Analytical solutions for steady groundwater flow through aquifer folds and faults

Movement of the earth’s crust creates complex shear zones, faults, folds, and deformation bands. These features impact fluid flow both by changing flow paths and by altering the hydraulic properties of the crust [1‐6]. Analytical investigations of groundwater flow through faults with altered hydraulic properties have been made by Haneberg [7] who considered one-dimensional flow through a fault zone with altered hydraulic conductivity. Anderson [8] considered two-dimensional Dupuit flow through a linear fault, and Anderson and Bakker [9] investigated flow through a faulted multi-aquifer system, based on the Dupuit approximation. Numerous numerical studies of flow through faulted or deformed aquifers exist, including Bense and Van Balen [10] who investigated regional flow through a faulted system, including the effects of fault relay ramps.

Even without changes to the hydraulic conductivity, the deformed geometry increases head losses and alters pressure and stress distributions. Cornelissen and Jansen [11] investigated the distribution of pore pressure in the vertical plane along a displaced normal fault, characterized by both a dip and offset; they used a numerical solution to the Schwarz–Christoffel transformation to solve the flow problem. Similarly, the impacts of the deformation of a simple homogeneous and confined aquifer on groundwater flow are considered here. The changing gradients and flow paths through the deformed region are examined without considering local changes in the hydraulic conductivity. Three idealized examples of deformed geometries are investigated, as presented in Fig. 1. The first example shows a fault similar to that studied by Cornelissen and Jansen [11], with a \(90^{ \circ }\) dip.

A solution for steady groundwater flow through a deformed aquifer is derived by applying the singular point method of Chaplygin and Zhukovsky [12]. The method was originally developed to solve problems of free jets in ideal fluids by the hodograph method using a circular sector as a reference plane. The method is well suited to solve free boundary problems in groundwater flow [13‐17]. The method is applied here to confined groundwater flow using a rectangular reference plane. The singular point method makes use of the basic principles of groundwater mechanics, including the method of images and the analysis of stagnation points, to develop the conformal mapping of the physical plane onto the plane of the complex potential.

Two-dimensional, potential flows for multiple wells in rectangular domains with various combinations of equipotential and impermeable boundaries have been long-investigated. Only a few examples within the field of groundwater mechanics are referenced here. Muskat [18] used the conduction sheet analog method to simulate direct-line, staggered-line, and five-spot line drive flood networks for oil recovery operations. Subsequently, the method of images with real analysis was used by Bear [19] and Mantoglou [20] to solve groundwater flow problems in a rectangular domain, by creating a doubly periodic lattice of image wells. Similar work using the method of images and complex analysis was performed more recently by Ding and Wang [21] who investigated multiple wells in a rectangular domain bounded by impermeable boundaries. Lu et al. [22] used complex analysis, the method of images and conformal mapping to examine steady state pumping in rectangular aquifers with five different combinations of equipotential and impermeable boundaries. Wang et al. [23] solved the same suite of problems using real analysis and double Fourier transforms. Of the five combinations of boundary conditions considered by Lu et al. [22], four can be solved in parametric form using the Schwarz–Christoffel transformation. Many of those solutions can also be written directly without a parametric relationship [24]. Only the problem with parallel equipotentials and parallel impermeable boundaries cannot be solved by this approach, and the results must explicitly include an infinite sum of image wells.

The three main objectives of this paper are as follows: The paper starts with a problem statement and a description of the chosen rectangular reference pane. Next, general solutions are presented of the mapping of the reference plane onto the physical domain and the complex potential domain. Finally, the solution is applied to three specific groundwater flow problems: Flow through a vertical fault, flow through a fold, and flow through a fault relay ramp.

