Introduction

Hot dry rock (HDR) is a new kind of renewable and environmental-friendly energy to be exploited for electricity generation (Zarrouk and Moon 2014). Typically, this geothermal energy is often stored in high temperature and low permeable crystalline rocks at a depth of at least 2000 m with an in situ temperature typically above 150 °C (Maxwell 2014). The enhanced geothermal system (EGS) is commonly used for petrothermal systems if their permeability is artificially increased due to hydraulic stimulation. To extract the heat from HDR, EGS is also applied with artificial water injection due to the low and no permeability of HDR reservoir. This system involves a dual-well system by pumping cold fluids through a well into HDR, then extracting the heated fluids through another well (Jelacic et al. 2008). Many EGS projects have been tested in the USA, UK, Japan, Germany and Australia, while there is no successful commercial plant due to the high cost of system development (Panel 2006; Chen and Jiang 2015; Xie and Min 2016). In China, the study of HDR started in 2010 (Lu et al. 2017). It has been estimated that the total HDR energy in China is 20.9 × 106 EJ, which is equivalent to the standard coal of 714.9 × 1012 t (Wang et al. 2012). Gonghe basin, Songliao basin and Yangbajiang have been selected as the potential areas for further HDR study (Guiling et al. 2020).

As tools for geothermal resource exploration, geophysical methods are often applied for geological analysis and modelling in geothermal fields. When considering the density and susceptibility contrast, gravity and magnetic data are applied for delineating the geological structure, such as buried faults, the depth of Moho interface and the Curie point depth (e.g. Uwiduhaye et al. 2018; Xi et al. 2018; Zaher et al. 2018; Zhao et al. 2019). The electromagnetic (EM) methods [e.g. MT (magnetotelluric)] could also delineate the faults and potential geothermal areas. The high conductivity or low resistivity anomalies could reflect the distribution of hot-saline fluids, resistive host rocks, or even partial melting (Peacock et al. 2012; Kana et al. 2015; Samrock et al. 2015). At the exploitation stage, the seismic method (e.g. the P-wave or S-wave analyse) could also be used for the faults and granite mapping, as well as the fracture monitoring (Khair et al. 2015; Li et al. 2018; Fan et al. 2020).

The Gonghe geothermal field is in the north-eastern margin of the Qinghai–Tibet Plateau. Several works on the geothermic mechanism were conducted in this area. Zhao et al. (2020) outlined the possible buried faults and calculated the regional depth of Moho interface and Curie point depth by large-scale gravity and magnetic data, with the assumption of the heat source in the upper crust. Gao et al. (2018) used 3D MT imaging method to characterise 3D resistivity models with the conclusion of heat source at the middle crust in the Gonghe Basin. Furthermore, Zhang et al. (2018a) measured the latest heat flow and conducted the 1D theoretical crustal thermal structures simulation. The unusual heat flow here was caused by the cooling of a shallow magma chamber. For the theoretical study, Xu et al. (2018) simulated the power generation potentiality for EGS and provided the strategy for the HDR exploitation, along the profile of DR4–QR1–DR3–DR2 wells in Qiabuqia (Gonghe town).

Current geophysical studies in this area are mainly focused on the regional structure. Small-scale geophysical data are thereby required for the local geological analysis so as to provide prior information for the EGS site determination and HDR modelling. In this paper, we aim to use gravity and magnetic data in Gonghe town and its surrounding area to conduct the density and susceptibility modelling. 2D manual inversion was conducted for the analysis of the geological structure, in-depth to 10 km. 3D joint inversion shows the models within 3500 m depth. With lithology record and temperature log along DR4-QR1-DR3-DR2 wells, the HDR distribution and application for energy production was then described.

