1. INTRODUCTION

Swelling rocks exhibit the characteristic property of in-creasing their volume when they interact with water. This process constitutes a significant problem in founda-tion engineering [1] and tunneling [2]. Tunneling in swelling rocks in particular is one of the engineering tasks that still entails major inherent uncertainties today. Among the significantly problematic types of swelling rocks are the anhydritic claystones of the Gypsum Keu-per formation, i.e. rocks composed of a clay matrix comprising finely distributed anhydrite. This kind of rock is widely distributed in North-Western Switzerland and South-Western Germany and has caused serious damage, operational problems and very high repair costs in a number of tunnels. Anagnostou et al. [2] provided an overview of the theoretical models for the swelling problem and of recent or ongoing research works. Fur-thermore, they discussed the processes underlying the swelling phenomenon from a qualitative point of view and identified fundamental questions which are im-portant not only for understanding the phenomenon, but also from a tunnel design approach. More specifically, there are still open questions concerning the role of chemical reactions and transport processes, the role of the clay matrix and the relationship between swelling pressure and strain.

The swelling of anhydritic rocks can be attributed to the transformation of anhydrite into gypsum crystals. This transformation takes place through the solution phase and occurs because, under the conditions prevailing in the field, the equilibrium concentration of anhydrite is

higher than the equilibrium concentration of gypsum. The equilibrium concentration or solubility of a mineral is the maximum amount of solute that dissolves in a sol-vent. It depends on the pressure, temperature and pres-ence of foreign ions in the solution (see, e.g. [3, 4]). The minerals with lower solubility represent the stable min-erals in a system. In the present paper, conditions are chosen under which gypsum is a thermodynamically stable mineral. These include: atmospheric pressure; ab-sence of foreign ions; temperature lower than the transi-tion temperature between anhydrite and gypsum, which ranges between 42 and 60 °C, according to the literature [5]. Table 1 gives, among other parameters, the equilib-rium concentrations of anhydrite and gypsum at a tem-perature of 20 °C [6]. This temperature lies within the relevant range for tunnels in Gypsum Keuper.

When anhydrite is in contact with pore water, it starts to dissolves into calcium and sulphate ions and once the equilibrium concentration of gypsum is reached, the lat-ter starts to precipitate. The growing gypsum crystals may cause loosening of the rock, increase its permeabil-ity and accelerate seepage flow with the consequence that leaching of gypsum may occur at a later stage be-cause the calcium and sulphate ions circulate with the water (advection). This has been observed in the field, for example in the Schanz railway tunnel [7]. It may, nevertheless, also happen that the gypsum crystals cause clogging of the pores, thereby reducing permeability and seepage flow rate.

The present study disregards such transport processes by considering a closed system and by focusing on the two

Simultaneous anhydrite dissolution and gypsum precipitation in a closed swelling rock system Serafeimidis, K. and Anagnostou, G. ETH Zurich, Zurich, Switzerland

Copyright 2012 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 46th US Rock Mechanics / Geomechanics Symposium held in Chicago, IL, USA, 24-27 June 2012.

This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: Anhydritic claystones are among the most problematic rocks for tunneling. Their swelling has led to serious dam-age and high repair costs in a number of tunnels. One cause of the swelling process is undoubtedly the formation of gypsum. An-hydrite dissolves into calcium and sulphate ions in the pore water and gypsum crystals subsequently precipitate. It is a markedly time-dependent process which in nature might take several decades to complete. Apart from the kinetics of the chemical reactions involved, advection and diffusion are also important factors in the evolution of swelling. The present paper focuses on simultane-ous anhydrite dissolution and gypsum precipitation while leaving aside the transport processes. Understanding such a so-called “closed system” represents the first step towards more complex models involving transport. The paper begins with the governing equations and presents estimates of the kinetic parameters. The model is calibrated with experimental results from the literature, and parametric studies are performed in order to investigate the role of the initial volumetric fractions of the constituents and the specific surface areas of the minerals involved. A simplified model of anhydrite hydration is proposed, which identifies the gov-erning process and possible orders of magnitude in terms of the swelling process duration.

more important chemical reactions: the anhydrite disso-lution and the gypsum precipitation.

