Groundwater Flow Equations in Saturated Porous Media PDF

# CHAPITRE II: EQUATIONS DES ÉCOULEMENTS DANS LES MILIEUX POREUX ## II.1 Équation générale de l'écoulement en milieu saturé The equation of groundwater flow is based on the combination of the law of conservation of mass of water in motion, and Darcy's law (1856), which incorporates the properties...

# CHAPITRE II: EQUATIONS DES ÉCOULEMENTS DANS LES MILIEUX POREUX ## II.1 Équation générale de l'écoulement en milieu saturé The equation of groundwater flow is based on the combination of the law of conservation of mass of water in motion, and Darcy's law (1856), which incorporates the properties of the medium. ## Loi de Darcy : The law of Darcy is: $Q = K\frac{A}{L}(h_1 - h_2)$ Where: * Q: the flow rate * K: hydraulic conductivity (Darcy's permeability) * A: the surface area of the cross-section * q: volume per unit surface unit of time * $h_1$ and $h_2$: piezometric levels respectively in two points 1 and 2 * L: Distance between two points 1 and 2. ## Figure 1: Schéma descriptif de la loi de Darcy Figure 1 describes the law of Darcy using a diagram. ## Figure 2: Volume élémentaire d'un milieu poreux montrant les volume transitant à travers les différentes faces Figure 2 shows a small volume of a porous medium showcasing the volume passing through different faces. ## Écoulements souterrains en milieu poreux saturé en conditions de flux stationnaires (régime permanent) In a steady state, a hydraulic system is in equilibrium and there is no variation in head over time. If the fluid is incompressible, p is constant and the equation of continuity for a small volume is determined by considering the variations of mass in a small volume element within a porous medium in a time interval (At). ## Loi de conservation de masse (principe de continuité) The law of conservation of mass applies only in a fixed volume where the change in fluid volume over a period is equal to the algebraic sum of the mass fluxes crossing the volume considered. The equation for conservation of mass is: $\frac{d(pq_x)}{dx} + \frac{d(pq_y)}{dy} + \frac{d(pq_z)}{dz} = n\frac{dp}{dt} + \rho \frac{du}{dt}$ Where: * $\rho$: density of the fluid * $q_x$, $q_y$, $q_z$: are the flow rates of the fluid in the x, y, and z directions respectively * $u$: is the porosity of the medium * n: is the coefficient of compressibility of the medium With the assumption that the fluid is incompressible, the above equation reduces to $\frac{d(pq_x)}{dx} + \frac{d(pq_y)}{dy} + \frac{d(pq_z)}{dz} = 0$ ## Équations for groundwater flow in saturated porous media in steady state Equation 1, 2, and 3 are for porous media in a steady state condition. $q_X = -K\frac{dh}{dx}$ (Eq 1). $q_Y = -K\frac{dh}{dy}$ (Eq 2). $q_Z = -K\frac{dh}{dz}$ (Eq 3). For an isotropic and homogeneous porous medium, the permeability (hydraulic conductivity) K_X=K_Y = K_Z and K(x, y, z) = constant, if the flow is permanent (no change of h over time) we get: $\frac{d^2h}{dx^2} + \frac{d^2h}{dy^2}+\frac{d^2h}{dz^2} = 0$ (Diffusion equation/ Laplace equation) (Eq. 8) This three-dimensional equation is general, provided that the aquifer is homogeneous and isotropic, however, it is a simplification for confined aquifers. ## Écoulements souterrains en milieu poreux saturé en régime transitoire If the fluid is compressible, the flow rate is a function of time, and the porous medium is also deformable, the demonstration is more complex. In this general case, the continuity equation for the porous medium is written as: $\frac{d(pq_x)}{dx} + \frac{d(pq_y)}{dy} + \frac{d(pq_z)}{dz} = n\frac{dp}{dt} + \rho \frac{du}{dt}$ (Eq. 9) The first term on the right hand side of Equation 9 describes the mass flux produced due to the expansion of the fluid due to a change in density. This is controlled by the compressibility of the fluid. The second term is the mass flux produced due to the compaction of the porous medium, influenced by the variation in the porosity, n, and is determined by the coefficient of compressibility of the aquifer (a). Changes in the density of the fluid and the porosity of the formation are both produced by a change in the hydraulic head. The volume of water produced by these two mechanisms for a unit decline of the hydraulic head is the specific