1 Introduction

For a long time, the dimensioning of underground openings in salt rocks was primarily based on mining experience. Only in the last century, analytical and numerical calculation methods of geomechanics have been increasingly used. This was not least due to some catastrophic collapses of mining fields (rock bursts) with a strong mining-induced energy release (Minkley 1998) and the loss of potash and rock salt mines by flooding. Both practical experience and geomechanical calculations are essential for an economical and sustainable salt extraction at high recovery rates and complement each other.

The fundamental requirements of safe dimensioning for potash or rock salt mining are the stability of the mining system, i.e. the strength of pillars and mining panels, and the integrity of the hydraulic barrier, i.e. the protection of the overlying salt back that shields the mine from ground water inflow.

Insufficiently dimensioned pillars and the brittle behaviour of the mined salt rocks played a crucial role in the collapse of mining fields (Minkley 2002). The tendency for brittle fracture decreases from carnallitite, hard salt, trona, sylvinite to rock salt. Here, brittle failure means that at failure, the load-bearing capacity of a pillar suddenly drops to (almost) zero, as opposed to a more ductile behaviour where the load capacity gradually decreases towards a residual strength. In the former case, the potential energy of the overburden is released in a collapse, leading to dynamical failure events. For ductile pillar behaviour, on the other hand, the pillar deformation absorbs enough energy to ensure a more or less harmonic convergence of the mining panel. Hence, rock bursts occurred primarily in potash mines where carnallitite was mined. In sylvinite and rock salt mines, only few rock bursts are known worldwide, and only if extremely high recovery rate and, accordingly, very slender pillars were realised.

For the dimensioning of room-and-pillar systems, i.e. the determination of pillar widths and panel extensions, several methods are commonly used. There are empirical formulae based on rock mechanical tests on pillar models, analytical derivations using simplified constitutive models, and numerical simulations of single pillars and larger mining panels. The analysis of in situ collapses provides a basis to check dimensioning approaches and to derive empirical relations for the ratio of pillar width to pillar height (slenderness ratio) that is required ensure the viability and stability of pillars in salt rocks.

In addition, geomechanical calculations based on suitable material laws are required to quantitatively assess the system stability and integrity at the specific geological conditions (Minkley et al. 2001). The material laws must be able to properly capture the spectrum of mechanical properties of salt rocks and discontinuities, ranging from visco-plastic with time-dependent softening to extremely brittle fracture (see e.g. Fig. 1).

Fig. 1
figure 1

Contour failure of carnallitite pillars. In the left panel, quasi-static deformations lead to an hourglass shape, while the pillar in the right panel dynamically expelled its outer layers (local magnitude \(M_{\mathrm{L}}=1.7\))

Simulations of mining fields also allow an analysis of the hydraulic barrier, i.e. the surrounding strata that shield the mine from ground water. These include the salt back and, depending on the geology, additional impermeable strata such as claystone. With increasing extraction rate, from room-and-pillar mining to complete longwall caving, the convergence-induced stresses in the barrier strata become larger. To maintain the integrity of the hydraulic protection layers, the tensile stress criterion, the shear stress criterion (dilatancy criterion) and the minimum stress criterion (frac criterion) have to be satisfied.

The mine has to be dimensioned in a way that the integrity criteria are fulfilled in a sufficiently thick saliferous protective layer (greater than several tens of metres). For a loss of integrity, it is sufficient that one of the criteria is violated. If the tensile or shear stress criterion is violated, the stresses induce plastic deformations in the rock, and the associated damage leads to an increased permeability. However, in these cases, cracks mainly arise in the near field of the mining horizon. The minimum stress criterion also acts in the far field, due to the mining-induced extensive and far-reaching stress redistribution. So-called hard mining edges have repeatedly led to the flooding of salt mines, because directly above mining edges shear stresses are concentrated and, in greater distance, extensional strains are induced which at the protective layer top (top of the saliferous formation) result in a drop of the minimum principal stress below the acting groundwater pressure, such that the water could penetrate the (initially impermeable) salt rocks. This process, known as pressure-driven percolation, proceeds in the direction of the major principal stress and was responsible for the flooding of several potash and rock salt mines, in extreme cases causing sinkholes on the surface.

In coupled mechanical–hydraulic models, the interaction between mining dimensioning and its effect on the surrounding hydraulic protective layers can be analysed quantitatively.

In this article, the geomechanical principles for dimensioning of statically and dynamically stable mining fields and for maintaining the integrity and tightness of the hydraulic protection layers during recovery of potash and rock salt deposits are presented and discussed. In the next section, we discuss empirical relations for the strength of pillars and contrast these with pillar collapses and rock bursts. We then turn to numerical computations in Sect. 3. Here, we first introduce the constitutive models for rocks and bedding planes and then present a few example calculations. In Sect. 4, we describe the integrity criteria in more detail and show a few examples of mine floodings and discuss the underlying causes.

2 Empirical and Analytical Dimensioning

2.1 Estimates of Pillar Load Capacity

Empirical dimensioning methods generally aim to compare the stresses resulting from the underground excavations with the load capacities of the pillars, \(\sigma _{\mathrm{ PC}}\). Hence, one can define a factor of safety,

$$S=\frac{\text {load \, capacity}}{\text {pillar \, load}}=\frac{(1-\eta )\sigma _{\mathrm{ PC}}}{\int \!\rho g \,{\hbox{d}}z}.$$
(1)

where the pillar load is determined by the depth z, overburden density \(\rho\) and gravitational acceleration g, as well as the extraction ratio

$$\eta= 1-\frac{A_{\mathrm{p}}}{A_{\mathrm{sys}}}$$
(2)

given in terms of the pillar cross section \(A_{\mathrm{p}}\) and the system area \(A{\mathrm{ sys}}\).

The pillar load capacity \(\sigma _{\mathrm{ PC}}\) most crucially depends on the salt variety and on the width-to-height ratio \(\alpha =W{/}H\), and a number of estimates exist in the literature. For example, Uhlenbecker (1968) has used laboratory tests on square pillar models to deduce a formula for carnallitite,

$$\sigma _{\mathrm{ PC}}= a\cdot \alpha ^2 + b\cdot \alpha + c +d\cdot \frac{1}{\alpha }.$$
(3)

where ad are empirical parameters depending on the carnallite content. This formula has been used extensively in room-and-pillar mining systems. On the other hand, Menzel (1970) analytically derived the load capacity in the framework of limit equilibrium theory, using a particular multicomponent failure envelope. The resulting expressions depend on the rock varieties involved and on a number of rock mechanical parameters and need to be evaluated numerically due to the chosen approximation scheme. The approach already shows the importance of cohesion between the pillar and surrounding rock, in particular for larger width-to-height ratios. However, the analytical derivation relies on simplified constitutive models that crucially do not account for material softening, which is more or less automatically included in empirical methods based on pillar model experiments.