Regional setting

Gonghe basin is in the middle-west part of China and the northeast margin of the Qinghai–Tibet Plateau. In this area, the topography is majorly dominated by the Qinghai–Tibet Plateau (Sun et al. 2007). Many hydrothermal springs have been discovered along the NE–SW and N–S extending faults (Liu et al. 2017) and the hot water comes from the deeper part of the basin (~ 1000 m), with a high concentration of arsenic and other minerals (Fang et al. 2009). Thermal survey results also reveal an excellent geothermal condition in this basin, in terms of geothermal gradient and terrestrial heat flow. The geothermal gradient could be over 6.8 °C/100 m, and the temperature at a depth of 3705 m could be 236 °C (Yue et al. 2015; Zhang et al. 2018b). The heat flow here is as high as 119.3 mW/m2 (Xu et al. 2019), far more than the world’s mean rate (65 mW/m2 over continental crust) (Pollack et al. 1993).

This basin is also named as the Gonghe Aulacogen (Zhang et al. 2004), with a complex geological background (e.g. He et al. 2017). Generally, it is tectonically controlled by the sinistral strike-slip framework of the Kunlun Orogen, Qinling Orogen to the south and Qilian Orogen to the north in Fig. 1a (Fang et al. 2005). The edges are surrounded by several faults, which formed in the Cenozoic era accompanying volcanic eruptions. For different directions, it is bordered by the foothills of the Qinghainan Fault (north), the Animaqing Suture Zone (extended from Kunlun Fault) (south), the Wahongshan Fault (West) and the Duohemao Fault (east). Besides, the Yellow River flows in the central part of the basin and spills this basin with an NE orientation. These NNW–SSE and NWW–SEE faults bounding the basin give rise to its rhomboid shape (Zhang et al. 2006).

Fig. 1
figure 1

a Simplified tectonic map of Gonghe–Guide area and its neighbouring region (after Zhang et al. 2018) including the location of the study area. (b) Topographic map of the study area. AB and CD indicate the position of the manual inversion profiles. G1–G2 indicates the position of stratigraphic distribution profile in Fig. 5

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Quaternary and Neogene sediments mainly cover the land in this area. The Cenozoic overburden thickness in this basin could be greater than 5000 m (Sun et al. 2011). Near the Gonghe town, the thickness of the sedimentary layers is around 1300–1500 m, where the hot water mainly originates. Moreover, the main igneous rocks, including granite and granodiorite, are more widely distributed (Zhang et al. 2006). The granite rocks were formed in Indosinian and Yanshanian periods and laid the basement at a depth of over 2000 m. The granite with temperatures more than 150 °C is the main target of the HDR reservoir (Yan 2015; Zhang et al. 2018b).

Materials and methods

Study area

Our study area is the Gonghe town (Qiabuqia) with its surrounding area. It is in the north-central part of the Gonghe Basin and located about 30 km south away from the Qinghai Lake. It covers approximately 820 km2 (Fig. 1a), and the elevation ranges from 2550 to 3350 m above the sea level. It is higher in the north part, at the northing of 4025 000–4027 000 m. The lowest area is in the south-eastern part near the Tiegai town.

Prior to the gravity and magnetic measurements, more than ten wells have been drilled here (e.g. DR4, QR1, QR3, DR2 and GR1). The temperature could reach around 203 °C at 3600 m depth in the well GR1 (not shown in the figure) (Zhang et al. 2018b). The lithology drilling records also indicate a two-layer system here. That is, the shallow part is the hydrothermal reservoir, with the sediments of around 900–1500 m thickness. It is underlain by granite, which is the target HDR area (Xu et al. 2018). Besides, the thermal testing for granite samples here shows the average thermal conductivity of 1.86 W/(m K) and the heat generation rate of 2.41–5.73 µW/m3 (Zhang et al. 2018a).

Gravity and magnetic data collection and interpolation

For the data collection, we used the CG-5 relative gravimeter and GSM-19T proton precession magnetometer to gather the gravity and magnetic data, respectively (Reudink et al. 2014). GPS was used to acquire positioning coordinates at each station. In total, the gravity dataset consists of 4200 stations with a spacing of approximately 500 m. The magnetic dataset has 7 360 stations with around 200 m spacing.

