Real-case benchmark for flow and tracer transport in the fractured rock

The tunnel is located in the Jizera Mountains, in the north of the Czech Republic (Fig. 1 left). It is excavated in a portion of the Bohemian massif—the Krkonoše-Jizera Composite Massif (Žák et al. 2009; Klomínský and Woller 2010). The tunnel length is 2600 m, with an azimuth 67°. Positions are expressed by distance from the WSW end. The first 890 m of the tunnel was excavated with the tunnel boring machine (TBM) method and remaining part utilizing drill-and-blast methodology (Fig. 1, center and right). The diameter of the TBM part is 3.6 m, while the size of the D&B part is similar but irregular. There are irregularly distributed intervals of bare rock and shotcrete (both in the TBM and in the D&B sections). The altitude of the lower (WSW) end is 657 m, and the upper (ENE) end is 697 m giving a slope of approximately 1.5 %. The highest elevation above the tunnel is at 820 m above mean sea level.

Most inflow to the tunnel is observed where overburden is shallow—in the interval from 50 to 100 m (0.5–1 L/s) and in the interval 2200–2450 m (1.5 L/s). Where the overburden is thicker in the deeper part of the tunnel, there are fully dry intervals, short intervals with some leakage (but not freely flowing water), and several places (faults/fractures) with medium to strong inflow (ones to tens mL/s); the total inflow into the deep part is below 0.5 L/s. All the inflow is collected in a canal built in the tunnel floor. The resulting hydrogeological conceptual model, discussed in several preceding works (e.g., Hokr et al. 2014), is illustrated in Fig. 2; it is composed of a shallow permeable zone (weathered rock), deeper hard rock crossed by several subvertical fractures or faults.

Here simplified geometries, derived from this conceptual model, are used for the benchmark model formulation and solution in this paper. These benchmark models are based on the coupling of three relevant subdomains: the shallow weathered granite zone, a single vertical “fracture” (or fault), and the compact granite block (notation is fully specified in “Fracture/matrix meaning” section)—Fig. 3. Permeability decrease with depth was simplified to two zones, the upper more permeable and the lower less permeable, representative of permeability distribution near the tunnel [as proved in the 3D model of Hokr et al. (2014)]. The anisotropy of flow (large-scale permeability) is a result of the geometric configuration of the model, with the 2D fracture domain in the model expressing the preferential direction explicitly. The derived set of benchmark problems covers different combinations of the three main features: deep versus shallow overburden, strong versus weak inflow, and a single fracture versus a broader fault zone, as listed in Table 1.

To constrain model boundary conditions and parameters, we use several kinds of measured data: flow rates of individual fractures, inflow into the collecting canal, apparent water ages derived from natural tracers, and climatic and hydrologic data on the surface.

Single discharge measurements were collected in several places along the tunnel. We note that while a particular sampling location cannot be guaranteed to collect all water inflow from a given feature, the temporal changes of the flow rate should be representative of the true changes in discharge. The average value of discharge was estimated by its contribution to the total inflow in the collecting canal, using flow rate measurements in the collecting canal. The single inflow rates can be directly assigned to the discharges from the “fracture” domain in the model. Matrix domain discharge was evaluated indirectly by measurement of the increase in canal discharge along segments with no significant individual inflow features (Hokr et al. 2012). The inflow values are presented as a part of the model variants table—Table 1.

Most of the individual inflow locations were sampled and analyzed for natural tracer concentrations. Although the model problem is motivated by analysis of natural tracers to determine the water “age” and interpret the rock transport parameters, for these benchmark models we disregard the complex relationship of water age to tracer concentration observed and take the groundwater ages as given. The model is analyzed by means of synthetic “fictitious tracer” pulse, which is easier for comparison of solution between different solvers.

