gradient. In case of homogeneous isotropic porous medium, the discharge per unit width of the aquifer may be written as,

(6.9) When principal directions are used as the coordinate system, the equation (6.7) can be written as, (6.11) Or (6.12) Where

is the principal coordinate.

The transformation from any arbitrary xy coordinate system to the principal system can be obtained using the following relationship

coordinate

(6.13) (6.14)

Lecture 7 : Storage Coefficients

Specific Storativity Specific storativity of a porous medium is defined as the volume of water released or added into a unit volume of the aquifer under unit declination in the piezometic head ( Φ). Thus it can be written as, (7.1) Where, S0 is the specific storativity of the aquifer, ΔV W is the amount of water release or added into the aquifer, V is the total volume of the aquifer and ΔΦ is the change in the piezometric head.

Fig. 7.1 Sketch to explain aquifer storativity of confined aquifer Aquifer storativity Consider a confined aquifer of horizontal area A and depth D as shown in Fig. 7.1. The initial piezometric head is at C. VW is the amount of water withdrawn from the aquifer. As a result

of pumping, the piezometric head is dropped down by an amount of ΔΦ. In this case, the equation (7.1) can be written as, (7.2) Or, (7.3) Or, (7.4) Where Ss is the aquifer storativity. Thus storativity for a confined aquifer is defined as the volume of water released from storage or added to the aquifer per unit horizontal area under unit declination or rise of peizometric head (Φ). It may be noted that like transmissivity (T) of an aquifer, the storativity is also an aquifer property. In case of confined aquifer, when Dupuit assumption of essentially horizontal flow in an aquifer is considered, the parameter T and Ss should be used. However, in the case of three dimensional flows, the hydraulic conductivity (K) and specific storativity (S0 ) need to be used. Specific yield In case of unconfined aquifer, storativity of an aquifer can be defined as the volume of water released or added to the aquifer from a unit area under unit declination or rise in water table. In this case, the storage coefficient is called as specific yield. Fig. 7.2 shows an unconfined aquifer with horizontal area ‘A’. The initial water table is at C. VW is the amount of water withdrawn from the aquifer. As a result of pumping, the water table is drop down by an amount of Δh. In this case, the specific yield can be written as,

Fig.7.2 Sketch to explain aquifer storativity of unconfined aquifer (7.5) It may be noted that a certain amount of water is always retained in the aquifer due to the capillary and hygroscopic forces which is known as specific retention (Sr). As such, Sy + Sr = η (7.6) Where, η is the porosity of the porous matrix. The specific yield is therefore always less than the porosity of the porous media. Specific yield is also sometime called effective porosity.

Lecture 5: Dupuit Approximation for Phreatic Aquifer

In case of phreatic aquifer, the phreatic surface is never horizontal and the equipotential lines are not vertical. As such, the hydraulic head (φ) is a function of spatial coordinates x, y, z and time t. Further on the phreatic surface, a non- linear boundary condition has to be specified. At the same time, the location of the phreatic surface is also not known. Fig. 8.1 (a) shows the actual phreatic surface with streamline. At point p, specific discharge is in a direction tangent to the

Fig. 8.1 Pressure distribution The specific discharge can be expressed as, (8.1) At phreatic surface, pressure is zero, hence φ = z. As per the observation made by Dupuit (1863), the slope of the phreatic surface is very small and is in the range of 1 in 1000 to 10 in 1000. As such, Dupuit suggested that sinθ can be replaced by tanθ. Thus the equation (8.1) can be written as (8.2) It may be noted that assumption of small θ is equivalent to the following assumptions 1. Equipotential lines are vertical, and 2. The flow is essentially horizontal For the Fig. 8.2, if we consider Dupuit assumption, the hydraulic head is a function of x and y only, i.e. h(x,y).

Fig. 8.2 Flow in phreatic aquifer Thus (8.3) And (8.4) The total discharge per unit width of the aquifer can be written as (8.5) (8.6) And (8.7) Discharge between two reservoirs Considering Dupuit assumption of horizontal flow, the discharge between two reservoirs can be calculated easily. Consider the phreatic aquifer shown in Fig. 8.3. The hydraulic conductivity of the homogeneous aquifer is K. The discharge per unit width of the aquifer in x direction can be obtained by

Fig. 8.3 Phreatic aquifer

(8.8) (8.9) Integrating between x = 0 to any distance x

(8.10) (8.11) (8.12) (8.13)

Equation (8.13) can be used to calculate discharge through the aquifer. This equation is also known as the Dupuit-Forchheimer discharge formula. Lecture 9 : Flow Through Unconfined Horizontal Stratified Aquifer