The IfG has performed numerous laboratory tests on the load capacity of carnallitite pillars: pillar models of varying width-to-height ratio were loaded in a servohydraulic testing machine. Initially, the pillar core will remain strongly confined and is able to sustain the load. At the peak strength, the pillar suddenly collapses: the contour disintegrates, while the pillar core is recompacted by the hanging wall pushing down. For illustration, the bottom panels of Fig. 2 show samples after the test, once with the core removed (left panel) and once with the contour debris removed (right panel). The top right panel shows the load capacity as a function of width-to-height ratio (Minkley 2002). To a good approximation, it is described by a rather simple linear relation. A similar expression holds for sylvinite; hence we have

$$\sigma _{\mathrm{ PC}}^{\mathrm{ sylv}}= 25\,\text {MPa}\cdot (1+0.75 \alpha ),$$
(4a)
$$\sigma _{\mathrm{ PC}}^{\mathrm{ carn}}= A+ B\cdot \alpha.$$
(4b)

The parameters A and B depend on the carnallite content (for 60–80 % carnallite, we have \(A=19.7\) MPa and \(B=9.4\) MPa). With these formulae and an appropriate factor of safety, e.g. \(S=2.5\) (Sievers 1972) or \(S=3\) (Uhlenbecker 1980), the mining system can be considered long-term stable.

Fig. 2
figure 2

Experimental investigations of the load capacity of carnallitite pillars. Top left servohydraulic test equipment, top right experimental results, bottom left disintegrated pillar contour, bottom right briquetted pillar core

2.2 Pillar Failures and Rock Bursts

To evaluate the dimensioning method discussed above, we can contrast them with past failures. In the history of potash mining, spanning more than 150 years, a number of rock bursts have occurred. For carnallite, the most brittle potash mineral, ten rock bursts are reported with magnitudes up to an intensity of \(M_{\mathrm{L}} = 5.6\) (Minkley 1998, 2002, 2004a). For the hard salt of the Staßfurt seam, two field collapses are known (Gimm and Pforr 1961). In a Trona (carbonatic salt rock) mine in Wyoming, there was a rock burst with a magnitude of \(M_{\mathrm{L}} = 5.1\) (Swanson and Boler 1995). In 1995, in the Russian salt mine Solikamsk where sylvinite was mined (with carnallitite exposed in the hanging wall), a rock burst with \(M_{\mathrm{L}} = 4.8\) caused a sudden surface subsidence of up to 4.5 m (Malovichko et al. 2001).

Even in rock salt mines, dynamically proceeding field collapses are reported for room-and-pillar mining with slim pillars (Berest et al. 2008). At the French site Varangéville, earthquake-like vibrations occurred on 31 October 1873 because at a depth of about 155 m, a mined rock salt area collapsed. The failing square pillars had a width-to-height ratio of \(\alpha = 1.1\) and the overall extraction ratio of the deposit was 82 % (Minkley et al. 2012a). In the Retsof rock salt mine in the USA, where a brittle fracture process occurred at a depth of 330 m, the extraction rate was again over 80 % (Yager et al. 2001). As a result of the collapse with a magnitude of \(M_\mathrm{L} = 3.3\), the world’s largest rock salt mine was flooded.

Figure 3 shows a comparison of the calculated load-bearing capacity for brittle carnallitite and trona pillars, as given by dimensioning methods discussed above, with the pillar loads in potash mining fields which failed and resulted in a collapse. The numbered dots represent the rock bursts summarised in Table 1. Point 7 represents a contour fracturing event comprising 36 carnallitic pillars in the Hattorf mine in 1969, leading to a local magnitude of \(M_{\mathrm{L}} = 2.4\). The direct trigger was an increase in the mining thickness from 4 to 6 m (Uhlenbecker 1980). Since the safety factor was sufficient (\(S = 1.5\) according to Uhlenbecker), the contour failures did not lead to a complete collapse.

Fig. 3
figure 3

Comparison of pillar loads at failure and calculated load capacities. Dots indicate rock bursts (numbers as in Table 1), and the curves correspond to the dimensioning methods according to Menzel (A), Uhlenbecker (B) and the IfG (C)

Table 1 Summary of rock burst data

The load capacities according to Uhlenbecker [curve B, Eq. (3)] and the IfG [curve C, Eq. (4)] roughly coincide for \(\alpha\,\lesssim\,4,\) and show good agreement with the failure cases. At higher \(\alpha\), the linear IfG curve tends to give a better fit than the nonlinear relation of Uhlenbecker. From back calculations, one can show that effectively there is no difference between square and long pillars, i.e. the load capacity is, to a good approximation, a linear function of the width-to-height ratio \(\alpha\). Of course, one has to keep in mind that the parameters given in Fig. 3 are average values for the collapsed mining fields, and variations in geology and mining parameters can blur the distinctions between square and long pillars. However, since uncertainties in geological conditions are unavoidable in mine dimensioning, the simple linear relation (4) provides a good basis for pillar dimensioning in brittle salt rocks such as trona or carnallitite, validated by laboratory and in situ observations.

On the other hand, the analytical estimate (curve A) significantly overestimates the factor of safety, in particular at higher \(\alpha\). The underlying reason is that the limit equilibrium analysis is not suitable for rocks with pronounced softening behaviour: limit equilibrium assumes that at maximum load, the entire pillar cross section reacts plastically, and thus, a stable, plastic stress field dominates in the pillar, regardless of the state of deformation. This assumption is approximately fulfilled in rock salt and sylvinite, but not in brittle material such as carnallitite, where the pillar contour zone is destroyed by brittle fracture long before the pillar core is included in the plastification. This is indicated by the large number of sudden contour breakouts (Fig. 1) at a correspondingly high level of stress and dynamic excitation, e.g. such as by blasting (Minkley and Rose 1989).

