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Hydrogeology Journal

, Volume 26, Issue 5, pp 1669–1681 | Cite as

Assessment of groundwater exploitation in an aquifer using the random walk on grid method: a case study at Ordos, China

  • Tongchao Nan
  • Kaixuan Li
  • Jichun Wu
  • Lihe Yin
Paper

Abstract

Sustainability has been one of the key criteria of effective water exploitation. Groundwater exploitation and water-table decline at Haolebaoji water source site in the Ordos basin in NW China has drawn public attention due to concerns about potential threats to ecosystems and grazing land in the area. To better investigate the impact of production wells at Haolebaoji on the water table, an adapted algorithm called the random walk on grid method (WOG) is applied to simulate the hydraulic head in the unconfined and confined aquifers. This is the first attempt to apply WOG to a real groundwater problem. The method can not only evaluate the head values but also the contributions made by each source/sink term. One is allowed to analyze the impact of source/sink terms just as if one had an analytical solution. The head values evaluated by WOG match the values derived from the software Groundwater Modeling System (GMS). It suggests that WOG is effective and applicable in a heterogeneous aquifer with respect to practical problems, and the resultant information is useful for groundwater management.

Keywords

China Random walk path method Heterogeneity Groundwater management 

Evaluation de l’exploitation des eaux souterraines dans un aquifère en utilisant la méthode de la marche aléatoire sur grille: un cas d’étude dans l’Ordos, chine

Résumé

La durabilité est. un des critères clefs d’une exploitation efficace de l’eau. L’exploitation des eaux souterraines et le déclin du niveau piézométrique sur le site de source d’eau d’Haolebaoj dans le bassin de l’Ordos dans le NW de la Chine ont attiré l’attention du public en raison des menaces potentielles pour les écosystèmes et les zones de pâtures dans la région. Afin de mieux étudier l’impact des puits de production à Haolebaoji sur le niveau piézométrique, un algorithme adapté nommé la méthode de la marche aléatoire sur grille (MAG) est. appliqué pour simuler la charge hydraulique dans les aquifères libres et captifs. C’est. la première tentative d’application de MAG sur un problème réel hydrogéologique. Cette méthode peut non seulement évaluer les valeurs de la charge hydraulique mais aussi les contributions de chaque terme source/puits. Il est. permis d’analyzer l’impact des termes source/puits comme si on disposait d’une solution analytique. Les valeurs de charge hydraulique évaluées par MAG correspondent aux valeurs dérivées du logiciel de modélisation du système des eaux souterraines (GMS). Il est. suggéré que la méthode MAG est. efficace et applicable à un aquifère hétérogène avec des problèmes pratiques, et que l’information qui en résulte est. utile pour la gestion des eaux souterraines.

Evaluación de la explotación de agua subterránea en un acuífero utilizando el método de camino aleatorio en cuadrícula: un estudio de Caso en Ordos, China

Resumen

La sostenibilidad ha sido uno de los criterios clave de la explotación eficaz del agua. La explotación del agua subterránea y la profundización de la capa freática en la fuente de agua de Haolebaoji en la cuenca de Ordos en el noroeste de China llamaron la atención del público debido a las preocupaciones sobre posibles amenazas a los ecosistemas y a las tierras de pastoreo en la zona. Para investigar mejor el impacto de los pozos de producción en Haolebaoji en la capa freática, se aplica un algoritmo adaptado llamado método de camino aleatorio en cuadrícula (WOG) para simular la carga hidráulica en los acuíferos confinados y no confinados. Este es el primer intento de aplicar WOG a un problema real de agua subterránea. El método no solo puede evaluar los valores de carga hidráulica sino también las contribuciones hechas por cada término de fuente/sumidero. Uno puede analizar el impacto de los términos fuente/sumidero como si uno tuviera una solución analítica. Los valores de carga hidráulica evaluados por WOG coinciden con los valores derivados del software Groundwater Modeling System (GMS). Se sugiere que WOG es eficaz y aplicable en un acuífero heterogéneo con respecto a problemas prácticos, y la información resultante es útil para la gestión del agua subterránea.

采用网格随机游动方法评价含水层地下水开采:中国鄂尔多斯的一个研究案例

摘要

可持续性是有的地下水开采和效地下水开采的关键标准之一。中国西北地区鄂尔多斯盆地Haolebaoji水源地的地下水开采及水位下降引起了人们的关注,由于人们担心对本地区生态系统及放牧场地造成潜在的威胁。为了更好地研究Haolebaoji生产井对水位的影响,应用了一种被称为网格随机游动方法的调整算法来模拟非承压和承压含水层的水头。这是把网格随机游动方法应用到现实地下水问题中第一次尝试。该方法不仅能评估水头值,而且能评估各个源汇项做出的贡献。正如人们采用解析方法一样,人们可以分析源汇项的影响。由网格随机游动方法得到的评估水头值与软件地下水模拟系统得到的水头值非常吻合。这表明,网格随机游动方法在针对异质含水层中的实际问题非常有效和实用,相关的信息对于地下水管理非常有用。

