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In hydrogeology or soil mechanics, the flownet is an important tool for solving two-dimensional irrotational flow problems in the presence of hydraulic structures like dams or sheet pile walls. Initially, the construction of the flownet evolved as a graphical method considering some physical constraints of representing the flow lines and the equipotential (piezometric) lines. Later analytical functions of complex variables, satisfying steady-state flow conditions by following the Laplace equation, were used to find some closed-form solutions for flow domains with simple boundary conditions. However, the process becomes extremely complicated when the flow domain is irregular in shape or unknown altogether. In this scenario, methods of conformal mapping can be used successfully to transform an unconfined flow domain (through an earth dam) or confined flow domain (underneath a concrete dam) into a simple rectangular domain or a half-space plane where the closed-form solutions can be obtained easily. Further, the solutions can be mapped inversely to the original flow domain. The Laplace invariant property of conformal mapping ensures the correctness of the solutions after inverse mapping from the simple domain to the original domain.
Earth Dam
Let us define two spatial coordinates, viz., a velocity potential function $\varphi(x,y)$ and a stream function $\psi(x,y)$. $\varphi(x,y)$ and $\psi(x,y)$ are chosen such that they satisfy the governing Laplace equation of the steady-state flow, i.e., $\nabla^2 \varphi(x,y) = 0$ and $\nabla^2 \psi(x,y) = 0$. The two functions $\varphi(x,y)$ and $\psi(x,y)$ are conjugate harmonic (i.e., they follow the Cauchy-Riemann equation), meaning that the two families of curves given by $\varphi(x,y) = \mathrm{constant}$ and $\psi(x,y) = \mathrm{constant}$ represent mutually perpendicular trajectories. Let us consider two complex planes $z$ and $w$, respectively, such that $$z=x+\mathrm{i}y\quad\mathrm{and}\quad w=\varphi+\mathrm{i}\psi\quad\mathrm{with\quad i}=\sqrt{-1} \tag{1}$$ Now, the Kozeny's solution for flow through the earth dams, as shown in Figure 1(b) can be obtained using the following simple transformation $$z = Cw^2\quad\Rightarrow x+iy = C(\varphi^2+i2\varphi\psi-\psi^2) \tag{2}$$$$\therefore x = C(\varphi^2-\psi^2) \quad \mathrm{and} \quad y = 2C\varphi\psi\tag{3}$$ where, $C$ is a constant, depending on the boundary conditions of the problem. Now, along the free surface CDE (commonly known as the phreatic line or line of seepage), $\psi=q$ and $\varphi=-ky$, where $k$ is the coefficient of hydraulic conductivity (or the coefficient of conductivity) of the soil. Therefore, Equation (3) yields $$C=-\dfrac{1}{2kq}\tag{4}$$ The equation for the line of seepage, referred to as Kozeny's basic parabola, can be obtained as $$x = -\dfrac{1}{2kq}(\varphi^2-\psi^2) = -\dfrac{1}{2kq}(k^2y^2-q^2) = \dfrac{1}{2} \left(\dfrac{q}{k}-\dfrac{k}{q}y^2 \right) \tag{5}$$ It should be noted that the real upstream slope of the dam or face BCH (as shown in Figure 1(a)) is modified as a parabola, face B'C'H' in the $z$-plane (as shown in Figure 1(b)). This is in anticipation of the shape of the obtained flownets in the $z$-plane (refer to Figure 2), and correction should be made to account for this discrepancy. The focal distance $x_0$ can be estimated by putting $y=0$ at equation 5 as $$x_0=\dfrac{q}{2k}\quad\Rightarrow q=2kx_0 \tag{6}$$ Therefore, Kozeny's basic parabola can be expressed as $$x = \dfrac{1}{2}\left( 2x_0-\dfrac{y^2}{2x_0}\right) = x_0-\dfrac{y^2}{4x_0} \tag{7}$$ Here, the value of $x_0$ can be estimated by considering the coordinate of the point $C (x_C,y_C)$ from Figure 1(a) as $$x_0 = 0.5 \left( x_C+\sqrt{x_C^2+y_C^2} \right); \quad x_C=h_w; \quad y_C=-b_d+f+0.5h_w(b_d-t_d)/h_d \tag{8}$$ Finally, the complex transformation can be expressed by using Equations (4) and (6) as $$z=f(w)=-\dfrac{w^2}{4k^2x_0}\tag{9}$$
Figure 1: (a) realistic geometric shape of an earth dam in $z$-plane, (b) modified geometric shape of an earth dam in $z$-plane, (c) boundary conditions of flownets in $w$-plane
Example
Consider an earth dam with a height equal to $20\ \mathrm{m}$, a top width equal to $12\ \mathrm{m}$, a bottom width equal to $32\ \mathrm{m}$, and the filter width equal to $6\ \mathrm{m}$ as shown in Figure 2. The height of the water level is $19\ \mathrm{m}$, and the coefficient of permeability of the homogenous isotropic soil is $3.5\times 10^{-8}\ \mathrm{m/s}$. A simple MATLAB script to generate the flownets (combinations of flowlines and equipotential lines) for the considered earth bank is provided in this repository, and the output of the code is shown in Figure 2.