Fig. 9.1 Phreatic aquifer with two horizontal strata Consider the horizontally stratified aquifer shown in Fig. 9.1 above. The hydraulic conductivity of the upper strata of the unconfined aquifer is K 2 and that for the bottom strata is K 1 . The depth of the bottom strata is a. Considering Dupuit assumption, the discharge per unit width at a distance x from the left reservoir can be expressed as, (9.1) Integrating from x = 0, h(x) = h1 and x = L, h(x) = h2 , we can obtain (9.2)

(9.3) (9.4)

(9.5)

Equation (9.5) can be used to calculate the discharge through the horizontal stratified unconfined aquifer. In case of n numbers of strata, the equation can be expressed as (9.6) Flow through unconfined vertically stratified aquifer

Fig. 9.2 Phreatic aquifer with two vertical strata Consider the vertically stratified aquifer as shown in Fig. 9.2 above. The left strata has the hydraulic conductivity of K 1 and the right strata has the hydraulic conductivity of K 2 . If we consider the first strata (9.7) Integrating from x = 0, h(x) = h1 and x = L1 , h(x) = h3 , we can obtain, (9.8) (9.9) Similarly for the second strata, we can obtain (9.10) Equating equations (9.9) and (9.10), we have

(9.11) (9.12) (9.13)

Equation (2.13) can be used for calculating discharge through the vertica lly stratified unconfined aquifer. In case of n numbers of strata, the equation can be expressed as (9.14)

Equation (9.14) can be used to calculate the discharge through the vertically stratified unconfined aquifer.

Lecture 6: Governing Equation for flow Through Porous Medium

In this lecture, we will derive the governing equation for flow in confined and unconfined aquifer. Initially we will derive the governing equation for two-dimensional flow in an aquifer and in the next lecture we will derive the generalized governing equa tion for flow in porous medium. Two-dimensional flow in confined aquifer'

Fig. 10.1 Confined aquifer with elementary control volume Let B be the thickness of the aquifer. Q x is the inflow per unit width in x direction Q y is the inflow per unit width in y direction Inflow in x direction is Qx dy Outflow in x direction is

(10.1) (10.2)

Net flow in x direction (10.3) (10.4) Similarly, net flow in y direction is (10.5) Total net flow through the control volume is (10.6) Now, as we know that aquifer storativity (10.7) (10.8) (10.8) Change in storage in time dt is

(10.9) Total net flow = change in storage in time dt (10.10) (10.11) Now putting, (10.12) We have, (10.13) This is the flow equation for anisotropic non-homogeneous confined aquifer for unsteady condition. In case of homogeneous aquifer, the equation becomes (10.14) In case of homogeneous isotropic aquifer Tx = Ty = T (10.15) If a source N(x, y, t) is present, the equation becomes (10.16) Here, the source term represents the pumping or recharge per unit horizontal area of the aquifer. If case of pumping a negative sign is used as mass is withdrawn from the control volume. On the other hand, the source term will be positive in case of recharge as we are adding mass to the system. For example, if Qm3 /sec is the pumping rate from the control volume, the value of N(x, y, t) will be

.

Two-dimensional flow in a phreatic aquifer

Fig. 10.2 Elementary control volume for phreatic aquifer Consider a phreatic aquifer as shown in the Fig. 10.2. Let inflow per unit width of the aquifer in x direction is Q x and that in y direction is Q y . Total inflow in x direction is Qx dy (10.17) Total outflow in x direction is

(10.18) Net flow in x direction is, (10.19) Similarly the net flow in y direction is (10.20) Let N(x, y, t) is the source or sink of the control volume per unit area. Source or sink flow is therefore N(x, y, t)dxdy Total net flow of the control volume is

(10.21)

(10.22) If Sy is the specific yield of the aquifer, then (10.23) (10.24) Now change in storage in time dt can be written as, (10.25) For maintaining continuity Net flow in time dt = change in storage in time dt (10.26) (10.27) Now (10.28) (10.29) Putting in (10.30) In case of non-homogeneous isotropic aquifer (10.31) In case of homogeneous isotropic aquifer (10.32) This is the basic continuity equation for 2-D groundwater flow in a phreatic aquifer with a horizontal impervious base. This equation is also known as the Boussinesq equation. The equation derived above is a nonlinear one, because of the product term equation may be converted to a linear equation using the following techniques.

