Introduction

Land subsidence is the geological phenomenon in which the regional ground elevation decreases as a result of compression of the soil under the effect of natural or man-made factors, and can lead to serious disasters in the process of urbanization. Land subsidence can be viewed from two aspects: the soil structure and the engineering characteristics. With respect to soil structure, factors such as soil properties, stratum structure, and consolidation state can affect the process of ground deformation (Xu et al. 2012; Burbey 2003; Shi et al. 2008; Conway 2016). While from the aspect of engineering characteristics, groundwater usage and surface loadings are the main factors that lead to land subsidence (Xu et al. 2016; Chen et al. 2003; Wang et al. 2009; Gong et al. 2009). During the exploitation of groundwater, for example, in a ground-source heat-pump (GSHP) engineering project, the pore-water pressure between soil particles will dissipate because of a decrease in water levels, possibly leading to a reduction of the effective pressure and skeleton compression, as well as narrowing of the pore channels, which ultimately results in land subsidence. At the same time, small grains and particles in the skeleton can separate from the skeleton and migrate with the water, resulting in recombination of grains and ground deformation (Bedrikovetsky et al. 2011; Chen et al. 2003; Zeitoun and Wakshal 2013).

In previous studies, physical parameters such as porosity have usually been assumed to be constant; however, in fact, they are unsteady and vary during the consolidation deformation process. Luo et al. (2007) established a three-dimensional (3D) seepage model for deep foundation-pit dewatering engineering projects in Shanghai, China, and this model was used to simulate and predict the state change of groundwater flow in deep foundation-pit dewatering engineering projects. Chen et al. (2001) established a subsidence model with 3D groundwater flows coupled to one-dimensional (1D) nonlinear consolidation. This model was then applied to engineering projects in Suzhou (eastern China), with the hydraulic conductivity varying (reduced from 0.00101 to 0.00067 m/d) during the consolidation of soft clay (Chen et al. 2003). Liu et al. (2014) established a subsidence model with changeable hydraulic conductivity that considered internal soil changes (including physical and mechanical properties) in the process of the strata deformation caused by water pumping. In this model, the Konzeny-Carman equation was adopted to describe the relationship between permeability and porosity.

Most previous research has focused on the consolidation deformation process caused by the reduction of the water level and the effective pressure (Cui and Tang 2010; Zhang et al. 2015; Xu et al. 2008; Liu and Huang 2013). However, ground deformation processes in a GSHP engineering project will have some special features: complicated hydrothermal conditions and a particle deposition effect. Groundwater flow and heat transport will go through a periodic change in GSHP engineering projects, which may affect the process of ground deformation. In addition, the biological, physical, and chemical factors, along with various types of environmental factors (such as temperature and seepage) can generate various particles such as silica, colloid and biofilm particles, with a large size range from 0.1–100 μm, that may be transported with groundwater and become deposited in pores (Bouwer 2002; Zheng et al. 2005). In the meantime, the flow characteristics of groundwater can be affected by the changing pore structure; moreover, the process of particle deposition can influence the mechanical property of the stratum. These aspects may lead to different features of the ground deformation process in the GSHP engineering project (Liu et al. 2016a, b; Low et al. 2008; Erban et al. 2014); however, few studies have considered this process.

The main objectives of this paper are: (1) to propose a ground deformation model to represent the dual-structure stratum at Wuhan; (2) to study the change law of the ground deformation and flow parameters while considering the particle deposition effect; (3) to analyze the characteristics of the ground deformation in GSHP engineering projects.

Characteristics of the dual-structure stratum at Wuhan

Wuhan is located in the eastern edge of the Jianghan Plain and on the banks of the Yangtze River (Fig. 1). Wuhan is covered with Quaternary loose sediments and is considered to be a typical dual-structure stratum (Fig. 2). In other words, the upper part is an aquitard layer with a relatively low permeability, while the lower part is a confined aquifer layer with a high permeability. This dual-structure stratum is distributed widely in the Wuchang, Hankou and Hanyang riverside areas, i.e., the first terrace of the banks of the Yangtze River and the Hanjiang River (Li 2010).

Fig. 1
figure 1

Study area

Full size image
Fig. 2
figure 2

Lithologic column of typical strata in Wuhan

Full size image

According to the particular geological structure and distribution of the groundwater, the stratum structure of the typical dual stratum in Wuhan can be divided into two groups.

  1. 1.

    Phreatic layer of muddy soil and clay. This layer has a low permeability and can be regarded as a confining bed.

  2. 2.

    Confined aquifer layer of medium-coarse or coarse sand. This layer has a high permeability and is always regarded as an aquifer.