Figure 2: flownets for a homogeneous isotropic earth dam
Flat-bottomed concrete gravity dam
In this section, the realistic boundary conditions for flownets underneath a flat-bottomed concrete gravity dam (see Figure 3(a)) are identified in the physical domain or $z$-plane, and then accordingly considered in the $w$-plane as shown in Figure 3(b). Thereafter the analytical expressions for the flownets will be solved in the $w$-plane, and then a transformation scheme will be used to map the solutions onto the physical plane. For this problem, the complex velocity $W$, can be expressed as $$W=\dfrac{dw}{dz}=\dfrac{\partial \varphi}{\partial x}+\mathrm{i}\dfrac{\partial \psi}{\partial x} = v_x-\mathrm{i}v_y \tag{10}$$ Further, $W$ can be expressed in terms of $z$ to obtain the transformation function as $$W = v_x-\mathrm{i}v_y = \dfrac{M}{\sqrt{b_1^2-z^2}}\tag{11}$$ where $M$ is the real constant and $2b_1$ is the bottom width of the dam (refer to Figure 3(a)). The main reason for choosing the function is that it follows the velocity conditions at the boundaries in the physical domain. According to Equation (11), $W$ is real along line $BC$, and imaginary along lines $A_{\infty}B$ and $CD_{\infty}$. This is consistent with the flow velocities being horizontal along $BC$ (i.e. $v_x\neq 0,v_y=0$) and vertical along $A_{\infty}B$ and $CD_{\infty}$ (i.e. $v_x=0,v_y\neq 0$). Now, Equations (10) and (11) yield $$dw = Wdz \quad \Rightarrow w = \int \dfrac{M}{\sqrt{b_1^2-z^2}} \ dz \qquad \therefore w = \varphi +\mathrm{i}\psi = M\sin^{-1} (z/b_1)+N\tag{12}$$ where $N$ is the constant of integration. The following boundary conditions are used to find the constants $M$ and $N$. $$\text{At point B:} \quad \psi = 0, \ \varphi = -k(h_1-h_2)=-kh, \ w=-kh, \ z = -b_1; \qquad \Rightarrow -\frac{M\pi}{2}+N=-kh$$$$\text{At point C:} \quad \psi = 0, \ \varphi = 0, \ w=0, \ z = b_1; \qquad \Rightarrow \frac{M\pi}{2}+N=0$$ Therefore, $M=kh/\pi$ and $N=-kh/2$, and the relations between $z$- and $w$-planes can be expressed as $$w = \frac{kh}{\pi}\sin^{-1}\left(\frac{z}{b_1}\right)-\frac{kh}{2}=\frac{kh}{\pi}\left[ \frac{\pi}{2}-\cos^{-1}\left(\frac{z}{b_1}\right)\right]-\frac{kh}{2} = -\frac{kh}{\pi}\cos^{-1}\left(\frac{z}{b_1}\right) \tag{13}$$$$\Rightarrow z = f(w) = b_1\cos(\pi w/kh) \tag{14}$$
Figure 3: (a) geometry of a flat-bottomed concrete gravity dam in $z$-plane, (b) boundary conditions of flownets in $w$-plane
Example
Consider a flat-bottomed concrete gravity dam with geometrical parameters as $a_1=20\ \mathrm{m}$, $a_2=18\ \mathrm{m}$, $a_3=4.5\ \mathrm{m}$, $2b_1=18\ \mathrm{m}$ and $b_2=1.5\ \mathrm{m}$, as shown in Figure 3(a). The heights of the water level at the upstream side and downside side are $h_1=15\ \mathrm{m}$ and $h_2=2\ \mathrm{m}$, respectively. The coefficient of permeability of the homogenous isotropic soil is $3.5\times 10^{-8}\ \mathrm{m/s}$. A simple MATLAB script to generate the flownets for the flat-bottomed concrete gravity dam is provided in this repository, and the output of the code is shown in Figure 4.