. The

1. Assuming where is the average constant transmissivily of the aquifer and ΔT is the deviation from the average. The flow equation can be reduced to a linear equation by putting

. In this case, (10.33)

2. In the second method the flow equation can be rewrite as (10.34) Putting, We have, (10.35)

Lecture 7: Governing Equation for Leaky Aquifer

A confined or unconfined aquifer is called a leaky aquifer when water is gained or lost through the bounded semipervious layers. In this lecture, we will derive the governing equation for the two dimensional flow in confined and unconfined leaky aquifer. Two-dimensional flow in leaky confined aquifer The piezometric head in the main aquifer is (φ) Fig. 11.1. The main aquifer is confined by the semipervious aquifer at top and bottom. The piezometric head in the top phreatic aquifer is (φ1 ) and that for the bottom confined aquifer is (φ2 ). The thickness of the main aquifer is B and the thickness of the top and bottom semi-pervious strata is B1 and B2 respectively. The hydraulic conductivity of the main aquifer is K and that for the top and bottom semipervious strata is K 1 and K 2 respectively.

Fig. 11.1 Elementary control volume for leaky confined aquifer For the 2D control volume shown in Fig. 11.1. Let

Q x is the inflow per unit width in x direction

Q y is the inflow per unit width in y direction Inflow in x direction is Qx dy Outflow in x direction is

(11.1) (11.2)

Net flow in x direction (11.3) (11.4) Similarly, net flow in y direction is (11.5) The flow enter into the control volume from the bottom semi-pervious layer is (11.6) The flow coming out from the control volume through the top semi-pervious layer is (11.7)

Total net flow of the control volume is (11.8) Now, as we know that aquifer storativity (11.9) (11.10) (11.11) Change in storage in time dt is (11.12) Total net flow = change in storage in time dt (11.13) (11.14) Now putting, (11.15) (11.16) We have, (11.17) (11.18) Putting B2 /K2 = σ2 and B1 /K1 = σ1 , we have (11.19) This is the flow equation for anisotropic non-homogeneous leaky confined aquifer for unsteady condition. In case of homogeneous aquifer, the equation becomes (11.20) For homogeneous and isotropic condition, (11.21) Putting Tσ1 = λ1 and Tσ2 = λ2 (11.22) The λ1 and λ2 are known as leakage factor of the semi-pervious layer. If a source N(x,y,t) is present, the equation becomes (11.23)

Two-dimensional flow in leaky unconfined aquifer Consider the leaky unconfined aquifer as shown in Fig. 11.2. The bottom of the unconfined aquifer has a semi pervious layer. The piezometric head in the bottom confined aquifer is (φ). The thickness of the bottom semi-pervious strata is B. The hydraulic conductivity of the main aquifer is K and that bottom semi-pervious strata is K 1 .

Fig. 11.2 Elementary control volume for leaky unconfined aquifer For the 2D control volume shown in Fig. 11.2. Let inflow per unit width of the aquifer in x direction is Q x and that in y direction is Q y . Total inflow in x direction is Qxdy (11.24) Total outflow in x direction is (11.25) Net flow in x direction is, (11.26) Similarly the net flow in y direction is (11.27) The flow enter into the control volume from the bottom semi-pervious layer is qvdxdy Let N(x,y,t) is the source or sink of the control volume per unit area. Source or sink flow is therefore N(x,y,t)dxdy Total net flow of the control volume is

(11.28)

(11.29) (11.30)

If Sy is the specific yield of the aquifer, then (11.31) (11.32) Now change in storage in time dt can be written as, (11.33) For maintaining continuity

Net flow in time dt = change in storage in time dt (11.34) (11.35) Now (11.36) (11.37) (11.38) Putting in (11.39) (11.40) Putting, σ = B/K1 (11.41) In case of non-homogeneous isotropic aquifer (11.42)

Lecture 8 : Governing Equation for Three-Dimensional Flow in Porous Medium

Fig. 12.2 Elementary control volume Consider the control volume as shown in the Fig. 12.1. The dimension of the rectangular parallelepiped box is dx, dy and dz. Let qx , qy and qz are the volumetric flow per unit area entering in to the control volume through (y - z), (x - z) and (x - y) faces respectively. The mass flux inflow along x direction is ρqxdydz. Outflow in the x - direction will be The excess of inflow over outflow of mass during time interval can be expressed as, (12.1) (12.2)

Similarly in y and z direction, we have (12.3) Total excess of mass inflow over outflow can be expressed as (12.4) Now, as per the principle of mass conservation, the excess of mass must be equal to the change in mass during dt. Change in storage can be written as Now,

. (12.5)

