Detailed engineering geological assessment of a shotcrete lined pressure tunnel in the Himalayan rock mass conditions: a case study from Nepal

In 2008, three dimensional stresses were measured at locations TT1, TT2, and TT3 in the test tunnel (TT) by using the 3D overcoring technique (SINTEF 2008). The details of the test location plan is shown in Fig. 4. The figure also shows location details of the measurement points and the topographic condition. Three measurements in the bore hole were performed at locations TT1 and TT2 at depths 6 m, 6.5 m, and 7 m from tunnel face, respectively. SINTEF (2008) highlighted that the stress measurement at TT3 was disturbed owing to water ingress and only one measurement was performed at 8 m depth. The standard overcoring technique as is being practiced around the world was used in the stress measurement.

The computer program called Determination of In situ Stress by Overcoring (DISO) developed by SINTEF (Lu 2006) was used to calculate the magnitude and the orientation of principal stresses from the strain readings and use of elastic parameters. The parameters, such as density, Young’s modulus of elasticity, and Poisson’s ratio were determined by laboratory testing. The DISO program is a statistical calculation tool that analyses stress parameters giving mean values of principal stresses with standard deviations and orientation of corresponding stresses. The computed stress magnitude and orientations are presented in Table 1. In the table, a very low magnitude of S₃ at TT2 compared to the other two locations is noticed. The unexpectedly low measured minimum principal stress at location TT2 was inferred to be attributed to the presence of nearby fracture zones.

The overcoring stress results indicated that the ‘OLD HRT’ alignment proposed in 2008 as unlined/shotcrete lined tunnel was safe for implementation, which was originally designed using the Norwegian confinement criteria. The assumption on this pre-construction phase design was that the minimum principal stress would most likely be sufficient to safeguard the rock mass from hydraulic jacking due to the water head of about 420 m. Still, it was recommended that the magnitude of minimum principal stress is measured at the downstream end of the headrace tunnel during excavation.

Following the recommendation during design, the magnitude of minimum principal stress was measured at locations 1 and 2 along the excavated headrace tunnel at the downstream end by using the hydraulic fracturing technique (Figs. 3 and 5). The measurement procedure followed was similar to that suggested by the International Society for Rock Mechanics (Haimson and Cornet 2003). However, an exception in this test was that the fracture orientation was not determined since the purpose of the test was just to measure the magnitude of minimum principal stress. The measurement details and test results are explained in detail by SINTEF (2013). The test equipment was placed at a desired borehole depth and two packers are inflated with water, closing a borehole section of about 1.0 m. At each test section, the initial fracture was induced in the isolated borehole in the in situ rock mass by pumping water. The water pressure required to induce the initial fracture is fracturing pressure ‘P_f’. The test was then followed by re-openings of the induced fracture. The pressure required to re-open the fracture is fracture re-opening pressure ‘P_r’. The applied water pressure, packer pressure, and water flow rate were continuously logged during the whole test period. The shut-in pressure was identified at a point where the flow was closing in the test section. In each test section in the borehole, at least three test cycles were performed. Finally, the shut-in pressure at each cycle was considered as the magnitude of minimum principal stress at the test location. All three pressures at different measured locations are listed in Table 2. The table also indicates some unsuccessful tests where the water pressure was not able to make a new distinct crack; rather water was lost through permeable pre-existing cracks or joints in the test section or entire borehole depth. Even excluding the unsuccessful tests, the minimum value of minimum principal stresses (P_si = S₃) at both locations were found far lower than the pressure from the 420 m water head. It was then concluded that the unlined/shotcrete lined tunnel at the OLD HRT was not safe for a 420 m water head and as a result the ‘NEW HRT’ alignment was proposed.