Raw gravity data were corrected through the standard corrections to achieve the complete Bouguer anomaly (Uwiduhaye et al. 2018). Free air and Bouguer corrections were calculated using the International Association of Geodesy 1967 formula (Tóth et al. 2005) and terrain correction was applied with the 90 m-SRTM data (Jarvis et al. 2008). The original magnetic data were also corrected, including the normal correction and diurnal correction (Dobrin and Savit, 1960; Nabighian et al. 2005). Meanwhile, reduction to magnetic poles was also conducted, as magnetic anomalies usually do not overlap with the structure (Over et al. 2018). The processed gravity and magnetic (anomaly) data were then interpolated across the study area onto a common 200-m grid, using ordinary local kriging and a neighbourhood of 20–30 points.

Regional–residual separation of gravity and magnetic data

The Bouguer gravity anomaly and magnetic anomaly are controlled by a regional trend resulting from the presence of deep and large structures. In most cases, there is no information on density or magnetic susceptibility heterogeneities in the deep zones (Gabtni and Jallouli 2017). Many methods have been applied to the regional–residual separation, such as second and higher polynomial fitting and upward continuation (Jacobsen 1987; Wang 2006; Zeng et al. 2007; Represas et al. 2013).

In this paper, we used the upward continuation to filter the gravity and magnetic data. This method attenuates high wavelength anomalies related to shallow geological bodies and provides a possibility to image deeper structures associated with regional gravity and magnetic patterns. To extract a regional field at a depth of z0, the height of upward continuation as 2z0 is required (Gabtni and Jallouli 2017). In this way, the residual anomaly could be then regarded as the response from the shallow geological bodies. When it goes to the HDR delineation, the upper boundary has been achieved is around 2500 m (Xu et al. 2018). Therefore, we chose 7 km as the height of upward continuation, then the residual anomaly could then reflect the density and magnetic susceptibility distribution in those parts of the area which are shallower than 3500 m.

2D manual modelling

The principal application of the magnetic and gravity data is to determine the depth of the anomalous sources of the observed anomalies, and the quantitative interpretation of the gravity and magnetic could be carried out through trial-and-error modelling assuming the 2D approach. Oasis Montaj software is used in gravity and magnetic processing, interpretation and visualisation. In this software, we could conduct 2D modelling via a module called gym-sys. The initial geophysical model could be set up according to the MT or seismic evidence, and then we need to adjust the shape or value of the model so that the calculated curve fits the raw data curves as much as possible (Oasis Montaj 2006).

We selected two profiles AB and Cd across the study area (Fig. 1b). The profile AB was in the NS direction and passed near the wells and the Gonghe town. The profile CD was in the EW direction and across the town. Prior to manual modelling, the initial model creation was mandatory. We referred to the MT inversion models by Gao et al. (2018). His results revealed that the surface down to around 2000–3000 m was a conductive zone corresponding to the Quaternary and Neogene sediments. A high resistivity zone below this area was down to ∼10 to 15 km depth, which was associated with non-weathered metamorphic and igneous rocks. Therefore, our initial model was then set up with three parts from the surface to the depth of around 12 km. The upper one was regarded as the Quaternary sediments, the middle one was Neogene sediments, and the bottom was the granite rocks. For the average density value, the Quaternary and Neogene sediments in Qinghai Province were measured as 1.80 and 2.34 g/cm3, respectively (Hu 1991). In the study area of Gonghe town, the experimental density and magnetic susceptibility of granites are 2.532–2.555 g/cm3 and 0.0456 × 10–3 ~ 13.847 × 10–3 SI separately (Xing 2017).

3D cross-gradient joint inversion

Unlike manual modelling, which needs labours to modify models, 3D inversion with optimisation algorithm could achieve the models automatically. Regarding the structural similarity, the rock structure or boundaries may coincide between different geophysical models (e.g. density, susceptibility), and thus the joint inversion could serve for multiple property coupling and integration (Gallardo and Meju 2011). The cross-gradient function could detect differences of physical property fields within large or small gradients without any discontinuities or singularities, so the cross-gradient joint inversion has also been applied and proven to be useful and stable, through various applications on multiple joint inversion problems (Pak et al. 2017). However, this kind of joint inversion relies on the algorithm of the single dataset inversion. In this paper, we used the gravity and magnetic joint inversion based on the smooth \({l}^{0}\) norm constraint of minimum support functional stabiliser (SL0-MS) (Portniaguine and Zhdanov 2002; Zhang et al. 2020). It shows the ability to generate focused images for geological structures with higher efficiency.

