The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

Numerical Analysis of the Life-time of an Abandoned Gypsum Mine

D. Betti, G.Buscarnera, R. Castellanza, R. Nova Milan University of Technology (Politecnico), Milan, Italy

Keywords: bonded geomaterials, collapse, chemo-plasticity, gypsum, mines, weathering

ABSTRACT A numerical analysis of the evolution with time of an abandoned gypsum mine flooded by water is presented. The lay-out of the problem is shown first, together with the experimental results obtained in laboratory tests on gypsum specimens retrieved from site. An elastic-plastic strain-hardening constitutive model, capable of taking weathering effects into account, is sketched next. A chemo-mechanical analysis of the progressive dissolution of a gypsum pillar in contact with water is then presented. It is shown that it is possible to describe the dissolution process until pillar failure under the action of the overburden is achieved. The remaining life-time of the abandoned mine can be therefore estimated and a decision about the possible remedial measures taken on the basis of rational considerations.

1 Introduction Mining activities in Europe came progressively to an end in the last century, being largely more convenient from an economical viewpoint either getting raw materials from overseas developing counties or recycling waste. Abandoned mines have become however a potential source of environmental hazard. Water, that was pumped away during mine exploitation, has now flooded the left in place empty chambers and attacked in various ways the rock pillars. In Lorraine (France), for instance, (El Shayeb et al., 2001), colonies of bacteria have proliferated and progressively eroded the siderite cementing iron oolites, therefore reducing the pillar bearing capacity. In a number of cases the weight of the overburden could not be sustained any longer and collapses occurred. The settlement trough induced at the soil surface by such deep failures reached hundreds of meters in width and as much as one meter in depth. Several houses were consequently destroyed and some people injured or even killed. Although caused by different physico-chemical phenomena, similar collapses were recorded in evaporitic rock deposits (e.g. Ghoreychi, 2006), with similar effects on facilities, houses or people. Water, or even air humidity, progressively dissolves gypsum pillars, so that pillar collapse under the overburden weight is certain, sooner or later. The engineering problem is therefore the estimation of the remaining life-time of the abandoned mine and the design of the possible remedial measures to delay such a phenomenon and reduce its potentially catastrophic consequences. The paper deals with a numerical analysis of an abandoned gypsum mine in the Orobic Pre-Alps (Northern Italy). The mine was flooded by water coming from adjacent calcareous strata after exploitation ended, more or less 30 years ago. A continuous water circulation affects the totally submerged two inferior mining levels, dissolving gypsum and jeopardising the stability of the entire mine. The lay-out of the problem is presented first, together with the results of an extensive series of laboratory tests on gypsum specimens retrieved from site. A constitutive model, capable to describe the behaviour of bonded geomaterials, taking degradation of mechanical properties due to weathering into account, is presented next. A chemo-mechanical analysis of the problem is then shown in which the usual static equations are coupled with the chemical reactions that govern the dissolution process. It is demonstrated that, on the whole, the model is capable to describe the progressive degradation of the pillars until collapse occurs. The remaining life-time of the mine can be then estimated and a decision concerning possible remedial measures can be taken on the basis of rational arguments.

2 Problem setting The abandoned mine we shall be referring to, the Carale mine at S.Brigida (Lombardy), is located in an evaporitic formation known as S. Giovanni Bianco formation, that was deposited in a marine environment (Carnic, Superior Trias). It is constituted by alternating gypsum and anhydrite deposits. Figure 1 shows two sections of the mine. The two lower excavation levels are now completely filled with water, that is flowing through from the adjacent uphill highly fractured dolomitic rocks, and is seeping downhill under the village located nearby. Since gypsum is

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highly soluble in water (rate of dissolution in water at ambient temperature about two orders of magnitude larger than that of limestone), the mine pillars are progressively reduced in diameter. At constant overburden load, the stress on pillars is therefore increasing until the failure level is achieved and mine collapse occurs.

Level 1: dry

Leve 2: partially flooded

Level 3 e 4: flooded

Level 1: dry

Leve 2: partially flooded

Level 3 e 4: flooded

Water INFLOW

Water OUTFLOW

(a) (b)

Figure 1. Sections of the Carale mine: (a) side view; (b) planar view of the 3th excavation level

The overburden pressure on the pillars varies with the pillar position from 2 to 6 MPa, depending on the depth of the pillar with respect to the free surface. The equivalent diameter of the pillar ranges from 4 to 8 m. As a first approximation, at least in the central part of the mine, it can be assumed that the mine geometry is periodic. A single pillar can be schematised as a rock cylinder within a larger cylindrical chamber filled with water.