Two more stress measurement campaign were conducted in 2014 at locations 3, 4, 5, and 6 (MSG 2014) and 2015 at location 7 (MSG 2015). The locations 4, 5, 6, and 7 were along the NEW HRT alignment and location 3 was at the erection tunnel to the top of the upper penstock shaft (Fig. 3). In these campaigns, both hydraulic fracturing and hydraulic jacking techniques were employed. The detail of stress measurement locations, including location and orientation of test holes, diameter and length of test holes, and size of the tunnel, are illustrated in Fig. 5. The tests were conducted in close agreement with the recommendations of the ISRM standard (Haimson and Cornet 2003), with the exception of the determination of the fracture orientation for the same reasons previously discussed. Numerous tests were conducted in each borehole at different measurement locations. The details of the testing procedure are mentioned in MSG (2014 and 2015). Packer pressure, injection pressure, and flow rate were continuously logged during the testing period for each measurement.

In addition to the previously mentioned three pressures, the jacking pressure ‘P_j’ as an additional estimation of the minimum principal stress was measured from the step rate tests. In most cases, the shut-in pressure is equal to the jacking pressure. Table 3 gives the summary of tests results consisting of fracturing pressure, re-opening pressure, shut-in pressure, and jacking pressure at each test section of different boreholes at locations 3, 4, 5, 6, and 7. The table also indicates some unsuccessful tests where no pressure was built up because of high permeability. In addition, some tests were also neglected for averaging of minimum principal stress due to unexpectedly high or low values.

Haimson and Cornet (2003) recommended a relationship to estimate the magnitude of maximum principal stress in the case of vertical borehole. Since the bore holes are sub-horizontal or inclined at different locations, the originally recommended relationship is modified keeping in mind that the fracture is induced in the direction perpendicular to minimum principal stress. This brings another fact that the stress parallel to the fracture is obviously the maximum principal stress. Following this hypothesis, the originally recommended relationship is modified and is represented by Eq. 7.

Equation 7 is used to estimate the magnitude of maximum principal stress where S_t is tensile strength of the rock tested in the laboratory by the authors. In summary, Table 5 shows the statistical distribution of magnitude of both maximum and minimum principal stresses at different locations, i.e., 1, 2, 3, 4, 5, 6, and 7. The values given in Table 4 are calculated from Tables 2 and 3.

In the outer reach (downstream end) of the OLD HRT of the Tamakoshi project, the topography has slopes in multiple directions (Fig. 6a). This condition of the topography is very frequent in the Himalayan region. The multiple slopes have different vertical rock covers, lateral rock covers, and valley slope angles for a selected location of an unlined/shotcrete lined pressure tunnel. It is hence necessary to define the representative geometrical parameters required for the confinement criteria in multiple valley slopes. In topography with slopes in different directions, representative values of both vertical rock cover and the term ‘Lcosβ’ should be minimum of all possible values. Mathematically, both terms can be represented by Eqs. 8 and 9, respectively, where the topographic correction should be applied.

In the equations, n is the total number of slopes in different directions. At both locations 1 and 2 of the OLD HRT, corresponding critical values required for Eqs. 1 and 3 are estimated by using Eqs. 8 and 9, respectively. In doing so, the topographic correction is also applied for each side of the slope.

The geometrical parameters are listed in Table 5. The corresponding factor of safeties given by Eqs. 2 and 4 are calculated using the respective input parameters as given in Table 5. For safe operation of an unlined pressure tunnel at normal operating condition of the power plant, the factor of safeties should be greater than 1.3 (Benson 1989). The table shows that both factors of safeties (FoS₁ and FoS₂) are greater than 1.3. This indicates that the proposed unlined/shotcrete lined pressure tunnel is safe against hydraulic jacking at locations 1 and 2, which are located at the most critical stretches of the OLD HRT alignment.

The design criterion defined by Eq. 5 is checked at both location 1 and location 2. The minimum principal stresses measured in 2013 are used to calculate the factor of safety (FoS₃) defined by Eq. 6. From statistical distribution of the measured values, it is found that there is only about 18% probability that the FoS₃ is more than one in location 1 and about 70% probability in location 2. On the other hand, the mean value of FOS₃ in terms of measured stress is slightly less than one for location 1 and slightly more than one in location 2. It is emphasized here that the minimum value of FoS₃ given by the measured stress is very low in both locations (Fig. 7).