Hence, to assess the load-bearing capacity of carnallitite pillars, the brittle material behaviour has to be taken into account. Fractures develop over time, starting at the contour where, due to the almost uniaxial stress state, the relatively low strength of carnallitite is exceeded first. A layer of damaged rock follows, form which, depending on the loading conditions, the fracture process proceeds, either more or less continuously, or, related to dynamical excitations, quite rapidly. In this case, a contour layer can be suddenly softened and expelled (see Fig. 1) (Uhlenbecker 1980).

Pillar stability is also significantly affected by discontinuities and bedding planes, either in the pillar or on the boundary between seam and roof or bed. The higher strength of the pillar core results from a triaxial stress state where the confining pressure is provided by the friction and adhesion at the hanging and footwall. If this contact is weak, considerable shear motion can occur and lower the confining pressure and consequently the load capacity of the pillar. This mechanism explains why even very squat pillars can fail. Dynamic processes in the rock salt can result if either salt matrix or bedding planes soften abruptly (Minkley et al. 2007).

Ultimately, rock bursts were caused by insufficient pillar sizes due to an inadequate understanding of the softening behaviour of the various salt rocks and the shear properties on bedding planes, in particular under dynamical conditions. These processes are very hard to capture in simple load capacity formulae. Furthermore, the methods for pillar dimensioning presented in Sect. 2 only apply to single pillars. Generalising the formulae to entire room-and-pillar systems is a simplified approximation which is only justified if the geological conditions and the mine layout are uniform. For actual mining fields with complex geometry and geology, the physical system equations are usually no longer solvable in a closed analytical form. Hence, a proper assessment of the stability of a mining panel usually requires numerical simulations using advanced geomechanical models, which we will discuss in the next section.

3 Numerical Simulation of Softening Processes in Salt Rocks

In Sect. 3.1, we present the constitutive models for the rock mass and bedding planes. In both cases, we use a nonlinear failure criterion and displacement-dependent softening. These models are based on decades of scientific work at the IfG and its predecessors and are permanently applied in mining, storage and repository applications worldwide. Then, we discuss some examples, first for single pillars in Sect. 3.2 and then for dynamical collapses of larger mining fields in Sect. 3.3.

3.1 Softening Models for Rocks and Weakness Planes

The visco-elasto-plastic constitutive model for salt rocks gives a realistic description of hardening, softening and dilatancy processes of saliferous and non-saliferous sediments. Furthermore, an adhesive shear model was developed which describes displacement- and velocity-dependent softening and failure mechanisms at bedding planes in the salt rock mass as well as weakness planes (i.e. fault zones, joints) in the overburden (Minkley 2002; Minkley and Mühlbauer 2007).

The visco-elasto-plastic model concept is sketched in Fig. 4. It is based on standard models of continuum mechanics and can describe both ductile and brittle material behaviour. As failure criterion, a generalised nonlinear Mohr–Coulomb criterionFootnote 1 is used (Minkley 1997), such that the yield boundary is given by

$$f(\sigma )= \sigma _1 - \sigma _{\mathrm{ D}}- \left( 1+ \frac{\sigma _{\mathrm{ MAX}}-\sigma _{\mathrm{ D}}}{\sigma _\phi +\sigma _3}\right) \sigma _3.$$
(5)

The yield curve starts at the uniaxial compressive strength \(\sigma _{\mathrm{ D}}\) and rises with increasing confining pressure, similar to the Mohr–Coulomb criterion. At larger \(\sigma _3\), it levels off and the differential stress \(\sigma _1-\sigma _3\) asymptotically approaches the constant value \(\sigma _{\mathrm{ MAX}}\), with the rate determined by the slope parameter \(\sigma _\phi\).

Fig. 4
figure 4

Sketch of the visco-elasto-plastic model. The Kelvin and Maxwell components form the Burgers element for primary and secondary creep. The generalise Mohr–Coulomb slider describes failure, including hardening, softening and dilatancy

If the yield curve is reached, there will be plastic shear deformation governed by the (non-associated) plastic potential

$$g(\sigma )= \sigma _1 - \sigma _3\left( 1+\frac{\sigma _\psi \tan \beta }{\sigma _\psi + \sigma _3}\right) .$$
(6)

The parameters \(\sigma _\psi\) and \(\tan \beta\) determine the dilatancy. Material hardening and softening are incorporated by a dependence of the yield curve and dilatancy parameters on the accumulated plastic deformation, i.e. \(\sigma _{\mathrm{ D}}=\sigma _{\mathrm{ D}}(\epsilon ^{\mathrm{ p}})\). Details of the formulation can be found in Minkley and Mühlbauer (2007).

The viscous, i.e. time-dependent, material behaviour—typical of salt rocks—is characterised by increasing deformation at constant load (\({\dot{\epsilon }}>0\) for \({\dot{\sigma }}=0\)) and decreasing stress under constant strain (\({\dot{\sigma }}\,<\,0\) for \({\dot{\epsilon }}=0\)). These phenomena are known as creep and relaxation. After loading, the creep rate is initially high and decreases over time, approaching a constant value at which the material can deform basically indefinitely. These phases are denoted as primary (or transient) and secondary (or stationary) creep, respectively. If the differential stress is high enough, damage accumulates and the creep rate increases again up to the point of creep rupture.

In the visco-elasto-plastic model, primary and secondary creep are described by a modified Burgers model. Straight after loading, the parameters \(G^{\mathrm{K}}\) and \(\eta ^{\mathrm{K}}\) of the Kelvin model (Fig. 4) determine the initial rate and hardening, while the secondary creep is only affected by the (modified) Maxwell element. Since many experiments show that secondary creep rates are not well captured by a linear stress dependence or a one-component Norton ansatz, in this model the stress dependence of secondary creep is modified with respect to the original Maxwell model by a hyperbolic sine,Footnote 2

$${\dot{\epsilon }}= \frac{\sigma _{\mathrm{eff}}}{3\eta ^{\mathrm{M}}_0} \cdot \sinh \left( m\sigma _{\mathrm{eff}}^n\right) .$$
(7)

where \(\eta ^{\mathrm{M}}_0\) is the unmodified Maxwell viscosity and the parameters m and n determine the stress dependence. The effective stress is \(\sigma _{\mathrm{eff}}=\sqrt{3 J_2}\), where \(J_2\) is the second invariant of the deviatoric stress tensor. Tertiary creep is characterised by a dilatant softening process that may lead to creep failure, described by the plastic slider element as discussed above.