In addition, the intermediate layer, which has permeability between the phreatic layer and the confined aquifer layer, is classified separately in places. This layer is mainly comprised of silty-fine sand in the Wuhan area; thus, it can also be regarded as an aquitard.

Methodology

Flow equation

The GSHP engineering system has two notable features: one feature is high-intensity pumping, which can cause a sharp decrease in the groundwater level; the other feature is the borehole filter that is placed in the confined aquifer layers that have high permeability. Thus, it is difficult to form a uniform groundwater head in the aquifers of a dual-structure stratum. In fact, the groundwater flow is mainly horizontal in the confined aquifer layer and vertical in the phreatic layer, and they each have an independent groundwater head.

The flow equation of the confined aquifer layer can be described as follows (Hochmuth and Sunada 1985; Li et al. 2012; Rupp and Selker 2006; Serrano and Workman 1998; Serrano 1995; Xu and Yu 2008):

$$ \frac{\partial }{\partial x}\left( KM\frac{\partial H}{\partial x}\right)+\frac{\partial }{\partial y}\left( KM\frac{\partial H}{\partial y}\right)+\frac{\partial }{\partial z}\left( KM\frac{\partial H}{\partial z}\right)+w={\mu}^{\ast}\frac{\partial H}{\partial t} $$
(1)

where K is hydraulic conductivity of the confined aquifer layer, M is the thickness of the confined aquifer layer, H is the groundwater head of the confined aquifer layer, w is the leakage recharge, μ∗ is the storage coefficient of the confined aquifer layer. All mathematical terms used in this paper are also given in the Appendix.

Equation (1) can be transformed into Eq. (2) to consider both skeleton compression and the thermodynamic effect (Bo et al. 2001; Zhang et al. 2006):

$$ \frac{\partial }{\partial x}\left( KM\frac{\partial H}{\partial x}\right)+\frac{\partial }{\partial y}\left( KM\frac{\partial H}{\partial y}\right)+\frac{\partial }{\partial z}\left( KM\frac{\partial H}{\partial z}\right)+w={\rho}_{\mathrm{w}}g\left(\alpha + n\beta \right)M\frac{\partial H}{\partial t}+n{\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(2)

where ρw is the density of water, g is the acceleration of gravity, α is the volume compression coefficient of the porous medium, n is the porosity, β is the volume compression coefficient of water, βT is the thermal expansion coefficient of water, T is the temperature of the groundwater; the groundwater and the porous medium are assumed to be in a state of thermal equilibrium.

The initial conditions are:

$$ \left\{\begin{array}{c}H\left(x,y,z,t\right)={H}_0\\ {}w\left(x,y,z,t\right)=0\end{array}\right.\kern0.5em t=0,\left(x,y,z\right)\in D $$
(3)

The boundary conditions are:

$$ \left\{\begin{array}{ll}H\left(x,y,z,t\right)={H}_0& t>0,\left(x,y,z\right)\to \infty \\ {}w\left(x,y,z,t\right)=Q& t>0,\left(x,y,z\right)\in \left({x}_0,{y}_0,{z}_0\right)\\ {}w\left(x,y,z,t\right)=\frac{h-H}{h-M}{K}^{\prime }& t>0,\left(x,y,z\right)\in D\end{array}\right. $$
(4)

where D is the area of study, H0 is the initial groundwater head, h is the groundwater head of the phreatic layer, K′ is the hydraulic conductivity of the phreatic layer.

The flow equation of the phreatic layer can be described as:

$$ \frac{h-H}{h-M}{K}^{\prime }={\mu}^{\prime}\frac{\partial h}{\partial t} $$
(5)

where μ′ is the specific yield of the phreatic layer.

Equation (5) can be transformed into Eq. (6) to consider both skeleton compression and the thermodynamic effect (Bo et al. 2001; Zhang et al. 2006):

$$ \frac{h-H}{h-M}{K}^{\prime }={\rho}_{\mathrm{w}}g\left(\alpha + n\beta \right)M\frac{\partial h}{\partial t}+n{\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(6)

The initial conditions are:

$$ \left\{\begin{array}{c}H\left(x,y,z,t\right)={H}_0\\ {}h\left(x,y,z,t\right)={H}_0\end{array}\right.\kern0.5em t=0,\left(x,y,z\right)\in D $$
(7)

The boundary conditions are:

$$ \left\{\begin{array}{c}H\left(x,y,z,t\right)={H}_0\\ {}h\left(x,y,z,t\right)={H}_0\end{array}\right.\kern0.5em t>0,\left(x,y,z\right)\to \infty $$
(8)

Thermal transport equation

There are four main factors that can influence heat transport in porous media:

  1. 1.