Basnet and Panthi (2018b) used FLAC^3D model (ITASCA 2017) to simulate the stress state of the Tamakoshi area covering both locations 1 and 2 of the OLD HRT. Figure 6 shows the simulated values of minimum principal stress along the headrace tunnel profile and along the cross section across location 2. The simulation was carried out considering both without and with the presence of weakness zone. The factor of safety (FoS₃) is calculated by using simulated values of the minimum principal stress for both cases at both locations (Fig. 7). As seen in the figure, FoS₃ is less than 1.3 in both locations and is influenced by the presence of weakness zone, which is mainly due to the de-stressing effect. The analysis result suggests that the minimum principal stress magnitudes are inadequate against hydraulic jacking at both locations. On the other hand, the figure clearly shows that the factors of safety given by the Norwegian confinement criteria are very optimistic compared to FoS₃.

Basnet and Panthi (2018a) analyzed both failure and successful cases of unlined high pressure shafts and tunnels of Norway. As an outcome of the analysis, they proposed both favorable and unfavorable ground conditions for the applicability of the Norwegian confinement criteria. Hence, the OLD HRT of the Tamakoshi project is compared with three of the failure cases consisting of an unlined tunnel at the Askara project, unlined shaft at the Byrte project and unlined shaft and tunnel at the Fossmark project; and with one successful unlined tunnel case called Nye Tyin (Table 6).

In Table 6, different important aspects consisting of topography, rock mass and jointing, presence of fault and/or weakness zones, in situ stress state, and hydrogeology are compared between the selected Norwegian cases and the OLD HRT alignment of the UTHP. As one can observe, most of the ground conditions prevailing at the downstream end of the OLD HRT of UTHP do not favor the possibility for an unlined high pressure tunnel (static head of 420 m).

The fact is that the unlined tunnel at the OLD HRT alignment is unfavorable because the minimum principal stress is inadequate to avoid the hydraulic jacking at the most critical studied locations. The low stress values are mainly attributed to the presence of multiple valley topography, weakness zones along major river valleys, and local shear zones crossing the tunnel alignment. The joints in this area are open or filled with loose material due to a decrease in confinement from the Gongar valley side. Open joints and the joints filled with loose material have high conductivity that makes an easier path for water to flow through the joints. In addition, the lower stress values compared to water pressure creates higher probability for hydraulic jacking of the pre-existing joints, which again eases the water flow through the jacked joints. Based on the above analysis and comparison, it is worth saying here that the design modification of OLD HRT was a necessity. Now the concern is whether the NEWLY SHIFTED HRT is safe to be used as an unlined/shotcrete lined tunnel or not. In this regard, thorough assessment and analysis are carried out below by considering all possible aspects, such as topography, rock mass quality, weakness and shear zones, and in situ stress state.

The overall view of the topography and geological features, such as jointing, weakness zones, and shear zones, at the Tamkoshi project area were observed from the left bank of the Tamakoshi valley and the right bank of the Gongar valley (Fig. 10). In addition, the detailed geological features were identified and mapped along the road cuts and natural rock exposures at the surface. During mapping, emphasis was given to assess the features that cross cut the NEW HRT alignment (Fig. 10a). The potential weakness zones were identified and their orientations were mapped. The weakness zones that are close to parallel to the foliation are identified as shear zones as defined by Panthi (2006) and Basnet and Panthi (2018b). In addition to the weakness zones, the jointing at road cuts and natural rock exposures were mapped in great detail. Potential ground water flow locations (local springs) were traced and identified. It is emphasized that in the areas where thick colluvium deposit and vegetation are present the ground water will percolate and expose at surface far below the potential leakage locations. Hence, a careful observation was necessary to identify the potential leakage locations at the surface.