There are two kinds of tracer-based water age evaluation methods available: fitting the water molecule stable isotopes evolutions by a lumped-parameter model (dispersion in particular) and transforming the ³H/³He concentrations into the “apparent” age by the decay formula, which assumes the use of the piston-flow model. In both cases, we use data from unpublished analyses, within International Atomic Energy Agency (IAEA) (Šanda 2013) and SÚRAO (Hokr 2014) projects. The stable isotopes, determined mean ages for the shallow water inflow, were similar for positions at 76, 125, and 142 m, so we consider a single common value of 42 months for both model variants relevant to these locations (M1 and M2 in Table 1). This value falls within a large uncertainly range of the apparent ³H/³He age at these locations. The age of the 1565 m (105 m depth) fault zone inflow is 25 years determined by ³H/³He quite reliably (representative for M4 in Table 1), while the age of the dropping 798 m (140 m depth, M3 in Table  1) single-fracture inflow is estimated to 10 years using only observed ³H concentration, with larger uncertainty. These ages are not considered to be fully accurate, only representative for the benchmark purposes of this paper.

The recharge rate cannot be exactly determined. Here climatic and hydrologic measurements were used for indirect recharge estimation. The weather station operated by TUL at the tunnel entrance measures standard set of data, but the precipitation gauge does not melt snow, so the year totals are not accurate. We consider a rounded value from nearby official stations, which is 1000 mm per year, of which 20 % is estimated as the recharge. For the recharge variability, the precipitation evolution itself would not be well representative. Instead, we consider stream flow rate measured near the location as an indicator. We assume that when outflow exceeds the base value, then infiltration/recharge is occurring. These data are assumed to appropriately represent snowfall and melting effects. The infiltration calculated using the discharge excess method is normalized to the overall 200 mm/year average, which is expressed by the following formula for monthly totalized/averaged values [mm]:where Q _outflow is the monthly total outflow from the watershed and \(Q_{\text{outflow}}^{\hbox{min} }\) is its minimum over observation period, expressing the base flow. This method is not expected to accurately measure the recharge rate at the site, but rather provide a representative value for total recharge and its seasonal variability for benchmark purposes.

Depending on model variant, the meaning of hydrogeological objects represented by 2D and 3D subdomains is different. The 2D structure can be either a fracture in the strict sense as an opening between two rock surfaces with the flow controlled by the cubic law or a planar representation of certain higher permeability zone, understood as a porous medium with Darcy’s law controlled flow in a block of small finite thickness. Both interpretations are mathematically equivalent with the given transmissivity value.

The 3D domain can either mean a matrix block (compact rock without any fractures more significant than mineral grain interfaces) or an equivalent continuum representation of hard-rock blocks between larger-scale faults including a network of fractures less significant than the one represented by the 2D model domain. For consistent and simpler presentation in the paper, we use the terms “fracture” and “matrix,” respectively, for 2D and 3D subdomains, without regard to their actual physical role.

The conceptual model is based on the general situation in Figs. 2 and 5, transformed into individual local blocks representing a particular tunnel position (Figs. 3, 4). We include topographically driven flow in the shallow zone and vertical flow in the matrix and fracture below, which is disturbed by the tunnel drainage. We considered two general models to represent deep or shallow tunnel segments (corresponding also to different scale with respect to the topography, as illustrated in Fig. 5):

We define one model (notation M1) based on the shallow tunnel configuration (with the measurement representing the shallow tunnel segment as a whole) and three models (notation M2–M4) based on the deep tunnel configuration, for three different depths representing particular measured discharge locations, also distinguishing different meanings of the 2D and 3D subdomains (fracture/faults, matrix/equivalent continuum)—Fig. 3b–d, Table 1. We apply symmetries for the numerical problem solution, one vertical plane for M1 (along the tunnel axis) and two for M2–M4 (along the tunnel axis and the fracture plane), with the resulting configuration on Fig. 4. The reported values (model and measurement) are considered for the full model in contrast with the symmetric portions in the numerical model.

The M2–M4 domain is a block of 100 m length along the tunnel, 300 m width perpendicular to the tunnel and 400 m high (Fig. 4 left). The dimensions are chosen so that the boundary does not interact much with the tunnel effect. For technical reasons of domain connection in the discretized model, the fracture is extended along the shallow zone up to the surface. The shallow zone depth is 20 m. The tunnel depths and the fracture domain thicknesses are specified in Table 1. The meaning of the thickness value of 1 m for a single fracture is that we regard the hydraulic conductivity value as the transmissivity value in [m²/s] (without regard on a real geometry) and the porosity is conceptualized as the ratio of mobile and tracer-accessible water volume to the total volume in the fracture. The permeability in M4 is representative of the permeability of the fracture network in the fault zone, and porosity can be conceptualized at the fracture network porosity for the fault zone.