    Thermal advection, i.e., heat transfer as a result of the flow of the pore fluid

  2. 2.

    Thermal conduction, i.e., heat transfer as a result of temperature gradients

  3. 3.

    Thermal dispersion, i.e., heat transfer as a result of nonuniform velocities of the solute

  4. 4.

    Source-sink term

Based on these factors, a thermal transport equation that considers the particle transportation and deposition process can be obtained (Wang et al. 2012).

Thermal advection

Heat difference between the inflow and outflow of the element (hexahedron mini-element model) via thermal advection per unit time can be described as (Bergman and Incropera 2011)

$$ \left\{\begin{array}{c}\left[{C}_{\mathrm{w}}{Tv}_{\mathrm{x}}\left(x,y,z\right)-{C}_{\mathrm{w}}{Tv}_{\mathrm{x}}\left(x+\Delta x,y,z\right)\right] dydzdt=-\frac{\partial }{\partial x}\left({C}_{\mathrm{w}}{Tv}_{\mathrm{x}}\right) dxdydzdt\\ {}\left[{C}_{\mathrm{w}}{Tv}_{\mathrm{y}}\left(x,y,z\right)-{C}_{\mathrm{w}}{Tv}_{\mathrm{y}}\left(x,y+\Delta y,z\right)\right] dxdzdt=-\frac{\partial }{\partial y}\left({C}_{\mathrm{w}}{Tv}_{\mathrm{y}}\right) dxdydzdt\\ {}\left[{C}_{\mathrm{w}}{Tv}_{\mathrm{z}}\left(x,y,z\right)-{C}_{\mathrm{w}}{Tv}_{\mathrm{z}}\left(x,y,z+\Delta z\right)\right] dxdydt=-\frac{\partial }{\partial z}\left({C}_{\mathrm{w}}{Tv}_{\mathrm{z}}\right) dxdydzdt\end{array}\right. $$
(9)

where vx, vy, vz are components of Darcy velocity in x, y, z directions respectively, Cw is the volumetric thermal capacity of groundwater, T is temperature.

The total heat difference between the inflow and outflow of the groundwater of the element via thermal advection per unit time can be expressed as:

$$ -\operatorname{div}\left({C}_{\mathrm{w}} Tv\right) dxdydzdt $$
(10)

Thermal conduction

Fourier’s law states that the heat flux through a material is proportional to the negative gradient of the temperature that the heat transverses. Thus, the heat flux via thermal conduction can be expressed as:

$$ {I}_{\mathrm{T}}=-{\lambda}_{\mathrm{T}}\operatorname{grad}T $$
(11)

where λT is heat conductivity of bulk porous media.

Thermal dispersion

The hydrodynamic dispersion coefficient can express the diffusion capacity of a substance to porous media under a specified flow rate, and it includes the effects of both diffusion and pore-scale groundwater velocity variations. It reflects the effects of both groundwater flow and pore structure on the transportation of a solute on a macro-scale. Mechanical dispersion includes longitudinal and lateral dispersion, representing the dispersion phenomenon along and perpendicular to the flow direction, respectively. Thermal dispersion can both enhance viscous dissipation and intensify the heat diffusion in porous media.

The heat flux of thermal dispersion can be described as:

$$ {I}_{\mathrm{D}}=-{\lambda}_{\mathrm{D}}\operatorname{grad}T $$
(12)
$$ {\lambda}_{\mathrm{D}}={C}_{\mathrm{w}}g{\alpha}_0\left|v\right| $$
(13)

where λD is the thermal mechanical dispersion coefficient, α0 is thermal dispersity.

The heat flux per unit area via thermal dispersion can be described as:

$$ I={I}_{\mathrm{T}}+{I}_{\mathrm{D}}=-\left({\lambda}_{\mathrm{T}}+{\lambda}_{\mathrm{D}}\right)\operatorname{grad}T $$
(14)
$$ \lambda ={\lambda}_{\mathrm{T}}+{\lambda}_{\mathrm{D}} $$
(15)

where λ is the thermodynamic dispersion coefficient.