As seen in the photos shown in Fig. 10b and c, the rock mass at the outer reach of headrace tunnel (downstream stretch) is schistose and fractured near surface and occurrence of weakness/shear zones are frequent. In addition, there exists complex topography with multiple slope directions due to river confluence and high relief toward both valleys (Tamakoshi valley on the right and Gongar valley on the left sides in Fig. 10c). The overall surface investigation showed that the outer reach of the headrace tunnel is more risk prone regarding the possibility of hydraulic jacking and consequent leakage compared to the upstream reach of the headrace tunnel where rock mass is relatively massive and less jointed. In this regard, a thorough rock engineering investigation is carried out along the NEW HRT alignment between the points T3 and T4 where static water head will be between 30 m and 115 m.

The texture of the rock, which is exposed at the surface, can be seen in the photos in Fig. 11a and b. Figure 11a shows that the rock mass is foliated, and Fig. 11b on the other hand shows relatively intact rock mass. The rock mechanical test was carried out at different stages of the project (Basnet and Panthi 2018b). The authors tested the rock samples collected during field mapping carried out in 2014 and 2016. The rock samples tested by the authors were extracted from the bore hole ‘ST-1’, but from different depths to represent overall rock mass (Fig. 11c). The rock cores have a diameter of about 50 mm, which is the same as that recommended by ISRM standard for different tests (ISRM 1978a, b; Bieniawski and Bernede 1979). The length and other dimensions of the samples were prepared according to the ISRM standards.

The tests were carried out in the rock mechanical laboratory at NTNU, Norway. The uniaxial compressive strength (UCS) and elastic modulus of nine rock cores were tested. Seven discs were prepared and tested to find tensile strength using the Brazilian test. In addition, other rock mechanical properties, such as unit weight and sound velocity, were also determined. Table 8 lists the statistical distribution of the tested parameters consisting of UCS (σ_ci), modulus of elasticity (E_ci), Poisson’s ratio (ν), unit weight (γ_r), sound velocity (V_s), and tensile strength (σ_t). In the table, ‘Δ’ is an angle between the schistosity plane and the loading direction. During the UCS test, failure occurred along the schistosity plane in most of the tested cores.

Panthi (2006) illustrated that the uniaxial compressive strength of intact rock specimen is smallest when the schistocity plane is inclined at around 30 degrees from the loading direction. In the tested samples of intact rock of the UTHP, the average angle is about 31 degrees, which means that the UCS values obtained from the laboratory test should be very close to the minimum possible strength. Table 8 shows that both compressive and tensile strengths of the tested intact rock of the UTHP are high enough compared to the water pressure at the tunnel. This indicates that the intact rock itself is not vulnerable to the hydraulic fracturing. However, the schistosity developed owing to weak minerals could make it vulnerable to hydraulic jacking.

The strength anisotropy of the rocks is mainly dependent on the mineralogical composition of the tested rock samples. Hence, altogether 11 rock samples were used for mineralogical analysis. The statistical distributions of the percentage of different minerals are listed in Table 9. The percentage of quartz exceeds 60% in most of the tested samples and the remaining minerals are represented by mica, biotite/muscovite, chlorite, graphite, and talc.

The rock mass quality along the headrace tunnel was mapped during tunnel excavation and rock mass quality registration was mainly carried out using the Q-system as defined by Barton et al. (1974). The six parameters of Q-value are listed in Table 10 with their statistical range for each of the particular rock mass quality classes registered along the headrace tunnel. As one can see in Table 10, the rock mass quality along the tunnel varies from good (Q-value exceeding 4.0) to very poor (Q-value less than 0.1) rock mass class. In the hard rock, the Q-values in the fractured or jointed rock mass fall between the ranges of 0.1 to 4.0. On the other hand, the Q-values of the crushed, weathered, and clay mixed rock mass of the weakness zones fall below 0.1, representing the extremely poor to exceptionally poor rock mass category.

Figure 12 shows rock mass quality mapped along the new alignment (NEW HRT) of the headrace tunnel between chainage 2 + 914 m (T3) to chainage 7 + 860 m (T4). As one can see, almost four-fifth of the length of the headrace tunnel has good quality rock mass. However, about 700 m of the mapped headrace tunnel has fractured rock mass and approximately 200 m of the tunnel stretch meets the rock mass of the weakness zones. It is especially noted that the rock mass at the downstream end of the headrace tunnel (Downstream from chaingae 7 + 300 m) where the static water pressure will reach its maximum of 1.15 MPa is mainly dominated either by fractured rock mass or by the rock mass of the weakness zones.