Consider the physical plane (z-plane) illustrated in Fig. 2, which presents a very general description of a deformed aquifer. This general geometry can be used to represent all three geometries illustrated in Fig. 1, as well as others, as special cases. Steady groundwater seepage occurs in the aquifer, entering on the left side of the deformed zone and exiting on the right side. All boundaries of the physical plane are impermeable and the total flow from left to right in the aquifer is constant and equal to U. The thickness of the aquifer is \(H_l\) on the left and \(H_r\) on the right of the deformed region. The base of the aquifer steps down by an amount a, while the top steps down by \(H_l+a-H_r\) (see Fig. 2); the locations of the steps in aquifer base and top are offset by the horizontal distance b. The vertices of the physical plane are numbered 1 through 6 in Fig. 2, with vertices 1 and 4 lying at \(x=\infty \) and \(x=-\infty \), respectively; \(z=x+\textrm{i}y\) is the complex coordinate.

The two vertices P and S are branch points in the z-plane, and each lies at the end of a slot of infinitesimal width. As will be shown, the general solution allows these points to shift between the numbered vertices. For example, the figure shows vertex S lying at the end of a slot on boundary segment 5–6; alternately, S could lie on segment 4–5, rotating the slot by \(-\pi /2\) and altering the shape of the physical plane. For clarity, the locations of points S and P will be treated as illustrated in Fig. 2 in the derivation of the solution; after the general solution is formulated, examples are provided that demonstrate alternate locations for these points.

Steady groundwater flow is governed by Laplace’s equation. A solution is sought in the form of a complex potential \(\Omega (z)=\phi + \text {i}\psi \), where \(\phi \) is the specific discharge potential and \(\psi \) is the stream function. The specific discharge potential is related to the hydraulic head h as \(\phi =kh\), where k is the hydraulic conductivity. The x- and y-components of the specific discharge vector are obtained from the specific discharge potential asor in complex form as

The solution \(\Omega (z)\) must satisfy the following conditions along the boundaries of the physical plane:The notation 1–2–3–4 refers to the entire upper boundary of the aquifer from vertex 1 to 2 to 3 to 4. Similarly, 4–5–6–1 refers to the bottom boundary of the aquifer (vertex 4 to 5 to 6 to 1). Furthermore, \(\Im \) indicates the imaginary part of a complex function, and \(\psi _0\) is an arbitrary constant value. In addition, the real part of the complex potential is set to \(\phi _0\) at a reference point, \(z_0\), to make the solution unique:where \(z_0\) is any location in the physical plane, where the head is known to be \(\phi _0\). The specific location chosen is not important—it only affects the value of equipotentials in the solution, but has no effect on the flow field.

An analytic solution to the stated problem is derived using conformal mapping. Introductions to conformal mapping in the context of groundwater flow can be found in, e.g., Verruijt [25], Strack [26], and Bakker and Post [27]. The derived solution will be in parametric form as \(z(\zeta )\) and \(\Omega (\zeta )\), where \(\zeta =\xi +\text {i}\eta \) is the rectangular reference plane with length L and height B, illustrated in Fig. 3. The reference plane is chosen such that the two vertical sides of the rectangle correspond to the vertical sections 5–6 and 2–3, respectively, in the physical plane. Similarly, the piece-wise horizontal section 6–2 corresponds to the bottom of the rectangle and the piece-wise horizontal section 3–5 corresponds to the top of the rectangle. The exact locations of vertices 1, 4, P, and S are initially unknown in the reference plane, as well as the aspect ratio B/L of the rectangular reference plane, and must be evaluated to obtain a complete solution.

The parameters are evaluated from a non-linear system of conditions presented along with the solution. The system of conditions is then simplified by considering the three special cases introduced in Fig. 1, and flow nets are presented for each.

The first example concerns flow through an aquifer with a vertical fault (Fig. 6). The thickness of the aquifer is equal to H on both sides of the fault (\(H_r=H_l=H=1)\), the step in the base is a, and there is no offset (\(b=0\)). Points 2 and P coincide (\(p=0\)) and points 5 and S coincide (\(s=0\)). Point 1 is located along the bottom boundary of the rectangular reference plane at \(\zeta _1=\delta \). Because of symmetry, point 4 is located at \(\zeta _4=L-\delta + \text {i}B\) along the top boundary of the reference plane. The conditions for evaluating the mapping parameters, (16) through (20), becomeConditions (26) and (27) reduce to the same expression.