Where S0 is the specific storage, VT is the total volume of the porous matrix Then, (12.6) Now rate of change of storage can be written as (12.7) Putting this in the flow equation, one can have (12.8) Considering water is incompressible (ρ = constant) (12.9) (12.10) Now, as per Darcy’s law qx , qy and qz can be written as (12.11) (12.12) (12.13) Putting in flow equation (12.14) (12.15) (12.16) In case of homogeneous aquifer, (12.17) In case of homogeneous isotropic aquifer (K x = Ky = Kz = K)

(12.18) (12.19) For steady state condition

the equation becomes (12.20)

For 2-D steady state condition, it will be (12.21) If we have distributed sources and sinks of N(x,y,z,t) in the aquifer, the flow equation becomes (12.22) Boundary and initial conditions For solving the steady state flow equation as derived above, appropriate boundary conditions are needed. It is one of the required components of the mathematical model. On the other hand, for solving transient flow equation, appropriate initial condition is also required. Boundary conditions are generally three types. They are Dirichlet boundary condition, Neumann boundary condition and mixed boundary condition. Dirchlet boundary condition In case of Dirichlet boundary condition, prescribe value of the variable h(x,y,z,t) is specified at the boundary of the problem domain. This is also known as type I boundary condition. The head may be constant or may vary in space or in time. Neumann boundary condition In case of Neumann boundary condition, the gradient of the variable ( ) is specified at the boundary of the problem domain. Here, n is the direction, x, y, and z. One of the most frequently use Neumann boundary condition is the no flow boundary condition, i.e. = 0 at the boundary. As discussed, in case of Neumann boundary condition, we have (12.23) Where, C1 is constant (12.24) Where qn is the Darcy's flux in the nth direction. As such in case of Neumann boundary condition, we can also specify the Darcy's flux at the boundary instead of the gradient of the variable. The Neumann boundary condition is also knows as Type II boundary condition. Mixed boundary condition We can also specify a mixed boundary condition in the form given below. (12.25) This is also known as Type III boundary condition and is the linear combination of Type I and Type II boundary condition. Initial condition For time dependent problem, an initial condition for the head field h(x,y,z,t = 0) has to be specified. This is known as initial condition.

Fig. 12.3 shows the flow domain between two observation wells in case of a confined aquifer. The lines CF and DE represent equpotential lines of potential φ0 and φL respectively. Therefore on these two boundaries, Dirchlet boundary is to be applied. On the other hand at top and bottom of the surfaces of the aquifer are impervious. The lines CD and EF represent the no flow boundary. As such the Neumann boundary condition is to be applied on these two sides. Fig. 12.4 shows the aquifer with boundary conditions.

Fig. 12.3 Flow domain between observation wells

Fig. 12.4 Confined aquifer with boundary condition

Lecture 9 : Solution of The Flow Equation

This lecture will discuss about the analytical solution of the flow equation for very simplified scenarios. Analytical solution of 1D steady flow problem in a confined aquifer

Fig. 13.1 A confined aquifer Consider the one dimensional flow in a homogeneous confined aquifer as shown in Fig. 13.1. The hydraulic head at x = 0 is φ0 and that at x = L is φL. The transmissivity of the aquifer is T. The expression for calculation of hydraulic head any distance x can be obtained by solving the one dimensional flow equation One dimensional flow equation in case of confined aquifer can be written as (13.1) By integrating (13.2) (13.3) (13.4) Integrating again, (13.5) (13.6) Now When

x=0

φ = φ0

x=L φ = φL Putting first boundary condition in equation (13.6), we have C2 = φ0 Again putting second boundary condition in (13.6), we have

(13.7) (13.8)

Putting C1 and C2 in equation (13.6), we have

(13.9) The equation (13.9) can be used to calculate hydraulic head for any value of x. The equation (13.9) can also be written as (13.10) The discharge Q is (13.11) Analytical solution of 1D steady flow problem in an unconfined aquifer

Fig. 13.2 An unconfined aquifer between two lakes Consider the one dimensional flow in a homogeneous unconfined aquifer as shown in Fig. 13.2. The hydraulic head at x = 0 is h0 and at x = L is hL. The hydraulic conductivity of the aquifer medium is K. There is uniform recharge of N per unit area. For one dimensional flow in a phreatic aquifer, the flow equation can be written as, (13.12) (13.13) (13.14) Integrating (13.15) Integrating again, (13.16) Now when

x=0

h = h0

x=L h = hL Putting in equation (12.15), we have (13.17) Now putting the values of the constants C 1 and C2 in equation (13.15), we have

(13.18) (13.19) Equation (12.19) can be used to obtain the water table profile. Fig. 13.3 shows the water table profile for K = 30 m/day, N = 0.2 m3 /day/m2 , L = 100 m, h0 = 30 m and h1 = 30 m.