The three different groups of rock mass indicated in Fig. 12 behave differently when exposed to high water pressure. The good quality rock mass are relatively resistive toward the hydraulic fracturing/jacking if the minimum principal stress is higher than the static water pressure. However, the fractured rock mass are vulnerable to hydraulic jacking and leakage due to the presence of joints either open or filled with silty clay and sand. Similarly, the rock mass of the weakness zones are unfavorable for the unlined tunnels due to very low magnitude of minimum principal stress caused by local de-stressing.

The rock mass in the Tamakoshi area is foliated and schistose. The rock exposures seen at surface have at least three joint sets including foliation joints. The typical rock exposures at the cliff area and the road cut are shown in Fig. 13a and b. The orientations of the joint sets at the surface are presented by Panthi and Basnet (2017). During the surface mapping, the joints at the surface were found open or filled with silt and clay. Long persisting joints identified at the surface possibly communicate the joints mapped inside the tunnel. The typical joint sets along the tunnel wall and tunnel face are shown in Fig. 13c and d, respectively.

Figure 14 presents the orientation of joints in stereographic projection (Fig. 14a) and joint rosettes (Fig. 14b, c and d). The overall jointing pattern along the mapped tunnel stretch between T3 and T4 is represented by three distinct joint sets and some random joints (Fig. 14a). The jointing patterns for three different categories of rock mass discussed above are presented separately in Fig. 14b, c, and d. As one can see in the figures, the good quality rock masses have mainly two prominent joint sets. The mapped joints found along the tunnel are mostly tightly healed and intact, excluding some crossed joints that are filled with silty clay with a thickness 1 to 2 mm. In addition, quartz and feldspar veins of up to 20 cm thick are also occasionally glued within the foliation joints. The fractured rock mass on the other hand has three to four distinct joint sets plus random joints (Fig. 14c). The rock mass in this category consists of mainly the joints filled with permeable silty clay of thickness exceeding 10 mm. This thickness reaches 100 mm in those areas of the rock mass where local shear bands are developed along the foliation plane. In the cases of the weakness zones, it was difficult to map distinct joint sets due to the fact that the rock masses here are completely crushed. Despite this, some distinct joint sets were mapped and their main orientations are presented in Fig. 14d.

The colluvium deposits shown in Fig. 10 are almost de-stressed and weathered material. The large cracks and big boulders observed at the surface as shown in Figs. 10 and 15a and b are mainly due to de-stressing caused by stress anisotropy. It is emphasized here that these de-stressed areas are mostly located at the outer reach of the headrace tunnel alignment. It is logical to infer that both colluvium deposits and de-stressed boulder blocks have very small contributions in the in situ stress state at the tunnel location. The de-stressing effect due to such deposits depends upon the depth of the deposits from the surface. It is very difficult to estimate the depth of many of these areas due to limited accessibility at the surface caused by steep topography. Nevertheless, the logged information from boreholes ‘ST-1’ and ‘RCD-P3’ indicated that the weathering depth continued maximum up to 25 m from the surface. The mapped material at borehole ‘RCD-H1’ located at the right bank of Bhaise Khola indicated that the completely weathered rock material continued all the way up to a 48 m depth from the surface (Panthi and Basnet 2017). On the basis of the bed rock exposures at the surface and information from the borehole, it is concluded that the weathering depth along the headrace tunnel varies between 10 m to 50 m in depth.

Potential weakness zones were marked at the surface as shown in Fig. 10. The major weakness zones along the Tamakoshi valley and upper part of the Gongar valley are represented by the crushed zones (Basnet and Panthi 2018b). In addition, one of the major weakness zones along the Gongar valley near the Tamakoshi valley is a shear zone. Some weakness zones were also marked at the surface along the headrace tunnel alignment as shear zones. A distinct shear zone that is exposed at the surface near the shaft area is shown in Fig. 15c. This zone demarcates two different rock qualities. On the left side of the demarcation line (red color) in Fig. 15c, the rock mass is jointed but relatively stronger than the rock mass on the right side of the line, which is crushed and disintegrated. Altogether, eight distinct shear zones were identified at the surface in the area between Bhaise Khola and Gongar Khola.