Avaliação da explotação de águas subterrâneas em um aquífero utilizando passeio aleatório no método da grade: estudo de Caso em Ordos, China

Resumo

Sustentabilidade tem sido um dos critérios chave da explotação efetiva de água. A explotação de águas subterrâneas e a depleção da superfície freática no aquífero de Haolabaoji na bacia de Ordos no NO da China tem despertado a atenção da administração pública devido a riscos potenciais aos ecossistemas e às áreas de pastagens do local. O impacto do bombeamento de poços em Haolebaoji sobre a superfície freática, foi investigado por intermédio de um algoritmo adaptado chamado método de passeio aleatório em grade (random walk on grid - WOG) para simular as cargas hidráulicas em aquíferos livres e confinados. Esta é a primeira experiência com WOG aplicado a um problema real de escoamento de águas subterrâneas. O método é capaz de avaliar cargas hidráulicas bem como o balanço hídrico dos termos de fonte/sumidouro. Os resultados obtidos por WOG correspondem aos resultados obtidos pela solução numérica da equação geral do escoamento de águas subterrâneas, pelo programa Groundwater Modeling System (GMS). Este fato evidencia que WOG é efetivo e adequado em aquíferos heterogêneos considerando problemas práticos, e a informação resultante útil no gerenciamento de águas subterrâneas.

Introduction

The Ordos basin is located in the eastern part of NW China; it is a large arid region and is one of the most important bases for China’s energy and heavy chemical industries. Besides, the Ordos basin is one of the largest groundwater basins in the world (Hou et al. 2017; Herczeg et al. 1991) and groundwater is the main water source for human uses, livestock and manufacturing. Human activities and increasing water demand for industrial development are becoming the main factors influencing the groundwater systems and ecosystems in the area. On the other side, because the basin is located in an arid and semi-arid region, recharge to the groundwater systems relies heavily on the limited precipitation and lateral inflow. In the last decade, a number of waterworks have been established and operated for water supply to energy production in the Ordos energy base, which includes coal mining, electricity generation and the coal-based chemical industry (Hou et al. 2006). Though the total reserve volume of groundwater storage is large, the recharge rate seems quite limited. Sustainability of groundwater systems and ecosystems are impacted by natural and anthropogenic factors. Due to lack of sufficient knowledge of hydrogeology and an ineffective groundwater management strategy, water resources in Ordos basin have experienced over-exploitation which has led to water-table decline, atrophy of lakes and grassland degradation. The growing concerns about potential threats to groundwater-based ecosystems and traditional graziery in the area are catching more and more attention from the public and water management authorities (Wang et al. 2012; Cai et al. 2014; Liu et al. 2016). Assessment of the impact of anthropogenic factors on the groundwater system is a requirement of sustainable development.

In general, to simulate and evaluate groundwater systems, a set of governing equations are used to describe physical processes like groundwater flow, solute or heat transport and so on. These governing equations are partial differential equations (PDEs) of elliptic, parabolic or mixed type, which are solved via analytical or numerical approaches. Analytical solutions are often unavailable for complicated systems, while numerical solutions fail when definite conditions (boundary and initial conditions) change. The random walk path method (RWP) is a powerful tool to solve elliptic or parabolic PDEs. Unlike conventional solution methods which solve for the solution in the entire domain with specific definite conditions, RWP attempts to express the pointwise solution in terms of definite conditions and source/sink terms. The latter can provide the solution to the equation but also the contributions of definite conditions and source/sink terms, which the conventional solutions are unable to show directly. As a result, RWP is a better choice if one is interested in the contributions of model components (boundary conditions, source/sink terms etc.) rather than specific values of solutions. Due to this attribute, it is also possible to analyze the impact of source/sink terms quantitatively through RWP.

The RWP solves PDEs via random walk simulation. The first scheme of RWP, the random walk on spheres method (WOS), was proposed by Muller (1956) to solve a Laplace equation with Dirichlet boundaries, which is efficient for elliptic equations but not for parabolic equations due to difficulty in simulating the first exit time (i.e. the moment the particle arrives at a Dirichlet boundary node). Later, the random walk on rectangles method (WOR) was used to overcome this difficulty (Milstein and Tretyakov 1999) but preferably applicable to problems with polygonal boundaries (Deaconu and Lejay 2006). The WOS and WOR work well in homogeneous media but encounter difficulties while being applied to problems in highly heterogeneous media. A relocation (including refraction) of the randomly walking particle is required every time the particle passes any interface of parameter discontinuity (Hoteit et al. 2002; Lejay and Martinez 2006; Mascagni and Simonov 2004). In highly heterogeneous media, relocation becomes impossible; hence, generally speaking, the existing RWP algorithms, i.e. WOS and WOR, cannot handle high heterogeneity. After summarizing the features of WOS and WOR, Nan and Wu (Nanjing University (China), unpublished paper “Random Walk Path solution to groundwater flow dynamics in highly heterogeneous aquifers”, 2018) recommended an adapted RWP algorithm, the random walk on grid method (WOG), to handle high heterogeneity and transient problems, and demonstrated its effectiveness for transient problems and highly heterogeneous media in synthetic cases. The applicability of WOG needs further investigation in real groundwater problems.