Fig. 13.3 Water table profile By differentiating (13.19), we have (13.20) (13.21) (13.22) (13.23) (13.24) Equation (13.24) can be used to calculate the discharge at any distance x

Lecture 10: Solution of 2D Steady Flow Problem in a Confined Aquifer

In this section we will discuss about the solution of the steady flow equation using finite difference method. Before discussing the method, let us have a look at the basic finite difference schemes. It is hoped that all of you are aware about the Taylor series. A function φ(x + Δx) can be expanded using the Taylor series, (14.1) (14.2) Dividing by Δx, we have (14.3) 0(Δx) is the remaining terms of the series when Δx → 0 then 0(Δx) → 0 Then the derivative of the function φ at x is (14.4) This is known as forward difference approximation of the function φ at x. Similarly, (14.5) (14.6) (14.7) This is known as backward differentiation. Again subtracting equations (14.1) and (14.5) we have (14.8) (14.9) This is known as central differentiation Again adding equations (14.1) and (14.5) we have (14.10) (14.11) Solution of steady state flow equation for confined homogeneous and isotropic aquifer The steady state flow equation for homogeneous and isotropic confined aquifer can be written as (14.12) Where, T is the transmissivity of the aquifer (m2 /day), φ is the hydraulic head (m), N is the pumping or recharge value (m3 /day/m2 ).

Fig. 14.1 2D confined aquifer with boundary conditions Now consider the aquifer as shown in Fig. 14.1. The 2D confined aquifer is homogeneous and isotropic and has no flow boundary at two sides and constant head boundary on two other two sides as shown in the Fig. 14.1. For applying the finite difference scheme, the aquifer has to discretize as sown in Fig. 14.2 below. Let Δx and Δy are the size of a discretize grid.

Fig. 14.2 Discretized 2D confined aquifer with boundary conditions Applying finite difference approximation as discussed in earlier section, we can write that (14.13) (14.14) Putting (14.13) and (14.14) in equation (14.12) and after simplifying, the finite difference approximation of the steady 2-D flow equation at cell (i, j) may be expressed as: (14.15) Where A = (T/(Δx)2 ) and A = (T/(Δy)2 )

Now writing the finite difference approximation of the flow equation in each grid center we will have a set of simultaneous equations. The number of equations is equal to the number of unknown head values. The set of simultaneous equations can be solved by any suitable method. Alternatively, these discretized equations can also be solved using the embedded optimization approach using Excel solver available in MS Excel. In order to apply the excel solver, the simultaneous equation solving problem has to be converted as an optimization problem. The optimization model may be written as (Bhattacharjya 2011) Minimize (14.16) Subject to (14.17) (14.18)

Fig. 13.3 Solution of 2D flow problem in a confined aquifer Where, i = 1,2,3,.......,I and j =1,2,3,........,J, I and J are the total number of columns and rows of the discretized aquifer. The nonlinear optimization model can be solved using the Excel solver. Bhattacharjya (2011) has presented the solution procedure of the 2D flow problem using excel solver. The solution of the problem will give the hydraulic head at each grid center of the discretized aquifer. The Fig. 14.3 shows a sample solution of a 2D confined aquifer in the form of contour map.

Module 3: WELL HYDRAULICS Lecture 11: Governing equation for radial flow in an aquifer The flow towards a well, situated in homogeneous and isotropic confined or unconfined aquifer is radially symmetric. Fig. 15.1(a) shows the cone of depression caused due to constant pumping through a single well situated at (0,0) in a confined aquifer. Fig. 15.1(b) shows the cone of impression caused due to constant recharge through the well. In case of homogeneous and isotropic medium, the cone of depression or cone of impress ion is radially symmetrical. The governing equation derived earlier in Cartesian coordinate system for confined and unconfined aquifer can also be derived for radial flow in an aquifer. In this lecture, we will derive the governing flow equation for confined and unconfined aquifer in polar coordinate system. The main objective of this conversion is to make the 2D flow problem a 1D flow problem. The resulting 1D problem will be simpler to solve.