3 Experimental results A number of specimens were retrieved from site and tested in unconfined compression in the laboratory. Before testing, each specimen was exposed to the action of flowing water for different time lengths. The effect of water on gypsum specimens is clearly illustrated in Figure 2a. The corresponding measured values of the limit load are plotted in Figure 2b. The trend of the compressive strength is similar, passing from 16 MPa to zero, but its exact value depends on the diameter chosen as representative for the degraded specimen. After 7 days, when the diameter is still more or less constant along the height of the specimen, the compressive strength is reduced to about 5 MPa. Correspondingly, the initial stiffness modulus is reduced to 0.7 GPa, from the original value of 2.5 GPa.

(a) (b)

Figure 2. Effect of exposing gypsum specimens to flowing water: (a) specimen degradation for different time lengths; (b) measured values of limit load (initial specimen diameter 25 mm) vs. time of exposure to a water flux

In order to reproduce the pillar collapse in the abandoned mines, a small scale (1: 2000) weathering test has been conducted. A gypsum specimen (H = 2.5 mm) has been subjected to dissolution in flowing water under a constant axial stress of 6 MPa (Figure 3a). This axial stress is consistent with the in situ axial stress of the actual pillars. The results of the experimental test in term of settlement vs. time are reported in Figure 3b. After 21 days the dissolution of the gypsum induces the collapse of the small scale pillar, as it is clearly visible in Figure3c.

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Time [days]

Settl

emen

t[m

m]

(a) (b)

(c)

Figure 3. Constant load tests on a small scale pillar subjected to water flux

The chemical equation governing the dissolution process for gypsum is:

2 24 2 2 4 22 3CaSO H O H O Ca SO H O+ −• + ⇔ + + (1)

According to Liu et Nancollas (1971) the rate of gypsum dissolution in pure water is linearly dependent on the difference between the concentration of the saturated solution and the current concentration of gypsum in water, i.e.:

( ( ))sdm KA C C tdt

= − (2)

where m is the gypsum mass dissolved in water, A the contact area between water and gypsum, Cs the gypsum saturation concentration in water and C is the current concentration, while K is an experimental constant. In the following, dissolution is considered as a volumetric process. Taking into account Eqs. (1) and (2) and disregarding the sulphate production, it is assumed that:

[ ] [ ]( )4 2 2 2

1 2 s

CaSO H OK H O Ca Ca

t+ +∂ •

⎡ ⎤ ⎡ ⎤= − −⎣ ⎦ ⎣ ⎦∂ (3)

[ ]( )2

2 22 2 s

CaK H O Ca Ca

t

++ +

⎡ ⎤∂ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦∂ (4)

[ ] [ ]( )2 2 2

3 22s

H OK H O Ca Ca

t+ +∂

⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦∂ (5)

where the quantities within square parentheses indicate the concentrations of the different substances (in grams per cubic metres). For stoichiometric reasons (Eq (1)), the three constants are not independent, but:

[ ]

2

2 14 22

M

M

M CaK K

M CaSO H O

+⎡ ⎤⎣ ⎦=•

[ ]

[ ]2

3 14 22

M

M

M H OK K

M CaSO H O=

• (6)

where MM is the molar mass of the quantity within parentheses.

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The value of the saturation concentration of the calcium ions in water can be derived by knowing that 2.5 kg of gypsum are dissolved in 1 cubic metre of water. Since the molar mass of gypsum is 172.168 g/mol, the saturation concentration in 1 cubic metre of water is equal to 14.52 mol/m3. The value of the constant K1 has been experimentally determined in the following way. A mass of 0.5 g of gypsum was put in a 200 cm3 container, filled with water. The container was shaken for 30 minutes and the residual amount of gypsum deposited on the base of the container weighed. The residual weight of gypsum is only 5.6% of the original value. By integrating Eqs (3) and (4) it is possible to determine the value of the constant that is consistent with the measured residual value, i.e. K1=6.877mm3/(gs). The dissolution of a solid element in space and time is governed by the following equation:

2ii i i

C D C Rt

∂= ∇ +

∂ (7)

Where Ci and Di are the concentration and the diffusion coefficients of the i-th species, respectively, whilst Ri is the corresponding reaction term, i.e. the r.h.s. term of Eqs. (3-5). To simplify the problem, we shall assume that K3=0, that means that the amount of water produced by the reaction of gypsum with water is negligible with respect to the amount of water flowing from the exterior to the interior of the gypsum solid. It is also assumed that the diffusion coefficients of gypsum and calcium ions are nil, i.e. they are assumed to remain where they are produced. The only term that need to be experimentally determined is therefore the molecular diffusion coefficient of water within the solid, Dg. The following experiment was conducted. A series of gypsum specimens, with a diameter of 25 mm were put in a container and exposed to flowing water. The calcium ions are considered to be transported away with the flux so that gypsum is always in contact with pure water. After 94 days, in the average, the specimens are completely dissolved. The experiment is simulated as shown in Figure4. A column of water, that is assumed to be pure throughout the test, is adjacent to the specimen boundary. By numerically integrating Eq (7) with a finite difference scheme, it is possible to determine the value of the gypsum concentration step by step within the entire specimen. The appropriate value of the diffusion coefficient is found by trial and error, imposing that the gypsum concentration at the nodes corresponding to the specimen symmetry axis becomes zero after 94 days. The value of the diffusion coefficient derived is Dg.=0.7407*10-5 mm2/s. The FD code used for solving the problem is known as RT3D (Clement, 1997). Having determined the necessary parameters, it is now possible to simulate the diffusion of water within the specimen when only a portion of its external surface is in contact with water. Figure 5 shows a series of snapshots of the concentration of calcium sulphate within the specimen at different instants of time.

symmetry axis

time [days] (a) (b)

Figure 4. Determination of the diffusion coefficient: (a) mesh used for the numerical analysis; the black column is made of pure water throughout the test, whilst, initially the specimen is constituted by dry gypsum; (b) Variation of

gypsum concentration with time at the nodes corresponding to the specimen symmetry axis

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[CaSO42H2O] [kg/m3]

Figure 5. Snapshots of the concentration of calcium sulphate [CaSO42H2O] within the specimen at different time instants. Pure water is in contact with gypsum only in the central part of the external boundary of the specimen.

4 Constitutive model for bonded geomaterials allowing for weathering induced bond degradation

The mechanical behaviour of bonded soils can be successfully described by constitutive models based on the theory of hardening plasticity or its extensions. In this paper, the general constitutive framework for describing the mechanical behaviour of a bonded geomaterial subject to mechanical and/or chemical degradation is presented. For the sake of simplicity, it is assumed that degradation, whatever its cause may be, affects internal variables only and not other material properties, e.g. elastic moduli or friction angle. As a first fundamental hypothesis, it is assumed that the material behaviour can be considered as isotropic. This clearly limits the type of materials to which the proposed constitutive theory can be applied. This choice has been made for the sake of simplicity, to limit the complexity of the model from a conceptual point of view, and to keep to a minimum the number of material parameters necessary for its calibration. It is further assumed that, under the hypothesis of small transformations, the strain rate is decomposed additively

in an elastic, reversible part, eε& and a plastic, irreversible part pε& : e p= +ε ε ε& & & (8)

The elastic behaviour of the material is defined by postulating the existence of a strain energy function ( )eψ ε such that

( )ee

ψ∂=∂

σ εε

(9)

It is therefore clear that elastic moduli are not affected by accumulated plastic strains (e.g. mechanical destructuration), by weathering, or by any other non-mechanical action. Irreversibility is introduced by requiring the state of the material, defined in terms of the effective stress tensor σ and the set of the internal variable vector q to lie in the convex set:

{ }( , ) ( , ) 0f= ≤σΕ σ q σ q (10)

The internal variables q ; are assumed to depend on the history of the material element, via the accumulated

plastic strain tensor pε ; and a set of (scalar) internal variables ϑ (possibly reduced to one) taking into account the degradation caused by all the non-mechanical effects (chemical degradation, temperature, viscosity, etc.):

( , )= pq q ε ϑ (11) The evolution of the internal variables is thus provided by the following generalized hardening law:

∂ ∂= +∂ ∂

pp

q qq εε

&& & ϑϑ

(12)

As in the classical theory of plasticity, plastic strain rates are defined by the following nonassociative flow rule:

( , )p gγ ∂=∂

ε σ qσ

&& (13)

where ( , )g σ q is the plastic potential, while γ ≥& is the plastic multiplier. The loading/unloading conditions are prescribed in terms of the so-called Kuhn–Tucker complementarity conditions:

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0γ ≥& ( , ) 0f ≤σ q ( , ) 0fγ ⋅ =σ q& (14) which imply that plastic loading can occur only for states on the yield surface. Upon plastic loading the following consistency condition must hold:

( , ) 0f =σ q& (15) Working on the equations introduced above the following expression of the plastic multiplier is obtained:

1

p

f fK

γ⎛ ⎞∂ ∂ ∂

= ⋅ ⋅ + ⋅ ⋅⎜ ⎟∂ ∂ ∂⎝ ⎠e qD ε

σ q&& & ϑ

ϑ (16)

where ( ) / 2x x x≡ + denotes the positive part of x , and:

pf gK H∂ ∂

= +∂ ∂

eDσ σ

; f gH

⎛ ⎞∂ ∂ ∂= − ⋅ ⋅⎜ ⎟∂ ∂ ∂⎝ ⎠

p

qq ε σ

(17)

4.1 Constitutive functions In the following a basic description of the constitutive functions characterizing the model will be given. Details can be found in Nova et al. (2003). The mathematical expression for both the yield function and the plastic potential is the expression originally proposed by Lagioia et al. (1996) where the influence of the 3rd invariant has been considered through the expression proposed by Gudehus (1973). A point of major importance in the overall development of the model is the hardening law. In the expression of the loading function, the material variable ps plays the role of the preconsolidation pressure. Under the assumed hypothesis of material isotropy, the evolution of ps is associated with the invariants of the plastic strain rates, which reflect the macroscopic effects of irreversible fabric modification. It is assumed that:

( )p ps s s v s sp pρ ε ξ ε= +& (18)

where sρ and sξ are material constants. The quantity pm accounts for the effects of interparticle bonding. The degradation of interparticle bonds can occur for both mechanical and non-mechanical effects, like grain rearrangements or chemical dissolution. The following evolution equation for the internal parameter pm is therefore adopted:

( )p p mm m m v m s

pp pρ ε ξ ε= − + + && ϑϑ

(19)

where mρ and mξ are material parameters controlling the rate of mechanical degradation of bonding. The

second term, playing the role of the quantity ∂∂q &ϑϑ

in Equation (12), quantifies the non-mechanical debonding.

The main consequences of the non-mechanical hardening term are described by figure 6, which shows the evolution of the yield surface with the degree of weathering. A simulation of an oedometric test in which debonding is artificially induced by seeping an acid through the specimen is schematically shown in Figures 6 b and 6c (after Nova et al., 2003)

(a) (b) (c) Figure 6: (a) Yield surface evolution with the degree of weathering; (b) experimental and calculated stress path in the triaxial plane; (c) measured and predicted variation of axial strain with time.

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5 Chemo-mechanical analysis The codes RT3D and GeHoMadrid (Fernandez Merodo et al.,1999) were combined together (Parma, 2004) to treat chemo-mechanical problems such as that of interest for Carale mine. The architecture of the combined code GHM-RT3D can be schematised as follows. Given the initial and boundary conditions, concerning the concentrations of the different chemical species, RT3D calculates the variations of such quantities for the first time step. In particular it is calculated point by point the current value of gypsum concentration and the degradation parameter Xd,, defined as:

[ ][ ]

4 2

4 2 0

21

2d

CaSO H OX

CaSO H O•

= −•

(20)

where the denominator of the last member is the gypsum concentration at the beginning of the process. The constitutive law of the soil is updated and the new distribution of stresses and strains is calculated with GeHoMadrid assuming:

2( ) (1 )d dX Xϑ = ϑ = − (21) It is therefore possible to determine the settlement evolution during time of a laboratory specimen under constant load or even that of a mine pillar under the weight of the overburden. The simulation of the constant load test on a small scale pillar, subjected to water flux, is presented In Figure 7. The deformed mesh (x10) is presented in Figure 7a, while the settlement induced by the dissolution process is shown in Figure 7b and the redistribution of axial stresses in the specimen sections (for different instants of the dissolution process) is reported in Figure 7c. It is interesting to note that the coupled chemo-mechanical code is really time dependent: in fact the calculated specimen failure occurs at 20 days as in the experimental test.