The heat difference between the inflow and outflow of the element via thermal conduction and mechanical dispersion per unit time can be described as:

$$ \left\{\begin{array}{c}\left[{I}_{\mathrm{x}}\left(x,y,z\right)-{I}_{\mathrm{x}}\left(x+\Delta x,y,z\right)\right] dydzdt=-\frac{\partial {I}_{\mathrm{x}}}{\partial x} dxdydzdt\\ {}\left[{I}_{\mathrm{y}}\left(x,y,z\right)-{I}_{\mathrm{y}}\left(x,y+\Delta y,z\right)\right] dxdzdt=-\frac{\partial {I}_{\mathrm{y}}}{\partial y} dxdydzdt\\ {}\left[{I}_{\mathrm{z}}\left(x,y,z\right)-{I}_{\mathrm{z}}\left(x,y,z+\Delta z\right)\right] dxdydt=-\frac{\partial {I}_{\mathrm{z}}}{\partial z} dxdydzdt\end{array}\right. $$
(16)

The total heat difference between the inflow and outflow of groundwater of the element via thermal conduction and mechanical dispersion per unit time can be expressed as:

$$ -\operatorname{div}(I) dxdydzdt $$
(17)

Source-sink term

The heat flux of the inflow and outflow of groundwater in the system as well as the source-sink term can be described as:

$$ {qC}_{\mathrm{w}}{T}^{\ast } dxdydzdt $$
(18)

where q is the volume flux of the source, T∗ is the temperature of the source.

Thermal transport equation

Heat changes caused by changes of temperature in an element per unit time are given by:

$$ {C}_{\mathrm{a}}\frac{\partial T}{\partial t} dxdydzdt $$
(19)

where Ca is the volume heat capacity of the aquifer.

The heat difference between the inflow and outflow result in the change of heat in the element; thus, the following can be deduced based on the law of conservation of energy:

$$ {C}_{\mathrm{a}}\frac{\partial T}{\partial t}=\operatorname{div}\left(\lambda \operatorname{grad}T\right)-\operatorname{div}\left({C}_{\mathrm{w}} Tv\right)+{qC}_{\mathrm{w}}{T}^{\ast } $$
(20)

Thus, it can be concluded that:

$$ \left[{nC}_{\mathrm{w}}+\left(1-n\right){C}_{\mathrm{r}}\right]\frac{\partial T}{\partial t}=\operatorname{div}\left\{\left[{\lambda}_{\mathrm{D}}+n{\lambda}_{\mathrm{w}}+\left(1-n\right){\lambda}_{\mathrm{r}}\right]\operatorname{grad}T\right\}-\operatorname{div}\left({C}_{\mathrm{w}} Tv\right)+{qC}_{\mathrm{w}}{T}^{\ast } $$
(21)

where Ca, Cw, Cr are heat capacities of the bulk porous medium, water and soil skeleton respectively, λD is the thermal mechanical dispersion coefficient of the porous media, λw, λr are heat conduction coefficients of the water and soil skeleton respectively, v is Darcy velocity.

Based on the law of particle deposition, the relationship between the porosity and the deposition mass of particles can be expressed as (Liu et al. 2016a, b):

$$ n=\frac{\rho_{\mathrm{s}}{n}_{\mathrm{i}}-S}{\rho_{\mathrm{s}}+C} $$
(22)

where ρs is the density of the suspended particles, ni is the initial porosity, S is the concentration of particles deposited in pores, C is the concentration of suspended particles.

Thus, it can be concluded that:

$$ {\displaystyle \begin{array}{l}\left[\frac{\rho_{\mathrm{s}}{n}_{\mathrm{i}}-S}{\rho_{\mathrm{s}}+C}{C}_{\mathrm{w}}+\frac{\rho_{\mathrm{s}}\left(1\hbox{-} {n}_{\mathrm{i}}\right)+C+S}{\rho_{\mathrm{s}}+C}{C}_{\mathrm{r}}\right]\frac{\partial T}{\partial t}=\\ {}\kern4em \operatorname{div}\left\{\left[{\lambda}_{\mathrm{D}}+\frac{\rho_{\mathrm{s}}{n}_{\mathrm{i}}-S}{\rho_{\mathrm{s}}+C}{\lambda}_{\mathrm{w}}+\frac{\rho_{\mathrm{s}}\left(1\hbox{-} {n}_{\mathrm{i}}\right)+C+S}{\rho_{\mathrm{s}}+C}{\lambda}_{\mathrm{r}}\right]\operatorname{grad}T\right\}-\operatorname{div}\left({C}_{\mathrm{w}} Tv\right)+{qC}_{\mathrm{w}}{T}^{\ast}\end{array}} $$
(23)

For particles at low concentration (ρs ≫ C), Eq. (23) can be simplified to:

$$ {\displaystyle \begin{array}{l}\left[\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){C}_{\mathrm{w}}+\left(1-{n}_{\mathrm{i}}+\frac{S}{\rho_{\mathrm{s}}}\right){C}_{\mathrm{r}}\right]\frac{\partial T}{\partial t}=\\ {}\kern4em \operatorname{div}\left\{\left[{\lambda}_{\mathrm{D}}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){\lambda}_{\mathrm{w}}+\left(1-{n}_{\mathrm{i}}+\frac{S}{\rho_{\mathrm{s}}}\right){\lambda}_{\mathrm{r}}\right]\operatorname{grad}T\right\}-\operatorname{div}\left({C}_{\mathrm{w}} Tv\right)+{qC}_{\mathrm{w}}{T}^{\ast}\end{array}} $$
(24)

This is the thermal transport equation of the aquifer that considers the particle deposition process.

