Survey

Transcript

CHAPTER 2 Principles of Groundwater Flow Simon Mkhandi Department of Civil Engineering. P.O.Box 35131, Dar es Salaam, Tanzania. Akpofure E. Taigbenu Department of Civil & Water Engineering, National University of Science & Technology, P.O. Box AC 939. Bulawayo, Zimbabawe. Brian K. Rawlins Department of Hydrology, University of Zululand, P Bag X1001, KwaDlangezwa 3886. South Africa The following competencies can be achieved on completion of this chapter: Define some aquifer parameters and groundwater flow variables. Explain storage mechanisms in confined and unconfined aquifer. Derive the equations that govern flow in groundwater systems. Provide solutions to simple flows in phreatic aquifers. Highlight some features of surface water – groundwater interaction from those solutions. 2.1 Groundwater flow Flow in ground water is in general complex if the details of the flow within the interconnected pores (interstices) are taken into consideration. Those details of the flow at microscopic level through the irregular interstices with irregularly shaped soil particles are overlooked by considering the flow on a macroscopic scale. To achieve this macroscopic perspective to the flow, a representative elementary volume (REV) serves as a point in the continuum approach to the flow analysis. The REV is a conceptual volume containing soil grains and pore space within which the spatial and temporal distribution of flow variables (density, velocity, temperature, pressure, etc) can be assessed in a continuum sense. When this approach is not adopted, then a point could be in the solid phase when it falls on a soil grain or in the liquid or gaseous phase when it falls in a pore space (void). Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 Considering the total flow Q normal to an area A (containing both soil grains and pore spaces), then the flow velocity normal to the flow, referred to as the specific discharge, is given by q Q A (2.1) Taking the average volumetric porosity n (see definition in chapter 1) to be equal to the average areal porosity, the actual area over which flow takes place is nA , the rest being occupied by the solid soil grains. The average velocity V through connected interstices is given by V Q q nA n (2.2) The actual available flow area within the pores is reduced by fluid particles that are attached to the soil grains by capillary forces and ionic bonding to the soil grains and as well as dead-end pores that may found within the porous matrix, so that the effective flow area is now ne A , where ne is the effective porosity. The expression for the average velocity that is given by eq. (2.2) is now modified to V Q q ne A ne (2.3) Darcy (1856) provided the relationship between the specific discharge q and the gradient of the energy head that drives the flow. This relationship is now popularly known as the Darcy law. Using an apparatus similar to that shown in Figure 2.1, Darcy found that the specific discharge is proportional to the gradient of the hydraulic head. That is q g h h1 h2 L L (2.4) where g denotes the hydraulic gradient and h p / z is the hydraulic head. It is noted in eq. (2.4) that h h1 h2 represents the energy loss arising mainly from frictional losses as the fluid meanders through the tortuous paths in the interstices. The kinetic energy or dynamic head component is usually not considered in the energy loss calculations because it is considered negligible compared to the pressure head p / and elevation or potential head z due primarily to the sluggish nature of groundwater flow where velocities are generally small. The constant of proportionality in eq. (2.4) is referred to as the hydraulic 2-2 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 conductivity K and it reflects the fluid transmitting capacity of the porous medium. Introducing the hydraulic conductivity K , eq. (2.4) becomes q K h1 h2 L (2.5) The way that Darcy law given by eq. (2.5) has been written indicates a flow that is predominantly in one direction. Subsequently Darcy law is extended to flow in higher dimensions of flow (2 and 3 dimensions). Figure 2.1: Setup to demonstrate Darcy law When the flow occurs in higher dimensions, in general three dimensions, then Darcy law becomes qx h / x q q y K h K h / y q h / z z (2.6) The specific discharge is now a vector that has three components q x , q y and q z in the x , y , and z directions. The minus sign in eq. (2.6) is intended to correctly indicate that flow proceeds in the direction of decreasing hydraulic head. The hydraulic conductivity K is now a tensor that, in general, has 9 elements. That is 2-3 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 K xx K K xy K xz K xz K yz Kzz K xy K yy K yz (2.7) It is a symmetric tensor that can be diagonalized into its principal directions x * , y * , and z * , so that K becomes K x* x* K 0 0 0 K y * y* 0 0 0 Kz*z* (2.8) Further discussion on the hydraulic conductivity tensor is carried out on the basis of its behaviour for different types of formations that can be encountered. 2.1.1 Hydraulic Conductivity The hydraulic conductivity K is a parameter that occurs in most of the relations in Darcy law, and it is sometime defined as the flux per unit area per unit hydraulic gradient. It is a scalar with the dimension of L/T. It represents the capacity of the formation to transmit water, and as such it depends on the porous matrix and as well as the fluid properties. The factors of the porous matrix on which depends the hydraulic conductivity include the soil grain size, shape, size distribution, tortuosity, porosity, etc, while the fluid properties include density and viscosity . These factors are expressed in the relationship for the hydraulic conductivity that is given by K k g kg (2.19) where / is the kinematic viscosity of the fluid, and k is the intrinsic permeability that depends on the properties of the porous matrix. There are a number of empirical formulae relating the intrinsic permeability and the properties of the porous matrix (Bear, 1979). The hydraulic conductivity is determined in the laboratory with permeameters of which are the constant-head permeameter that is suited for noncohesive soils like sand, gravel and rocks, and the falling-head permeameter that is generally suited for cohesive soils with low hydraulic conductivity. Typical values of hydraulic conductivity of various rocks are presented in Table 2.1. 