In this study, WOG is applied to analyze the impact of water exploitation on groundwater system at the Haolebaoji site, which is a typical site of massive groundwater exploitation in Ordos basin. The objective of this study is to analyze the contributions of artificial exploitation and natural variations in surface-derived recharge to water-table decline. In this paper, the study area is briefly introduced, then the methodology of WOG and the analysis approach are explained in detail and the analysis results are presented and discussed.

Study area

The study area is the neighboring land to the Haolebaoji water source site. Haolebaoji site is located in Ordos City, Inner Mongolia, in the northern part of Ordos basin (Fig. 1). It is about 23 km from east to west and 22 km from north to south, and 450 km2 or so in area. With a temperate continental semi-arid climate, it is very hot in summer and cold in winter (−31–38 °C). The average annual temperature is 6.2 °C; average annual potential evaporation and precipitation are about 2,200–2,500 mm and 350 mm, respectively. The precipitation mainly occurs during July to September in the form of short storms. The average annual wind speed is about 1.9–4.2 m/s. The east and north sides of the study area are higher, about 1,370 m above sea level; the west and south sides are lower, about 1,290 m above sea level. The main water bodies are two inland lakes, i.e. Subei Lake and Kuisheng Lake, which are mainly recharged by groundwater and discharged by evaporation. The Quaternary sediments and Cretaceous formation outcrop in the study area. The Quaternary sediments are mostly distributed around Subei Lake with thickness from 0 to 20 m. The Cretaceous strata are mainly composed of sedimentary sandstones and the maximum thickness is about 1,000 m in the Ordos Basin (Yin et al. 2009). Unconfined and confined aquifers can be observed in the basin. The unconfined aquifer is composed of the Quaternary sediments and Cretaceous sandstones with thickness from 11 to 64 m. The underlying confined aquifer is more than 300 m thick. The two aquifers are separated by a discontinuous aquitard composed of mudstone layers. There exists strong hydraulic connection between the two aquifers according to water level data in boreholes. Some researchers have treated the two aquifers as an integrated unconfined aquifer (Yin et al. 2009; Wang et al. 2010). Fundamental geological structure and groundwater information are reported in Figs. 2 and 3. According to the principles of sustainability, the exploitation rate should be regulated so that the groundwater system is able to reach a new state of equilibrium; hence, this study is mostly interested in the steady states that the groundwater system can reach.
Fig. 1

Location of the study area and distribution of potential production wells at Haolebaoji water source site

Fig. 2

Spatial distribution of wells in the study area and hydrogeological maps of the a unconfined aquifer and b confined aquifer. After Inner Mongolia Second Hydrogeology Engineering Geological Prospecting Institute (2010). Adapted by Liu et al. (2016)

Fig. 3

Geologic sections A–A′ and B–B′ in the study area. After Inner Mongolia Second Hydrogeology Engineering Geological Prospecting Institute (2010). Adapted by Liu et al. (2016)

Materials and methodology

Available data

Wang (2010) collected hydrogeological and meteorological data at Haolebaoji site, including initial head distributions before exploitation (in year 2004), borehole head series during 2008 to 2009, precipitation data, evaporation data and so on. The authors established a numerical model of the two-aquifer system using Groundwater Modeling System (GMS 2007). The numerical model was calibrated to data of year 2004 and verified by data of year 2009. The predicted head contours in 2009 matched the measured heads quite well. Besides, according to figures 3-13, 3-14 and 3-15 in Wang (2010), the time series of simulated head in 12 observation wells were also comparable to the observed ones. The results suggested that the calibrated model was able to characterize the aquifer system to an acceptable extent.

The calibrated hydraulic parameters and basic hydrological data (precipitation, evaporation, water table at boundaries etc.) are adopted in this study mainly for two reasons. Firstly, the model in Wang et al. (2010) seems the latest model one can currently access. Although a project was initiated in 2016 to further investigate and characterize the basin aquifers, the project is still in progress and it will be 1 or 2 years before a new calibrated model becomes available. Secondly, the study reported here is focused on analyzing the exploitation impact from a new perspective, different from the traditional numerical simulation, but the results need be compared to traditional simulation results based on the consistent model and data. Though the model in Wang et al. (2010) is imperfect, it reflects sufficient complexity of a practical groundwater system. Besides, its replica model is rebuilt and rerun using MODFLOW-2000 through GMS software in this study to validate the simulation result.