The length L of the rectangular reference plane and the location of point 1 are set to \(L=1\) and \(\delta =0.25\), respectively. The choice of \(\delta \) determines the vertical jump a of the base at the fault (a smaller value of \(\delta \) results in a larger jump a). Finally, the height of the rectangle B is adjusted such that the condition at point 5 (24) is met and by symmetry the condition at point 2 (23) using a standard root-finding algorithm, which gives \(B=0.63963\). Once the value of B is determined, the step in the base is computed from condition (26) as \(a=0.5\).

To better understand the solution, contours of x and y are drawn in the reference plane for three values of B in Fig. 7. The corresponding boundary of the domain and resulting flow net are shown as well. The middle row of Fig. 7 is the solution to the stated problem, computed with the value for B such that points 2 and P coincide as well as points 5 and S (Fig. 7c, d). The top row corresponds to the case that the height of the reference plane is B/2. Points P and S are visible in the reference plane as stagnation points; point P is located on the bottom boundary of the reference plane between points 1 and 2, and point S is located on the top boundary of the reference plane between points 4 and 5 (Fig. 7b). As a result, sections 1–2 and 4–5 of the boundary of the domain in the physical domain extend horizontally beyond the fault and then loop back over a distance \(p=0.170\) (Fig. 7a). The bottom row corresponds to the case that the height of the reference plane is 2B and points P and S are on the right and left boundaries of the reference plane, respectively (Fig. 7f). As a result, sections 2–3 and 5–6 extend vertically into the aquifer in the physical domain over a distance \(s=0.133\) (Fig. 7e).

The solution of Example 1 is modified to simulate flow through a fold, where the top of the aquifer jumps at a different x-location than the bottom of the aquifer (Fig. 8). All parameters are the same as in Example 1, except that the parameter b is chosen non-zero. The conditions for evaluating the mapping parameters are the same as for Example 1, except that (25) becomesThe value of B is again computed such that \(z'=0\) in point 5. First, consider the case that \(\delta =0.25\) and \(b=0.4\). For this case the top of the aquifer jumps down after the bottom. The shape of the boundary and flow net are shown in Fig. 9a. The resulting value for \(B=1.6838\) and the jump in the base is \(a=1.174\). Second, consider the case that \(\delta =0.15\) and \(b=-0.4\). For this case the top of the aquifer jumps down before the bottom of the aquifer. The shape of the boundary and flow net are shown in Fig. 9b. The resulting value for \(B=0.4469\) and the jump in the base is \(a=0.5212\).

As a final example, consider flow through a fault relay ramp in the horizontal plane (Fig. 10). In the horizontal plane, H represents the width of the aquifer, which is chosen constant (\(H_l=H_r=H)\). The offset a is more than the width H, and the offset b is positive. The branch points P and S do not coincide with vertices 2 and 5, as they did in the previous two examples. Point 1 is again chosen at \(\zeta =\delta \) and point 4 at \(\zeta =L-\delta + \text {i}B\), so that the flow field is symmetric. The shape of the fault relay ramp and the flow field are computed for \(H=1\), \(b=2\), \(\delta =0.15\), and \(B=0.6\). The locations of points P and S in the reference plane are obtained from conditions (16) and (17). Point P is located between points 1 and 2 on the bottom of the rectangular reference plane (\(\zeta _P=0.2770\)), while point S is located between points 4 and 5 on the top of the reference plane (\(\zeta _S=0.7230 + 0.6\text {i}\)). The values of a, p, and s are obtained from Eqs. (18–20), resulting in \(a=1.9\) and \(p=s=1.129\). Finally, the shape of the fault relay ramp and a flow net are shown in Fig. 11.

A new potential flow solution was derived for steady groundwater flow through a deformed aquifer. The solution, developed by conformal mapping using the singular point method and a rectangular reference plane, includes several parameters that may be chosen to simulate flow through a variety of deformed aquifer features including faults, folds, and relay ramps. The singular point method allows the development of a very general solution using the basic tools of groundwater mechanics including superposition, the method of images, and stagnation point analysis.