Fig. 15.1 (a) Cone of depression (b) Cone of impression

Confined aquifer



(a) Radial flow to a well

   

(b) Section A-A in case of confined aquifer Fig. 15.2: A confiner aquifer Let us consider a case of radial flow to a single well (Fig.15.2) in a confined aquifer. The Fig. 15.2 (a) shows the radial flow towards a well and a control volume of thickness dr. The Fig. 15.2 (b) shows the vertical section AA of the aquifer along with cone of depression. The aquifer is homogeneous and isotropic and have constant thickness of b. The hydraulic conductivity of the aquifer is K. The pumping rate (Q) of the aquifer is constant and the well diameter is infinitesimally small. The well is fully penetrated into the entire thickness of the confined aquifer. This is necessary to make the flow essentially horizontal. The potential head in the aquifer prior to pumping is uniform throughout the aquifer.



Fig. 15.3 Control volume in case of confined aquifer

Consider the control volume shown in figure 15.3. The inflow to the control volume is Q r The outflow from the control volume is The net inflow to the control volume is Applying principle of mass conservation on the control volume Inflow - outflow = Time rate of change in volumetric storage Time rate of change in volumetric storage

(15.1)

(15.2) (15.3)

where So is the specific storage Replacing V by 2πrdrb, we have (15.4) (15.5) Where Ss is the aquifer storativity which is equal to S o / b Putting (15.5) in (15.3), we have (15.6) As per Darcy's law (15.7) Putting in equation (15.6) (15.8) Simplifying, (15.9) (15.10) This is the flow equation for radial flow into a well for confined homogeneous and isotropic aquifer. In case of steady state condition, the governing equation becomes, (15.11) Unconfined aquifer



(a) Radial flow to a well

 

(b) Section A-A in case of unconfined aquifer Fig. 15.4 An unconfined aquifer

Let us consider a case of radial flow to a single well (Fig. 15.4). The unconfined aquifer is homogeneous and isotropic. The hydraulic conductivity of the aquifer is K. The pumping rate (Q) of the aquifer is constant and the well diameter is infinitesimally small. The well is fully penetrated into the aquifer and hydraulic head in the aquifer prior to pumping is uniform throughout the aquifer.



Fig. 15.5 Control volume For the control volume shown in Fig. 15.5 above, The inflow to the system is Q r The outflow from the system is The net inflow to the system is Applying principle of mass conservation on the control volume Inflow - outflow = Time rate of change in volumetric storage

(15.12)

Time rate of change in volumetric storage

(15.13) (15.14)

where So is the specific storage Replacing V by 2πrdrh, we have (15.15) (15.16) Where Sy is the specific yield which is equal to S o / h . Now putting equation (15.16) in equation (15.14), we have (15.17) As per Darcy's law (15.18) Putting in equation (14.17) (15.19) Simplifying,

(15.20) (15.21) (15.22)

(15.23) This is the flow equation for radial flow into a well for unconfined homogeneous isotropic aquifer. In case of steady state condition, the governing equation becomes, (15.24) Or,

(15.25)

Lecture 16 : Solution of steady flow problem of confined and unconfined aquifer

In this lecture we will obtain the solution of the steady state flow problem in confined and unconfined aquifer. Confined aquifer In case of steady flow in confined aquifer, the flow equation becomes Or,

(16.1)

Or,

(16.2)

Integrating,

(16.3)

Or, Now, Darcy's law can be expressed as

(16.4)

(16.5) Therefore, the equation (16.4) can be written as

(16.6) (16.7) Now integrating, we have (16.8) (16.9)

 

Fig. 16.1: Confined aquifer Now consider the Fig. 16.1.

 

Putting it in equation (16.9)

 

And,

 

From these two equations, we have

  



(16.10)

(16.11)

(16.12) (16.13) Knowing hydraulic head at the well, the equation (16.13) can be used to calculate steady state hydraulic head for any values of r. This equation can also be used for estimation of aquifer transmissivity. For calculating aquifer transmissivity, the equation can be written as,

(16.14)

Unconfined aquifer In case of steady flow in unconfined aquifer, the flow equation becomes Or,

(16.15)

Or,

(16.16)

Integrating,

(16.17)

Or, Now, Darcy's law can be expressed as

(16.18)

(16.19) Therefore, the equation (15.18) can be written as (16.20) (16.21) Now integrating, we have (16.22) (16.23) (16.25)

1. Fig. 16.2 An unconfined aquifer with boundary condition Consider the unconfined aquifer shown in Fig. 16.2.

Putting it in equation (16.25)

(16.26) And, (16.27) From these two equations, we have (16.28) (16.29) Knowing hydraulic head at the well, the equation (16.29) can be used to calculate steady hydraulic head for any values of r. This equation can also be used for estimation of aquifer conductivity . The equation can be written for calculating aquifer conductivity as, (16.30) Lecture 17 : Solution unsteady flow problem of confined aquifer