Settlement analysis

The storage coefficient can be calculated as:

$$ {\mu}^{\ast }={\mu}_{\mathrm{s}}M={\rho}_{\mathrm{w}}g\left(\alpha + n\beta \right)M $$
(25)

where μ∗ is the storage coefficient, μs is the specific storage, ρw is the water density, n is the porosity, β is the water volume compressibility, α is the solid volume compressibility.

For particles at low concentration (ρs ≫ C), the relationship between the porosity and the deposition mass of particles can be described as (Liu et al. 2016a, b):

$$ n={n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}} $$
(26)

According to the Kozeny-Carman equation (Xu et al. 2008), the permeability is given by:

$$ k=\frac{n^3}{c{\left(1-n\right)}^2{s}^2} $$
(27)

where c and s denote the Kozeny constant and specific surface area based on the solid volume, respectively.

Thus, the hydraulic conductivity can be expressed as:

$$ K=\frac{\rho_{\mathrm{w}} gk}{\mu }=\frac{\rho_{\mathrm{w}}{gn}^3}{cs^2\mu {\left(1-n\right)}^2} $$
(28)

According to the features of aquifer compressibility:

$$ \alpha =0.434\frac{\left(1-n\right){C}_{\mathrm{C}}}{\sigma^{\prime }} $$
(29)

where CC is the compression index, σ′ is effective stress.

The flow equation of the confined aquifer layer can be described as:

$$ \frac{\partial }{\partial x}\left[M\frac{\rho_{\mathrm{w}}{gn}^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial x}\right]+\frac{\partial }{\partial y}\left[M\frac{\rho_{\mathrm{w}}{gn}^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial y}\right]+\frac{\partial }{\partial z}\left[M\frac{\rho_{\mathrm{w}}{gn}^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial z}\right]+w={\rho}_{\mathrm{w}} gM\left[0.434\frac{\left(1-n\right){C}_{\mathrm{C}}}{\sigma^{\prime }}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right)\beta \right]\frac{\partial H}{\partial t}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(30)

where w is given by:

$$ w=\frac{h-H}{h-M}{K}^{\prime }=\frac{h-H}{h-M}\frac{\rho_{\mathrm{w}} gn{\prime}^3}{cs^2\mu {\left(1-{n}^{\prime}\right)}^2} $$
(31)

Substituting Eq. (31) into Eq. (30):

$$ \frac{\partial }{\partial x}\left[\frac{Mn^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial x}\right]+\frac{\partial }{\partial y}\left[\frac{Mn^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial y}\right]+\frac{\partial }{\partial z}\left[M\frac{\rho_{\mathrm{w}}{gn}^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial z}\right]+\frac{n{\prime}^3\left(h-H\right)}{cs^2\mu {\left(1-{n}^{\prime}\right)}^2\left(h-M\right)}={\rho}_{\mathrm{w}} gM\left[0.434\frac{\left(1-n\right){C}_{\mathrm{C}}}{\sigma^{\prime }}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right)\beta \right]\frac{\partial H}{\partial t}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(32)

Ignoring the compressibility of water (β = 0):

$$ \frac{\partial }{\partial x}\left[\frac{Mn^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial x}\right]+\frac{\partial }{\partial y}\left[\frac{Mn^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial y}\right]+\frac{\partial }{\partial z}\left[M\frac{\rho_{\mathrm{w}}{gn}^3}{cs^2\mu {\left(1-n\right)}^2}\frac{\partial H}{\partial z}\right]+\frac{n^{\prime 3}\left(h-H\right)}{cs^2\mu {\left(1-n\right)}^{\prime 2}\left(h-M\right)}={\rho}_{\mathrm{w}} gM\left[0.434\frac{\left(1-n\right){C}_{\mathrm{C}}}{\sigma}\right]\frac{{}^{\prime}\partial H}{\partial t}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(33)