Some distinct weakness zones were also mapped along the tunnel during excavation (Fig. 12). One such zone encountered while tunneling is shown in Fig. 15d. The shear zones identified in the surface topography, such as SZ#3, SZ#4, SZ#8, and SZ#9, can be easily extrapolated to match the respective shear zones mapped along the NEW HRT alignment. Similarly, the possible location of SZ#2 was mapped at the upper penstock shaft and the OLD HRT. The shear zones are oriented more or less parallel to the foliation joints, while the crushed zones are striking parallel to the river valleys and steeply dipping (Basnet and Panthi 2018b). The orientation of all crushed and shear zones are presented in a stereographic projection as shown in Fig. 16.

According to Panthi (2006), rock weathering has great influence on the strength and deformability properties of the rock mass. The extent of influence depends on the degree of weathering. The degree of weathering observed along the headrace tunnel alignment is categorized into five different weathering classes according to ISRM (1978c), and their frequency distribution is presented in Fig. 17.

In the figure, ‘N’ is total number of mapped tunnel segments. As one can see in the figure, the rock mass along the tunnel alignment falls within the slightly weathered class (II) category representing good to very good rock mass quality. The fractured rock mass mainly falls within the moderately weathered class (III) category and the rock mass in the weakness zone falls in the highly weathered to extremely weathered class (IV) category. According to Panthi (2006), the degree of weathering greatly influences the strength and modulus of elasticity of the intact rock. Figure 17 shows the respective reduction percentage for each weathering grade based on Panthi (2006).

The water bearing areas at the project area are mostly localized along the rivulets and rivers. A typical rivulet crossing the headrace tunnel alignment has distinct joint sets, as can be seen in Fig. 15g. The headrace tunnel crosses over eight such rivulets (Fig. 10). These rivulets are the main source of ground water that seeps into the tunnel through permeable joints. A typical water inflow into the headrace tunnel is shown in Fig. 15h. Through these inflow paths, water may also leak during plant operation if the ground water pressure is below that of the hydro static water head. Hence, tunnel inflow registration gives crucial information regarding potential water leakage out of the tunnel during the operation phase. In Fig. 12, some typical inflow locations are indicated along the tunnel with the approximate amount of tunnel inflow registered during tunnel mapping. Most of the inflow was registered at the area of fractured rock mass with the highest inflow of about 25 l/s (Fig. 15h).

In FLAC^3D software, the differential equations that describe the mechanical behavior of the rock mass are solved with the explicit finite difference method (ITASCA 2017). The major calculation steps in FLAC^3D involve: (a) nodal forces are calculated from the loading, such as stresses, body forces, etc., (b) the equations of motion are used to derive new nodal velocities and displacements, (c) new strain rates are derived from nodal velocities, (d) constitutive equations are used to calculate new stresses from the strain rates and stresses at the previous time, and (e) the equations of motion are again used to derive new nodal velocities and displacements from stresses and forces. These calculation steps are repeated at each cycle until the maximum out-of-balance force approach to zero indicating that the system is reaching an equilibrium state. Nevertheless, the geometry, material properties, boundary, and initial conditions should be well defined before the execution of these calculation steps. The Tamakoshi project area consists of a single rock type, i.e., schistose gneiss. For simplicity, the rock mass is considered as a homogeneous and isotropic material even though the rock mass possess some degree of anisotropic behavior because of the schistosity. The constitutive equations derived for a linearly elastic model is used where the material is expected to exhibit linear stress-strain behavior. The authors strongly believe that these assumptions are representative enough in this study since the objective is to find the in situ stress state for a large area.