The Authors presented the first results concerning the kinetics of these chemical reactions in [8]. More specifi-cally, they investigated the effects of the initial size and shape of the minerals on the time evolution separately for anhydrite dissolution and gypsum precipitation. They also proposed a simplified criterion allowing estimates to be made of whether anhydrite dissolution or gypsum precipitation is the limiting mechanism for the duration of the anhydrite-gypsum transformation process. The present paper investigates the coupled process of simul-taneous anhydrite dissolution and gypsum precipitation in a closed system, develops a criterion for the limiting mechanism more rigorously and presents a simplified formula for estimating the duration of the anhydrite hy-dration process.

Section 2 of the paper presents the computational model with an emphasis on the reaction kinetics of anhydrite dissolution and gypsum precipitation. The governing equations are presented in a dimensionless form which shows the significant parameters of the problem under investigation and is also advantageous with respect to the parametric studies in the later sections. Section 3 checks the predictive capacity of the computational model on the basis of existing experimental data from the literature. Section 4 presents parametric studies for two cases of initial porosity: a very high initial porosity (which is more characteristic of an aqueous solution) and a low initial porosity (which applies to a porous medi-um), respectively. The reason for discussing these two cases is to illustrate some general patterns of behaviour. Sections 5 and 6 present simplified prediction equations for the limiting mechanism (anhydrite dissolution or gypsum precipitation) and the duration of the anhydrite to gypsum transformation.

2. COMPUTATIONAL MODEL

2.1. Mass balance The system under consideration consists of minerals and water under isothermal conditions. In the most general case, the constituents of the solid phase are anhydrite (A), gypsum (G) and inert minerals (S), i.e. minerals which do not participate in the chemical reactions. The water (W) generally contains calcium and sulphate ions (I). The mass of the constituent i per unit volume of the mixture is denoted by mi:

i i totm M V , (1)

where the subscript refers to the constituent (i = A, G, W, S or I); Mi [kg] denotes the mass of the i-th constituent at a given time and Vtot [m

3] is the volume of the mixture which (for small volume changes) can be taken approx-imately equal to the initial mixture volume Vtot,0.

Table 1. Model parameters (potential deviations will be men-tioned in the text)

Parameter Anhydrite Gypsum

Densities AGkg/m3 2960 2320 Equilibrium concentrations ceq,Aceq,G mol/m3 21 15.5

Orders of reactions aA aG 2 2

Reaction rate constants kA, kG kg/m2/s 3·10-6 5·10-7

The volume fractions of the mixture constituents are de-noted by i:

/i i tot i iV V m , (2)

where Vi andi is the volume and the density of the con-stituent i, respectively. The symbolw thus denotes the porosity of the medium. Obviously, the sum of the vol-ume fractions i should equal unity at all times. The ion concentration, which is important for calculating the re-action rates (see Section 2), can be expressed as a func-tion of the ion and water masses per unit volume:

c MI

VW

Wm

Im

W. (3)

In a closed system, the masses only change due to chem-ical reactions. In the current study, the dissolution of anhydrite decreases the anhydrite mass and increases the ion concentration, while the precipitation of gypsum in-creases the gypsum mass and decreases the ion concen-tration and the water content. The masses involved in these chemical reactions are:

4 4

0.136 0.136

CaSO Ca SO

kg mol kg mol

. (4)

and

4 2 4 2 2 2

0.136 0.036 0.172

Ca SO H O CaSO H O

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. (5)

The mass balance equation for the ions and the water therefore read as follows:

0 0 0

136

172I I A A G Gm m m m m m , (6)

0 0

36

172W W G Gm m m m , (7)

where mA0, mW0, mG0 and mI0 denote the initial masses per unit volume.