Time [days]

Settl

emen

t [m

m]

radius [mm]

Axi

al s

tres

s [M

Pa]

0 days10 days 14 days 20.3 days

(a) (b) (c)

Figure 7. Simulation of small scale pillar test: a) deformed mesh (x10); b) settlement vs. time; c) redistribution of axial stresses in the specimen sections at increasing time of the dissolution process.

Figure 8 shows the calculated evolution of the degradation parameter and of the second invariant of the corresponding plastic strain deviator within a pillar 5 m wide under the action of a constant load of about 120 MN, such that the initial pressure on the upper base of the pillar is 6 MPa. Figure 9a shows the calculated evolution of the axial stress within the pillar, while in Figure 9b it is plotted the trend of the displacement of the node on the symmetry axis and upper base. After about 240 years from the beginning of the process convergence is lost. The degraded pillar is in fact no more capable of sustaining the imposed load and a catastrophic collapse takes place.

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(a)

(b)

Figure 8 Evolution of a) the degradation parameter and b) the second invariant of plastic strain deviator at different time instants.

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3

radius [m]

vert

ical

str

ess

[MPa

]

initial vertical stress

62 years

125 years

195 years

235 years

0

2

4

6

8

10

12

14

16

18

20

0 50 100 150 200 250 300

time [years]

disp

lace

men

t [m

m]

a) b)

Figure 9. a) Evolution of axial stress within pillar; b) trend of the displacement of the pillar base with time

6 Conclusions A chemo-mechanical numerical approach is carried out to simulate the degradation of a gypsum pillar in an abandoned mine. After an outline of the problem, the chemical behaviour of a gypsum specimen is reproduced by integrating the appropriate chemical equations. Combining the chemical model with a proper constitutive law (an elastic-plastic strain-hardening law, capable of taking bond degradation into account) it is possible to define a chemo-mechanical model to reproduce the experimental test of a specimen surrounded by water under a constant load. The conceptual path followed for the gypsum specimen is ten extended to the case of an actual mine pillar. It is shown that it is possible to estimate with the presented analysis the remaining life-time of the abandoned mine. Planning possible remedial measures is now pssible on the basis of rational considerations.

7 Acknowledgements The financial support of the municipality of Santa Brigida (BG – Italy) is gratefully acknowledged.

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8 References Clement T.P. 1997. RT3D: a modular computer code for simulating reactive multispecies transport in three-dimensional

groundwater aquifers, Pacific Northwest National Laboratory,Richland, Washington D.C (USA).

El Shayeb, Y., Kounaili, S., Josien, J.-P., Guerniffey, Y. 2001. Towards the determination of surface collapse type over abandoned mines in the Lorraine iron basin. In: Sarkka, &, Eloranta, &. (eds.), Rock mechanics – a challenge for society, Lisse, Swets & Zeitlinger, 819–824.

Fernandez Merodo J.-A., Mira P., Pastor M., Li T. 1999. GeHoMadrid User Manual. Internal Report. CEDEX, Madrid (Spain).

Ghoreychi, M., 2006. Coupled processes involved in post-mining, INERIS, Eurock 06 (France).

Gudehus, G. 1973. Elastoplastische Stoffgleichungen fur trockenen Sand, Ingenieur_Archiv, 42, 151-169.

Lagioia, R., Puzrin, A. M., Potts, D. M. 1996. A new versatile expression for yield and plastic potential surfaces, Comp. & Geotechnics, 19, 171-191.

Liu S.T., Nancollas G.H. 1971. The kinetics of dissolution of calcium sulphate di-hydrate, Journal of Inorganic Nuclear Chemistry, 33,, 2311-2316.

Nova, R., Castellanza, R., Tamagnini, C., 2003. A constitutive model for bonded geomaterials subject to mechanical and/or chemical degradation, Int. J. Numer. and Anal. Meth. Geomech., 27, 705-732

Parma M. 2004. Modellazione dell’accoppiamento chemo-meccanico nei materiali cementati, Master Thesis, Milan University of Technology (Politecnico).

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