Theoretical and Numerical Investigation of the Cavity Evolution in Gypsum Rock
Abstract
When water flows through a preexisting cylindrical tube in gypsum rock, the nonuniform dissolution alters the tube into an enlarged tapered tube. A 2D analytical model is developed to study the transportcontrolled dissolution in an enlarged tapered tube, with explicit consideration of the tapered geometry and induced radial flow. The analytical model shows that the Graetz solution can be extended to model dissolution in the tapered tube. An alternative form of the governing equations is proposed to take advantage of the invariant quantities in the Graetz solution to facilitate modeling cavity evolution in gypsum rock. A 2D finite volume model was developed to validate the extended Graetz solution. The time evolution of the transportcontrolled and the reactioncontrolled dissolution models for a single tube with timeinvariant flow rate are compared. This comparison shows that for timeinvariant flow rate, the reactioncontrolled dissolution model produces a positive feedback between the tube enlargement and dissolution, while the transportcontrolled dissolution does not.
Key Points
 A 2D analytical model, the extended Graetz solution, is developed to study the transportcontrolled dissolution in a tapered tube
 An example is used to show how the extended Graetz solution can be applied to model cavity evolution in gypsum
 A comparison is made with reactioncontrolled dissolution model to show their behavioral difference regarding evolution over time
1 Introduction
Gypsum is one of the most soluble common rocks. It is about 10–30 times more soluble than limestone (Bögli, 1980), and it commonly has a lower mechanical strength. The dissolution of gypsum forms caves, sinkholes, disappearing streams, and other karst features that are also found in limestones and dolomites. Gypsum karst is known in almost all areas underlain by gypsum, and usually extends down to depths of at least 30 m below the land surface (Johnson, 1996). It is therefore important to have a better knowledge of the dissolution and evolution of gypsum cavities for the prediction of sinkhole or subsidence occurrence and, eventually, the choice of engineering sites.
The dissolution rates of a solid in an aqueous solution without further chemical reaction of the dissolved ions are controlled by two processes: (a) The chemical reaction at the surface, which depends on the chemical composition of the solution at the surface and (b) mass transport across a diffusion boundary layer of thickness (Dreybrodt, 2012; Rickard & Sjöeberg, 1983). In the following context, (a) and (b) are referred as reactioncontrolled and transportcontrolled, respectively. Mixed reaction/transportcontrolled dissolution of gypsum and other minerals has been observed and studied (Budek & Szymczak, 2012; Detwiler & Rajaram, 2007; Jeschke et al., 2001; Raines & Dewers, 1997). Reaction rates are often measured using crushed minerals in wellmixed laboratory systems that are designed to eliminate mass transport limitations. Such rates and associated laws may not be directly applicable to model the largescale subsurface processes, since the dissolution processes in subsurface environments are different with regard to the hydrodynamic conditions, surface properties, and material heterogeneity (Li et al., 2006). For example, subsurface gypsum dissolution kinetics are generally considered to be transportcontrolled (Barton & Wilde, 1971; Christoffersen & Christoffersen, 1976; James & Lupton, 1978; Keisling et al., 1978; Kemper et al., 1975; Liu & Nancollas, 1971; Navas, 1990; Ohmoto et al., 1991), so the reaction rate coefficient measured using fast rate rotating disk tests or batch tests are not directly applicable to modeling gypsum dissolution in a tube.
The transportcontrolled dissolution rate coefficient k_{t} is often determined by the hydrodynamic conditions. Analytical models have been developed to calculate k_{t} for corresponding hydrodynamic conditions, such as dissolution on a rotating disk (Levich, 1962) and dissolution in a cylindrical tube (Graetz, 1883).
According to Graetz (1883), the Sherwood number is higher near the entrance and asymptotically approaches 3.66 along the tube. The higher Sherwood number indicates a higher mass transport rate, as a result of which more solid dissolves near the entrance.