Figure 4: flownets underneath the flat-bottomed concrete gravity dam
Flat-bottomed concrete gravity dam with sheet pile
Most often, a hydraulic structure, e.g., concrete gravity dams founded on permeable soil layer(s), are provided with cut-offs (such as sheet piles) to decrease the seepage forces and the uplift forces resulting from the water flowing underneath the structure. In these cases, the complicated flownets can not be transformed from $w$-plane onto the physical plane directly by using some simple mathematical expressions of comics (i.e., curves developed at the intersection of a straight plane and a cone) as shown in the previous cases. To solve such problems, an additional auxiliary complex variable space $t$-plane can be used. The upper (or the lower) half of the auxiliary plan could be utilized to be mapped onto the $z$ and $w$-planes by the following transformations $$z = f_1(t) \quad \text{and} \quad w = f_2(t) \tag{15}$$$$\therefore z = f_1(f_2^{-1}(w)) = f(w) \tag{16}$$ Schwarz-Christoffel transformation technique is used to find the conformal mapping (i.e. the functions $f_1(t)$ and $f_2(t)$) of the dam profile in the $z$-plane and the complex seepage potential in the $w$ plane from the lower (or the upper) half space of $t$-plane, and finally the solved analytical flownets in the $w$-plane can be transferred onto the physical plane by the function $z=f(w)$.
Inclined sheet pile
Abbas Z. Ijam provided the solution technique to find the flownets underneath a flat-bottomed concrete gravity dam with an inclined sheet pile (as shown in Figure 5(a)). The inclination of the pile with the vertical axis is represented by $\theta$, and therefore, the parameter $gamma$ can be expressed as $\gamma=\dfrac{1}{2}-\dfrac{\pi\theta}{180}$.
Transformation from $t$-plane onto $z$-plane: points $C$ and $E$ are placed at $-1$ and $+1$, whereas the points $B$ and $D$ are placed at $t_B$ and $a$, respectively, in the $t$-plane (see Figure 5(b)). The required mapping can be expressed by the following Schwarz-Christoffel transformation as $$z = M\int (1+t)^{-\gamma}(t-a)(1-t)^{\gamma-1} \ dt+N \qquad \Rightarrow \frac{dz}{dt} = M(1+t)^{-\gamma}(t-a)(1-t)^{\gamma-1} \tag{17}$$ where $M$ and $N$ are constants. Since the domain in the $z$-plane is comprised of a radial slit, the mapping of the flow domain can be expressed as $$z = c(1+t)^{1-\gamma}(1-t)^{\gamma}\tag{18}$$$$\Rightarrow \dfrac{dz}{dt} = -c(1+t)^{-\gamma}(1-t)^{\gamma-1}(t+2\gamma-1)\tag{19}$$ where $c$ is a constant. Equations (17) and (19) yield $M=-c$ and $a=1-2\gamma$. Further, the constant term $c$ can be obtained by using $t=a$ and $z=s\exp(-\mathrm{i}\gamma\pi)$ in the case of Point D in Equation (18) as $$c = \dfrac{s\exp(-\mathrm{i}\gamma\pi)}{(1+a)^{1-\gamma}(1-a)^{\gamma}} \tag{20}$$ where $s$ is the length of the sheet pile (refer to Figure 5(a)). Therefore, the complex function representing the mapping of the $t$-plane onto the $z$-plane is obtained as $$z = f_1(t) = s\exp(-\mathrm{i}\gamma\pi)\left(\dfrac{1+t}{1+a}\right)^{1-\gamma}\left(\dfrac{1-t}{1-a}\right)^{\gamma} \tag{21}$$ Using $t=t_B$ and $z=-b_1$ in the above equation for point B, $$-b_1 = s\exp(-\mathrm{i}\gamma\pi)\left(\dfrac{1+t_B}{1+a}\right)^{1-\gamma}\left(\dfrac{1-t_B}{1-a}\right)^{\gamma}\qquad\Rightarrow -\frac{b_1}{s} = \left\lbrace\dfrac{t_B+1}{2(1-\gamma)}\right\rbrace^{1-\gamma}\left(\dfrac{t_B-1}{2\gamma}\right)^{\gamma} \tag{22}$$ The parameter $t_B$ can be obtained with any numerical iterative scheme based on Equation (22).