2-4 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 Table 2.1: Typical values of Hydraulic Conductivity (from Bouwer, 1978) Geological classification Unconsolidated materials: Clay Fine sand Medium sand Coarse sand Gravel Sand and gravel mixes Clay, sand, gravel mixes (e.g. till) K (m/d) 10-8 – 10-2 1–5 5 – 20 20 – 102 102 – 103 5 – 102 10-3 – 10-1 Rocks: Sandstone Carbonated rock with secondary porosity Shale Dense solid rock Fractured or weathered rock (Core samples) Volcanic rock 10-3 – 1 10-2 – 1 10-7 < 10-5 Almost 0 – 300 Almost 0 - 103 2.1.2 Range of Validity of Darcy law Darcy law provides a linear relationship between the specific discharge q and the hydraulic gradient g , and it is valid when the average flow is laminar in nature. That is the case for most regional groundwater flow problems of practical interest where the flow velocities are small so that inertia (driving force) is small compared to the resisting force due to viscous action. The ratio of inertia force to viscous force is accounted for by a dimensionless parameter known as the Reynolds number Re . It is expressed as Re qd (2.9) where d is a typical size of the solid grains. The typical size is sometimes taken as the d10 size that can be obtained from a particle size analysis. It is the sieve diameter that allows 10% by weight of the grains to pass through. Darcy law is considered valid when the Reynolds number does not exceed 10, and in most practical flows, this condition is satisfied, except in pockets of the flow region (discharge areas) where it may not be satisfied. 2-5 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 2.1.3 Isotropy and Anisotropy Isotropy and anisotropy refer to the direction dependence of the hydraulic conductivity. A formation is said to be isotropic if the hydraulic conductivity is the same in all three directions x , y , and z . That is Kxx K yy Kzz . Conversely, when Kxx K yy Kzz , then the formation is said to be anisotropic. Anisotropy arise from depositional patterns, pressure from overburden soil, flow channels due to chemical weathering, and structural fissures. These factors give rise to preferential flow directions in the formation. 2.1.4 Homogeneity and Heterogeneity Geological formations are said to be homogeneous when the hydraulic conductivity does not vary from one location to another. In that case, K xx , K yy and K zz are not dependent on x , y , or z . A medium is said to be heterogeneous or nonhomogeneous when the hydraulic conductivity varies with location. There are two types of heterogeneity. The first type refers to heterogeneity in which the variation in hydraulic conductivity can be represented by some functional distribution. That is K xx K xx ( x , y , z) , K yy K yy ( x, y, z) and Kzz Kzz ( x , y , z) . This type of heterogeneity occurs when the geological processes that give rise to the formation produce a random distribution in formation hydraulic conductivity. The second type of heterogeneity occurs when there exist regions of homogeneous formations that are separated by sharp discontinuities in hydraulic conductivity. When examining the flow in this type of formation, the compatibility relations with respect to the hydraulic head and flux at the interface of sharp discontinuities in hydraulic conductivity have to be examined. Type 2 heterogeneity formations may present it self as stratified formations in a particular direction. 2.2 Mechanisms of storage in Aquifers 2.2.1 Confined Aquifers Confined aquifers are known to be pressured aquifers because water in their pores is under pressure from the confining strata and overburden soil. At any typical location (Figure 2.2), the stresses will be borne by the soil grains (matrix of 2-6 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 the aquifer) and water in the pores. If we denote the total stress as , then the following relationship, first proposed by Terzaghi (1925), is given as (Lambe & Whitman, 1979) p (2.10) where is the stress borne by the soil grains and referred to as the intergranular or effective stress, and p the pore water pressure. Both the intergranular stress and pore water pressure act in a compensating manner when there is no change in the total stress on the formation. That is the case when there is withdrawal or addition of water into storage due to pumping activities from wells located in the aquifer. In that case, the change in is zero, that is d 0 , and equation (2.10) becomes d dp (2.11) Figure 2.2 Stresses at a plane in a soil. When withdrawal is taking place, the reduced pore water pressure results in a corresponding increase in the intergranular or effective stress, and vice-versa. Because pressure changes can be quite significant, particularly in confined aquifers, the compressibility of water and the soil matrix become quite important. Under ordinary pressures within the range of unit atmosphere, water is considered incompressible, but for larger values of pressures, water can no longer be treated as incompressible. When pore water pressure is reduced, water expands (attempts to occupy a larger volume), and this results in expulsion of water from the pores, thus enhancing depletion from storage. At the same time, the increased intergranular stress gives rise to the soil matrix collapsing (like a 2-7 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 sponge), resulting in reduced porosity that aids expulsion of water from the pores. On the other hand, when the aquifer is being recharge through wells, increase in storage is similarly enhanced by the compressibilities of the pore water and soil matrix. In this case, the increased pore water pressure causes water in the pores to contract, enhancing storage of water. At the same time, reduced intergranular stress causes the soil matrix to expand, giving rise to marginal increase in porosity that further enhances addition of water into