During rebuilding of the groundwater model, adopted fundamental data include: potential evaporation (2,134 mm/year), precipitation (~400 mm/year), coefficient of rainfall recharge (Fig. 4), hydraulic conductivity (Fig. 5), specific storage/yield in the aquifers (Fig. 5), pumping rate of domestic and agricultural use (~14,790 m3/d), infiltration recharge from agricultural irrigation (~550 m3/d), potential pumping rate of industrial exploitation (~3,000 m3/d × 22 wells = 66,000 m3/d; Wang 2010). Following Wang (2010), the pumping and infiltration of domestic and agricultural water are treated as areal average on the whole domain due to lack of detailed data. Since only steady states without and with industrial exploitation are considered in this study, the replica model simulates steady states only.
Fig. 4

Spatial distribution of the coefficient of rainfall recharge in the study area (modified from Wang et al. 2010)

Fig. 5

Zonation of hydraulic conductivity (K) and specific yield (u) or storage (SS) and discretization of the a unconfined and b confined aquifers (data from Wang 2010)

The conductance and head data used in defining general-head boundaries were undocumented, thus it is hard to retrieve the identical boundary data. However, it is noticed that under the impact of aforementioned exploitation activities, most of drawdowns at boundary were negligible and the others were smaller than 3 m, according to figures 4-4 and 4-6 in Wang (2010). Hence, it is possible to apply Dirichlet boundaries to the model. In this study, the Dirichlet boundary (prescribed-head boundary) is used for all boundary cells in top and bottom layers.

Theoretic basis of the random walk path method

The fundamental principle of RWP has roots in the famous Feynman-Kac theorem, which renders the probabilistic representation of the solution to a diffusion-type PDE with a piecewise smooth Dirichlet boundary.
$$ \left\{\begin{array}{l}\nabla \cdot \left(K\left(\mathbf{x}\right)\nabla h\left(\mathbf{x}\right)\right)+w\left(\mathbf{x}\right)=0,\kern1.5em \mathbf{x}\in D\\ {}\kern3em h\left(\mathbf{x}\right)=\varphi \left(\mathbf{x}\right),\kern1em \mathbf{x}\in {\varGamma}_{\mathrm{D}}\end{array}\right. $$
(1)
for a bounded, continuous function w(x) in domain D and a continuous function φ(x) on the boundary. In context of groundwater flow, K(x) is the location-dependent hydraulic conductivity and h(x) is hydraulic head. w(x) is the source/sink term. D is the aquifer domain. φ(x) is the head at Dirichlet boundary. Thanks to Feynman-Kac theorem, the solution to Eq. (1) can be represented as
$$ h\left({\mathbf{x}}_0\right)=\left\langle {\int}_0^{t_{\mathrm{e}}}\frac{1}{2}w\left(\mathbf{Z}(t)\right)\mathrm{d}t\right\rangle +\left\langle \varphi \left({\mathbf{Z}}_{\mathrm{e}}\right)\right\rangle $$
(2)
where x0 is the starting point of the random walk path, at which the pointwise solution is going to be solved. Z(t) is a random walk path starting at t = 0 and x0 which ends once hitting the boundary; the moment of hitting boundary is te called the exit time, and the corresponding exit position is denoted as Ze (a random location on D).
For time-varying parabolic PDE,
$$ \left\{\begin{array}{l}\nabla \cdot \left(K\left(\mathbf{x}\right)\nabla h\left(\mathbf{x}\right)\right)+w\left(\mathbf{x}\right)=S\frac{\partial h\left(\mathbf{x},t\right)}{\partial t},\kern1em x\in D\\ {}\kern6em h\left(\mathbf{x},0\right)=\psi \left(\mathbf{x}\right)\\ {}\kern5em h\left(\mathbf{x},t\right)=\varphi \left(\mathbf{x},t\right),\kern1em \mathbf{x}\in {\varGamma}_{\mathrm{D}}\end{array}\right. $$
(3)
The idea is similar to Eq. (1), except that time is treated as another dimension and the initial condition is taken as a “prescribed-head boundary” (“Dirichlet” boundary \( {\varGamma}_{\mathrm{D}}^{\ast } \)). ΓD and \( {\varGamma}_{\mathrm{D}}^{\ast } \) are sometimes termed terminal conditions together, where the random walk terminates. The solution is (Lejay and Marie 2013)
$$ h\left({\mathbf{x}}_0,t\right)=\left\langle {\int}_{0\wedge {t}_{\mathrm{e}}}^t\frac{1}{2}w\left(\mathbf{Z}(s)\right)\mathrm{d}s\right\rangle +\left\langle \varphi \left({\mathbf{Z}}_{\mathrm{e}},{t}_{\mathrm{e}}\right){\mathbf{1}}_{{\mathbf{z}}_{\mathrm{e}}\in {\Gamma}_{\mathrm{D}}}+\psi \left({\mathbf{Z}}_{\mathrm{e}}\right){\mathbf{1}}_{{\mathbf{z}}_{\mathrm{e}}\in {\Gamma}_{\mathrm{D}}^{\ast }}\right\rangle $$
(4)
where \( {\mathbf{1}}_{{\mathbf{Z}}_{\mathrm{e}}\in {\varGamma}_{\mathrm{D}}} \) is a zero-one indicator function, i.e. \( {\mathbf{1}}_{{\mathbf{Z}}_{\mathrm{e}}\in {\varGamma}_{\mathrm{D}}}=1 \) when Ze ∈ ΓD is true and 0 otherwise; similar for \( {\mathbf{1}}_{{\mathbf{Z}}_{\mathrm{e}}\in {\varGamma}_{\mathrm{D}}^{\ast }} \).
One is able to numerically evaluate Eq. (4) and obtain pointwise solutions to Eqs. (1) and (3).
$$ {h}_{(0)}\approx {\mathbf{d}}_{(0)}\bullet {\mathbf{h}}_{\left(\mathrm{T}\right)}+{\mathbf{c}}_{(0)}\bullet {\mathbf{w}}^{\ast} $$
(5)
where h(0) is the pointwise head to be evaluated [the subscript “(0)” standing for point x0], h(T) is a fixed vector containing heads at all terminal nodes, d(0) is the occurrence frequency the particle exits the domain at every terminal node, or the weight at every terminal node [the subscript “(0)” indicating the vector is for x0], w is an effective source vector with its entry \( {w}_j^{\ast} \) representing head variation at node j due to source/sink terms (its form will be explained in detail in the next subsection); and c(0) is a “source count” vector, recording how many times on average each source node has been passed by. Its entry cj corresponds to \( {w}_j^{\ast} \). The inner product of vectors c(0) ∙ w can be treated as head variation (like drawdown and rise) induced by source/sink terms. And d(0) ∙ h(T) is the probability-weighted average of heads at terminal nodes, reflecting the impact of boundaries on the solution.
The ways of numerically implementing RWP include walk on spheres, walk on rectangles and walk on grids (WOG) and are extended to various boundary conditions such as Neumann, Robin, general-head boundaries—Nan and Wu, unpublished paper (see previous details), 2018). To the best knowledge of the authors, WOG is currently the only scheme of RWP which can handle high heterogeneity; hence, in this study, WOG was chosen to investigate the impacts of source/sink terms in the problem. Figure 6 shows the flowchart of the WOG scheme.
Fig. 6