The domain of M1 is of similar size in vertical (400 m valley side and 580 m hill side) and transversal (400 m) directions, but much longer in the tunnel direction, 1000 m (Figs. 3a, 4 right).

The boundary conditions for either steady-state or transient flow are specified in Fig. 4. The infiltration rate (2nd type b.c.), either constant or variable, is prescribed on the top surface. The zero piezometric head (1st type b.c.) is prescribed on the vertical side of the shallow zone in the position opposite to the tunnel—representing the undisturbed equilibrium hydraulic state and controlling the flow direction in the shallow zone, i.e., out from the model, as would be the effect of topography in a real case. The remaining lateral boundaries are with no flow, based on symmetry. The tunnel wall is defined as the atmospheric pressure (1st type b.c.). The piezometric head on the bottom side is derived from the local topography and representative dimensions between the infiltration and drainage zones, i.e., the elevation difference of 200 along 1000 m distance corresponds to 80 m head difference along the 400 m model height. The head h = −80 m corresponds to the pressure head p = 320 m. The conditions are defined consistently on adjacent boundaries of the 3D matrix block and the 2D fracture plane (cases of top flow rate, tunnel pressure, bottom head, and lateral head).

The tracer concentration is prescribed on the recharge boundary and the zero concentration gradient (representing an advection-dominated outflow, i.e., no external disturbance) on the discharge boundaries of the tunnel and the “valley” side of the shallow zone. Zero mass flux is prescribed on the remaining boundary. For the purpose of benchmark, we consider a fictitious case of a pulse tracer injection. It is a precursor of the problem with real evolution of natural tracer concentration in the related paper (Gardner et al. DECOVALEX 2015 at http://​www.​decovalex.​org/​resources.​html#special-issues). We approximate a Dirac pulse injection by a short period (0, t ₁) of prescribed concentration (used c = 100), where t ₁ is typically a numerical time step. The effect of this discrete pulse time period is negligible for most simulations; however, in the case of the M1 problem, where the tunnel approaches the boundary surface, resulting in very short travel times right at the contact, the effect is apparent to the early time breakthrough.

In general, hydraulic conductivity, specific storativity, porosity, molecular diffusion coefficient, and dispersivity must be defined for each subdomain, respectively. The parameters are partly prescribed and partly subject of inverse problem solution, depending on problem variant. The full list of model inputs is in Table 2. Some of the parameters are common throughout the paper, while the others are specific for particular solution steps or model variants and are presented within a solution procedure below. We note that the parameter distribution has been greatly simplified, to keep the problems simple enough for comparison purposes.

The problem variants and their parameter sets are organized in the following structure:

We solve standard equations of porous media/fracture flow and solute transport. The fracture is represented by the same equation but with the appropriate meaning of the coefficient. The formulation of the equations for a set of subdomains of mixed dimensions is stated below. This is common for both Flow123d and OpenGeoSys solution although the actual numerical scheme is different. For PFLOTRAN solution, the 3D domain only is considered and the fracture is represented by an equivalent 3D domain. Vice versa, all the softwares are based on models of more generality than presented here and the presented equations are special cases of them.

We define the multidimensional problem domain Ω as \(\varOmega_{1} \cup \varOmega_{2} \cup \varOmega_{3}\) and denote the geometric dimension d = 1, 2, 3. The Darcy’s law and mass balance equation are formulated aswhere \(\vec{u}\) are the flux densities [m^4−d  s⁻¹], K are the hydraulic conductivities [m s⁻¹], h are the piezometric heads [m], S are the specific storativities [m⁻¹], and Q are sources/sinks [m^3−d s⁻¹], p is the pressure head [m], z is the elevation [m], and the subscript d denotes the belonging to a given subdomain. A compatible physical dimension is introduced through a geometric parameter δ _d , meaning the 1D domain cross-section area [m²], the 2D domain thickness [m], and δ ₃ = 1. The source/sink term includes a transfer to/from a domain of higher dimension. It is defined aswhere the fluxes between the different dimensions are defined proportional to head difference, consistently with the Darcy’s law. For more details on the subdomain connection, we refer to the software documentation (TUL 2015). The flux densities of subdomain-dependent physical dimension [m^4−d  s⁻¹] are related to average pore velocities \(\vec{v}_{d}\) of subdomain-independent dimension [m s⁻¹] as \(\vec{u}_{d} = n_{d} \delta_{d} \vec{v}_{d}\), where n _d are the porosities.