storage. This elastic behaviour of water and soil matrix of the confined formation are accounted for by the specific storativity S0 which is defined as the amount of water released from (or added to) storage in a unit volume of aquifer and per unit decline (or rise) in pressure (or piezometric head) S0 Uw U h (2.12) where U w refers to change in volume of water in storage, U the volume of aquifer, and h is the change in piezometric head. Another parameter, commonly used for confined aquifers, is the storativity or storage coefficient S which is defined as the amount of water released from (or added to) storage per unit horizontal area of aquifer and per unit decline (or rise) in piezometric head. S Uw A h (2.13) When the flow is assumed to be essentially horizontal and the confined aquifer is assumed to have uniform thickness b , then the relationship between the specific storativity S0 and the storativity S is: S S0b . If that is not the case, then both aquifer parameters should not be related, as the specific storativity applies to a three dimensional flow, while the storativity applies to two-dimensional horizontal flow. The storativity ranges between 10-6 and 10-3, of which 40% is due to the compressibility of water and the rest to the compressibility of the porous matrix. 2.2.2 Unconfined Aquifers The mechanism for storage in unconfined aquifers is slightly different from that in confined aquifers. For the unconfined aquifer, a water table serves as its upper boundary. (The water table is an approximate concept that, in our current discussions, neglects the capillary fringe and unsaturated column of soil that have 2-8 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 influence to some extent on the flow in the saturated region.) The unconfined nature of the aquifer and the pressures encountered preclude significant compressibility effects in the water phase and that of the soil matrix. If there are any elastic effects on storage, they occur at the onset of pumpage. During this initial time, the unconfined aquifer behaves as a confined one with a corresponding storativity S . As pumping continues, the water table is lowered because of drainage of water from the pores at the water table due to pressure differences at different points on the water table. (Drainage from the pores is usually not complete because of some portion of water in the pores that will be held to the soil grains by capillary and ionic forces that are usually much larger than the drainage forces providing the flow.) This draining of the sediments is accounted for by the specific yield S y which is the amount of water released from (added to) storage per unit area of aquifer and per unit decline (rise) of water table. The specific yield is related to the porosity of the aquifer by the relationship S y Sr n (2.14) where S r is the specific retention, indicative of the amount of water retained in the pores after drainage, and n is the porosity. Because of the above relationship, the specific yield is also sometimes referred to as the effective porosity. Issues of delayed yield of an unconfined are sometimes raised in the groundwater literature, and they refer to the rate of fall of the water table in relation to how quickly the pores are drained. When the water table falls quickly, the drainage of the pores may lag behind, but when the water table falls more slowly, pores tend to completely drain without any time lag. It is this phenomenon that raises the issue of the time dependence of the specific yield, which largely depends on the abstraction rate, hydraulic conductivity and porosity of the aquifer. Typical values of the specific yield S y are in the range of 0.01 and 0.3, and are generally much larger than the storativity. Figure 1.1 shows the general trend in the relationship between S y and n for various soils. 2.3 Derivation of equations of Flow Transient flow in groundwater systems can be described by equations that are derived by combining Darcy law earlier discussed and the statement of mass conservation or mass balance (continuity equation). This latter statement states 2-9 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 that mass inflow rate minus mass outflow rate equals rate of change of mass storage. A suitable volume of the flow is defined so that fluid and solid grains are allowed to enter and exit it while the demarcated volume remains fixed in space and time. This suitable volume is referred to as the control volume (CV), and it is chosen so that the information on the property or variable of the flow being sought can readily be achieved. The CV can be of finite dimensions in which case the flow property or variable cannot be described at each point or it can be of infinitesimal dimension so that information on the property of the flow is obtained at each point of the flow. When the CV is infinitesimal, the resultant equations are in terms of differential equations that still have to be solved by mathematical tools. In many instances, it may not be possible to solve these differential equations, necessitating their being simplified further to become amenable to known solution techniques. Using the infinitesimal cube shown in Figure 2.3 with dimensions of x , y , and z as CV, mass balance in, out and within it is examined. Across the face ABCD, the mass inflow rate is given by QABCD q x y z (2.15) where is the fluid density. The mass outflow in the x direction across face EFGH is obtained by taking the first term of the Taylor’s expansion of the quantity QABCD . That is QEFGH ( q x ( qx ) x ) y z x (2.16) In the x direction, the net mass flux is given by QABCD QEFGH ( qx ) x y z x (2.17) Making similar calculations for the net mass flux in the other directions, the net outflow through all the faces is given by ( q x ) ( q y ) ( qz ) x y z y z x 2-10 (2.18) Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 Figure 2.3: Infinitesimal control volume used to derive flow equations. The rate of change of mass storage within the CV is given as ( n) x y z t (2.19) On the basis of the statement of mass conservation earlier stated, eqs. (2.18) and (2.19) can now be combined to