We have already derived the flow equation for unsteady flow in confined aquifer. The equation can be written as, (17.1) Theis (1935) obtained the solution of the equation. His solution was based on the analogy between groundwater flow and heat conduction. Considering the following boundary conditions,

The solution of the equation for t ≥ 0 is (17.2) Where, s(r,t) is the draw down at a radial distance r from, the well at time t,

W(u) is the exponential integration and is known as well function. The well function W(u) can be approximated as (17.3) Theis Analytical solution As mentioned already, Theis analytical solution was based on the analogy between groundwater flow and heat conduction. In case of heat conduction, the change in temperature (v) at a point p(x,y) at any time t due to an instantaneous line source (x) coinciding with the Z axis can be obtained using the following equation given by Carslaw (1921). (17.4) Here, k is the Kelvin's coefficient of diffusivity For continuous source or sink x(τ) (17.5) For constant source x(τ)=x

(17.6) Considering (17.7) When,

(17.8) Then, (17.9)

(17.10) (17.11) The equation (17.11) derived for calculation of change in temperature can also be applied for calculation of drawdown at any point (x,y) at any time t. The coefficient of diffusivity is analogous to the coefficient of transmissivity of the aquifer divided b y the specific storage (Ss) of the aquifer. The continuous strength of the source and sink is analogous to the discharge rate divided by the specific storage. The equation (17.11) in case of drawdown in confined aquifer can be written as (17.12)

(17.13) Putting (17.14) Equation (17.14) can be used to calculate the drawdown at a distance of r at any time t when water is pumped at a constant rate of Q from the well. This solution is valid homogeneous isotropic aquifer having infinite areal extent and uniform thickness. Alternate analytical solution of radial flow equation The flow equation we have derived early (17.15) Let us consider (17.16) Thus,

(17.17) And (17.18) Now we can write that (17.19) We can also write (17.20) (17.21) Now as defined earlier in (17.16), we have (17.22) (17.23) We can also write (17.24) From equation (17.16) (17.25) Putting (17.25) in (17.24)

Now putting (17.23) and (17.26) in (17.21) (17.27) (17.28) Now, (17.29) From (17.16) (17.30) Putting (17.30) in (17.29) (17.31) Putting (17.19), (17.28), and (17.31) in (17.15) (17.32) (17.33)

(17.34) (17.35) Taking (17.36) The solution of the differential equation is (17.37) (17.38) Where C is a constant (17.39) (17.40) (17.41) Now next step is to find the value of the constant C. As per Darcy's law at the well face, the discharge from the well is (17.42) (17.43) Where rw is the radius of the well and b is the thickness of the confined aquifer. From (17.19), we have

Putting in equation (17.43), we have (17.44) (17.45) Putting (17.37), we have (17.46) (17.47) For most of the well the radius of the well is very small and the e u = 1 . As such the constant C can be written as (17.48) Putting the value of C in (17.41) we have (17.49) (17.50)

Equation (17.50) can be used to calculate the drawdown at a distance of r at a time t when water is pumped at a constant rate of Q from the well. This solution is valid for homogeneous isotropic aquifer having infinite areal extent and uniform thickness. Lecture 18: Time drawdown relations for constant, discrete and variable pumping scenarios

Constant pumping scenarios Fig. 18.2 shows the time drawdown curve for the pumping pattern shown in Fig. 18.1 In this case water is pumped continuously at a constant rate Q1 . The time drawdown curve can be obtain using (17.50).



Fig. 18.1 Pumping pattern

 

Fig. 18.2 Time drawdown curve The above solution is obtained by considering that the pumping is constant and continuous. However, in real world situation, the pumping will be varying with time and may be for a finite duration only as shown in Fig. 18.3 . Discrete pumping scenarios The radial flow equation derived for confined aquifer, and the assumed boundary conditions are linear in nature. As such, the basic principles of linearity can be applied here. The two basic principles of linearity are: (i) principle of superposition and (ii) principle of proportionality. These two principles can be applied to solve the problems related to finite duration pumping.

 Fig. 18.3 Discrete pumping Consider a case of pumping Q1 from t = 0 to t = t 1 as shown in Fig. 18.3.

This discrete pumping value can also be represented as shown in Fig 18.4.



Fig. 18.4 Another representation of discrete pumping In this case, the draw down at any time t can be calculated as,

(18.1) (18.2) (18.3) In this case for t 1 = 10 min, the time drawdown curve will as shown in Fig. 18.5

 Fig. 18.5 Drawdown pumping relationship in case of discrete pumping Variable pumping scenarios Similarly, for the variable pumping case shown in Fig. 18.6, the method of superposition technique can be applied.