The flow equation of the phreatic layer can be described as:

$$ \frac{h-H}{h-M}\frac{n^{\prime 3}}{cs^2\mu {\left(1-{n}^{\prime}\right)}^2}={\rho}_{\mathrm{w}} gM\left[0.434{\frac{\left(1-n\right){C}_{\mathrm{C}}}{\sigma}}^{\prime }+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right)\beta \right]\frac{\partial h}{\partial t}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(34)

Ignoring the compressibility of water (β = 0):

$$ \frac{n{\prime}^3\left(h-H\right)}{cs^2\mu {\left(1-{n}^{\prime}\right)}^2\left(h-M\right)}={\rho}_{\mathrm{w}} gM\left[0.434\frac{\left(1-n\right){C}_{\mathrm{C}}}{\sigma}\right]\frac{{}^{\prime}\partial H}{\partial t}+\left({n}_{\mathrm{i}}-\frac{S}{\rho_{\mathrm{s}}}\right){\beta}_{\mathrm{T}}M\frac{\partial T}{\partial t} $$
(35)

Ground deformations caused by pumping and injection consist of deformations of the phreatic layer and the confined aquifer layer, that is:

$$ {S}_{\mathrm{T}}={S}_1+{S}_2 $$
(36)

where ST is the total deformation of the ground, S1, S2 are deformations of the phreatic layer and the confined aquifer layer, respectively, and are given by:

$$ {S}_1={M}_{S_1}\frac{\left({H}_0-{H}_1\right){\gamma}_{\mathrm{w}}m}{2{E}_{\mathrm{S}}} $$
(37)
$$ {S}_2={M}_{S_2}\sum \limits_{\mathrm{i}=1}^{\mathrm{n}}\left(\Delta {\sigma}_{\mathrm{Si}}\frac{m_{\mathrm{i}}}{E_{\mathrm{Si}}}\right) $$
(38)

where \( {M}_{S_1} \) and \( {M}_{S_2} \) are empirical coefficients of the settlement, H0 is the initial water head of the confined aquifer layer, H1 is the water head of the confined aquifer after extraction, γw is the unit weight of water, m is the thickness of the phreatic layer, ES and ESi are the compression modulus of the phreatic layer and the confined aquifer layer, respectively, ΔσSi is the effective stress increment of each layer, mi is the thickness of each layer.

In this manner, the ground deformation can be calculated according to the relationship between the water head and the change of the effective stress, combining Eqs. (24), (33) and (35) into (38) using a second-development of COMSOL Multiphysics. COMSOL Multiphysics is a universal engineering software based on finite-element analysis, which has a large set of functions for analyses and solutions, and provided a second-development interface at different levels. In this study, the mathematical model was solved through a modification of the built-in functions, which mainly includes Darcy’s law and the Biot poroelasticity constitutive model.

Study area

Engineering background

The Baibuting Garden community adopted GSHP technology for the heating and cooling of buildings. The GSHP system consists of three pumping wells and three recharge wells, and the pumping rate is 110 m3/h (both for the withdrawal wells and the recharge wells). The wells are 47 m deep, and 0.15 m wide in diameter. The well group is distributed in a region of 80 m × 80 m, as shown in Fig. 3.

Fig. 3
figure 3

Spatial distribution of the well group

Full size image

Geological condition

According to the geotechnical investigation on Baibuting Garden, the upper layer is mainly a soft plastic silty clay of Quaternary Holocene, the middle layer is a dense silty sand of Quaternary Holocene, and the lower layer is siltstone and pelitic siltstone of Silurian. The strata distributions are listed in Table 1.

Table 1 The distribution of soil layers
Full size table

To obtain the initial temperature distribution of the groundwater, two test wells were drilled near W1 and W2. The results of the groundwater temperature investigation at various depths and times are shown in Figs. 4 and 5. Figure 4 shows that the groundwater temperature increases slowly (it is almost constant) as depth increases below the depth of 10 m, whereas Fig. 5 illustrates the change law of the groundwater temperature over depth to 30 m at different times, and it shows that the effect of surface temperature can reach the depth of approximately 20 m, below which the temperature tends to be stable. The figure shows that the changes of surface temperature mainly affect the regions of the ground above 20-m depth. In other words, the temperature below 20-m depth can be regarded as constant. Based on this observation, the groundwater temperature was chosen to be 18 °C in this study area.

Fig. 4
figure 4

Initial temperature of the groundwater

Full size image
Fig. 5
figure 5

Groundwater temperature-depth profiles as a function of time (No. 1 testing well)

Full size image

The engineering site lies in the first terrace of the Yangtze River, and the groundwater is greatly affected by the river. Because of seasonal variation in the water level of Yangtze River, the groundwater flow will change over time; thus, it is advisable to study the effects of groundwater flow. Three modes are considered in this study, i.e., reverse flow, forward flow and cross flow, as shown in Fig. 6. In this study, the flow rate is 3 × 10−6 m/s (100 m/year).