For this method, the minimisation of the cross-gradient joint inversion is specified as (Gallardo and Meju 2004; Pak et al. 2017):

$${\text{min }}\varphi = \varphi _{{\text{g}}} + \varphi _{{{\text{mag}}}} ,{\text{ subject~~to}}~\tau = 0$$
(1)

where \(\boldsymbol{\varphi }\) stands for the objective function for the joint inversion, \({\boldsymbol{\varphi }}_{g}\) and \({\boldsymbol{\varphi }}_{mag}\) are the objective functions of single gravity or magnetic inversion, \({\varvec{\uptau}}\) is the cross-gradient of the density and susceptibility model data sets.

\({\boldsymbol{\varphi }}_{g}\) and \({\boldsymbol{\varphi }}_{mag}\) are expressed as

$${\boldsymbol{\varphi }}_{g}={{(\mathbf{W}}_{{d}_{g}}{\mathbf{A}}_{g}\left({{\varvec{m}}}_{g}\right)-{\mathbf{W}}_{{d}_{g}}{\mathbf{d}}_{g})}^{\mathrm{T}}{(\mathbf{W}}_{{d}_{g}}{\mathbf{A}}_{g}\left({{\varvec{m}}}_{g}\right)-{\mathbf{W}}_{{d}_{g}}{\mathbf{d}}_{g})+{{\alpha }}_{g}{\left({\mathbf{W}}_{{e}_{g}}^{SL0}{\mathbf{W}}_{{m}_{g}}{{\varvec{m}}}_{g}-{\mathbf{W}}_{{e}_{g}}^{SL0}{\mathbf{W}}_{{m}_{g}}{\mathbf{m}}_{g\bullet apr}\right)}^{\mathrm{T}}({\mathbf{W}}_{{e}_{g}}^{SL0}{\mathbf{W}}_{{m}_{g}}{{\varvec{m}}}_{g}-{\mathbf{W}}_{{e}_{g}}^{SL0}{\mathbf{W}}_{{m}_{g}}{\mathbf{m}}_{g\bullet apr}).$$
(2)
$${\boldsymbol{\varphi }}_{mag}={{(\mathbf{W}}_{{d}_{mag}}{\mathbf{A}}_{mag}\left({\mathbf{m}}_{mag}\right)-{\mathbf{W}}_{{d}_{mag}}{\mathbf{d}}_{mag})}^{{\varvec{T}}}{(\mathbf{W}}_{{d}_{mag}}{\mathbf{A}}_{mag}\left({\mathbf{m}}_{mag}\right)-{\mathbf{W}}_{{d}_{mag}}{\mathbf{d}}_{mag})+{ \alpha }_{mag}{\left({\mathbf{W}}_{{e}_{mag}}^{SL0}{\mathbf{W}}_{{m}_{mag}}{\mathbf{m}}_{mag}-{\mathbf{W}}_{{e}_{mag}}^{SL0}{\mathbf{W}}_{{m}_{mag}}{\mathbf{m}}_{mag\bullet apr}\right)}^{{\varvec{T}}}({\mathbf{W}}_{{e}_{mag}}^{SL0}{\mathbf{W}}_{{m}_{mag}}{\mathbf{m}}_{mag}-{\mathbf{W}}_{{e}_{mag}}^{SL0}{\mathbf{W}}_{{m}_{mag}}{\mathbf{m}}_{mag\bullet apr}).$$
(3)
$${\varvec{\tau}}\left({\mathbf{m}}_{g},{\mathbf{m}}_{mag}\right)=\nabla {\mathbf{m}}_{g}\times \nabla {\mathbf{m}}_{mag }.$$
(4)