The tracer transport is governed by the advection–diffusion equationwhere c is the concentration [kg s⁻¹], \(\overline{\overline{D}}\) is the diffusion–dispersion tensor [m² s⁻¹], and n is the porosity [1]. Again, the source/sink term Q _d ^(c) comprises the interaction between the subdomains. The tensor \(\overline{\overline{D}}\) is defined bywhere α _L and α _T are the longitudinal and transversal dispersivities [m], D _m is the molecular diffusion coefficient [m² s⁻¹], and τ is the tortuosity [1].

Three different codes based on different numerical schemes are used for the comparison study in this paper. Although we focus on solution of the specific problem of this paper (the governing equations and the problem formulation specified above), the functionalities of the codes are wider, in quite different features between each other. Concerning the solution in this paper, the main features are compared in Table 3, including the reference to the participating authors’ team and introducing a shorter reference for the codes used in the text below (OGS, F123, and PFT). The code details and references are described in the subsections below.

While F123 and OGS share the common feature of mixed-dimensional domain (3D and 2D), PFT simulates only 3D subdomains. The numerical schemes differ in discrete representation of the 3D and 2D interaction in the mixed-dimensional models. F123 uses a mixed-hybrid FEM with independent unknowns in each domain, and OGS uses a standard FEM with shared unknowns at the domain boundary, which has, e.g., a consequence in the fracture boundary flux evaluation (a comment below, in “OpenGeoSys” section). PFLOTRAN solves a 3D domain only, using an integral finite volume method, in contrast to the finite element methods used by F123 and OGS. All the three solutions differ in the spatial discretization geometry—unstructured tetrahedral, structured or unstructured hexahedral (Table 3). Although a typical size of an element in the critical area around the tunnel is similar in all the used meshes (examples in Fig. 6), the total number of elements differs significantly (see Table 4); also a different ratio of nodes/elements is related to either tetrahedral or hexahedra choice. Most of these numerical features have a potential impact on the solution.

Next, the solutions differ with the temporal discretization. For the transient hydraulics, all the teams used 1-month time step in accordance with the input data resolution. For the tracer transport, OGS and PFT applied adaptive time-stepping schemes (starting from seconds up to several months) while F123 calculated with a prescribed time step constant through the simulation interval (1 month).

We do not in particular focus on the inverse algorithm behavior for the problem and mention them only for completeness, as they have secondary role in the evaluation. The inverse problem of fitting hydraulic conductivities in the steady state is especially simple: First, it should have a unique solution as the number of parameters and the number of fitted observations are the same. Secondly, each of the observations (flow rates) is dominantly sensitive to one of the two parameters: the fracture flow rate on the fracture transmissivity and the matrix flow rate on the matrix conductivity. Therefore, it is easy to iterate manually to an optimal combination of the parameters—a particular algorithm is demonstrated on the data of this paper in its follower (Hokr and Balvín 2016). The inverse codes actually used—UCODE coupled to Flow123d and DAKOTA coupled to PFLOTRAN—therefore are supposed to result “exact” values of inversion, not affected by an optimization method implemented.

As the first step, we solve a simpler case of steady-state flow as an inverse problem, i.e., we compare hydraulic conductivities calculated by the models which fit the measured boundary (tunnel) flow rates. The resulting values are in Table 5. Comparing the variants M2–M4 between each other, we see that the evaluated hydraulic conductivities are directly controlled by the order of magnitude of the tunnel inflow rates, for the respective subdomains, the fracture, and the matrix. The results correspond to conceptual assumptions: The most of the infiltrating water is discharged in the shallow zone outer boundary, creating almost horizontal flow with the head difference of 15–20 m which resembles the terrain slope excluded from the model geometry. The lower part of the model is controlled by combination of the vertical flow and the tunnel drainage in a reasonable balance (Fig. 7).