yield ( q x ) ( q y ) ( qz ) y z x ( n) t (2.20) If the fluid density does not vary spatially but can vary temporally, then is treated as a constant with respect to the spatial coordinates x , y and z , and taken out from the left hand side of eq. (2.20). That is q y qz q x y z x 1 ( n) t (2.21) Note that eq. (2.21) can be written in short hand notation as q 1 ( n) t (2.22) Applying Darcy law to the expression for the specific discharge vector, and assuming that the principal coordinates coincide with the coordinates x , y and z , then eq. (2.22) becomes 2-11 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 [K h] 1 ( n) t (2.23) or, in long hand, h h h Kxx Kzz K yy x x y y z z 1 ( n) t (2.23) Eq. (2.23) is the governing equation for three dimensional groundwater flow in a compressible porous formation. The storage term on the right hand side of the equation can be related to the mechanism for storage change in confined and unconfined aquifers discussed in section 2.2. That relationship is derived here for a confined aquifer, and simply stated for an unconfined aquifer. Expanding the storage term yields 1 ( n) 1 n p n t p p t (2.24) The term / p that indicates the change of the density of water to pressure changes can be related to the compressibility of water. The compressibility of water is defined as the volumetric change of water with respect to increase in pressure. That is 1 U w 1 Uw p p (2.25) where U w is the volume of water in the pore spaces or void, and the minus sign in eq. (2.25) indicates that the volume of water decreases with increase in pressure and vice-versa. The term n / p can be related to the elastic behaviour of the porous or soil or formation matrix. This elastic behaviour should not be confused with that of the soil grains that are themselves essentially inelastic. Changes in the intergranular or effective stress do not give rise to changes in the sizes of the individual soil grains but the size of the porous matrix. The compressibility of the porous matrix is defined as 1 Ub U b ' (2.26) where U b is the bulk volume of the soil matrix, and it is related to the volume of the soil grains U s , which remains the same despite changes in pore pressure and intergranular stress, as: U s (1 n)U b constant. 2-12 Noting the compensating Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 character of changes in pore water pressure and changes in effective stress when there is no change in the total stress on the formation, that is p ' , then the compressibility of the porous matrix given by eq. (2.26) can be expressed as 1 [U s / (1 n)] 1 [U s / (1 n)] 1 n Us ' Us p (1 n) p (2.27) Now we are in a position to rewrite the storage term of eq. (2.24) in terms of the elastic properties of water and the porous matrix that are given by eq. (2.25) and (2.27). That is 1 ( n) p (1 n) n t t (2.28) Recall the expression of the hydraulic head: h z p / g . The temporal derivative of the pressure is expressed approximately as: p / t g h / t , so that eq. (2.28) becomes 1 ( n) h g[ (1 n) n ] t t (2.29) The factor premultiplying the temporal derivative of the hydraulic head in eq. (2.29) is the specific storativity S0 for the confined aquifer. That is, S0 g[ (1 n) n ] so that eq. (2.29) becomes 1 ( n) h S0 t t (2.30) The differential equation that governs three-dimensional flow in a confined aquifer can now be expressed as h h h Kxx Kzz K yy x x y y z z S0 h t (2.31) When the aquifer is isotropic but heterogeneous, then K Kxx Kyy Kzz and eq. (2.31) becomes h h h K K K x x y y z z S0 h t (2.32) Note that in this case, K is in general a function of x , y and z . For an isotropic and homogeneous aquifer, the hydraulic conductivity is a constant and can be taken out of the spatial derivatives. Eq. (2.31) becomes 2h 2h 2h x2 y2 z2 S0 h K t (2.33) 2-13 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 If the flow is steady and/or water and porous matrix is incompressible, the right hand side of eq. (2.33) vanishes and the resultant governing equation becomes 2h 2h 2h 0 x2 y2 z2 or 2h 0 (2.34) This is the well known Laplace equation. It governs steady groundwater flow in an isotropic, homogeneous, incompressible confined formation. For an unconfined aquifer where storage change is largely due to drainage of water from the pores due to the lowering of the water table and minimally due to elastic behaviour of water and the soil matrix, the storage term is negligibly small. That is 1 ( n) 0 t (2.35) For the unconfined aquifer, the differential equation that governs three dimensional flow is given by h h h Kxx Kzz K yy x x y y z z 0 (2.36) or (K h) 0 (2.36) When the unconfined aquifer is isotropic and homogeneous, the flow equation reduces to Laplace equation earlier presented. It must be said here that although the unconfined aquifer is largely considered as inelastic so that the mass conservation equation (2.22) is q 0 and its governing equation (2.36) in three dimensions appears simpler than its confined flow counterpart (2.31), one must not loose sight of the fact that a water table or phreatic surface serves as the upper boundary of the unconfined aquifer. The location of the phreatic surface which is sought and its nonlinear movement makes the solution to the unconfined flow problem more challenging than the confined case. Without elaborating on the derivation of the equation that governs the movement of the phreatic surface, the conditions that have to be satisfied on it 2-14 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 are simply stated. On the phreatic surface, the pressure is atmospheric ( p 0 ), so that the hydraulic head is h( x , y , z, t ) z or F ( x, y , z, t ) h( x, y, z, t ) z 0 (2.37) where F ( x, y, z, t ) is the equation that described the phreatic surface. The movement of the surface is described by the relationship Sy F q F 0 t (2.38) which is expanded to 2 h 2 h h h h Sy Kxx K yy Kzz 0 x t z y z 2 (2.39) Observe how highly nonlinear the equation governing the movement of the phreatic surface is. It is a great challenge to even numerical techniques in solving the unconfined flow problem in three dimensions with this nonlinear equation on the phreatic surface. For an isotropic, homogeneous unconfined aquifer, eq. (2.39) becomes h h h h h Sy K 0 t x x y x 2 2 2 (2.40) It is noted that in all the flow equations so far derived, it has been assumed that there are no sources or sinks arising from abstracting and recharge wells or distributed recharge from accretion in the case of unconfined aquifers. When those sources/sinks are present, an additional term has to be included in the flow equation. To include such a term in eq. (2.31) yields h h h Kxx Kzz Q K yy x x y y z z S0 h t (2.41) where Q is the rate of volume of water contributed by the source/sink by unit volume of aquifer. It has a dimension of T 1 . 