The misfit is a non-dimensional feature that signifies the differences between the observation and the calculated result (e.g., Parker and McNutt 1980; Revets 2009). The model is said to be optimized when the misfit becomes absolutely minimum. Different misfit functions such as l₁-norm, l₂-norm are used in practice. In the present analysis, the l₁-norm is adopted for the measurement of misfit because of the simplicity in the calculation measure combined with its considerable robustness against outliers (Yin and Cornet 1994; Ask 2006 and Revets 2009). The misfit functions for both hydraulic fracturing and overcoring methods are first defined separately. For hydraulic fracturing data, the misfit is defined by Eq. 15.

Where S_mes1 and S_cal1 are the measured and calculated (simulated) principal stresses, respectively, S_d is the uncertainty in principal stress measurements (standard deviation in the present case) and M is the total number of measured stress data by the hydraulic fracturing method. The principal stresses are used in Eq. 15 instead of the normal stress in the originally proposed equation by Ask (2006) and Figueiredo et al. (2014). The reason behind this is that the fracture delineation was not recorded in the tests conducted in the UTHP. Both minimum and maximum principal stresses are used to calculate the overall misfit (Eq. 15) for all hydraulic fracturing data. Similarly, for overcoring test data, the misfit is defined by Eq. 16.

Where S_mes2 and S_cal2 are the measured and calculated (simulated) stresses, respectively, S_d is the uncertainty in stress measurements (standard deviation in the present case) and N is the total number of measured stress data by the overcoring method. In Eq. 16, the uncertainty associated with the borehole direction is neglected because overcoring tests were conducted within a maximum borehole length of 8 m, which is relatively short. The measured stresses are expressed in six different components in order to consider both magnitude and orientation of the stresses. The principal stresses presented in Table 1 are therefore transformed to the global coordinate system. The statistical values of transformed stresses are presented in Table 15. The stress data are then used to calculate the overall misfit (Eq. 16) for all overcoring data of the UTHP.

Since the hydraulic fracturing and overcoring methods are of different nature and concern different rock volumes, a global misfit function for the combined data set should be defined by introducing the weighing factors (Ask 2006 and Figueiredo et al. 2014). The weighing factors take into account the volume or the area involved by the measurement for each of the methods. The general global misfit function is hence expressed by Eq. 17.

Where the respective weighting factors for each test method are expressed by Eqs. 18 and 19.

Where ω^HF, A^HF, and A^REV represent the weighting factor, the measurement area, and area involved in representative elementary volume (REV) for the hydraulic fracturing data set, respectively. Corresponding notations for the overcoring data set are ω^OC, V^OC (measurement volume) and V^REV (REV volume). Minimized values of individual misfit functions, ψ_min^HF and ψ_min^OC, is the median of Eqs. 15 and 16, respectively for l₁-norm (Tarantola 2005). The area involved during hydraulic fracturing measurements depends on the injected volume and is usually of the order 1 m² (Ask 2006). On the other hand, the volume involved in the overcoring test is equal to the average volume of the hollow rock cylinder (Ask 2006 and Figueiredo et al. 2014). The respective area and volume involved in both tests in the UTHP are within the ranges recommended by Ljunggren et al. (2003). Further, the REV for the rock mass is assumed equal to 1 m³ volume (i.e., 1 m² area) as suggested by Ask (2006) and Figueiredo et al. (2014). The combined global misfit function for the minimization of the rock stress model of the UTHP is thus expressed by Eq. 20.

The global misfits obtained from Eq. 20 are finally minimized to define the best fit rock stress model.