1.1 Problem Description
The transportcontrolled dissolution models for cylindrical tubes (the Graetz solution) are used to study the evolution of subsurface cavities in limestone (Dreybrodt, 1996; Groves & Howard, 1994a, 1994b; Howard & Groves, 1995; Kaufmann & Romanov, 2008; Palmer, 1991; Rehrl et al., 2008; Sudicky & Frind, 1982) and other soluble materials (Raines & Dewers, 1997). Pipe network models constructed using these cylindrical tubes have also been used to study the dissolution in porous media and wormhole formation (Bernabé, 1995, 1996; Birk et al., 2005; Bryant et al., 1993; Budek & Szymczak, 2012; Buijse et al., 1997; Cohen et al., 2008; Fredd & Fogler, 1998; Hoefner & Fogler, 1988).
The Graetz solution has been proven to be useful in modeling the transportcontrolled dissolution in cylindrical tubes. However, its validity domain is not necessarily well understood. For instance, a dissolution rate coefficient calculated from the asymptotic value (3.66) of the Sherwood number is often used, ignoring the higher rate in the entrance region, which would induce an error in the mass conservation calculation. According to Budek and Szymczak (2012), the nonuniform dissolution along the tube reshapes the cylindrical tube into an enlarged tapered tube. The tapered tube has larger area in contact with the fluid and induces radial flow, which no longer satisfies the assumptions of the Graetz solution. Yet, in some models, the Sherwood number for the cylindrical tube was used for the tapered tube without formal justification.
A 2D analytical model is developed to explicitly study the effect of tube enlargement and induced radial flow in a tapered tube. By adopting the coordinate transform introduced by Zerkle and Sunderland (1968), the 2D mass conservation equation is solved in the same dimensionless form as the Graetz solution. Thus, the Graetz solution can be extended to model the dissolution in an enlarged tapered tube. This also provides a formal justification that the Sherwood number for a cylindrical tube can be used for the entrance region and the following region of the tapered tube. With the help of a quasi steady state approximation, the extended Graetz solution can be used to model the geometric evolution of the tube. A 2D finite volume model is developed to validate the extended Graetz solution. A comparison is made between the transportcontrolled dissolution and reactioncontrolled dissolution models with regard to their behavior during the dissolution process.
2 Dissolution in an Enlarged Tapered Tube
As the geometry of the tube evolves due to dissolution, the flow and transport problems become hard to model due to the moving boundary. However, in an underground reactive transport system, the timescale required for the pressure and concentration fields to reach equilibrium is much shorter than the timescale required for significant alteration of the tube geometry. This allows one to use quasi steady state concentration to calculate the rate of change in the radius and update the geometry for the next step of the transport calculation. This simulation method is referred as the quasi steady state approximation (Detwiler et al., 2001; Detwiler & Rajaram, 2007; Hanna & Rajaram, 1998; Szymczak & Ladd, 2009, 2011, 2013), and is used in the following derivations for flow and mass transfer.
The higher dissolution rates near the inlet cause nonuniform dissolution along the tube. More solid dissolves near the inlet, transforming an initially cylindrical tube (Figure 1a) into an enlarged tapered tube (Figure 1b), as discussed by Budek and Szymczak (2012). A cylindrical coordinate system (r, z) is used in the following discussion as shown in Figure 1b. Assume the profile of the tube is a function of the axial coordinate: , with the inlet radius being R_{1}. The exact profile is not prescribed for now, since it does not affect the derivation, as will be discussed in section 2.1.2.
2.1 Analytical Solution for TransportControlled Dissolution in a Tapered Tube
2.1.1 Flow Velocity Field
For the underground gypsum dissolution problem, and D are in the order of and (Jeschke et al., 2001), so the Schmidt number is in the order of 10^{3}. The hydrodynamic entrance region is much shorter than the mass transfer entrance region and can be neglected. It is thus reasonable to use the fully developed velocity profile (equations 3b) in the mass transport models.
2.1.2 Mass Conservation of the Solute in a Tapered Tube
The coordinates transform from z−r to transforms the solution domain from an irregular shape bounded by ( ) to a rectangle bounded by ( ). This makes the solution easier, as shown in the following derivation. This coordinate transform is adopted from the article by Zerkle and Sunderland (1968), in which the heat transfer in a tapered tube was modeled.