For (2)–(4), the subscripts \(g\),\(mag\) are related to the gravity (density) and magnetic (susceptibility), respectively. \(\boldsymbol{\varphi }\) is the objective function, \({\mathbf{W}}_{d}\) is the covariance matrix of the observed dataset, \(\mathbf{A}\) is the kernel for forward, \(\mathbf{m}\) is the model matrix, \(\mathbf{d}\) is the observed dataset, \({\alpha }\) is the damping factor, \({\mathbf{W}}_{e}^{SL0}\) is the weighting matrix constrained by smooth \({l}^{0}\) norm method, \({\mathbf{W}}_{m}\) is the covariance matrix of models, \({\mathbf{m}}_{apr}\) is the prior model matrix, T stands for the transpose, \(\nabla\) is the gradient, \({\varvec{\tau}}\) is the cross-gradient of the density and susceptibility models.

Using first-order Taylor expansion:

$$\mathbf{t}\left(\sigma ,\kappa \right)=\mathbf{t}\left({\sigma }^{0},{\kappa }^{0}\right)+\left[\begin{array}{cc}{\mathbf{B}}^{g}& {\mathbf{B}}^{m}\end{array}\right]\left[\begin{array}{c}\sigma -{\sigma }^{0}\\ \kappa -{\kappa }^{0}\end{array}\right].$$
(5)

\(\sigma\), \(\kappa\) represent the density and susceptibility models, and \({\mathbf{B}}^{g}, {\mathbf{B}}^{m}\) are the corresponding cross-gradient partial derivative, that is, \({\mathbf{B}}^{g}=\frac{\partial \mathbf{t}\left(\sigma ,\kappa \right)}{\partial \sigma }\) and \({\mathbf{B}}^{m}=\frac{\partial \mathbf{t}\left(\sigma ,\kappa \right)}{\partial \kappa }\).

The variational form for that would be

$$\delta {\text{P}}_{g}^{{\alpha_{g} }} \left( {{\mathbf{m}}_{g}^{{\omega_{g} }} ,{\mathbf{d}}_{g}^{{\omega_{g} }} } \right) = 2\left( {\delta {\mathbf{m}}_{g}^{{\omega_{g} }} } \right)^{T} \left( {{\mathbf{F}}_{{\omega_{g} }}^{T} \left( {{\mathbf{A}}_{g}^{{\omega_{g} }} \left( {{\mathbf{m}}_{g}^{{\omega_{g} }} } \right) - {\mathbf{d}}_{g}^{{\omega_{g} }} } \right) + \alpha_{g} \left( {{\mathbf{m}}_{g}^{{\omega_{g} }} - {\mathbf{m}}_{g \cdot apr}^{{\omega_{g} }} } \right) + \mu_{g} {\mathbf{B}}_{g}^{T} {\mathbf{t}}\left( {{\mathbf{m}}_{g}^{{\omega_{g} }} ,{\mathbf{m}}_{mag}^{{\omega_{mag} }} } \right)} \right)$$
(6)
$$\delta {\text{P}}_{mag}^{{\alpha_{mag} }} \left( {{\mathbf{m}}_{mag}^{{\omega_{mag} }} ,{\mathbf{d}}_{mag}^{{\omega_{mag} }} } \right) = 2\left( {\delta {\mathbf{m}}_{mag}^{{\omega_{mag} }} } \right)^{T} \left( {{\mathbf{F}}_{{\omega_{mag} }}^{T} \left( {{\mathbf{A}}_{mag}^{{\omega_{mag} }} \left( {{\mathbf{m}}_{mag}^{{\omega_{mag} }} } \right) - {\mathbf{d}}_{mag}^{{\omega_{mag} }} } \right) + \alpha_{mag} \left( {{\mathbf{m}}_{mag}^{{\omega_{mag} }} - {\mathbf{m}}_{mag \cdot apr}^{{\omega_{mag} }} } \right) + \mu_{mag} {\mathbf{B}}_{mag}^{T} {\mathbf{t}}\left( {{\mathbf{m}}_{g}^{{\omega_{g} }} ,{\mathbf{m}}_{mag}^{{\omega_{mag} }} } \right)} \right)$$
(7)