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dissolution is a second order reaction (i.e., A = 2), while the reaction rate constant kA amounts to 2.4 - 5.4·10-6

kg/m2/s [8]. The results of Kontrec et al. [6] also indicate a second order reaction, but a considerably lower reac-tion constant kA of 0.54·10-6 kg/m2/s. These differences can be explained by temperature effects, uncertainties over the extrapolation of the test results for zero flow velocity and uncertainties concerning the size and shape of the particles [8]. 2.4. Gypsum precipitation In dissolution processes, the surface area A appearing in Eq. (8) is clearly defined and expresses the surface of the mineral per unit volume which is in contact with water. In the case of precipitation, however, a difficulty arises with the surface area A or with the term GFG in Eq. (11) if the precipitating mineral is initially absent from the system. According to the theory of nucleation, which can be studied elsewhere (e.g. [10, 11]), as soon as the su-persaturation of a mineral is reached, nuclei start to form and crystal growth subsequently takes place on these nuclei. The time required for the first stable nucleus to be formed is defined as the induction time. A distinction can be drawn here between homogeneous and heteroge-neous crystal growth. In the former the crystals grow on the same mineral, while in the latter they grow on other minerals.

Homogeneous growth rarely occurs in nature [11] and it presupposes the existence of a number of nuclei which operate as centers for crystallization. The surface A is then the surface of the nuclei, and the mass change rate is given by Eq. (11). In order to circumvent the afore-mentioned difficulty during the modeling of homogene-ous nucleation, Steefel and Lasaga [14] suggest that once the solution becomes supersaturated with respect to an initially absent mineral, the latter instantaneously ac-quires a radius of 10 m which then develops over time. Nonetheless, the initial surface area does not depend on-ly on the initial radius but also on the number of nuclei formed and thus on the nucleation rate [19]. A common way to deal with this difficulty is to assume a constant number of nuclei and to vary their radius. For an initially absent mineral, the radius of the nuclei is set to zero, although their number is not. A constant update in the radius during modeling of the crystal growth is essential.

During heterogeneous nucleation, crystals, apart from nuclei, can also grow on foreign bodies such as the sur-faces of the different constituents of the rock. In this case the induction time vanishes, while nucleation and crystal growth occur simultaneously [12]. The surface area A in Eq. (8) is then the total surface area available for crystal growth, which is nonetheless very difficult to determine, introducing great uncertainties in the model. A number of powerful analytical techniques allow us to investigate and quantify the mineral surface [20]. However, in order

to fully understand the processes taking place at the in-terface between water and mineral, investigations must be performed in different scales. Direct observations of the changes on the surface of a mineral due to dissolu-tion and precipitation at the atomic scale appear to be extremely difficult, however, if not impossible.

Simultaneous mineral dissolution and precipitation takes place in many geochemical systems. Normally, dissolu-tion takes place more slowly than precipitation and therefore constitutes the limiting mechanism in the sys-tem. In such cases, assumptions concerning the precipi-tation rate are not necessary and the system will be close to the equilibrium concentration of the precipitating mineral [21]. Therefore, the aforementioned difficulties do not necessarily constitute a problem from the practi-cal point of view. We will investigate later in the present paper the conditions under which this is also true for the anhydrite-gypsum-water system.

The precipitation of gypsum is investigated assuming the presence of an initial mass of gypsum or of an inert min-eral on which gypsum growth takes place. In that way the complex process of nucleation is omitted. Further-more, the simplifying assumption is made that gypsum growth does not occur on the anhydrite surface. In reali-ty, it is entirely possible (and it has also been observed in nature) that gypsum crystals may also grow on an anhy-drite surface, thus sealing the anhydrite and delaying or completely halting its dissolution [22]. The simplifying assumption of the present paper, i.e. disregarding gyp-sum growth on anhydrite, thus overestimates the anhy-drite dissolution rate and underestimates the duration of the hydration process.

For the case of growth on spherical or cubical gypsum particles a similar equation to Eq. (13) applies:

2

3,

0 00 ,

Ga

eq GG GG G G

G eq G

c cdm mk F

dt m c

, (14)

while for gypsum crystal growth on inert minerals of spherical or cubical particles Eq. (13) takes following form:

2

3,

,

1Ga

eq GG GG S S

S eq G

c cdmk F

dt c

, (15)

where FS and s denote the specific surface area and the volume fraction, respectively, of the inert mineral on which gypsum growth takes place.