2.3.1 Horizontal Groundwater Flow If the flow is considered in the confined or unconfined to be essentially horizontal, then the piezometric head for the confined aquifer and water table elevation is then dependent on x , y and t . In such a case, streamlines are horizontal. The 2-15 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 flow equations obtained earlier are simplified further. It is common to refer to the approach that considers the flow to be horizontal as the hydraulic approach in contrast to the hydrodynamic one that addresses the full three-dimensional equations. The hydraulic approach is widely adopted because of the natural state of most aquifers where the lateral dimensions are of many order of magnitude larger than the vertical dimension so that variations in the flow in the vertical dimension are negligible. For the unconfined aquifer, the hydraulic approach is also commonly referred to as the Dupuit-Forchheimer approach or assumption because of the observations of small slopes of the water table in unconfined aquifers that were made by these two investigators. The hydraulic or essential horizontal flow approach and Dupuit-Forchheimer assumptions are widely adopted in solving regional groundwater flow problems largely because they approximate most practical flow situations and the resultant equations are simpler to solve. To obtain the differential equation for horizontal flow in aquifers, the equation in three dimensions is integrated over the flow depth. For an anisotropic, heterogeneous confined aquifer of uniform thickness b , the differential equation is h h Kxxb K yyb x x y y S0b h t or h h T T x xx x y yy y S h t (2.42) where Txx and Tyy are the transmissivities in the x and y directions, and S is the storativity. If the medium is isotropic and homogeneous, then T Txx Tyy and eq. (2.42) becomes 2h 2h x2 y2 S h T t (2.43) If the flow is steady and/or water and soil matrix are considered to be inelastic, then the right hand side of eq. (2.43) vanishes. In that case, we have 2h 2h 0 x2 y2 or 2h 0 (2.44) 2-16 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 This is again Laplace equation in two dimensions. It governs steady horizontal flow in isotropic, homogeneous confined aquifers. For horizontal flow in unconfined aquifers, integrating the differential equation in three-dimensions over the flow depth from a horizontal impervious bottom to the water table yields the equation h h Kxx h K yy h x x y y Sy h t (2.45) Note that the nonlinear movement of the phreatic surface has been incorporated in eq. (2.45) by the depth integration of eq. (2.36) over the flow thickness. If the unconfined aquifer is isotropic and homogeneous, then the flow eq. (2.45) becomes h h h h x x y y Sy h K t or 2 h2 2 h2 x2 y2 2S y h K t (2.46) Eq. (2.46) is nonlinear in the water table distribution h , and it is commonly referred to as the Boussinesq equation. When the flow is steady, then eq. (2.46) becomes 2 h2 2 h2 0 x2 y2 or 2 h2 0 (2.47) Note that for steady horizontal unconfined flow, the Laplace equation is expressed in terms of h 2 instead of h as in the confined flow case. 2.3.2 Physically Imposed Conditions All the differential equations earlier derived for flow in aquifers do have unique solutions only when appropriate conditions are applied on the boundaries of the flow region and, for transient flows, as well as conditions at the initial time. The conditions applied on the boundary are referred to as boundary conditions and they reflect known relations for the hydraulic head and specific discharge when 2-17 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 certain boundaries adjoin the aquifer of interest. It is with these known values for the hydraulic head and/or discharge on the flow boundary that the distribution for unknown quantities on the boundary and within the flow region can be calculated. These conditions are discussed with respect to the horizontal approximation to flow in aquifers. Boundary of prescribed hydraulic head: This is a boundary on which the aquifer is in hydraulic contact with a large water body like a river or lake. The hydraulic head on the boundary equals the known water elevation in the river or lake. This condition is stated as: h H1 ( x , y , t ) (2.48) where H1 ( x, y, t ) is the water elevation in the river or lake. Boundary of prescribed discharge: Along such a boundary, the normal discharge flux in or out of the aquifer is known. That is Qn KA h f1 ( x, y, t ) n (2.49) where f 1 ( x, y, t ) is the known flux and A is the flow area. When the boundary in impervious, f 1 ( x, y, t ) 0 implying that h / n 0 . There are other types of boundaries that can be encountered, but the above two are the most common. The initial condition specifies the spatial distribution of the water table an unconfined aquifer or the piezometric head for a confined aquifer at the initial time. It is this distribution that is used to calculate the hydraulic head at subsequent times. The solutions to the flow equations in three and two dimensions in confined and unconfined aquifer can be achieved by analytical and numerical methods. The approach based on the latter is the subject of another module on Groundwater Modelling. Analytical methods are applied in this and subsequent chapters. They can be applied only for simplified flow problems, and in certain situations the above flow equations do not have to be solved directly but by prudent choice of a CV and application of continuity and Darcy law, the analytical solutions can be 2-18 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 obtained. Analytical solutions to a few unconfined flows are addressed in the remainder of this chapter. 