The flowchart of the walk on grids method (WOG)

Formulation of the problem

One may notice that RWP is valid in confined aquifers with various boundary conditions. It is a little complicated in an unconfined aquifer since nonlinearity in the unconfined aquifer makes Feynman-Kac theorem inapplicable. Luckily, one is still able to use RWP for unconfined aquifers as long as the water table is more or less steady. That is, RWP can be applied to the nearly steady state of the aquifer system, which is exactly what sustainability requires in the long term. The system of two aquifers is treated as a single integrated aquifer and three-dimensional steady-state flow in it is considered. The basic governing equation for the flow is:
$$ \left\{\begin{array}{l}\nabla \cdot \left(K\left(x,y,z\right)\nabla h\left(x,y,z\right)\right)+w\left(x,y,z\right)=0,\kern1em \left(x,y,z\right)\in D\\ {}\kern6em h\left(x,y,z\right)={H}_{\mathrm{D}}\left(x,y,z\right),\kern1em \left(x,y,z\right)\in {\varGamma}_{\mathrm{D}}\\ {}\kern5em K\left(x,y,z\right)\frac{\partial h\left(x,y,z\right)}{\partial n}=-q\left(x,y,z\right),\kern1em \left(x,y,z\right)\in {\varGamma}_{\mathrm{N}}\end{array}\operatorname{}\right. $$
(6)
where K(x, y, z) is the location-dependent hydraulic conductivity, w(x, y, z) is the source/sink term, D is the aquifer domain, ΓD and ΓN are Dirichlet and Neumann boundaries, respectively, n is the outward length normal to the boundary ΓN. More specifically, the lateral boundaries are taken as prescribed or Dirichlet boundaries; the bottom is a no-flux boundary and the top is a Neumann boundary with known flux (evaporation and effective precipitation).
Using an orthogonal grid and finite difference, one can write the head at one node as the weighted summation of the heads at the six neighboring nodes and source term bonus, that is,
$$ h\left(x,y,z\right)=p\left(x+\right)\bullet {h}_{x+\Delta x}+p\left(x-\right)\bullet {h}_{x-\Delta x}+p\left(y+\right)\bullet {h}_{y+\Delta y}+p\left(y-\right)\bullet {h}_{y-\Delta y}+p\left(z+\right)\bullet {h}_{z+\Delta z}+p\left(z-\right)\bullet {h}_{z-\Delta z}+{w}^{\ast}\left(x,y,z\right) $$
(7)
$$ p\left(x\pm \right)={K}_{x\pm \Delta x}/{K}_{\mathrm{sum}},\kern0.5em p\left(y\pm \right)={K}_{y\pm \Delta y}/{K}_{\mathrm{sum}},\kern0.75em p\left(z\pm \right)={K}_{z\pm \Delta z}/{K}_{\mathrm{sum}} $$
(8)
$$ {R}_{xy}=\Delta x/\Delta y,{R}_{xz}=\Delta x/\Delta z,{w}^{\ast}\left(x,y,z\right)=w\left(x,y,z\right)\bullet {\left(\Delta x\right)}^2/{K}_{\mathrm{sum}} $$
(9)
$$ {K}_{\mathrm{sum}}={K}_{x+\Delta x}+{K}_{x-\Delta x}+{R}_{xy}^2\bullet {K}_{y+\Delta y}+{R}_{xy}^2\bullet {K}_{y-\Delta y}+{R}_{xz}^2\bullet {K}_{z+\Delta z}+{R}_{xz}^2\bullet {K}_{z-\Delta z} $$
(10)
where Kx + ∆x and Kx − ∆x denote effective hydraulic conductivities between the current node and two neighboring nodes in the x direction. Similar for those in y and z directions.
The final solution will be expressed in the form of Eq. (5) or equivalently,
$$ {h}_{(0)}\approx {\mathbf{d}}_{(0)}\bullet {\mathbf{h}}_{\left(\mathrm{T}\right)}+{\mathbf{c}}_{(0)}^{\ast}\bullet \mathbf{w} $$
(11)
with \( {\mathbf{c}}_{(0)}^{\ast}={\mathbf{c}}_{(0)}\bullet {\left(\Delta x\right)}^2/{K}_{\mathrm{sum}} \). One strong point of this method is that the pumping rates and head values at lateral boundaries are not necessarily required in the calculation provided they are definite. These specific values are only needed when one wants to evaluate head values.

