Modified Theis equation by considering the bending effect of the confining unit

doi:10.1016/j.advwatres.2004.08.007

Advances in Water Resources

Volume 27, Issue 10, October 2004, Pages 981-990

https://doi.org/10.1016/j.advwatres.2004.08.007 Get rights and content

Abstract

The Theis equation has been widely used to study the transient movement of groundwater as a result of pumping in a confined aquifer. It is well known that the observed drawdown at early times has an obvious departure from the theoretical drawdown based on the Theis equation. The Theis equation was derived under the assumption that total stress in the aquifer was constant and the mechanical behavior of the confining unit was neglected. However, most geological formations, especially those which are well consolidated, have rigidity and therefore may bend like a plate to a certain extent. The increase in the effective stress in the aquifer due to pumping may not contribute entirely to the compression of the aquifer, but may be partially cancelled out by bending of the overlying aquitard. This means only a part of the total stress is used to compact the aquifer, or the aquifer cannot produce as much water as estimated from the Theis equation. This paper investigated the impact of the bending effect of the confining unit on drawdown. An analytical model which couples flow in the aquifer and bending of the confining unit was presented. The theory is based on elastic plates and solutions were given to the drawdown of groundwater level and deflection of the overlying formation. The drawdown estimated from the new equation was compared with that from the Theis equation. It can be concluded that drawdown from the Theis equation is less than the drawdown predicted by including the bending effect of the confining unit. Both a hypothetical example and a field pumping test in Shandong Province, China, were used to demonstrate the bending effect of the confining unit in the analysis of pumping test data. This paper demonstrated that the initial disagreement between observed drawdown and the Theis solution could be caused by the bending effect of the confining unit, a phenomenon not well addressed in traditional pumping test analysis. A quantitative understanding of this phenomenon can provide improved guidelines for analyzing drawdown data in a confined aquifer.

Introduction

Groundwater flow induced by pumping is one of the fundamental issues in groundwater hydraulics. It is well known that the first solution of transient drawdown around a pumping well was given by Theis [15]. It was directly obtained from an analogous solution in heat conduction theory. The solution was further vindicated by Jacob [7]. Since then, the Theis equation has been widely used in the groundwater community [4]. Under the assumption that a fully penetrating well pumps water from an isotropic, homogenous, and infinite confined aquifer without inflow from surrounding formations, Theis’s solution gives the drawdown of water level, s, at a radial distance, r, and at the time, t, as $s = \frac{Q}{4 π T} \int_{u}^{\infty} \frac{e^{- y}}{y} d y$

Usually (1) is written in the form $s = \frac{Q}{4 π T} W (u)$ where Q is pumping rate, T is transmissivity, u = r²S/4Tt is a dimensionless group parameter, S is storage coefficient (dimensionless) and W(u) is exponential integral.

The formula (2) has been widely used to analyze pumping-test data to determine the parameters S and T of aquifers and subsequently modified for semi-confined aquifers by including leakage and compression of the aquitard (e.g., [8], [9]).

For most pumping tests, during the early time the observed drawdown is larger than that calculated from the Theis equation. Such a difference is believed to be caused by the difference between the real aquifer conditions and the assumptions behind the Theis equation. As pointed out by Kruseman and de Ridder [12], the theoretical equation is based on the assumptions that the well discharge remains constant and that the release of the water stored in the aquifer is immediate and directly proportional to the rate of decline of the pressure head. In fact, there may be a time lag between the pressure decline and the release of stored water, and initially the well discharge may vary as the pump is adjusting itself to the changing head. Some model the difference as being caused by well storage (e.g., [5], [14]).

The difference may be caused by the mechanical reaction of the confining unit in response to pumping. The mechanical relation between the confined aquifer and its confining unit was not mentioned by Theis [15]. This relation was discussed briefly by Jacob [7] who assumed that the total stress (summation of effective stress and pore pressure) in the aquifer remained constant while the piezometric surface of groundwater changes. A portion of the pumped water results from water expansion and aquifer compression in response to an increase in effective stress caused by pumping.

With compression of the confined aquifer due to groundwater lowering, the confining unit moves down and nonuniform subsidence occurs. Jacob’s assumption implied that the movement of the confining unit is controlled by the cone of depression and the loading above the aquifer and that the confining unit moves like a very soft plate, or a series of loosely connected blocks with no mechanical force between the blocks (Fig. 1(a)).

However, most geological formations, especially those which are well consolidated, have rigidity and therefore may behave like a plate to a certain extent (this is denoted as bending effect of the confining unit in this paper). The confining layer will not bend downwards as freely as, if it were, loosely connected blocks. Or, the friction at the interfaces of blocks can support the deformation to some degree (Fig. 1(b)). In consequence that the bending of the overlying formation will reduce the loading transferred to the aquifer. This bending effect was noted by Chen and Wu [3] during a pumping test in a confined aquifer. In a pumping test analysis for a particular aquifer, they speculated that the increase in the effective stress in the aquifer due to pumping may not contribute entirely to the compression of the aquifer, but may partially cancelled out by the overlying aquitard, or the plate. They believed that this bending effect could be one of the reasons responsible for larger observed drawdown than that estimated from the Theis equation.