 Fig. 18.6 Variable pumping pattern In this case, the pumping rate is Q1 from t = 0 to t = t 1 and Q2 from t = t 1 onwards The drawdown at any time t can be calculated as,

(18.4) (18.5) In this case if t 1 = 10 min, the time drawdown curve will look like the Fig.18.7 below.



Fig. 18.7 Drawdown pumping relationship in case of variable pumping

Lecture19:Solution unsteady flow problem of unconfined aquifer and leaky confined aquifer

This section will deal with the solution of unsteady flow problem in case of unconfined aquifer and also in case of leaky confined aquifer. Unconfined aquifer The time drawdown relationship is complex in case of unconfined aquifer. When drawdown is small compare to the saturated thickness of the aquifer, the solution method applied for the case of confined aquifer can also be applied to the case of unconfined aq uifer. However, when drawdown is significant, the solution method applied to the case of confined aquifer will not be applicable as it will violate the assumptions that has been made in case of confined aquifer. In this case, the water released from the storage will not be discharged instantaneously with the declination of the hydraulic head. In case of unconfined aquifer, in the initial period after the start of the pumping, water is released instantaneously from the storage. This situation is similar to the time drawdown relationship for the case of confined aquifer and can be approximated using the Theis type curve. After some period of time from the start of the pumping, the rate of drawdown will be slow due to the gravity drainage replenishment from the pores of the unsaturated zone. The gravity drainage of water from the unsaturated zone proceeds in a variable rate. Finally, an equilibrium condition is achieved between gravity drainage and rate of decline of water table at later stage. This portion of t he time drawdown relationship can also be approximate with the Theis type curve. Neuman (1975) gave the following solution for unconfined aquifer with fully penetrated well and constant discharge considering delayed yield.



(19.1)

Where 

(19.2) (19.3)

(19.4) It may be noted that u is applicable for the early drawdown data whereas uy is applicable for later drawdown data. W(u,uy η) can be calculated using the curve generated by Neuman (1975). Wells in a leaky confined aquifer A confined aquifer will be called a leaky aquifer when water is withdrawn from the confined aquifer, there is a vertical flow from the overlaying aquitard as shown in Fig. 19.1 . After the starts of the pumping, the lowering of piezometric head in the aquifer builds hydraulic gradient within the aquitard. As a result of the hydraulic gradient, downward vertical groundwater flow takes place through the aquitard.



Fig. 19.1 A leaky confined aquifer The drawdown of the piezometric surface can be obtained by (Hantush 1956, Cobb et al. 1982) (19.5) Where (19.6) (19.7) (19.8 Where T is the transmissivity of the leaky confined aquifer, K' is the vertical hydraulic conductivity of the aquitard, and b' is the thickness of the aquitard. Lecture 20: Partially penetrated well and change in hydraulic properties near a well

Partially penetrating well In a well when the intake of the well is less than the thickness of the well, then the well is called partially penetrated well. In case of partially penetrated well, the flow lines are not truly horizontal near the well. The flow lines are curved upward or downward near the well. However, at a distance far away from the well, the flow lines are horizontal. As a result of non-horizontal nature of the flow lines near the well, the length of the flow lines are more than the case of a fully penetrated well. Thus the drawdown in case of partially penetrating well is more than the fully penetrating well. Fig. 20.1 shows a partially penetrated well.



Fig. 20.1 Partially penetrated well

The drawdown of the partially penetrated well can be written as (20.1) Where, S is the drawdown of the fully penetrated well and Δs is the additional drawdown due to partial penetration. For the Fig. 20.2 given below,



Fig. 20 .2 Partially penetrated well

The additional drawdown, Δs can be calculated as (Todd and Mays, 2011) (20.2) Change in hydraulic properties near a well Consider a case of a pumping well as shown in Fig. 20.3 below.





Fig. 20.3 Pumping well

The discharge of the well can be expressed as (20.3) (20.4) (20.5) Here So

r2 > r1 V1 > V 2

Therefore, velocity near the well is more than the velocity away from the well. Due to the high velocity in the vicinity of the well , the fine particles that are present in the aquifer formation are moved with the flow of water. As a result of this phenomenon, the permeability of the aquifer medium will be more in the vicinity of the well.



Fig. 20.4 Recharge well

Now, in case of recharge well (Fig. 20.4), the impurities that are present in water are also move along with the injected water to the aquifer medium. As the velocity of flow in the vicinity of the well is higher, the impurities present in the water will move along with the