The Haolebaoji model is discretized using a grid of 200 × 200 × 2 in x-y-z direction, with Δx = 137.9840 m, and Δy = 122.4630 m. In the z direction, the top layer is the unconfined aquifer (“UNC” for brevity) and the bottom layer is the confined aquifer (“CON”). Δz varies following the thickness of the unconfined and confined aquifers. The ground surface of the northwest corner of the study area (108°52′E, 39°28′N) is selected as the origin. For simplicity, the cell index is used to represent the real location in space—for instance, (100, 110, 2) stands for the CON cell at (100Δx, 110Δy), and (30, 40, 1) stands for the UNC cell at (30Δx, 40Δy). In the Monte Carlo simulation of random walk, 105 realizations are used.

Results and discussion

Evaluation and validation of WOG results

The probability of moving in each direction can be evaluated based on model geometry, hydraulic conductivity and boundary type. Figure 7 shows the spatial probability of moving to x + direction or to the other layer for UNC and CON. Three major facts can be found. First, bounds of hydraulic conductivity zonation are shown by discontinuous variation of probability p(x+). This is also observed in the probability of other horizontal directions, i.e. p(x–), p(y+), p(y–) (not shown). The phenomenon of discontinuity is not obvious for p(). Second, variations in p() are dominated by aquifer thickness (Δz; especially in UNC), and the impact of hydraulic conductivity zonation seems less important. Third, p(z+) in UNC is much higher than p(z–) in CON. That is, at most of locations, the particle has a high probability of entering CON from UNC but low probability of coming back. This leads to the suggestion that most movements of random walk may occur in CON and more likely exit the domain through the boundary of CON. It can be verified through boundary weight (equivalent to exit probability at boundary).
Fig. 7

The probability of a moving to x + direction [denoted by p(x+)] in UNC; b moving to x + direction in CON; c moving to z + direction [i.e. entering CON from UNC, p(z+)]; d moving to z- direction, i.e. entering UNC from CON, p(z–)

The boundary weight d(0) of two layers for x0 = (104, 20, 1), x0 = (104, 20, 2), x0 = (157, 109, 1) and x0 = (157, 109, 2) are reported in Fig. 8. It is seen that for a same x0, d(0) for CON is much larger for d(0) for UNC. One interesting fact is that the results with x0 = (104, 20, 1) and x0 = (104, 20, 2) look very similar, which will make the boundary contributions to heads at (104, 20, 1) and (104, 20, 2) very close. Figure 8e–h shows a similar result for x0 = (157, 109, 1) and x0 = (157, 109, 2). However, a major distinction from Fig. 8a–d is that the differences between parts e and f of Fig. 8 and between parts g and h of Fig. 8 are much less remarkable than those between parts a and b of Fig. 8 and between parts c and d of Fig. 8. The reason is straightforward: near (104, 20, 1) or (104, 20, 2), p(z+) > > p(z–); but near (157, 109, 1) or (157, 109, 2), p(z+) and p(z–) are comparable, making the third phenomenon in Fig. 7 weaker.
Fig. 8

Calculated boundary weight d(0): a for UNC boundary with x0 = (104, 20, 1), b for CON boundary with x0 = (104, 20, 1), c for UNC boundary with x0 = (104, 20, 2), d for CON boundary with x0 = (104, 20, 2), e for UNC boundary with x0 = (157, 109, 1), f for CON boundary with x0 = (157, 109, 1), g for UNC boundary with x0 = (157, 109, 2), h for CON boundary with x0 = (157, 109, 2). Black star denotes the location of x0 in the x-y plane