storage, "Ss". Therefore, the rate of change of the storage of the fluid mass inside the control volume is given by the following equation: $ pS_s\frac{dh}{dt} $ (Eq 10) where: $S_s$: specific storage coefficient of the aquifer. Therefore, Equation 9 becomes: $\frac{d(pq_x)}{dx} + \frac{d(pq_y)}{dy} + \frac{d(pq_z)}{dz} = pS_s\frac{dh}{dt}$ (Eq 11) By applying Darcy's Law (Eq 1, 2 and 3), we get: $\frac{d}{dx}(K_x \frac{dh}{dx}) + \frac{d}{dy}(K_y\frac{dh}{dy}) + \frac{d}{dz}(K_z\frac{dh}{dz}) = S_S\frac{dh}{dt} $ (Eq. 12) If the porous medium is isotropic and homogeneous, then Equation 12 simplifies to: $\frac{d^2h}{dx^2} + \frac{d^2h}{dy^2}+\frac{d^2h}{dz^2} = \frac{S_S}{K}\frac{dh}{dt}$ (Eq. 13) The specific storage is given by: $S_s = pg\alpha + n\beta$ (Eq. 14) where: * $S_s$: is the coefficient of specific storage ($m^{-1}$) * $p$: is the density of the fluid (water) ($kg/m^3$) * $g$: is the acceleration due to gravity ($m/s^2$) * $\alpha$: is the coefficient of specific compressibility of the porous medium, defined by: $\alpha = \frac{1}{V} \frac{dV}{d\sigma}$ where: * $V$: total volume of the porous medium * $dV$: the change in volume of the porous medium due to a change in porosity * $\sigma$: effective stress * $n$: is the porosity of the porous medium * $\beta$: is the coefficient of compressibility of the fluid given by: Substituting Equation 14 into equation 13, we obtain: $\frac{d^2h}{dx^2} + \frac{d^2h}{dy^2}+\frac{d^2h}{dz^2} = \frac{pg(\alpha + n\beta)}{K}\frac{dh}{dt} $ (Eq. 15) ## Equations for transient flow in porous media - Equations 11, 12, 13, and 14 are transient flow equations for saturated porous media. Equation 12 is for anisotropic medium, while Equations 13 and 14 are for homogeneous and isotropic media. The solution h(x, y, z, t) (hydraulic head or piezometric head) describes the value of the piezometric head at all points in a three-dimensional flow field and at any time. A solution requires knowledge of the three hydrogeologic parameters K, $\alpha$, and $n$, and those of the fluid, $\rho$ and $\beta$. A simplification is obtained in the specific case of a confined horizontal aquifer, where h(x, y, t) is constant for any value of z. The flow is then two-dimensional. ## II. 2. Écoulement dans les milieux poreux stratifiés ## II.2.1 Coefficient de perméabilité moyenne verticale Vertical flow within such a porous medium is considered a flow in series, while horizontal flow is considered a flow in parallel. The equivalent hydraulic conductivity (Darcy's permeability) of such layered porous media can be easily determined. - The flow rate in each layer is equal to the total flow rate. - The total head loss is equal to the sum of the individual head losses in each layer The head drop in the two-layered system is given by the equation below, and the hydraulic conductivity is obtained by substituting the known values: $Q_1 = K_1.i_1.L_1.1$ (Eq. 17) $Q_2 = K_2.i_2.L_2.1$ (Eq. 18) The continuity equation is : $Q=Q_1+Q_2$ (Eq. 19) The flow rate through the two-layered system is: $Q = K_v (\frac{h}{L_1 + L_2})$ (Eq. 20) As there is no head loss, we get: $Q=Q_1+Q_2$ (Eq. 21) The total head drop is: $h = \frac{L_1.Q_1}{K_1} = \frac{L_2.Q_2}{K_2} = \frac{(L_1 + L_2).Q}{K_v}$ (Eq. 22) Therefore, the equivalent vertical permeability is: $K_v = \frac{(L_1 + L_2)Q}{L_1.Q_1} = \frac{L_1 + L_2}{L_1/K_1 + L_2/K_2} = \frac{\sum\limits_{i=1}^{n} L_i}{\sum\limits_{i=1}^{n} L_i/K_i}$ (Eq 23-25) ## Figure 3: Schéma d'un écoulement en série (Perméabilité moyenne verticale) Figure 3 illustrates the concept of vertical flow in a layered medium (layered permeability). ## II. 2.2 Coefficient de perméabilité moyenne horizontale The total flow rate is equal to the sum of the flow rate in each layer. The total head loss is equal to the head loss in each layer. This allows us to express the equivalent hydraulic conductivity as: $Q= Q_1 +Q_2$ (Eq. 26) $Q_1 = V.S = K_1.i.L_1.1$ (Eq. 27) $Q_2 = K_2.i_2.L_2.1$ (Eq. 28) For a horizontal flow with a constant gradient, this equation becomes: $Q = K_1.i.L_1.1 + K_2.i_2.L_2.1$ (Eq. 30) $Q = i(K.L_1 + K_2.L_2).1$ (Eq. 31) Therefore, the equivalent horizontal permeability is: $Q = K_H.i (L_1 + L_2).1$ (Eq. 32) $K_H = \frac{K_1.L_1 + K_2.L_2}{L_1 + L_2} $ (Eq. 33) and for an arbitrary number