Fig. 6
figure 6

Flow modes of groundwater: a. reverse flow, b. forward flow, c. cross flow

Full size image

Operational mode

In this study, the operational process of the GSHP is simplified to four steps in a cycle. The heating and cooling periods are both 90 d; between these periods, there is a 90-d inoperational period. For well flushing, the following two modes were chosen:

  • Continuous mode. Run continuously for 2 years, set W1, W2 and W3 as the pumping wells and W4, W5 and W6 as the recharge wells.

  • Flushing mode. The same as the continuous mode in the first year, and then the pumping and recharge wells were swapped in the second year.

Parameters

To simplify the computation, similar layers are merged. Generally, these lithologic layers can be divided into the phreatic layer and the confined aquifer layer. More narrowly, the miscellaneous fill layer and clay layer can be classified as the phreatic layer, and the fine sand layer can be classified as the confined aquifer layer. The geological parameters of each layer are presented in Table 2. The temperature of the recharge water in summer and winter are 30 and 5 °C, respectively.

Table 2 Geological parameter
Full size table

Results

Different recharge volumes

To study the deformation law of the ground for different recharge volumes, three modes (i.e., no recharge, half recharge and total recharge) are calculated. Figure 7 shows the ground deformation for different recharge volumes on day 90, and the ground deformation data were collected on the east–west axis shown in Fig. 3. Figure 8 shows the deformation law of the central point for different recharge volumes during a whole year.

Fig. 7
figure 7

Ground deformation for different recharge scenarios on the east–west axis (day 90)

Full size image
Fig. 8
figure 8

Deformation versus time for different recharge scenarios (in central point)

Full size image

Figures 7 and 8 show that maximum deformation was located near the pumping well and was severely affected by the recharge volume. The settlement reached 96 mm near the pumping well for the no recharge condition, whereas for the total recharge condition it was only 60 mm. The deformation decreases as the recharge volume increases; it can even uplift the ground by up to 29 mm for the total recharge mode. Figure 8 also shows that land subsidence has a periodic change rule for the pumping and injection process for a year, i.e., the ground deformation is similar during the heating and cooling periods. During the two inoperational periods, the ground deformation is almost the same. In addition, during the inoperational period, the land subsidence at the central point is approximately 21.5 mm, and this value is minimally affected by the recharge volume.

Different operational modes

To study the ground deformation law for different operational modes, the continuous mode and flushing mode are calculated; the result is illustrated in Fig. 9. The ground deformation data were collected on the east–west axis shown in Fig. 3. It is obvious that the ground deformation is significantly influenced by the operational mode. The ground deformation law is very different for the two operational modes—for example, the largest ground settlement site is near −40 and +40 m for the continuous mode and flushing mode, respectively; however, the maximum settlement value changes from 77.6 to 77.8 mm, i.e., it was almost unchanged.

Fig. 9
figure 9

Ground deformation under different operational modes on the east–west axis (day 450)

Full size image

Different flow modes

To study the ground deformation law for different flow modes, four flow modes—no flow, forward flow, reverse flow and cross flow—were considered, the result of which is illustrated in Fig. 10. The ground deformation data were also collected on the east–west axis shown in Fig. 3. The results show that ground deformation is barely affected by the groundwater flow in the GSHP operation stage. In other words, the initial background groundwater flow regime has little effect on the ground deformation during the operational stage. Consequently, the effect of groundwater flow during the operational stage can be neglected.

Fig. 10
figure 10

Ground deformation under different flow modes on the east–west axis (day 90)

Full size image

Considering particle deposition

The ground deformation and deformation change (%) after operation for two cycles for the continuous mode are illustrated in Figs. 11 and 12, respectively, and the ground deformation data were also collected on the east–west axis shown in Fig. 3. Deformation change (%) is defined as follows:

$$ \mathrm{Deformation}\ \mathrm{change}\left(\%\right)=\frac{S_{\mathrm{nd}}-{S}_{\mathrm{cd}}}{S_{\mathrm{nd}}}\times 100\% $$
(39)

where Snd is the ground settlement that neglects the particle deposition effect, Scd is the ground settlement that considers the particle deposition effect.