2.4 Flow in unconfined aquifers 2.4.1 Dupuit-Forchheimer assumptions for phreatic aquifers The Darcy law coupled with the statement of mass conservation can be used to solve simple systems of either lateral or vertical groundwater flow. It is common though for systems to have both vertical and horizontal flow components. These systems can be simplified so that the flow is in only one direction before Darcy law is applied. Vertical flow components are often neglected where groundwater moves primarily in a lateral direction and the flow is then considered to be purely horizontal and also to be uniformly distributed with depth. These assumptions, collectively called the Dupuit-Forchheimer assumption are used when solving unconfined flow systems. A simple illustration of the use of the Dupuit-Forchheimer assumption is outlined in Figure 2.4 where a stream is situated in an alluvial floodplain where the water in the stream is in direct hydraulic contact with the groundwater in the floodplain. A horizontal impermeable layer occurs at a relatively small distance below the stream bed. Water is flowing from the stream into the aquifer and the water table falls away from the boundary. The flow from the stream will be vertical beneath the stream and horizontal through the banks of the stream. To solve this situation, the vertical flow components are ignored and all flow is considered to be horizontal and the flow is also considered to be evenly distributed with depth (the Dupuit-Forchheimer assumptions). Applying Darcy’s law as used above to this situation between the water table at the edge of the stream channel (site 1) and another location 300m away from the stream channel using the water table surface as the streamline yields an estimate of flow velocity of the water away from the channel. V3 10.5 10 0.005m / d 300 The Dupuit-Forchheimer assumption indicates that all of the water below the water table moves at this velocity so the calculation of the discharge rate from the 2-19 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 stream is carried out using the average height of the water table between the two sections (10.25m) as the flow thickness. If the width of the flow system is taken as 1m then the seepage from the stream per unit length of stream can be calculated as 0.005 10.25 0.05125m3 / day . If both sides of the stream experience the same conditions, the total seepage from the stream can be determined by doubling this value to 01025 . m3 / day . Figure 2.4: Seepage from a stream in an unconfined aquifer with an impermeable bottom at a relatively shallow depth The Dupuit-Forchheimer theory loses accuracy in this case if the depth to the impermeable layer increases since the vertical flow component gains increased importance. Experimentation by Bower found that the theory gives reasonably accurate seepage values if the distance of the impermeable layer below the stream was not more than twice the height of water in the stream. 2.4.2 Steady unconfined flow between two reservoirs The flow situation illustrated in Figure 2.5 shows flow through a homogeneous isotropic phreatic aquifer situated between two surface water bodies (reservoirs) of infinite extent. Flow proceeds from the higher reservoir towards the lower one. Unlike a confined aquifer situation where the piezometric surface decreases linearly between the reservoirs, in a phreatic system such as this, the difficulty arises from the fact that the water table between the two reservoirs represents a 2-20 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 flow line. The shape of the water table therefore determines the flow distribution that at the same time governs the shape of the water table. Figure 2.5: Steady flow in an unconfined aquifer between two reservoirs To obtain a solution, Dupuit assumed that (1) the hydraulic gradient is equal to the slope of the water table, and (2) the flow is horizontal and uniform everywhere in a vertical section. These assumptions, although permitting a solution to be obtained, limit the application of the results. For unidirectional flow in Figure 2.5, the discharge per unit width Q at any vertical section can be given as: Q Kh dh K dh2 constant dx 2 dx (2.50) where h is the height of the water table above an impervious base and x is the direction of flow. Integrating, Qx K 2 h c 2 (2.51) Applying the condition at x 0 where h h0 , then the Dupuit equation becomes Qx K 2 [h0 h 2 ] 2 (2.52) which indicates that the water table has a parabolic profile. For flow between two bodies of water of constant heads h0 and h L separated by a distance L as in Figure 2.5, the water table slope at the upstream boundary of the aquifer is 2-21 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 dh Q dx Kh0 But the boundary h h0 is an equipotential line because the fluid potential in a water body is constant; consequently, the water table must be horizontal at this section, which is inconsistent with eq. (2.53). In the direction of the flow, the parabolic water table described by eq. (2.52) increases in slope. By so doing, the two Dupuit assumptions, previously stated, become increasingly poor approximations to the actual flow; therefore, the actual water table deviates more and more from the computed position in the direction of flow as indicated in Figure 2.5. The fact that the actual water table lies above the computed one can be explained by the fact that the Dupuit flows are all assumed horizontal, whereas the actual velocities of the same magnitude have a downward vertical component so that a greater saturated thickness is required for the same discharge. At the downstream boundary a discontinuity in flow forms because no consistent flow pattern can connect a water table directly to a downstream free-water surface. The water table actually approaches the boundary tangentially above the water body surface and forms a seepage face. Consequently, the water table does not follow the parabolic form but for shallow slopes it closely predicts the water table position except near to the outflow point. The above discrepancies indicate that the water table does not follow the parabolic form of eq. (2.52), nevertheless, for flat slopes, where the sine and tangent are nearly equal, it closely predicts the water table position except near the outflow. The equation, however, accurately determines Q or K for given boundary heads. 