Equation 12 is the same as the governing equation of the Graetz solution, except that equation 12 uses the transformed coordinates. The dimensionless form of solution developed by Graetz (1883) can be applied to solve equation 12.
The dimensionless axial coordinate Z (equation 7) has the denominator , which is a function of flow rate instead of tube geometry given that . The expressions for bulk concentration and Sherwood number are the same as those in the Graetz solution. The above derivation shows that the Graetz solution can be extended to solve the governing equation of the transportcontrolled dissolution in a tapered tube. It is referred as extended Graetz solution in the later discussion.
The above analysis also produces an unexpected conclusion. Since a tapered tube and a cylindrical tube have different hydrodynamic conditions, the Sherwood numbers should be different for these two types of tubes. Intuitively, the tapered tube geometry induces radial flow and has larger contact area with the flowing fluid, which should enhance dissolution. However, the above analysis shows that the tapered tube does not affect the dissolution rate, and that Sherwood number and bulk concentration for a tapered tube are the same as those for a cylindrical tube.
2.2 Extended Graetz Solution
Since the extended Graetz solution can be used to model transportcontrolled dissolution in a tapered tube, it is useful for modeling gypsum dissolution especially in the later stage when the enlarged tapered tube has been formed. Since the geometry of the tube and the velocity of the fluid are both evolving, the formulation using radius and velocity needs to be updated for every time step, as discussed by Budek and Szymczak (2012). To take advantage of the invariant quantities (such as length of the tube L and flow rate along the tube Q) in the system, the following governing equations are proposed.
2.2.1 Bulk Concentration
Equation 16 explicitly uses the Sherwood number S_{h} instead of the dissolution rate coefficient k_{t}, because k_{t} is a function of the evolving radius R(z). This form of the extended Graetz solution is easier to apply for the dissolution in an enlarged tapered tube than using k_{t} and updating k_{t} for the evolving geometry. Equation 16 also shows that for transportcontrolled dissolution in a tube, the bulk concentration is not a function of the geometry but a function of flow rate. Moreover, for an enlarging tube with timeinvariant flow rate, the bulk concentration will also be timeinvariant despite the evolving geometry.
2.2.2 Rate of Tube Enlargement
As shown in equation 21, when the flow rate is a timeinvariant, the tube does not enlarge linearly with time. Budek and Szymczak (2012) proposed a similar equation for the radius profile by assuming the Sherwood number S_{h} = 4 for the initially cylindrical and later tapered tube. However, their prediction neglected the high dissolution rate in the entrance region and uses the same Sherwood number for cylindrical and tapered tubes without formal justification. Equation 21 advances their model in two aspects. First, the Sherwood number accounting for the entrance region is used for more accurate bulk concentration calculation and radius prediction. Second, the Sherwood number has been proven to be applicable for the tapered tube.
3 Discussion
3.1 Application of the Extended Graetz Solution
The extended Graetz solution provides an analytical approach to model the transportcontrolled dissolution in a tapered tube. With the help of the quasi steady state approximation, the evolution of the tube can be modeled using the extended Graetz solution as shown with an application to model gypsum dissolution in a preexisting tube.
In the example in Figure 2, groundwater flows through multiple ground layers, of which the gypsum layer has preexisting tubes. Since the hydraulic conductivity of a tube is much higher than the other permeable layers, the flow rate in the tube is limited by the other less permeable layers and can be assumed as timeinvariant even when the tube is enlarging due to dissolution. The assumption of timeinvariant flow rate is simplified compared with what actually happens in the field, but it decouples the effect of the hydrodynamic conditions. In addition, in laboratory conditions, timeinvariant flow rate is often used to study the dissolution of rocks (James & Lupton, 1978; Smith et al., 2013).
Assume that groundwater with zero gypsum concentration flows in a preexisting cylindrical tube in gypsum rock, with 0.5 m length and 2 mm radius. The gypsum has a dry density of (Einstein et al., 1969) and solubility of (Raines & Dewers, 1997). The diffusion coefficient is used as the coefficient of the solute (Raines & Dewers, 1997). The reactioncontrolled dissolution rate coefficient (Jeschke et al., 2001) is used to calculate the Damköhler number and to justify that the gypsum dissolution in the example is indeed transportcontrolled. The parameters are summarized in Table 1.