As known from numerous observations of deformation processes in salt rock mining, the stability of single load-bearing elements, as well as of the whole mining system, does not only depend on the characteristics of the rock mass, but also affected by reactions of discontinuities and weakness planes in the rock. Thus, the mechanical properties of bedding planes need to be understood. Contrary to hard rocks, discontinuities and bedding planes in saliferous rocks are mainly characterised by adhesive friction (Minkley 1989). Softening is caused by loss of adhesive friction and smoothening of surface asperities during shear deformation. The adhesive friction model developed by the IfG (Minkley 2004b) starts from a shear strength approach similar to the Mohr–Coulomb model,

$$\tau _{\mathrm{ MAX}}= \mu \sigma _{\mathrm{ N}}+ c,$$
(8)

with normal stress \(\sigma _{\mathrm{ N}}\) and cohesion c. However, the friction coefficient

$$\mu=\mu_{\mathrm{ K}}\cdot (1+\Delta \mu )$$
(9)

is composed out of a term for kinetic friction,

$$\mu_{\mathrm{ K}}=\tan \left( \phi _{\mathrm{ R}} + i_0 \hbox {e}^{-K_2\sigma _{\mathrm{ N}}}\right) ,$$
(10)

and a factor for adhesive friction,

$$\Delta \mu=\Delta \mu _{\mathrm{vel}} \hbox {e}^{-K_1\sigma _{\mathrm{ N}}}.$$
(11)

The coefficient \(\Delta \mu _{\mathrm{vel}}\), in turn, is given by

$$\Delta \mu _{\mathrm{vel}}=f_{\mathrm{vel}}\Delta \mu _{\mathrm{ MAX}}.$$
(12)

The adhesive friction coefficient \(\Delta \mu _{\mathrm{ MAX}}\) results from normal-load-dependent activation of adhesive forces, and

$$f_{\mathrm{vel}}= \frac{1}{2}\left( 1+\tanh \left[ b_{\mathrm{s}} \ln \frac{v}{v_{\mathrm{crit}}}\right] \right)$$
(13)

describes the velocity dependence of adhesive friction. Here, v is the shear velocity, and \(v_{\mathrm{crit}}\) and \(b_s\) are model parameters. The function \(f_{\mathrm{vel}}\) interpolates between static and dynamic processes (with \(v\ll v_{\mathrm{crit}}\) and \(v\gg v_{\mathrm{crit}}\), respectively). \(\phi _{\mathrm{ R}}\) is the residual friction angle, while the dilatancy angle \(i_0\) represents the surface roughness and causes widening of the shear gap with progressive shear displacement. \(K_1\) and \(K_2\) are curvature parameters. If the shear strength is exceeded, the adhesive parameters are reduced as a function of shear displacement.

The resulting model behaviour is characterised by a shear strength peak with adhesive friction component, a drop in shear strength after the peak is exceeded, and a marked dependence of the frictional force on the velocity of the shearing process. The left panel of Fig. 5 shows the results of a benchmark simulation of two sliding blocks: for a dynamical event, i.e. a hight shear velocity, the frictional force is higher, but drops markedly after the adhesive peak is overcome. This drop can facilitate collapses, because kinetic energy can be quickly released. On the other hand, in a slow shear process, the shear stress does not drop form the residual friction plateau. The right panel shows how an in situ bedding plane is extracted and tested in the laboratory.

Fig. 5
figure 5

Verification of the adhesive shear model on bedding planes between carnallitite and rock salt

In Fig. 6, we illustrate the different shear strength components: the peak strength includes the adhesive friction component given by \(\Delta \mu _{\mathrm{ MAX}}\) [see Eq. (9)], kinetic friction depends on the angle of residual friction and the dilation angle [see Eq. (10)], and finally residual friction only involves the residual friction angle. The relative contribution depends on the velocity of the shear process as well as the displacement-dependent softening to reproduce the complex shear strength behaviour found in situ.

Fig. 6
figure 6

Shear strength of interfaces between rock salt and carnallitite

The visco-elastic-plastic constitutive model and the shear model for bedding planes are implemented as plugins for the numerical codes UDEC/3DEC by Itasca (2014). (The visco-elastic-plastic model is also implemented in the codes FLAC/FLAC3D for systems where bedding planes are not important.) These programs use a distinct element formulation to allow shear displacement between rock blocks and a finite-difference method for deformations of the rock mass itself. Hence, discontinuous processes (sliding on bedding planes, crack formation) can be captured, as well as the essential mechanisms of fluid transport through salt rocks, as we will discuss in Sect. 4. The explicit time-step algorithm (Cundall and Board 1988) is particularly suitable for the modelling of nonlinear processes and instability problems.

The implementations are verified by recalculation of strength and creep tests on rock samples in the laboratory as well as actual mining situations. To determine the shear strength of bedding planes, special specimens were prepared and tested with the servohydraulic MTS direct shear testing system of the IfG (Fig. 5, right panel). Material parameters depend on the salt rock variety and location, and the IfG has an extensive database of parameter sets for salt rocks and bedding planes. In summary, the presented modelling approach successfully describes the behaviour of various salt rocks, from slow creep processes to sudden rock bursts.

3.2 Static (In)Stability

Equipped with the numerical methods discussed in the previous section, we can analyse the behaviour of pillars and larger mining systems.

In Fig. 7, we have simulated the softening of a squat carnallitite pillar with \(\alpha =4\) (C60, i.e. 60 % carnalite content). The zones along the contour are damaged (the colours show dilatancy, i.e. damage), while the pillar core is intact: here, the confining pressure is sufficient to keep the load-bearing capacity above the load. If we ignore the elements with dilatancy \(\epsilon ^{\mathrm{p}}>2\)% (as they would fall out), we get the familiar hourglass shape of in situ pillars (see Fig. 7b, cf. also the lower panels of Fig. 1).

Fig. 7
figure 7

Contour damage in a carnallite pillar due to material softening in the simulation (left panel) and a mine (right panel). After removing the damaged elements, the simulated pillar matches the hourglass shape of in situ pillars well. a Simulated plastic deformation. Elements with dilatancy of 2 % or higher are severely damaged and would fall off the pillar. b In situ pillar with hourglass shape

If the pillars are too slender, they will fail. In Fig. 8, we show a simulation of a carnallite pillar that was made successively less squat (to mimic a secondary and tertiary mining phase) under constant load. The primary pillar with \(\alpha =4.2\) shows a stationary state with constant deformation rate. After a reduction to \(\alpha =3.3\), a new steady state with a higher creep rate is reached. The final reduction to \(\alpha =2.9\) leads to an accelerating deformation rate and a creep rupture.