The differences between the codes in a range of percents, in case of Flow123d versus OpenGeoSys, are good for such kind of problem and well verify the solution. The difference of PFLOTRAN from the others is acceptable concerning general uncertainties in hydrogeology, but can seem high considering a benchmark-kind of a problem and the linear equation solution. There are several features of the problem compromising a precise numerical solution. One is the tunnel (small-scale geometric feature related to the whole problem and the appropriate mesh refinement) and second is the fracture of either lower dimension or smaller thickness, besides the possibly secondary effect of the contrast of parameter magnitudes. Moreover, the contact between the shallow zone and the fracture along the edge is a situation similar to a singularity (Hokr et al. 2016). Then the difference between the code concepts, i.e., PFLOTRAN with the 3D subdomains only versus Flow123d and OGS with 3D–2D coupling, becomes significant for the solution results.

In this case, we compare the direct solutions by the codes, instead of fitting the measured values, which would be too complex for comparison. Yet, the input values have been iterated by some trial-and-error steps to obtain common inputs being quantitatively relevant (Table 2). Three values are evaluated: water inflow rate from the shallow zone domain into the tunnel (discharge from the model) and two values of the piezometric head: at the top of the model ([−1000; 0; 180]—see Fig. 4)—and above the place of the tunnel and the domain interface intersection (approx. [−150; 0; 27]). While the tunnel inflow rate is meant as a counterpart of the measurement (Table 1), the head values are intended to check a qualitative relevance against a concept of water table following the topography, i.e., varying within the shallow zone.

The illustration of solution, as a distribution of the head, is in Fig. 8. The comparison of the selected quantities is presented in Table 6. The piezometric head at the top (the left column) is in very good accordance between the two model solutions by F123 and OGS, while there are significant differences for the position above the shallow tunnel section (the middle column). In this place, both the tunnel pressure/head boundary condition input and the surface head postprocessing can be sensitive to mesh geometry in connection with position of discrete unknowns (e.g., conversion between pressure and head with input of element/node vertical coordinate). The calculated tunnel inflow rate is consistently changed with the calculated head in the representative place. All the evaluated model values sufficiently agree for the average infiltration case of 200 mm/year.

The calculated head is tens of meters below the surface at the top of the hill, while in a correct range near the shallow tunnel section. It was no worth to attempt more precise calibration, due to the major simplification of neglecting the unsaturated zone. The tunnel inflow for “steady-state” infiltration is little below the lower bound of the measurement (3 ml/s/m) and exceeds slightly the upper bound (15 ml/s/m) for the asymptotic limit of a heavy infiltration period. Thus, we can see the model concept and data well relevant to reality, within the limitation of the simplifications. In particular, the topography-parallel water table is not actually possible as the water flux in the shallow zone rises uniformly collecting the infiltration while the head gradient had to be constant.

This variant has been selected for test due to significant tunnel inflow rate variability in the shallow tunnel part (compared to M3 and M4), which is observable both in the real data and in the model.

As the variable input (infiltration) fluctuates around its average of 200 mm/year (used for the steady-state initial condition), the tunnel inflow rate should also oscillate around its single-value counterpart in the steady-state model presented above.

The results are plotted in Fig. 9. We can see that the tunnel inflow rate oscillations are directly related to the infiltration inputs, which means that the reaction of the system is relatively quick compared to the one-month resolution of the inputs. On the other hand, the transition curve from one infiltration rate to another infiltration rate is relatively steep and therefore sensitive to the temporal discretization. So the graph for a longer period with month resolution is composed of individual peaks and pits. Consequently, the differences of the peak/pit values between the models can be explained by a different temporal discretization or different precision of the numerical scheme.

The benchmark confirms to be well related to the real conditions, as the range of the tunnel inflow rate is very similar to the measured counterpart, although some of the individual peaks or plateaus do not fit between (any of) the models and the measurement (Fig. 9). It is appropriate to the used model geometric simplification, whereas in the real case the hydraulic storage properties, including the unsaturated zone, can be much more complicated. We also note that no calibration has been used for the model parameters, besides the steady-state hydraulic model fitting to the average tunnel inflow (“Steady-state hydraulics of M2–M4 with inversion” section).

Although the benchmark model is motivated by fit of tunnel water age and the pulse transport temporal analysis is a preparatory step for calibration with real tracer time evolution, the comparison in this paper has been done as the direct solutions of the three models for given sets of parameters. We recall from the “Model variants and their parameters” section that the inputs were previously estimated to produce quantitatively relevant transit times. We evaluate two main results: the breakthrough curve, i.e., the concentration evolution in the water discharging from the fracture domain (M2–M4) or from the shallow zone (M1) into the tunnel (a flow rate-weighted average of the whole circumference/section length), and the mean transit time estimate, calculated from these breakthrough curve results (Eq. 7).