The experimental results of Liu and Nancollas [23] and Kontrec et al. [6] show that gypsum precipitation obeys a second order rate law, while the reaction rate constant kG amounts to 3.75 - 5.19·10-7 kg/m2/s [8]. The experi-ments of Smith and Sweett [24] indicate a considerably higher reaction rate (5.35·10-6 kg/m2/s), which seems to be due to different testing conditions [8].

2.5. Governing system of equations Assuming that the particles are spherical or cubical and that the gypsum crystals grow on an inert mineral, the following dimensionless equations can be derived from equations (13) and (15) for the volume fractions of an-hydrite and gypsum, respectively:

2

3

0

1 AaA A

A

dc

d

, (16)

and

2

3

,

1 1Ga

G G

S eq G

d c

d c

, (17)

where

,eq A

cc

c , ,

,,

eq Geq G

eq A

cc

c , (18)

t

kAF

A0

A

A0

(19)

and

0 0

G S S A

A A A G

k F

k F

. (20)

The symbol represents a dimensionless time, while the dimensionless constant expresses how quickly gypsum precipitation occurs relative to anhydrite dissolution, i.e. it is a measure of the relative speed of the two processes. The higher the value of the quicker the gypsum pre-cipitation takes place relative to the anhydrite dissolu-tion. The time-development of the anhydrite hydration is governed by anhydrite dissolution if the value of is high and by the gypsum precipitation if the value of is low.

Equations (16) and (17) are coupled via the dimension-less ionic concentration c . From equations (3), (6) and (7) we obtain c as a function of the volume fractions of anhydrite and gypsum:

0

00

,

136172

GA A G

W A A

W eq A W

c cc

, (21)

where the porosity

W

W 0

36

172

G

W

G, (22)

the initial porosity

W 01

A0

S (23)

and

00

,eq A

cc

c . (24)

Eq. (16) only applies, of course, under the following conditions:

, 0 0

172, 0,

36W

eq A A G W GG

c c

. (25)

The last inequality follows from the condition W > 0 and the equations (2) and (7). It must also be fulfilled (in addition to c > ceq,G) in order that Eq. (17) applies.

Equations (16) and (17), with the above-described condi-tions and the concentration c according to Eq. (21), rep-resent a system of two non-linear ordinary differential equations for the evolution of the volume fractions of anhydrite and gypsum over time. It is a simple matter to verify that the solutions of this system can be expressed as follows:

0

,

, ,00

, ,

, , ,

, , , , , , , ,

A G Weq A

eq G eq A G GA

eq A eq A A A WW

c

c

c ccf

c c

, (26)

where the variable is dimensionless time; the parame-ter expresses the relative speed of the two reactions; the parameters A0, W0 and c0/ceq,A describe the initial mixture composition and the last four parameters on the right hand side represent material constants.

3. MODEL CALIBRATION

The model of Section 2 was tested by considering the experimental results of Kontrec et al. [6], obtained for a case involving simultaneous anhydrite dissolution and gypsum precipitation. Kontrec et al. [6] carried out a back analysis of their experiments on the basis of a very similar computational model but presented no details of the model or of the assumed parameters.

As there is no information available on the particle shapes used, spherical particles are assumed for the cal-culations. The best fitting curve is found to be for an ini-tial radius of rA = 0.27 m for anhydrite and rG = 4.3m for gypsum particles. These values correspond to specif-ic surface areas which are in very good agreement with the values given in [6] for some experiments where an-hydrite dissolution and gypsum precipitation were inves-tigated separately. In the test under consideration, the initial anhydrite and gypsum masses in the solution were equal (mA0 = mG0 = 2.312 kg/m3). The initial ion concen-tration c0 was equal to 16 mol/m3, i.e. slightly higher than the equilibrium concentration of gypsum. Concern-ing the kinetic constants, second order laws for both an-hydrite dissolution and gypsum precipitation

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