2.4.3 Example of unconfined flow An unconfined aquifer has a hydraulic conductivity of 1.8m/day and a porosity of 0.3. The aquifer is a bed of sand of uniform thickness of 35m. Two observation wells A and B, separated by a distance of 200m, indicate water levels of 22.5m and 24m below the land surface, respectively. Determine (a) the steady discharge per unit width, (b) the average pore velocity, and (c) the water table elevation midway between the two wells. 2-22 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 Solution . m/ d h0 35 22.5 12.5m , L 200m , hL 35 24 11m , K 18 (a) The steady discharge is Q K (h02 hL 2 ) 18 . (12.52 112 ) 0157 . m3 / d / m width of aquifer. 2L 2 200 (b) The average pore velocity is V Q nA The flow area can be taken as the average flow area between sections A . m2 / m width of aquifer. The average and B. That is A (12.5 11) / 2 1175 . / (1175 . 0.3) 0.0134m / d . pore velocity is found to be V 0157 (c) The water table midway is found by making use of eq. (2.52). That is hmidway 2 h02 hmidway 2Qx 2 0157 . 100 12.52 138.8m2 K 18 . 138.8m 118 .m 2.4.4 Steady unconfined flow between two reservoirs with uniform recharge This particular unconfined flow is illustrated in Figure 2.6a. Examining continuity of flow in the CV shown in Figure 2.6b of length dx , inflow = Q Wdx and outflow = Q dQ . Under steady conditions, inflow equals outflow, and as such dQ Wdx or dQ W dx (2.54) By Darcy law: Q Khdh / dx K / 2dh2 / dx , so that eq. (2.54) now becomes d 2h2 2W 2 dx K (2.55) Integrating eq. (2.55) twice yields h2 W 2 x c1 x c2 K (2.56) 2-23 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 where c1 and c2 are constants of integration. These constants are evaluated from the conditions of the water table elevation at the two reservoirs, that is h h0 at x 0 and h hL at x L . Implementing these conditions yield c2 h02 and c1 hL 2 h0 2 W L L K (2.57) Introducing these constants into eq. (2.56) yields W hL 2 h0 2 h x( L x) x h02 K L 2 (2.58) (a) (b) Figure 2.6: (a) Steady unconfined flow with vertical recharge; (b) Control volume used for analysis. 2-24 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 A water divide could be encountered at which flow occurs from it in either direction to the reservoirs and the height of the water table is a maximum. The location of the divide is found from the fact that the flux is zero, that is dh / dx 0 at x x D . Differentiating the water table profile given by eq. (2.58) with respect to x yields the expression for the location of the divide L K (h0 2 hL 2 ) xD 2 W 2L (2.59) The flow to the two rivers or reservoirs which represents the base flow is estimated by evaluating the Darcy flux at x 0 and x L . The Darcy flux is Q K dh2 1 h 2 h02 W ( L 2 x ) L 2 dx 2 KL (2.60) The base flow to the river at x 0 is 1 hL 2 h02 Q0 WL 2 KL (2.61) while the base flow to the river at x L is 1 hL 2 h02 QL WL 2 KL (2.62) When the water levels in the two reservoirs are the same with elevation h0 , the above equations are simplified. The water table distribution of eq. (2.58) becomes h2 W x ( L x ) h0 2 K (2.63) The water table profile is known as the ellipse equation because the predicted water table profile is a section of an ellipse. The water divide, at which the water table elevation is highest, occurs midway between the reservoirs and the maximum elevation of the water table is obtained by substituting x L / 2 into eq. (2.63) to obtain hmax 2 WL2 h02 4K (2.64) Equations (2.63) and (2.64) are useful for analyzing flow toward subsurface agricultural drains in humid regions where W is approximately constant (see Figure 2.7). In particular, eq. (2.64) can be used to estimate the drain spacing L required to ensure that hmax does not exceed a value that has been determined to 2-25 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 be satisfactory for crop growth. In this case, the rivers or reservoirs are considered as part of a system of ridges and drains in a farm. 2.4.5 Example of unconfined flow with uniform recharge A long canal was constructed running parallel to a river 1.5km away. Both fully penetrate a sand aquifer with hydraulic conductivity of 1.2m/d. The area is subject to a rainfall of 1.8m/year and evaporation of 1.3m/year. The elevation of water in the river is 33m and in the canal it is 28m. Determine (a) the water divide (where specific discharge is zero), (b) the maximum water table elevation, (c) the steady discharge per kilometre into the canal, and (d) the steady discharge per kilometre into the river Solution . m / d , W 18 . 13 . 0.5m / yr 137 . 103 m / d h0 33m , L 1500m , hL 28m , K 12 (a) The water divide is given by eq. (2.59) xD L K (h02 hL 2 ) 12 . (332 282 ) 750 661m 2 W 2L 0.00137 2 1500 (b) The maximum water table elevation is found from substituting x D 661m for x in eq. (2.58) hmax 2 0.00137 282 332 661 (1500 661) 661 332 1587.74m2 12 . 1500 hmax 1587.74 39.85m (c) The steady discharge per kilometre into the canal is obtained from eq. (2.62) 1 282 332 . 2.055 0169 0.00137 1500 103 m3 / d / km 2 12 . 1500 2 Qcanal Qcanal 111 . 103 m3 / d / km (d) The steady discharge per kilometre into the canal is obtained from eq. (2.61) Qriver 1 282 332 . 2.055 0169 0.00137 1500 103 m3 / d / km 2 12 . 