The calculated mean transit time is strongly dependent on the used simulation time interval. The dependence and especially the detection of time interval necessary for convergence of the evaluated integrals is a subject of continuing paper (Hokr and Balvín 2016). Here the comparison is made for particular time intervals agreed between the teams, referring to a concept of “10 times MTT”. Typically, the concentration in the M2–M4 simulations in the final time is between two and three orders of magnitude lower than the peak value, but the MTT value is still relatively far from the limit value of the infinite interval.

The MTTs are presented in Table 7 together with the reference data of the “measured water age.” The breakthrough curves are plotted in four individual graphs in Fig. 10, together with the graphically illustrated MTT values, namely to show the pulse asymmetry by the relatively large distance (in time axis) between the peak position and the MTT. Both the curves and the means fit very well for M2, especially considering usually quite observable numerical approximation errors like the numerical diffusion and the significant differences of the three numerical schemes. For M3 and M4, the similar good fit is observed between Flow123d and OpenGeoSys while PFLOTRAN produces curve peaks little sooner and the decrease rates larger than the other two codes; this corresponds to visibly smaller MTT. The reasons are analyzed in more detail in the next section. The fit for M1 is good for the peak position, which results from immediate transit of mass from the surface just above the tunnel. But there are significant differences in the decrease rate of the tail. The results of OGS and F123 are closer to each other, even with reasonable agreement of MTT, while PFT results to MTT closer to the reference water age value. Although the curve relation is opposite to M3–M4, the reason could be similar in dispersion evaluation and numerical diffusion (below).

For all the M2–M4 variants, the MTT is larger than the reference value (but quite little for M2), which comes simply from the inverse model used for the porosities estimate which used shorter simulation interval than the comparison here. On the other hand, the procedure how the reference values were obtained is also related to a shorter interval: For M2, the stable isotope data processed by a lumped-parameter model are from about 10-year periods (120 months, compared to 600-month simulation), and for M3 and M4, the ³H/³He “apparent” age (based in fact on the piston flow) strongly underestimates any contribution of flow paths longer than about 50 years of transit time (600 months, compared to 6000 months for the pulse simulation-based MTT evaluation). These arguments also partly answer why some of the porosities obtained by the inverse model can seem unrealistically small: The model tries to “accelerate” the transport, to decrease the MTT value appropriate for a smaller tracer-capturing period.

PFLOTRAN shows a sharper peak, shorter MTT, and different tailing characteristics for the M3 and M4 models, which have longer transit times in general. In order to explore the reason for the difference between the PFLOTRAN simulations and the other codes in the experiment, and its relation to the code architectures and numerical schemes (features listed at the beginning of “Numerical schemes and software” section), we investigated the effect of dispersion on the breakthrough curve and MTT.

The results from M3 simulations for a dispersion coefficient of zero and a run with a longitudinal dispersion coefficient of 5 m (as in Table 2) are shown in Fig. 11. Additionally, the Flow123d simulation has been done on a refined mesh of about 4–5 times larger numbers of nodes and elements. For higher dispersivity values, we see a broader peak and a longer tail and longer transit times, as expected. The difference between the models gets significantly smaller with decreasing dispersion and with mesh resolution. The effect of mesh refinement was significant for smaller dispersion only, as the DG scheme of F123 is able to compensate the numerical diffusion for less advection-dominated problems.

Therefore, we believe a highly likely source of discrepancy of the models is a combined effect of the numerical implementation of the hydrodynamic dispersion tensor and of the numerical diffusion/dispersion which is a complex process, especially when considering different numerical schemes and meshing topologies. E.g., the dispersion coefficients can be sensitive to approximation of velocity in the mesh.

While there is discrepancy between the models, the mean ages produced are generally within a factor of two. Given the complexity of the benchmark model, and the large differences between code architectures, this spread is probably representative of the type of structural model uncertainty that can be expected for the transit time distribution in fracture flow models. It should be noted that the hydraulic comparison was much closer than that of the transit time distribution.