0.5  Length of the tube  
2.0  Initial tube radius  
Flow rate  
2.5  Solubility (equilibrium concentration)  
Solute diffusivity  
Reactioncontrolled dissolution rate coefficient  
Density of dry gypsum  
Kinematic viscosity of water 
The dimensionless quantities are calculated to determine the controlling mechanisms in the dissolution process. The Reynolds number ( ) for the flow in the tube is 1, indicating that the flow is laminar. The Péclet number ( ) is 1,000, indicating that the mass transport is controlled by the diffusion in the radial direction. The Damköhler number ( ) is 179.2, which indicates that the reaction rate is much higher than the radial diffusion rate and that the dissolution is controlled by diffusion in the radial direction (transport) instead of surface reaction. The dimensionless quantities verify that the extended Graetz solution can be applied to model the dissolution in the gypsum tube.
The timescale for the concentration in the tube to reach steady state can be estimated as: or . The timescale for the tube to enlarge 10% can be estimated as: . The timescale required for the concentration field to reach equilibrium is much shorter than the timescale required for significant alteration of the tube geometry. This comparison shows that the quasi steady approximation can be used in this example to model the cavity evolution in gypsum.
By applying the quasi steady state approximation, the bulk concentration can be calculated for the tube using equation 16. The normalized bulk concentration is plotted against the distance from the inlet z and dimensionless axial coordinate Z. With the assumption of timeinvariant flow rate, the radius profiles for can be calculated using equation 21 and are plotted in Figure 3b.
Since the flow rate is timeinvariant, the dimensionless axial coordinate Z and the concentration profile along the tube are also timeinvariant despite the evolving geometry. The calculated concentration profile in Figure 3a is applicable for . The nonuniform dissolution in the tube transforms the initially cylindrical tube into a tapered tube, as shown in Figure 3b. The nonuniform dissolution is caused not only by the concentration difference ( ), but also by the higher mass transport rate in the entrance region. This example shows that the extended Graetz solution can be applied to model the dissolution of gypsum in a tube, especially for later stages when the tube is no longer cylindrical.
3.2 Numerical Validation
The tapered tube with radius profile for t = 200 days in Figure 3b was used as the geometry of the modeled tube. All the other parameters are the same as listed in Table 1, except for the radius profile. A structured nonorthogonal finite volume grid was used to discretize the domain as shown in Figure 4a. The fully developed laminar flow described by equation 3 was used in the 2D model, as shown by the streamlines in Figure 4b. Since the finite volume model uses a nonorthogonal grid, the minimum correction approach was used for nonorthogonal flux correction (Moukalled et al., 2016).
A steady state concentration field was obtained by the model, as shown in Figure 5b. The concentration profile is normalized and plotted in r−z coordinates. The solute diffuses from solidliquid interface to the center of the flow, as the diffusion boundary layer thickens along the tube. The normalized bulk concentration ( ) and Sherwood number (S_{h}) along the tube can be calculated from the concentration profile produced by the 2D numerical model according to their definitions. and S_{h} are plotted against the dimensionless axial coordinate Z as defined in equation 17 and compared with the extended Graetz solution as shown in Figure 5a.
As discussed in section 2.1.2, the dimensionless axial coordinate Z is not a function of the tube geometry, but a function of flow rate. Hence, the numerical results have the same dimensionless axial coordinate as in the example in section 3.1. As shown in Figure 5, the extended Graetz solution produces the normalized bulk concentration and Sherwood number profile for a tapered tube, which match the numerical simulation. This comparison validates the extended Graetz solution for modeling the transportcontrolled dissolution in a tapered tube.
An explanation for the bulk concentration and Sherwood number of a tapered tube being the same as those of a cylindrical tube is proposed as follows: since the flow is laminar, the radial flow near the wall of the tapered tube follows streamlines that are parallel to the tube wall, as shown in Figure 4b. The diffusion flux is perpendicular to the direction of velocity, which is the same as for a cylindrical tube. The larger tube radius has larger contactarea; however, it has longer diffusion length as counterpart for the dissolution. The overall result is that the for the same flow rate, the tapered geometry does not enhance dissolution for the transportcontrolled dissolution, so the Sherwood number and bulk concentration do not change with the tube geometry.