In (6) and (7), \({\mathbf{F}}_{{\omega }_{type}}={\mathbf{W}}_{{d}_{type}}{\mathbf{F}}_{type}{\mathbf{W}}_{{m}_{type}}^{-1}{{(\mathbf{W}}_{{e}_{type}}^{SL0})}^{-1}\) with the corresponding type of gravity or magnetic methods. \({\mathbf{F}}_{type}\) is the Fréchet partial derivative matrix of \({\mathbf{A}}_{type}\left({\mathbf{m}}_{type}\right)\).

With the optimisation algorithm, such as the conjugate gradient method, the density and susceptibility models would be achieved when it meets the maximum iteration times or the RMS requirement.

Results

Spatial distribution of gravity and magnetic data

For the original kriged data, the gravity anomaly value (Fig. 2a) ranges from – 400 to – 360 mGal. In the north and southeast edge, it is higher (− 380 ~  − 365 mGal), while the central part is dominated by a lower negative anomaly (− 395 ~ − 400 mGal). As for the magnetic anomaly in Fig. 2b, it has a different tendency with the range from − 10 to 195 nT. The zone with high value (50–90 nT) is concentrated in the middle part with an NNE–SSW trend.

Fig. 2
figure 2

Spatial distribution in the study area of (a) gravity kriged data; (b) magnetic kriged data. Through the upward continuation with the height of 7 km: (c) gravity regional data, (d) magnetic regional data, (e) gravity residual data and (f) magnetic residual data

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The regional gravity anomaly in Fig. 2c is similar to the original one but with a smoother trend. The value is in a low rate (− 395 ~ − 387 mGal) in the Gonghe and Shazhuyu towns. The regional magnetic anomaly shown in Fig. 2d also keeps a similar trend, and the edges are more apparent in the south part of Gonghe town.

The spatial distribution of residual anomalies in Fig. 2e, f appear differently from the original kriged and regional data. For the gravity result, the value is from − 5.4 to − 1.4 mGal. The whole area is majorly covered by the high anomaly (> − 3.1 mGal), while it is lower in the west and southeast margins (< − 4.0 mGal). The residual magnetic anomaly ranges in − 8 ~ 22 nT. Similarly, there is still a low-value zone (< − 3 nT) in the north-eastern part of Gonghe town. However, the remaining area is in a medium value (− 2 ~ 6 nT).

2D manual inversion modelling

Figure 3 shows the 2D manual joint inversion models of gravity and magnetic data in AB and CD profiles. For these two models, the geophysical responses match the observed data well, as the RMS is 3.246 (magnetic in AB), 0.79 (gravity in AB), 1.551 (magnetic in CD), 1.99 (gravity in CD).In AB profile, the magnetic curve fluctuates, and it is higher in the northern part. In contrast, the gravity curve goes down and then up smoothly. In CD, the magnetic value also changes, and two peaks appear, while the gravity one in this profile keeps stable at the rate of – 394 mGal.

Fig. 3
figure 3

2D manual joint inversion models of gravity and magnetic data in AB and CD profiles of the study area

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For both manual models, the density of the first and second layers are both estimated as 1.8 g/cm3 and 2.34 g/cm3, respectively. The susceptibility is assumed as zero, in terms of the contrast between the sediments and granites. The thickness of the sedimentary layers is then around 1500 m. Although the bases are composed of granites, they have different properties. In profile AB, there are five blocks. The density is set with a small change as 2.48–2.58 g/cm3. This is reasonable because there is no apparent gravity anomaly along the profile. The susceptibilities are in a range of 0.005–0.0138 SI. As for CD, there are three blocks. The major one is with a lower susceptibility (0.0063 SI) but a higher density (2.58 g/cm3), The other two blocks are in a smaller size and their bottom depth is around 8000 m. In addition, deep faults could be outlined between different blocks, which could reach to 12 km depth, as shown in AB.