1500 2 Qriver 0.943 103 m3 / d / km 2-26 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 2.5 Flow through stratified phreatic aquifers The previous solutions have all assumed that groundwater flow is in a homogeneous and isotropic aquifer. In reality, such aquifers rarely exist and the hydraulic conductivity is not the same in all directions. This is particularly so for unconsolidated alluvial aquifers where anisotropy tends to be more common. Variations in hydraulic conductivity are caused by the fact that where alluvial material is deposited under water, the particles are rarely spherical and so when the particles settle out they tend to come to rest with their flat sides down. This tends to have the effect of leading to horizontal hydraulic conductivities being greater than the vertical hydraulic conductivities. Another cause for anisotropy is that alluvium typically consists of layers of different materials, each of which possesses a unique value for hydraulic conductivity. If these layers are horizontal, then any single layer with a relatively low hydraulic conductivity (e.g. clay) retards vertical flow through the entire alluvial bed. Horizontal flow, however, can occur easily through any layer having a relatively high hydraulic conductivity. In a typical field condition with alluvial deposits it is therefore common to find a hydraulic conductivity in the horizontal direction K x that is greater than the vertical hydraulic conductivity K y . Ratios of Kx / K y usually fall in the range of 2 to 10 in alluvial materials but values of up to 100 or more may occur where clay layers are present. For consolidated geological materials, anisotropic conditions are governed by the orientation of strata, fractures, joints, solution openings or other structural conditions. These do not necessarily possess a horizontal alignment. Figure 2.7: System of ridges and drains in a farm receiving uniform recharge 2-27 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 2.5.1 Flow in a Horizontally stratified Phreatic Aquifer The case of flow in a horizontally stratified aquifer between two reservoirs at water elevations h0 and hL is shown in Figure 2.8. The flow in the lower layer is Q1 K1b dh dx (2.65) and that in the upper layer is given as Q2 K2 (h b) dh dx (2.66) The total flow per unit length of the aquifer at any section is: Q Q1 Q2 constant (steady flow). By carry the integration with respect to x and implementing the condition that at x 0 h h0 , yields Qx K1b(h0 h) K2 [ h02 h2 b(h0 h)] 2 (2.67) Applying the condition that at x L h hL , then the Dupuit discharge is given as Q K1b (h0 h) K2 [(h0 b) 2 (hL b) 2 ] L 2L (2.68) The expression for the discharge indicates that the steady discharge is as a combination of confined flow in the lower stratum and unconfined flow in the upper one. Figure 2.8: Flow in a Horizontally stratified Phreatic Aquifer 2-28 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 2.5.2 Flow in a Vertically stratified Phreatic Aquifer Using a vertically stratified unconfined formation with two strata that lies between two rivers or reservoirs at water levels h0 and hL (Figure 2.9), the steady discharge is estimated using Dupuit assumptions that the flow is essentially In the first region 0 x L1 , the expression for the water table is horizontal. similar to eq. (2.52). That is 2Qx K1 h 2 h02 (2.69) Inserting the condition that at x L1 , h h1 , though still to be evaluated, into eq. (2.69) yields 2QL1 K1 h12 h02 (2.70) For the second region L1 x L , the expression for the water table is similarly written h 2 h12 2Q( x L1 ) K2 (2.71) Inserting the condition that at x L , h hL into eq. (2.71) yields hL 2 h12 2QL2 K2 (2.72) Substituting eq. (2.70) into (2.72) yields L L hL 2 h02 2Q 1 2 K1 K2 or Q h02 hL 2 L L 2 1 2 K1 K2 (2.73) 2-29 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 Figure 2.9: Flow in a Vertically stratified Phreatic Aquifer The expression for the discharge is accurate, but not for the water table distribution that are given by eqs. (2.69) and (2.71) due the failure of the Dupuit approximation to account for the seepage face. The expression for the discharge can be extended to the case when there are N strata as h02 hL 2 Q N L 2 i i 1 Ki 2.6 (2.74) Exercises 1. Water flows between a vertical sand column of 120cm length and 200cm 2 area. The difference between water levels at the inflow and outflow reservoirs is 30cm. If the porosity of the sand is 0.36 and its hydraulic conductivity is 20cm/d, determine (a) the total discharge through the column, (b) the specific discharge, (c) the average flow velocity, and (d) the hydraulic gradient along the column. 2. Define the storativity for confined and unconfined aquifers. A volume of water of 40 × 106m3 has been pumped from a phreatic aquifer through wells that are more or less, uniformly distributed over an area of 100km 2 of 2-30 Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2 the aquifer. If the aquifer’s specific yield is 0.2, determine the average drawdown of the water table over the area. 3. A long canal was constructed running parallel to a river 2.5km away. Both fully penetrate a sand aquifer with hydraulic conductivity of 3.1m/d. The area is subject to a rainfall of 2.6m/year and evaporation of 0.9m/year. The elevation of water in the river is 45m and in the canal it is 40m. Determine (a) the water divide (where specific discharge is zero), (b) the maximum water table elevation, (c) the steady discharge per km length of canal into the canal, and (d) the steady discharge per km length of river into the river. The following standard groundwater textbooks have been consulted in the creation of this chapter. They all cover in varying degrees of detail the material covered in this chapter. Students are recommended to consult these for further information and clarification of the course content. In addition, students are encouraged to make use of the Internet where a wide variety of resources can be found. 2.7 References 1. Bear, J (1979) Hydraulics of Groundwater, McGraw-Hill Book Company, New York. 2. Bowen, R (1986) Groundwater, Elsevier Applied Science Publishers, London. 3. Bower, H (1978) Groundwater hydrology, McGraw-Hill, Tokyo. 4. Driscoll, F.G (1986)Groundwater and Wells, Johnson Screens, St Paul. 5. Fetter, C.W (1994) Applied hydrogeology, Prentice Hall, Englewood Cliffs. 6. Freeze, R. Allan and Cherry, J.C (1979) Groundwater, Prentice–Hall, Inc., New Jersey. 7. Todd, D.K (1980) Groundwater Hydrology, 2ed. John Wiley, New York. 8. Meinzer, O.E (1949) The Physics of the Earth IX, Hydrology, Dover Publications, New York. 2-31