Various interesting phenomena related to pumping in aquifer–aquitard systems have been investigated by previous researchers using approaches beyond standard groundwater theory. For example, to explain the so-called Noordbergum effect (the rising of hydraulic head in the overlying or underlying aquitard in response to pumping in the main aquifer), both analytical and numerical studies which couple fluid flow with aquifer deformation based on Biot’s theory of consolidation were developed to examine the water level fluctuation in the aquitard during pumping (e.g., [11], [16]). Helm [10] presented analytical solutions to investigate the horizontal aquifer movement in response to pumping. A comprehensive review of various studies related to aquifer tests is beyond the scope of this paper and interested readers can find such a good review of such studies in Kruseman and de Ridder [12], Kim and Parizek [11], and Helm [10].

For a real pumping test in a complicated multiplayer aquifer system, various effects such as the Noordbergum effect, horizontal aquifer movement, and the bending effect will interact with each other and influence the drawdown in the pumped aquifer. This paper will focus only on the bending effect of the confining unit and how it will modify the drawdown calculated by the Theis equation. Instead of using a complicated poroelastic solid theory or Biot’s theory of three-dimensional consolidation, this paper employs a much simpler approach based on the theory of thin elastic plates to investigate the bending effect of the confining unit. An analytical solution of the same format as the classic Theis equation will be presented for drawdown of groundwater level in the aquifer. It will be demonstrated that Theis’s solution is a special case in which the confining unit has no rigidity or the aquifer matrix is incompressible. The drawdown from the Theis equation is less than that from the modified Theis equation which includes the bending effect of the confining unit. The difference increases when the radial distance is small and time is short. The findings of this paper can provide additional insights in understanding the difference between observed drawdown and the drawdown based on the traditional Theis equation when analyzing early drawdown data in a confined aquifer.

Section snippets

Stress analysis

Consider a horizontal confined aquifer extending towards infinity under a confining unit (Fig. 2). Water is extracted by a fully penetrating well. The total stress σ, the effective stress σ′ and the pore pressure p in the aquifer have a relation presented by the principle of effective stress (e.g., Bear [2]) $σ = σ^{'} + p$ Eq. (3) describes the mechanical relationship of groundwater and aquifer medium. This equation assumes that the aquifer is saturated or almost saturated by water and the soil grains

Model and solutions

For the aquifer well system showed in Fig. 2, the transient radial flow of the confined aquifer can be described with the following equation $T (\frac{\partial^{2} s}{\partial r^{2}} + \frac{1}{r} \frac{\partial s}{\partial r}) = \frac{\partial}{\partial t} (Δ V) + q (r, t)$ where q is the sink or source term. According to (8), (10), (14) can be rewritten as $T (\frac{\partial^{2} s}{\partial r^{2}} + \frac{1}{r} \frac{\partial s}{\partial r}) = \frac{\partial w}{\partial t} + μ_{w} \frac{\partial s}{\partial t} + q (r, t)$

Considering (6), (7), and (10), (4) can be rearranged into $Δ σ = \frac{γ_{w}}{μ_{m}} w - γ_{w} s$

The change of total stress in the aquifer is also the additional loading which causes the bending of the confining unit. According to the

Comparison of the modified Theis solution with the Theis solution

The preceding analysis shows that the drawdown is a function of three dimensionless parameters: 4at/r², r⁴/c, μ_w/S. For convenience of the following discussion, more dimensionless group parameters are defined as follows $\begin{matrix} s_{D} = s / (Q / 4 π T) \\ t_{D} = 4 at / r^{2} \\ r_{D} = r^{4} / c \end{matrix}$

Fig. 3 shows plots of s_D versus t_D for a given μ_w/S when r_D = 1 and 0.01. The curve from Theis’ solution, which is equivalent to the case when r_D = ∞, is also presented in the figure. It can be seen that the dimensionless drawdown from the Theis equation

Case studies on the bending effect

For the convenience of discussion in the previous section, dimensionless time and drawdown are used (Fig. 3, Fig. 4), but the physical implication of these dimensionless numbers is not always straightforward. A hypothetical example and a real pumping test are chosen to discuss further the significance of the bending effect.

The impact of the bending effect on the estimated parameters will be investigated when the bending effect is significant but ignored. Drawdowns will be created using the

Summary

In the traditional theory of confined groundwater flow toward a well, the total stress in the aquifer is assumed to be constant so that drawdown is directly proportional to the increment of effective stress. The assumption may not be appropriate when the bending effect of the confining unit is considered. This effect will reduce the total stress transferred to the aquifer and reduce the amount of water that can be expelled from storage.

An analytical model that couples flow in the aquifer and

Acknowledgment

The authors are grateful to Hongbin Zhan for his suggestions on the early draft of the paper. We appreciate Andrew Barry, Dong-Sheng Jeng, Todd Rasmussen and an anonymous reviewer whose constructive comments led to a significant improvement of this paper.

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Modified Theis equation by considering the bending effect of the confining unit

Abstract

Introduction

Section snippets

Stress analysis

Model and solutions

Comparison of the modified Theis solution with the Theis solution

Case studies on the bending effect

Summary

Acknowledgment

J. Hydrol.

Advanced mechanics of materials

Dynamics of fluids in porous media

Introduction to hydrogeology

A simple and powerful method of parameter estimation using simplex optimization

Ground Water

The flow of water in an elastic artesian aquifer

Trans. Am. Geophys.