Fig. 11
figure 11

Ground deformation under continuous mode on the east–west axis

Full size image
Fig. 12
figure 12

Deformation change (%) under continuous mode on the east–west axis (with deposition)

Full size image

Figure 11 shows that the particle deposition process has an effect on the deformation of the ground, and the effect will significantly increase as time increases. The results showed that the maximum ground settlement near the pumping well is 77.6 mm when the particle deposition effect is neglected. When the particle deposition is considered, this value is 76.0 mm on day 90, 75.3 mm on day 270 and 69.4 mm on day 630. Figure 12 shows that the maximum deformation change (%) is distributed over an area (x) of 30–40 m. In addition, the curves of the deformation change (%) are highly related to the curves of the particle deposition, and the position of maximum change (%) is also the location where most particles are deposited. The deformation change (%) at this position is as high as 43.3%.

The ground deformation law for the continuous mode during the shut-off period is illustrated in Figs. 13 and 14, and the ground deformation data were also collected on the east–west axis shown in Fig. 3. Figure 13 shows that the particle deposition process has some influence on the deformation of the ground during the shut-off period. The maximum ground deformation of 21.5 mm is at the center when the effect of particle concentration is neglected. When the deposition effect is considered, the position of the maximum ground deformation moves to x = −4.8 m, and ground deformation at the center is relatively small due to particle deposition. Additionally, based on Fig. 14, the curves of the deformation change (%) are highly related to the curves of particle deposition.

Fig. 13
figure 13

Ground deformation under continuous mode on the east–west axis at day 180

Full size image
Fig. 14
figure 14

Deformation change (%) under continuous mode on the east–west axis at day 180

Full size image

The ground deformation law for the flushing mode on days 450 and 630 are illustrated in Figs. 15 and 16, respectively, and the ground deformation data were also collected on the east–west axis shown in Fig. 3. The maximum ground deformation lies near the pumping well, i.e., at x = −40 m (Fig. 15). The particle deposition quantity in the stratum is greatly reduced because of well flushing compared with the previous year. This change in particle deposition quantity leads to a slight change in ground deformation during days 450–630, and the maximum deformation change (%) changes from 6.7 to 9.1%. Figure 16 shows that the maximum deformation change (%) is distributed around x = 5.2 m, which is the location where the most particles are deposited.

Fig. 15
figure 15

Ground deformation under flushing mode on the east–west axis

Full size image
Fig. 16
figure 16

Deformation change (%) under flushing mode on the east–west axis (with deposition)

Full size image

A study on the effect of the groundwater flow in the GSHP operation stage has been discussed in the previous section; thus, its influence in the shut-off stage is discussed in this section. The ground deformation and the deformation change (%) during the shut-off period for different flow modes are illustrated in Figs. 17 and 18, respectively, and the ground deformation data were also collected on the east–west axis shown in Fig. 3. The results show that the flow mode of the groundwater can have an effect on the ground deformation, but the effect is very weak. For the condition of forward flow, the ground deformation near the pumping well decreases, whereas near the recharge well, the value increases, due to the transport of particles towards the pumping wells. In contrast, the ground deformation near the pumping well increases, and that near the recharge well decreases for the reverse flow; this behavior can be attributed to particle transport towards the recharge wells. The ground deformation law for cross flow is similar to that for reverse flow, i.e., the ground deformation near the pumping well increases, whereas near the recharge well, the value decreases. It is concluded that the particle concentration in the central area is relatively high; thus, particles may migrate under the effects of groundwater scouring. Overall, the effect of groundwater flow on the ground deformation is relatively small.

Fig. 17
figure 17

Ground deformation under different flow modes on the east–west axis (day 180)

Full size image
Fig. 18
figure 18

Deformation change (%) under different flow modes on the east–west axis (day 180)

Full size image

Conclusions

In this paper, a ground deformation model that considers the characteristics of a GSHP engineering project and the effect of particle deposition was proposed for the dual-structure stratum in Wuhan. The main conclusions are as follows.

  1. 1.

    Based on the characteristics of a typical dual-structure stratum in Wuhan, a ground deformation model that considers the changing porosity and declining permeability caused by particle deposition in the ground was developed to determine the value of ground deformation.

  2. 2.

    A mathematical model of thermal transport in a confined aquifer layer was established, and the effects of thermal advection, thermal conduction and thermal dispersion were considered. In addition, the effects of particle deposition on the formation parameters were also considered in the thermal transport equation.

  3. 3.

    Particle deposition during the long-term pumping and recharge process has an effect on the deformation of the ground, with the effect significantly increasing over time. The curves of the deformation change (%) are highly related to the curves of particle deposition. Moreover, different flow modes of the groundwater can have an effect on ground deformation, but the effect is very weak.