3D joint inversion results

The inversed density and magnetic susceptibility models are depicted in Fig. 4. These models show the distribution of density and susceptibility deviation. The density deviation is to the 2.58 g/cm3, which is the assumed widespread granite density in the 2D manual inversion. For the magnetic susceptibility, it is the absolute value, when considering the susceptibility of the sedimentary layers is 0.

Fig. 4
figure 4

Through 3D gravity and magnetic joint inversion: (a) 3D density model, (b) 3D magnetic susceptibility model

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For these two models, they have different distributions and their own characteristics. For the density model in Fig. 4a, it could be divided into two parts in the vertical perspective. The upper zone (0–1400 m) is rather uniform in the low-value band (< − 0.3 g/cm3). With the depth increases, the transition zone occurs at around 1400–2000 m. The density increases in the middle part. Below this area, the distribution of density goes to be stable and two blocks appear. One is in a higher density (> − 0.1 g/cm3) and mainly locates in the north-eastern part covering Gonghe town and Dongba. The size of this block decreases little when it comes to 3000–3500 m. The other block with a low density (< − 0.3 g/cm3) is in the south-western part, and its value is consistent when it goes deeper.

However, the susceptibility model in Fig. 4b has an entirely different trend. Its value ranges from 0 to 5 × 10–3 SI. In the top part within 0–600 m, it distributes dispersedly, and the value in the central part is relatively lower. Beneath this part, major block with the susceptibility of more than 3.5 × 10–3 SI exists in the central part of the area. The block is elongated in the NNE–SSW trend as in the kriged magnetic map. This block expands when it goes deeper and nearly covers the whole area at 3000 m depth. However, the block size decreases at the sections with 3000 m to about 3500 m depth.

Discussion

Several studies discuss the geothermal mechanism here. For the heat source, it might be the magma capsules or melted granites in the upper crust (Zhao et al. 2020). The instantaneous heat conductive process to transport the heat. (Wannamaker et al. 2004; Gao et al. 2018). From the resistivity models, there are several vertical narrow conductive zones between the near-surface conductive zone and the middle crust conductive zone as hidden faults. These faults are served as the channels for transporting heat from the heat source to the shallow zone (around the depth of 3000 m) (Gao et al. 2018), The difference of the density and susceptibility would be then affected by the melting bodies with the intrusion since Late Triassic (Zhang et al. 2018a).

When it goes to the shallow part, the 3D resistivity model from MT works shows significantly conductive anomalies at depths above ∼3000 m, related to HDR or hydrothermal reservoirs. The Quaternary sediment layer serves as the cap for the geothermal system (Gao et al. 2018). The low-value anomaly in gravity data coincides with the location of the quaternary sedimentary deposits (Pan et al. 2009). It was formed after the basin uplift in Mesozoic (Fang et al. 2005). These layers have low permeability and can prevent heat from escaping. The sedimentary layers in the study area could be outlined in 1000–1500 m from our achieved models, but it is hard to point out the specific line between Quaternary and Neogene sedimentary rocks.

The strata distribution along the DR4–QR1–DR3–DR2 wells in Fig. 5a shows the hydrothermal and the HDR reservoirs from the Earth's surface to the 3300 m depth of the study area (Xu et al. 2018). The hydrothermal reservoirs are located at a depth of around 200 m and 700–1000 m. The granite rocks are in the deeper parts, but those with hot temperature (> 150 °C) are defined as HDR. For these wells, the temperature all increases sharply in Fig. 5b, with a gradient of around 40–50 °C/km. Notably, it reaches 150 °C at 2336 m depth for DR4, and 2104 m for DR3. The deep granite would be then defined as HDR.