Transformation from $t$-plane onto $w$-plane: the required Schwarz-Christoffel transformation for mapping between the $w$-plane and the lower half of the $t$-plane is given by $$w = M_1\int(t-t_B)^{-1/2}\cdot(t-1)^{-1/2} dt +N_1 = \mathrm{i}M_1 \sin^{-1}\left(\dfrac{2t-t_B-1}{1-t_B}\right)+N_1 \tag{23}$$ where the constants $M_1$ and $N_1$ can be obtained via following boundary conditions as $$\mathrm{At\ point\ E:} \quad t = 1 \ \mathrm{and}\ w=0; \qquad \Rightarrow \dfrac{\mathrm{i}M_1\pi}{2}+N_1=0$$$$\mathrm{At\ point\ B:} \quad t = t_B \ \mathrm{and}\ w=-kh; \qquad \Rightarrow -\dfrac{\mathrm{i}M_1\pi}{2}+N_1=-kh$$$$\therefore\quad \mathrm{i}M_1=\dfrac{kh}{\pi} \quad \mathrm{and} \quad N_1 = -\dfrac{kh}{2} \tag{24}$$ Hence, the complex function representing the mapping of the $w$-plane onto the lower half of the $t$-plane can be expressed as $$w = \dfrac{kh}{\pi}\sin^{-1}\left(\dfrac{2t-t_B-1}{1-t_B}\right)-\dfrac{kh}{2}\qquad \Rightarrow t = f_2^{-1}(w) = \dfrac{1}{2}\left\lbrace(1-t_B)\cos\left(\dfrac{\pi w}{kh}\right)+(1+t_B)\right\rbrace \tag{25}$$ Finally, flownets in the $z$-plane can be obtained with the help of Equations (21) and (25), where the parameter $t_B$ is estimated iteratively using Equation (22).
Figure 5: (a) geometry of a flat-bottomed concrete gravity dam with an inclined sheet pile in $z$-plane, (b) an auxiliary complex variable space $t$-plane, (c) boundary conditions of flownets in $w$-plane
Vertical sheet pile
The seepage analysis of the concrete gravity dam with a vertical sheet pile, as shown in Figure 6(a), can be carried out as a special case for a concrete dam with an inclined sheet pile, where the angle created by the sheet pile with the vertical axis is zero. In this case, $\theta=0$, $\gamma=1/2$ and $a=0$. Therefore, the transformation between the physical $z$-plane and the complex $w$-plane can be expressed as $$z = f_1(t) = s\sqrt{t^2-1} \tag{26}$$$$t = f_2^{-1}(w) = \dfrac{1}{2}\left\lbrace(1-t_B)\cos\left(\dfrac{\pi w}{kh}\right)+(1+t_B)\right\rbrace \tag{27}$$$$t_B = -\dfrac{\sqrt{b_1^2+s^2}}{s} \tag{28}$$
Figure 6: (a) geometry of a flat-bottomed concrete gravity dam with a vertical sheet pile in $z$-plane, (b) an auxiliary complex variable space $t$-plane, (c) boundary conditions of flownets in $w$-plane
Example
Consider the flat-bottomed concrete gravity dam of the previous example is provided with a vertical sheet pile of length equal to $12\ \mathrm{m}$. A simple MATLAB script to generate the flownets for the dam with a vertical sheet pile is provided in this repository, and the output of the code is shown in Figure 7.
Figure 7: flownets underneath the flat-bottomed concrete gravity dam with a vertical sheet pile at its bottom