Figure 9 reports the source count c(0) in UNC and CON with x0 = (104, 20, 1), x0 = (104, 20, 2), x0 = (157, 109, 1) and x0 = (157, 109, 2). It can been seen that for a same horizontal location, c(0) in CON tends to be larger than c(0) in UNC, which is more or less similar to the behavior of d(0). When an areal or pointwise source/sink exists in the model, its contribution will be reflected with c(0) at the same locations, as shown in Eq. (5).
Fig. 9

Source count c(0): a for UNC with x0 = (104, 20, 1), b for CON with x0 = (104, 20, 1), c for UNC with x0 = (104, 20, 2), d for CON with x0 = (104, 20, 2), e for UNC with x0 = (157, 109, 1), f for CON with x0 = (157, 109, 1), g for UNC with x0 = (157, 109, 2), h for CON with x0 = (157, 109, 2). White circles denote the locations of industrial pumping wells in the x-y plane

After c(0) and d(0) are calculated, one only needs to substitute source terms w and boundary heads h(T) into Eq. (5) to evaluate h(0). According to the heads at the boundaries of the two aquifers and the source data (including precipitation, evaporation, areal exploitation for agricultural and domestic use, infiltration from agricultural irrigation) at the top in year 2004, heads at 39 locations in both UNC and CON (78 locations in total) inside the domain are evaluated and compared to results of GMS replica model. The locations of 39 nodes include 22 industrial well locations and 17 arbitrarily picked locations, such that these locations are distributed more uniformly in the domain. The head difference Δh (i.e. head from WOG subtracted by head from GMS) at 78 locations are reported in Fig. 10a,b. It is found that all head differences are within [−2, 2] and most are within [−1, 1]. Thus the results are acceptable, suggesting that WOG is an effective tool to calculate the groundwater flow; furthermore, head differences in CON are smaller than those in UNC, which may result from two reasons. First, sampling errors are larger in UNC because random walk paths are more concentrated in CON, making c(0) and d(0) more smooth and accurate in CON. Increases in the realization number will improve the result. Second, errors in aquifer thickness have stronger impact in UNC. In Fig. 7c, it is found that the moving probability can be greatly influenced by the thickness of UNC, an influence which may play a more significant role in heads in UNC.
Fig. 10

Head differences between WOG and GMS results: a in UNC, without industrial pumping; b in CON, without industrial pumping; c in UNC, with industrial pumping; d in CON, with industrial pumping

When pumping is conducted, it can be reflected by adding pumping rates to source terms w in Eq. (5). It is now assumed that every industrial well starts pumping water from CON with the same rate, 3,000 m3/d, until a new steady state is reached. By adding the pumping rate to each industrial well location and plugging the new w into Eq. (5), one can directly evaluate the corresponding heads at the same locations under pumping using a single equation. Similarly, head differences between WOG and GMS under industrial pumping are reported in Fig. 10c,d. It can be seen that most of the head differences in UNC and CON are still within [−2, 2], supporting the effectiveness of WOG. Furthermore, the head differences in Fig. 10c,d are remarkably larger than those in Fig. 10a,b, which suggests that WOG results are of lower accuracy when predicting heads under industrial pumping. It is not surprising, since c(0) and d(0) used in head prediction do not take into account the change of UNC thickness; however, the acceptable accuracy of WOG results implies that the major hydraulic connection almost remains unchanged.

In Fig. 10, most of large head deviations occur far away from the boundary, which reflects the controlling impact of boundary to WOG solutions near the boundary. It is noteworthy that if any active well is located near the boundary, the boundary type may change. A change of boundary type will make the previous c(0) and d(0) invalid and new c(0) and d(0) values have to be recalculated.

Assessment of pumping at potential well locations

Now let one consider the influence of pumping at 22 industrial wells on the water table. When the pumping rates are large enough, the lowest points of the water table are surely located at the same horizontal locations as the pumping wells. That is, one only needs to keep an eye on the groundwater level in pumping wells, which makes the problem much simpler. Thus one only needs to check how 44 pumping nodes (22 in UNC and 22 in CON) influence the head in 22 nodes in UNC (corresponding to planar locations of industrial wells). In Eq. (11), the impact of industrial wells can be separated so that
$$ {h}_{(k)}\approx {h}_{(k),\mathrm{NP}}+{\sum}_{i=1}^{22}{\mathrm{c}}_{(k),i}^{\ast, \mathrm{U}}{q}_i^{\mathrm{U}}+{\sum}_{i=1}^{22}{\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}}{q}_i^{\mathrm{C}} $$
(12)
where \( {h}_{(k). NP}={\mathbf{d}}_{(k)}\bullet {\mathbf{h}}_{\left(\mathrm{T}\right)}+{\mathbf{c}}_{(k)}^{\ast}\bullet {\mathbf{w}}_{\mathrm{NP}} \), representing the head at (xk) without industrial pumping; wNP includes all source/sink terms except industrial pumping; \( {\mathbf{c}}_{(k)}^{\ast}={\mathbf{c}}_{(k)}\bullet {\left(\Delta x\right)}^2/{K}_{\mathrm{sum}} \); \( {q}_i^{\mathrm{U}} \) and \( {q}_i^{\mathrm{C}} \) stand for pumping rate from UNC and CON at the i-th well location; \( {\mathrm{c}}_{\left(\mathrm{k}\right),\mathrm{i}}^{\ast, \mathrm{U}} \) and \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \) are effective source counts at the k-th well location influenced by the i-th well location. Loosely speaking, they represent the sensitivity or response matrix of the water table to pumping rates at 44 locations. Figure 11 shows the values of \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{U}} \) and \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \). It can be seen that \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{U}} \) is much larger than \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \), saying that the water table is very sensitive to pumping in UNC. That is, pumping in CON with a same rate will produce smaller decreases in the water table, which is somehow common sense in hydrogeology. Furthermore, \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \) tells us quantitatively how pumping in CON affects the water table.
Fig. 11