of layers, this equation becomes: $K_H = \frac{\sum\limits_{i=1}^{n} K_i.L_i}{\sum\limits_{i=1}^{n}L_i}$ (Eq. 34) ## Figure 4: Schéma d'un écoulement en parallèle (Perméabilité moyenne horizontale) Figure 4 illustrates the concept of horizontal flow in a layered medium (layered permeability). ## II.3. Solution pour l'écoulement radial en régime permanent ## II.3.1. Pompage en régime permanent - Méthode de Dupuit ## II.3.1.1. Cas d'une nappe libre Consider a point M on the free surface with coordinates x and z. If we denote the curvilinear abscissa along the free surface by s, the hydraulic gradient at M is dz/ds and the discharge is tangent to the free surface. Dupuit's hypothesis assumes that the free surface has a small slope. The lines of flow are straight and horizontal in this case. The hydraulic gradient in M is given by: $I_M = \frac{dz}{ds} $ (Eq. 35) The Darcy velocity is: $v = K \frac{dz}{ds} = KI$M (Eq. 36) ## Figure 1: Puits dans une nappe libre Figure 1 shows a well in an unconfined aquifer with an impermeable substratum. The flow rate is: $Q = SxV \approx SxVx$ (Eq. 37) Where: * S: is the surface area through which the flow rate is calculated. * V: is the Darcy velocity. $S = 2\pi.x.z$ (Eq. 38) The flow rate is: $Q = 2\pi.x.z.K\frac{dz}{dx}$ (Eq. 39) <start_of_image> Schematic of a well in an unconfined aquifer with an impermeable substratum. Now integrating the flow rate equation: $\int_{r}^{R}\frac{Q}{2\pi.K.z}dx = \int_{h}^{H}dz $ (Eq. 40) The integration gives: $Qln \frac{R}{r} = \pi.K(H^2 - h^2)$ (Eq. 41) $Q = \pi K\frac{H^2 - h²}{ln\frac{R}{r}}$ (Eq. 42) ## Figure 2: Puits dans une nappe libre (approche de Dupuit) Figure 2 depicts the free surface of an unconfined aquifer in the Dupuit approach. ## II. 3. 1. 2. Cas d'une nappe captive The flow rate is calculated at a constant head. We can then rewrite the equation for the flow rate as: $Q = SxV \approx SxVx $ (Eq. 43) Where: * S: is the surface area through which the flow rate is calculated. * V: is the Darcy velocity. The surface area S is: $S = 2\pi xxe$ (Eq. 44). The Darcy velocity is: $V = K\frac{dz}{dx}$ (Eq. 45). We can then express the flow rate as: $Q = 2\pi xeK \frac{dz}{dx}$ (Eq. 46) Integrating the flow rate equation between the two radii, we obtain: $\int_{r}^{R}\frac{Q}{2\pi xeK} dx = \int_{h}^{H} dz$ (Eq. 47) $\frac{R}{r}Q = 2\pi eK (H-h)$ (Eq. 48) The flow rate is finally written as: $Q = 2\pi eK\frac{(H-h)}{ln\frac{R}{r}}$ (Eq. 49) ## Figure 3: Puits dans une nappe captive Figure 3 shows a well in a confined aquifer with an impermeable substratum. ## Remarks: * Determining the radius of influence R requires different approaches. The simplest method is when the piezometric surface level is measured during pumping. Alternatively, empirical formulas provide an estimate of R. * The empirical formula of Sichardt : $R = 3000(H-h)\sqrt{K}$ (Eq. 50) * Determining the radius of influence R during transient conditions requires knowledge of the discharge rate, transmissivity, and the time elapsed since pumping started. The following formula can be used to evaluate this: $R = 1.5 \frac{KxH\Delta t}{n}$ (Eq. 51) Where: * $K$: is the hydraulic conductivity ($m/s$) * $t$: is the duration of transient flow (seconds) ## Equations for the potential function of the free surface in unconfined aquifers The product KH is called transmissivity and is denoted by T. We can write the equation of the free surface as: $Q = 2\pi.x.z.K\frac{dz}{dx}$ (Eq. 52) Integrating this equation gives the equation of the streamline: $\int_{r}^{x}\frac{Qdx}{\pi.K.x} = \int_{h}^{z}2z.dz$ (Eq. 53) $\int_{r}^{x}\frac{Q}{\pi.K.x}dx = z^2-h^2$ (Eq. 54) $\frac{Q}{\pi.K}ln\frac{x}{r} = z^2 - h^2$ (Eq. 55) $\frac{Q}{\pi.K}ln\frac{x}{r} = z^2 - h^2$ (Eq. 56) $z^2 = h^2 + \frac{Q}{\pi K}ln\frac{x}{r}$ (Eq. 57) This method allows us to calculate the radius of influence R and the potential function of the free surface by knowing the flow rate, the transmissivity of the aquifer, and the initial and final piezometric heads. Here, Q is the flow rate of the well, K is the hydraulic conductivity of the aquifer, h is the initial piezometric head, and R is the radius of influence of the pumping well. This concludes the analysis and equations of flow in porous media presented in the document.