Fig. 8
figure 8

Simulated creep rupture of a carnallite pillar: the pillar width-to-height ratio was progressively reduced until the pillar failed

To show the relevance of bedding planes, we have plotted the load capacity of a squat (\(\alpha =4\)) carnallitite pillar in Fig. 9. Footwall and hanging wall consist of rock salt, and the cohesion on the interface between carnallitite and rock salt determines the load the pillar can sustain: in the case of a constant cohesion of \(c = 1\) MPa at the contact, the pillar deforms without failure, and the load is roughly constant. On the other hand, a cohesionless surface leads to a complete loss of load-bearing capacity already at moderate deformation. The underlying reason is that the friction on the bedding planes provides the necessary confining pressure in the core of the pillar.

Fig. 9
figure 9

Load-bearing capacity of C\(_{60}\) carnallitite pillar with \(\alpha =4\) with and without cohesion

In the case of static loading, it can usually be assumed that at the contact planes of the pillars to the roof and footwall, cohesive forces are present in addition to frictional resistance. In contrast, under dynamic loads, adhesive friction or cohesion at the contact planes may be suddenly lost (Minkley 1989). Otherwise, rock bursts in carnallitic mining fields with predominantly relatively squat (\(\alpha\,\gtrsim\) 3–4) square pillars would not have occurred.

3.3 Dynamic Analysis

After a series of dynamical simulations both after (Minkley 2002) and before field collapses (Minkley and Menzel 1999), the general mechanism of rock bursts in potash mining is well understood. The fracture process is initiated by the local failure of a pillar, either by dynamic excitation due to blasting or triggered spontaneously by a creep rupture. Based on this local instability, the fracture process spreads like a chain reaction and, within a few seconds, an unstably dimensioned mining field with up to several thousand pillars can be destroyed. An essential prerequisite for the physical explanation of these dynamic processes is the adequate mechanical description of the softening processes, in the salt rock itself and on the bedding planes (see Sect. 3.1).

We first discuss a idealised three-dimensional model of progressive pillar failure in a mine field. The geomechanical model, sketched in Fig. 10, consisted of 400 carnallitic square pillars sandwiched between thick rock salt strata. The overburden is modelled by a 600-m-thick sandstone layer with a regular orthogonal fracture system. The discrete representation of small tectonic joints together with a pore water pressure in the overburden allows to realistically describe the subsidence behaviour above mine fields under quasi-static and dynamic boundary conditions. The mechanical behaviour of the joints is described by the shear model presented in Sect. 3.1.

Fig. 10
figure 10

Geomechanical model. The right panels shows the mining field without the overburden

All strata were described by the visco-elasto-plastic material model (see Sect. 3.1). In the overburden, creep is suppressed by a very high viscosity parameter. The failure strength of the sandstone and shear strength of the interfaces are based on laboratory testing of drill core material from the bunter sandstone (Minkley and Wüste 1979). The compressive failure strength and the softening of the pillar material were determined by uniaxial and triaxial compression tests. Figure 11 shows the boundary curves for a carnallitite with an average carnallite content of 80 % (C80). The upper curve gives the initial yield limit, while the lower curve gives the yield function for the softened material with a plastic deformation of 10 %.

Fig. 11
figure 11

Strength and softening of carnallitite. The curves show the yield limit without deformation (upper curve) and with softening after plastic shear deformation of 10 %. The yield curves are computed according to the visco-elasto-plastic model, Eq. (5)

In addition to the softening properties of the carnallitite rock mass, the stability behaviour of compact pillars under dynamical load conditions is crucially determined by the transition from static to kinetic friction at the contact interfaces between the carnallitite pillars and the rock salt in the hanging wall and footwall. The shear model is capable of describing both the displacement-dependent damage for joints in the hard rocks and the loss of adhesion or cohesion which is essential for saliferous interfaces. The shear parameters were determined from salt blocks with natural bedding planes (Fig. 6) in the IfG using the MTS shear testing system (Fig. 5).

In the model, the fracture process was triggered by the removal of three pillars in the centre of the mining area (right panel of Fig. 10). As a result, there was a chain-reaction pillar failure: Fig. 12 shows the maximum principal stresses in the pillars at different times as the fracture process proceeds outward and a high-pressure zone is formed around the area of the already collapsed pillars, leading to the progressive pillar failure until the field edge is reached. In Fig. 13, the stress evolution in the centre of the pillars is plotted: the pillar load rises, the drops sharply as the pillars are destroyed by brittle fracture. After the initial removal event, the whole field collapsed in about 2 s, and the pillar debris has filled the excavations. After about one second, the overburden has subsided and rests on the footwall, so that the pillar loads rise again.

Fig. 12
figure 12

Time evolution of the chain-reaction pillar collapse in a \(1\,\hbox {km}^{2}\) mining field. At \(t=0\) three pillars in the centre were removed to initiate the collapse. a \(t=0.9\,\hbox {s}\), b \(t=1.4\,\hbox {s}\), c \(t=3.1\,\hbox {s}\)

It should be noted that a chain-reaction-like failure of compact square pillars becomes possible only if adhesive friction is lost at the contact surfaces of the carnallitite pillar to the over- and underlying rock salt. Sliding tracks on bedding planes between pillar and surrounding rock have been observed as a sign of intense shear motion after loss of static friction in various rock burst areas of potash mining.

Fig. 13
figure 13

Pillar loads in the collapsing panel (see Fig. 12). The numbers correspond to pillars from the centre to the edge, as indicated. The collapse front spreads outward with a speed of up to 800 m/s

A practical example of a dynamic fracture process is the back calculation of the rock burst of Völkershausen (Minkley 2004a), which happened on 13 March 1989, in the Werra potash mining district in Thuringia. The geomechanical calculation is based on a geological model section along a WE profile which crosses the collapse area. The layer stack of the Zechstein ranges from the Upper Fulda clay (T6r) below the base of the bunter sandstone cap rock to the footwall of lower Werra anhydrite (A1). The material behaviour of the various layers was described in the geomechanical model with the visco-elastic-plastic material model, with individually adapted properties for hardening/softening, dilatancy and creep. The calculations for the static initial state before the rock burst under simultaneous action of softening and creep simulated a period of 20 years.