Effective source counts \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{U}} \) and \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \) at the k-th well location influenced by the i-th well location for all 22 industrial locations. a \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{U}} \): pumping from UNC; b \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \): pumping from CON

If one needs to keep the head within a reasonable range in the long term and the boundary is unaffected, one can find the feasible pumping plan with a simple arithmetic procedure of optimization. Of course, when heads at multiple positions are required, the problem becomes a multi-objective optimization problem but still easy to solve.

Different from the conventional response matrix method (e.g. Gorelick 1983) which mainly relies on analytical solutions to simplified models, WOG is applicable to general complex problems in heterogeneous aquifers like Haolebaoji site. Effective source counts \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{U}} \) and \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \) from WOG provide a straightforward measure for optimizing the spatial distribution of exploitation rates. The much smaller values of \( {\mathrm{c}}_{(k),i}^{\ast, \mathrm{C}} \) than \( {\mathrm{c}}_{\left(\mathrm{k}\right),\mathrm{i}}^{\ast, \mathrm{U}} \) suggest that the more pumping happens in CON, the weaker influence the exploitation has on the water table. The pumping influence, not only at well locations but also the entire domain, can be evaluated. The effective source count is a direct representation of drawdown prices one has to pay for pumping groundwater. On one hand, it makes the groundwater management much easier technically and more efficient. On the other hand, the effective source count could work as a “visible price list”, reminding groundwater regulators and the public of environmental consequences to endure. Though the ecological requirements of water-table depth at Haolebaoji are still to be determined, effective source count from WOG can be used to ensure the criteria to be met once the criteria are determined.

Weakness of WOG solutions

Implementation of WOG relies on the Monte Carlo method, which is often computationally expensive—for example, in this study, solving a WOG solution with 105 realizations requires 10 min, though the solution can be used to other boundary and pumping scenarios. Speedup of numerical implementation will be a necessary step to make this method popular; furthermore, as explained in the preceding, WOG solutions are pointwise. It means that if one needs solutions at every point, one has to use WOG for every point. Last but not least, changes of boundary type and conductance in the model will make WOG solutions invalid and a recalculation is required for WOG.

Conclusions

For the first time, the WOG algorithm is applied to a real groundwater problem, at the Haolebaoji water source site, to investigate the impact of source/sink terms on hydraulic heads. The WOG scheme is selected to implement RWP owing to its ability to deal with heterogeneity. Considering the existence of the unconfined aquifer in the problem and the requirement of sustainability, this simulation is limited to the states of equilibrium without and with industrial pumping. The simulated head values are compared to the results from a GMS model based on data from documented studies. The correspondence between the results support the effectiveness of WOG applied to the real groundwater problem. Unlike the usual numerical methods, WOG provides information on the contributions by various source/sink terms, like different production wells and boundaries. With the contribution information, head values of interest can be evaluated even by hand when the quantities of these terms change; thus, it is extremely efficient when one needs to compare a large number of candidate pumping plans. To meet the requirements of sustainability, one just needs to simulate heads at positions of interest and optimize source/sink terms to make these head values fall in a reasonable range. This study supports that WOG is effective and applicable in a heterogeneous aquifer when applied to practical problems. An effective source count from WOG makes groundwater management much easier and more efficient. More importantly, it could work as a “visible price list”, reminding groundwater regulators and the public of local environmental consequences.

Notes

Funding Information

This work is funded by the project “Hydrogeological investigation at 1:50 000 scale in the lake-concentrated areas of the northern Ordos Basin” of China Geological Survey (Grant No. DD20160293), the National Natural Science Foundation of China (Grant No. 41602250) and the Fundamental Research Funds for the Central Universities (Grant No. 0206-14380032).

Compliance with ethical standards

Conflict of Interest

We declare that we have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Hydrosciences, School of Earth Sciences and EngineeringNanjing UniversityNanjingChina
  2. 2.Xi’an Center of Geological Survey, China Geological SurveyXi’anChina

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