Fig. 5
figure 5

The profile G1–G2 along DR2, DR3, DR4 and QR1 wells: (a) stratigraphic distribution based on drilling data (after Xu et al. 2018), (b) temperature logs of DR2, DR3, DR4 and QR1 (after Xu et al. 2018), (c) distribution of rock density based on the 3D density model, (d) distribution of magnetic susceptibility of the rocks according to the 3D magnetic susceptibility model

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Compared with the density and susceptibility models, there is consistency. The boundary between sedimentary layers and granite is located in somewhere between 900 and 1500 m depth. There is also a boundary between 0 and 1 × 10–3 SI, at nearly the same position in the susceptibility model, while the density model roughly matches for the boundary of − 0.3 and − 0.2 g/cm3. The water existing in sedimentary layers might reduce the density in a degree and cause the density mixture for the sedimentary parts. The boundary between non-HDR and HDR in 2200–2400 m fits the interface of − 0.1 and 0 g/cm3 well, but there is not any susceptibility boundary at this level.

From these comparisons, the boundaries of HDR matches the density boundary between − 0.2 and 0 g/cm3 well, while the edge of the high susceptibility nearly fits the upper boundary of the granite. In this way, the density of HDR would be estimated over 2.6 g/cm3, and the density of sedimentary rocks on the ground is between 2.1 and 2.3 g/cm3. For the magnetic susceptibility, the HDR is estimated more than 4 × 10–3 SI. By these standards, the prospective HDR districts are then circled at the area along Gonghe town and Dongba, and the volume would be then estimated as 6100 km3 above 3500 m depth. Meanwhile, 2000 m would be a preferable minimum depth for exploitation. With some pre-existing fractures, both 2D and 3D EGS modelling could be simulated along with the further engineering evaluation (MacFarlane et al. 2014; Xu et al. 2018).

For the HDR delineation here, temperature, stress and the hydrothermal alteration should be discussed for the change of density and susceptibility. Regardless of the minor change of mineral composition of granites, the porosity would be a major factor for the density change. However, the porosity decreases up to around 100 °C and then consecutively increase up to 800 °C (He et al. 2018). The temperature would not make the density increase. For the susceptibility, it changes slightly at 100–200 °C (Just and Kontny 2012). The overburden stress can cause the closure of microfractures and pore spaces, with an increased density of granite. At a depth of 2–3 km, the overburden stress in the study area is estimated at around 73–87 MPa (Weinert et al. 2020). Considering the effective stress in the sedimentary basin is generally lower than 20 MPa (e.g. Paul et al. 2010; Zhang 2013), the decreasing rate of porosity could reach around 2.5% (Jia et al. 2017). The hydrothermal alteration also affects the physical attribute here. Hou et al. (2019) used the integrated multicomponent geothermometry method to outline the conceptual model, with hot water flow in the fracture zone at the depth of around 2000 m. The alteration and weathering could then decrease the density of granite.

Conclusions

A gravity and magnetic survey was carried out in the Gonghe town in the potential Gonghe geothermal field. To better understand the areal and vertical distribution of the HDR, the gravity and magnetic data were inverted using 2D manual inversion and 3D cross-gradient joint inversion based on SL0-MS. The results indicate that this area is overlapped by the sedimentary layers, the thickness of which is around 1000–1500 m. Granites of different periods and intrusion processes are widely distributed and located under the sediments, with some deep faults.

As for the HDR delineation, it is mainly located along Gonghe town and Dongba. The density and susceptibility are inverted as over 2.6 g/cm3 and 4 × 10–3 SI, respectively. Its upper boundary is around 2000 m depth, and the volume is estimated as around 6100 km3 in those parts whose depths are less than 3500 m.

The appearance of the density and magnetic susceptibility models are affected by the lithology, stress and hydrothermal alteration. Therefore, as for the EGS site location, reservoir temperature and structural patterns need more attention. At the heat exploitation stage, other geophysical methods including the microgravity, seismic and more detailed MT surveys (e.g. 4D inversion of time-lapse MT data sets) would be more applicable for monitoring the injected fluid (Nam et al. 2017). For the theoretical study, as the geothermal concept is still disputable in this area, regional gravity and magnetic data would be much beneficial for the regional structure delineation and geological modelling. For example, the long wave deviations in gravity and magnetic fields could also help to reveal the deep structures.