The rock mass parameters, such as elastic modulus and Poisson’s ratio, are the main parameters that could potentially induce the changes in the stress field. Since, the rock mass is assumed to be homogeneous and a linearly elastic material, changing the elastic modulus will not induce changes in the stress field. On the basis of the comprehensive analysis, Basnet and Panthi (2018b) concluded that the Poisson’s ratio of the rock mass at the project area is at around 0.25. This value was hence used for this study as well. It is further highlighted that both normal and shear stiffness of the interfaces depend on the value of Young’s modulus (E₀) and thickness (t) of the weakness zone. In this study, t is fixed and E₀ is changed for the optimization process. The interface parameters presented in Table 14 for the weakness zones; CZ#1, CZ#2, CZ#3, SZ#1, and SZ#2 are used as fixed entities since these parameters were already optimized by Basnet and Panthi (2018b). The value of Young’s modulus for other weakness zones, SZ#3, SZ#4, SZ#5, SZ#6, SZ#7, SZ#8, and SZ#9, is denoted by E₀¹ (Table 16) and is changed for the optimization process in the present study. First of all, both magnitude and orientation of S_Tmax and S_Tmin are changed until the global misfit is minimized for the interface parameters given in Table 15. The optimization results from trials 1 to 28 are presented in Table 16. The influence of the 3D geometry and interfaces in the stress state and all three principal stresses are shown in Fig. 22. As one can see in the figure, the magnitudes of all three principal stresses are attenuated owing to the presence of deep valleys (i.e., both Gongar and Tamakoshi) and also the presence of weakness zones (i.e., both crushed and shear zones). It is highlighted that correct representation of 3D topographic geometry and engineering geological parameters of weakness zones are crucial in defining the best fit model.

Once the magnitude and orientation of tectonic stresses are optimized, the value of E₀¹ is changed between 1.0 GPa to 2.0 GPa for further optimization. It has been found that the global misfit is further reduced below the previously optimized value with the increase of E₀¹. However, it is not logical to increase the value of E₀¹ more than 2.0 GPa as far as physical validity is concerned. The value of E₀¹ is hence fixed as 1.6 GPa to make it compatible with the value of E₀ for the rest of the weakness zones. Finally, the optimized model with the parameters as shown in trial 31 in Table 16 is considered as a best fit model for the further interpretations.

The magnitudes of the principal stresses measured at all measurement locations and corresponding simulated stresses obtained from the model at different selected trials, including the optimized trial, are further plotted in Fig. 23. The measured values are plotted with the error bars equal to one standard deviation on each side of the mean values. It has been found that the difference between measured (mean) and simulated principal stresses is less than one standard deviation for more than 55% of the test data and is less than two standard deviations for more than 80% of the test data for best fit model. The authors consider this as an acceptable condition for the model to be valid.

After the model is validated, the magnitude of minimum principal stresses along the headrace tunnel are extracted and shown in the tunnel profile (Fig. 24). As one can see in the figure, there is considerable stress attenuation near the shear zones along the NEW HRT.

As has been mentioned before, some locations at the surface topography inside the model extent are destressed. To account for the impact of destressed area in the stress state, the model tries to simulate this by cutting topography with the depth equal to an assumed ‘destressed depth’ (DD). More specifically, the purpose of this simulation is to quantify the impact of destressed area on the magnitude of minimum principal stress. For this, the rock mass within the destressed depth is considered as a ‘destressed rock masses’ and is assumed to have negligible contribution to the magnitude of minimum principal stress to the locations below it. The model is therefore solved for equilibrium state after the volume of rock mass equal to a uniform depth of DD is removed from the geometry shown in Fig. 18b. The model is simulated for two different destressed depths, i.e., 50 m and 100 m. Hence, there are actually three different geometries altogether. First, geometry with present topography, i.e., DD = 0 m (Fig. 18b); second, geometry with DD = 50 m, and third, geometry with DD = 100 m. These three geometries are simulated separately as case A, case B, and case C, respectively (Fig. 25). The magnitude of minimum principal stress shown in Fig. 22 is the output of case A. The stress values along the NEW HRT are further extracted from Fig. 22 and presented in Fig. 25 as case A. Similarly, the magnitudes of minimum principal stress along the NEW HRT for case B and case C are shown in Fig. 25. The result shown in Fig. 25 suggests that the value of minimum principal stress along the headrace tunnel decreases with the increase in destressed depth. The figure also shows the static water pressure (P_w) along the NEW HRT alignment from T3 to T4. As one can see in Fig. 25, the value of minimum principal stress is less than water pressure at the tunnel locations of some weakness zones, i.e., at around chainage 3 + 600 m, at around chainage 4 + 450 m, and downstream from approximate chainage 7 + 300 m.