The initial impact for the rock burst was a blast on a carnallitite pillar at the second level (Fig. 14, top left, point 1). As the calculation shows, the collapse process stopped after 2.5 s on the eastern edge of the mining field (Fig. 14, top left, point 3). The seismological studies also came to the conclusion that the propagation of the fracture process through the entire mining area happened within a period of 2–3 s (Ahorner 1991). Thus, the chain-reaction pillar fracturing process in the dynamic calculation is in good agreement with the actual timing of the rock burst process.

Fig. 14
figure 14

Rock burst Völkershausen (1989). Upper left pillar stresses, upper right subsidence, lower panel geomechanical model

Figure 14 (upper right part) shows the calculated vertical displacement at different times after initiation of the fracture process. Immediate surface subsidence reached about 1 m, which is in the same order for both numerical back analysis and direct observations after the rock burst. The top left panel of Fig. 14 demonstrates the dynamic nature of the fracture process by showing the time evolution of the stress in selected pillars. While the stresses in the centre of the square carnallitite pillars increase up to 100 MPa before failure (points 1–3 in Fig. 14), the stresses in the eastern adjacent barrier pillars stabilise at about 30 MPa (point 4). The mining area east of the barrier pillar remains spared from the collapse, in agreement with the in situ findings. The pillar stresses oscillate around the initial static load level of 40 MPa (point 5 in Fig. 14, position of the points see Fig. 14, lower panel).

It has to be stressed again that for a long time the complex failure mechanism of load-bearing support elements in salt mining and the triggering of large-scale pillar collapses could not be described satisfactorily in geomechanical models. If only the conventional Mohr–Coulomb friction approach is used, geomechanical calculation models are not able to simulate the stability behaviour of compact carnallitite pillars under static and dynamic loading conditions. Only experimental investigations of the velocity-dependent shear behaviour of carnallite/rock salt interfaces provided the key to understand the discontinuum softening process at high loading rates: under static load conditions, friction and cohesion forces are effective on the contact planes of the pillars to the roof and footwall, without any potential for a drop of shear stresses. Only under dynamic stresses at high shear displacement rates, an additional friction resistance is activated. If this friction peak is overcome, there is a sudden transition from static to kinetic friction, associated with a large drop in shear resistance. During rock bursts in potash, softening processes are at work both in the pillar salt rock itself and also at the interfaces to the surrounding rock.

The softening models for saliferous interfaces and salt rocks that the IfG has developed in recent years provide a consistent and unambiguous explanation for the failure and collapse of compact carnallitite pillars within a few seconds by numerical simulation of the physical processes in geomechanical models.

4 Integrity of Saliferous Barriers

Next to mechanical collapse, the other major risk in salt and potash mining is flooding. Generally, mines require a hydraulic barrier, i.e. a sufficiently thick layer of rock salt, to prevent groundwater from flowing into the excavations, which usually results in a complete loss of the mine. In this section, we will first list the relevant criteria for the integrity of the barrier, with particular emphasis to the minimal stress criterion. Then, we discuss a few examples of flooded mines and the underlying geomechanical reasons.

4.1 Hydromechanical Conditions for Integrity Loss of Saliferous Barriers

The tightness of saliferous barriers is of crucial importance for potash and salt mining and also for the underground storage of hydrocarbons in caverns and waste disposal in salt rock. Undisturbed salt rocks have a very low, and unconnected, porosity (\(\lesssim\)2 % for bedded, less for domal salt). They are impermeable to fluids, as can be seen from a number of natural analogues, where water or gases (e.g. CO2 of methane) are trapped in salt strata under lithostatic pressure for millions of years (Minkley et al. 2015). However, excavations will change the stress and strain distribution, and one needs to check whether the hydraulic properties change. There are three relevant criteria for tightness of the hydraulic protection layer:

  • Dilatancy (or shear stress) criterion Plastic shear deformation can lead to damage in the rock, associated with a volume increase (dilatancy) due to the formation of microcracks, which dramatically increase the permeability.

  • Tensile stress criterion Similar to shear stresses, tensile stresses above the tensile strength of the rock mass will lead to damage and the formation of cracks which can serve as fluid pathways.

  • Minimal stress (or frac) criterion If the minimal stress in the rock mass is lower than the fluid pressure, the fluid can force its way along discontinuities, on the microscale (grain boundaries) or the macroscale (bedding planes).

To ensure hydraulic protection, one thus requires a barrier of sufficient thickness (\(\gtrsim\)10s of metres) around the excavation in which all three criteria are satisfied. However, the criteria differ in reach: mechanical damage generically occurs close to the excavations due to large locally induced shear and tensile stresses. The minimal stress criterion, on the other hand, can also act in the far field due to extended stress redistributions caused by mine convergence. We will thus discuss the fluid transport in salt rocks in some more detail.

4.2 Percolation

Fluid transport in salt rocks differs from the one in porous rocks such as sandstone: polycrystalline structure and viscous behaviour imply that there is very little porosity, and the pores do not form a connected pore space through which fluids could flow. Undisturbed salt rocks are thus composed out of impermeable grains which are joined along closed grain boundaries. Various examples of gas outbursts where gases were contained in salt rocks under high pressure for millions of years underline the tightness of intact rock salt.

Permeation test in the laboratory using coloured liquids confirm that fluid motion proceeds along the grain boundaries (see Fig. 15). However, for fluids to permeate along the boundaries, they have to overcome the normal pressure that tends to press the grains together. The normal pressure on the boundaries depends on the stress state in the rock mass—far away from excavations, the stress is basically isotropic, and the normal stress on grain boundaries equals the isotropic stress. Excavations, on the other hand, change the stress state, and normal pressure is now bounded by the minimal principal stress. This implies that (Minkley et al. 2012b)

  • fluids can move through the rock mass if the fluid pressure is higher than the minimal principal stress, and that

  • fluid motion proceeds in the direction of the maximal principal stress, orthogonal to the minimal principal stress.

This transport mechanism in known as pressure-driven percolation.

Fig. 15
figure 15

Pressure-driven percolation on grain boundaries of rock salt if \(p_{\mathrm{ fl}}>\sigma _{\mathrm{ min}}\)

It is important to appreciate the difference to the familiar Darcy flow through porous media, which occurs as soon as there is a pressure gradient, and in the direction of this gradient: for intact salt rocks, there is a percolation threshold, i.e. a fluid pressure below which there is no flow through the rock mass. The value of the threshold is given by the minimal principal stress. Furthermore, the flow direction is determined by the stress state of the rock, i.e. in the direction of the maximal principal stress. The IfG has conducted numerous laboratory experiments, in situ investigations and numerical simulations of fluid transport through salt rocks supporting this paradigm.

Thus, a complete treatment of fluid motion through salt rocks requires coupled hydromechanical calculations which take into account the discontinuous nature of the rock mass and the motion of fluids along the grain boundaries. For a simplified continuum-based approach, this can be approximated by comparing the minimal principal stress and the ground water pressure at the given depth below the surface. Hence, the minimal stress criterion proper states that the factor of safety

$$S=\frac{\sigma _{\mathrm{ min}}}{p_{\mathrm{ fluid}}}$$
(14)

has to be larger than one. For a less conservative estimate, one might replace the minimal principal stress by the stress orthogonal to the potential path between aquifer and mine (usually the horizontal stress) because of the directed nature of the fluid transport.

The mechanism of pressure-driven percolation generically has a larger reach than damage-based fluid transport since damage is usually localised around the mine openings while the excavations can change the large-scale stress state in the salt formation. In particular, subsidence changes over hard mining edges, i.e. transitions from mining panels to undisturbed rock or sufficiently large barrier pillars, can cause extensional strains which lower the horizontal stress and can open up fluid pathways (see Fig. 16).

Fig. 16
figure 16

Sketch of a “hard mining edge.” The extensional strain in the transition zone between large subsidence over the excavations and little to no subsidence over the barrier pillar lowers the horizontal stress, potentially allowing ground water to percolate down to the mine

4.3 Examples of Integrity Loss of Saliferous Barriers During Mining Activities

Around the world, violation of the minimal stress criterion has repeatedly caused floodings of mines. Here, we will present a few examples and discuss the underlying reasons. This survey shows that, regardless of the specific deposit, the same basic mechanism is at work.

4.3.1 Völkershausen 1989

A loss of integrity and tightness of hydraulic protective layers around the mining horizon in salt mining can occur both under static and dynamic loading conditions. In the latter case, the mechanism generically is a sudden unloading of barrier strata when a pillar system collapses and the over- and underlying strata do not completely follow.

In connection with the convergence jump in the mining horizon by the rock burst Völkershausen 1989 (see Sect. 3.3), \(\hbox {CO}_{2}\) was detected emanating from cracks in the floor of the mining chambers at the SE edge of the field. The subsaliferous bed (lower Werra anhydrite) contains CO2 originating from volcanic activities in the tertiary, with a pressure of about 7–7.5 MPa. The calculation of the secondary stress field after the rock burst indicates that the minimal principal stress in the base of the rock salt suddenly dropped from about 20–1 or 2 MPa, much lower than the gas pressure (see Fig. 17). The footwall barrier in the region of the Feldatal fault zone was only about 30 m, and a gas frac was inevitable.

Fig. 17
figure 17

Gas frac during the Völkershausen rock burst due to unloading of the footwall barrier

4.3.2 Solikamsk II

Dynamical loads can also cause failure of the hanging wall barrier. On 18 November 2014, a sinkhole of about 30 m diameter appeared above the Solikamsk II mine field (Fig. 18). In hindsight, increased microseismicity since September 2014 and lowered salt concentrations in a subsurface backfill basin due to freshwater dilution can be interpreted as indications of the sinkhole formation.

The sinkhole formed above the edge of an area which collapsed in a rock burst on 1 May 1995. The mining depth is about 330 m, and the rock burst caused surface subsidence of up to 4.5 m (Fig. 19). In 2005, the IfG analysed the rock burst and assessed the hydraulic barrier (Minkley et al. 2005). The minimal stress criterion was violated throughout the full protective barrier (Fig. 20, upper panel); thus, a simplified isotropic interpretation of the criterion would have implied an immediate flooding of the mine. However, when comparing the groundwater pressure with the horizontal stress in the protective rock salt-marl interbedding (Fig. 20, lower panel), the violation did not penetrate the full barrier: The horizontal confining pressure exceeded the groundwater pressure in a thin rock salt layer (thickness at the southern edge only a few metres) and so was able to block the pressure-driven percolation in the vertical direction for some time.

Fig. 18
figure 18

Sinkhole above the Solikamsk II mine (November 2014)

Fig. 19
figure 19

Rock burst Solikamsk II (1995): collapsed area (dashed line) and surface subsidence (contours). The line marks the two-dimensional model cut, and the red dot indicates the 2014 sinkhole (colour figure online)

Fig. 20
figure 20

[from Minkley et al. (2005)] (colour figure online)

Assessment of the hydraulic barrier in Solikamsk after the rock burst. Upper panel ratio of minimal principal stress to groundwater pressure, lower panel ratio horizontal stress to groundwater pressure. Values below one (pink) indicate a violation of the (minimal) stress criterion

The continuing subsidence after the rock burst led, via the associated extensional strains above the mining edges, to a further decrease in the horizontal confining pressure and finally to pathways for pressure-driven percolation to reach the mining horizon. Along these pathways, the rock salt was dissolved until the overburden could not span the gap anymore and the sinkhole appeared.

The IfG back analysis of the rock burst (Minkley et al. 2005) concluded that “…a hydrogeological hazard cannot generally be excluded in the future, which could potentially lead to a complete flooding and the loss of the whole mine.” This prediction has been confirmed 10 years later. The inflow started at a few \(1000\,\hbox {m}^{3}/\hbox {h}\), was at 500–800 \(\hbox {m}^{3}/\hbox {h}\) in January and early February 2015 and will likely increase in spring.

4.3.3 Weeks Island

Static loads can cause extensional strains that result in a violation of the minimal stress criterion, in particular due to subsidence over mining edges.

In the salt dome of Weeks Island, Louisiana, rock salt has been mined in depths 150–220 m on two floors from 1902 to 1977 (Bauer et al. 2000). In 1981, the rock salt mine was repurposed as part of the Strategic Petroleum Reserve (SPR). After the storage of oil, an inflow of water was noted, but initially considered non-threatening. Not before two sinkholes occurred in 1992 and 1995 (Fig. 21), the oil was drawn out again. Both sinkholes have formed above the perimeter of the mine, where both floors are stacked vertically and form a “hard” mining edge. Investigations to determine the causes with respect to geological peculiarities brought inconclusive results. Only after thorough investigations the flooding of the mine was ascribed primarily to geomechanical causes (Neal et al. 1995; Thoms and Gehle 1995). Over the mining edge, extensional strains occurred in the 110-m-thick overlying rock salt barrier due to the induced subsidence. This led to a reduction in the minimum principal stress in the area of the salt top, until the minimum stress criterion was violated and water could open the grain boundaries and find its way into the mine.

Fig. 21
figure 21

[from Neal et al. (1995)]

Mine-induced sinkholes over the US Strategic Petroleum Reserve storage facility at Weeks Island salt dome, Louisiana

4.3.4 Berezniki I

The same mechanism is responsible for the loss of integrity of the overlying geological saliferous barrier followed by drowning of the sylvinite mine Berezniki I in flat bedding (in 2007; the nearby mine Berezniki III drowned under similar circumstances in 1986). Similar to that what happened in the Weeks Island mine, a sinkhole occurred above the mine Berezniki I close to a hard mining edge which developed where the mining areas of two levels terminated at the shaft safety pillar. Northwest of the sinkhole, the largest subsidence above Berezniki I was in the order of 3.7 m. The IfG has conducted an back analysis to explain the integrity loss on the basis of coupled hydromechanical calculations. The geomechanical model describes the situation of the two-floor mining in about 250 m depth in the sylvinitic seam Kr II and the partially carnallitic seam AB using the visco-elastic-plastic material model for saliferous rocks (Sect. 3.1). The numerical model demonstrates how extensional strains are initiated above the hard mining edge as a result of the convergence and subsidence, so that the minimum principal stress drops below the water pressure in the overburden and pressure-driven hydraulic flow paths are created in the 90-m-thick overlying saliferous barrier (Fig. 22). Once these flow paths reach the dilatant zone around the mining chambers, water starts to flow into the mine and dissolve the barrier in the process. Finally, the resulting cavity in the salt becomes so large that the overlying overburden could no longer resist and a sinkhole was formed (Fig. 22).

Fig. 22
figure 22

Sinkhole Berezniki I and coupled hydromechanical simulations showing the flow path due to pressure-driven percolation above the hard mining edge

4.3.5 K2 Mine, 1985

On 29 December 1985, water was discovered in the International Minerals and Chemical Corporation (IMC) K2 potash mine in Saskatchewan (van Sambeek 1993). The inflow with a rate of about 1800 \(\hbox {m}^3/\hbox {h}\) was concentrated at the south-western edge of the mine in Panel 10B, coming from the rock salt roof. The rock salt barrier at the basis of the Dawson Bay Formation (limestone and shale) was only about 30 m thick. The K2 mine excavated sylvinite from the Esterhazy Member of the Upper Prairie Evaporite Formation in a depth of 975 m, using continuous long chamber mining. The extraction ratio within the panels initially was about 30–35 %, and by 1985 it had increased to 50–55 %. Up to the flooding, the mine was operating for 23 years. The mine operators tried in vain to save the mine by injection of concentrated calcium chloride into the Dawson Bay Formation. After a brief decrease in inflow rates, a new inflow appeared at the south-eastern edge.

Again, this fits well with the general geomechanical mechanism. Over mining edges or bearing pillars, the extensional loads due to subsidence lower the minimal principal stress and lead to violations of the minimal stress criterion. Hence, pressure-driven percolation occurs and generates fluid pathways from the aquifer to the mine.

5 Summary and Conclusions

In this report, we have summarised dimensioning methods for potash and salt mining, with the emphasis on the stability of the pillar system and the integrity of the hydraulic barrier. By comparing various dimensioning methods developed for potash mining with in situ failure cases, we see that the calculated load capacities for compact carnallitite pillars are partly overestimated. This is because

  • the softening behaviour of the rock mass and

  • the influence of contact surfaces at roof and footwall.

have not properly been taken into account.

The softening of salt rocks leads to a reduction in the pillar load capacities over time. This process manifests itself first at the contour, where it produces the familiar hourglass pillar shape, but can also progress further into the pillar core. The bedding planes provide the necessary confining pressure to ensure an undamaged pillar core. Dynamic system instability in salt mining occurs if the pillars are insufficiently dimensioned, and is characterised by a chain-reaction pillar fracturing process that leads to the collapse of mining fields in a few seconds with high seismic energy release. A crucial factor is the sudden reduction in the shear strength on the bedding planes under large and fast shear displacements, leading to a loss of confining pressure and hence of load-bearing capacity.

When solving numerous practical problems in potash and rock salt mining, it clearly turned out that a treatment without taking into account strain softening and the existing bedding planes will not provide a satisfactory explanation of the observed rock mechanical processes. This means that the constitutive models used in numerical simulations need to reflect these basic properties. Empirical formulae that provide estimates for the load capacity cannot capture the full pillar behaviour, much less that of large mining fields in a complex geological setting.

For a complete understanding of the geomechanical phenomena, the discontinuous nature of salt rocks is important, not only on the macro- but also on the microscale. As we saw, softening processes in salt rock and along macroscopic discontinuities determine the load- bearing behaviour of pillars and thus the dynamic system stability of mining fields. The understanding of the discontinuum mechanical properties on a micromechanical level, i.e. at the grain boundaries of the salt crystals, was identified as the key to a physical understanding of fluid pressure-driven percolation behaviour and thus the integrity of saliferous barriers.

The back analysis of a number of active salt mine floodings around the world demonstrates that, due to the occurring subsidence in the mining seam, extensional strains are induced in the surrounding saliferous horizons. Above these zones, the minimum stresses in the salt barrier are lowered below the ground water pressure, thus initiating a pressure-driven fluid percolation. A mine layout has to be designed such that the barrier integrity is maintained throughout the operation.

The mining practise provides the ultimate test whether the dimensioning methods based on experience and geomechanics adequately describe the mechanical properties of salt rocks. The analysis of failure cases in situ, in 1:1 scale, is an important basis for the development of conceptual models and a check of assumptions. Only through a comparative analysis of previous worldwide failure cases, the mechanisms of mine floodings and rock bursts are better understood, and the insights can be used for improved mine design.