3.3 Comparison to ReactionControlled Dissolution in a Tube
The above example in sections 3.1 and 3.2 shows how a preexisting tube evolves due to transportcontrolled dissolution: for a timeinvariant flow rate, the concentration profile is also timeinvariant, while the radius increases nonlinearly with time. This behavior is different from the case of reactioncontrolled dissolution. It is necessary to make a comparison to differentiate the two with regard to time evolution.
With the help of quasi steady state approximation, the reactioncontrolled dissolution model can be applied with the flow parameters listed in Table 1. The normalized bulk concentration profile and radius profile along the tube can be plotted, as shown in Figures 6a and 6b.
As shown in Figure 6, the bulk concentration along the tube is much lower and the radius increase is smaller compared with the case of transportcontrolled dissolution shown in Figure 3. This is reasonable given that the reactioncontrolled dissolution rate is lower than that of the transportcontrolled dissolution rate. The nonuniform dissolution along the tube is less pronounced in Figure 6b than in Figure 3b. For the reactioncontrolled dissolution, the nonuniform dissolution is caused by the concentration difference along the tube only, while for transportcontrolled dissolution, the higher dissolution rate in the entrance region also contributes to the nonuniform dissolution.
Apart from the differences regarding the bulk concentrations and radius profiles, the two dissolution models for preexisting tubes differ in the time evolution. The effluent concentrations (bulk concentration at the outlet of the tube ) are compared for the two cases, as shown in Figure 7. For timeinvariant flow rate, the transportcontrolled dissolution model produces a timeinvariant effluent concentration as shown in blue, while the reactioncontrolled dissolution model produces an effluent concentration that increases with time, as shown in red.
The overall dissolution rate for the tube is the total mass dissolved from the tube for a unit time. It indicates how quickly the mass has been removed due to dissolution. For a single tube with timeinvariant flow rate, the overall dissolution rate for the transportcontrolled dissolution case is timeinvariant despite the tube enlargement. However, the tube enlargement enhances the overall dissolution rate for the reactioncontrolled dissolution case, as indicated by the red curve in Figure 6. The increasing overall dissolution rate for the reactioncontrolled dissolution indicates a positive feedback between the tube enlargement and dissolution even when the flow rate is timeinvariant.
4 Conclusions
A 2D axisymmetric analytical model is developed to study the transportcontrolled dissolution in an enlarged tapered tube. The tapered geometry and induced radial velocity were explicitly considered in the governing equations. By adopting the coordinate transform used by Zerkle and Sunderland (1968), the mass conservation equation can be transformed to the same form as the Graetz problem. Thus, the Graetz solution can be extended to simulate the dissolution in a tapered tube. The Sherwood number for a tapered tube is shown to be the same as that of a cylindrical tube. This provides a formal justification for the transportcontrolled dissolution model developed by Budek and Szymczak (2012). An alternative form of the extended Graetz solution (equations 16 and 19) is proposed to take advantage of the invariant variables. This form also recommends to use the Sherwood number as a function of dimensionless axial coordinate Z instead of the asymptotic value, so that the high mass transport rate in the entrance region is considered.
An example is used to show the application of the extended Graetz solution in modeling the geometrical evolution of a preexisting tube in gypsum. With the help of quasi steady state approximation, the Graetz solution can calculate the concentration and radius profiles for different stages of dissolution. A numerical model is developed to validate the extended Graetz solution by simulating the dissolution in a tapered tube. The extended Graetz solution matches the numerical results very well. An explanation is proposed to rationalize the conclusion that the Sherwood numbers for the cylindrical tube and the tapered tube are the same.
A comparison is made between the transportcontrolled and the reactioncontrolled dissolution models to show their behavioral difference in time evolution during dissolution: for timeinvariant flow rate, the transportcontrolled dissolution model produces timeinvariant concentration profile, despite the evolving geometry, while the transportcontrolled dissolution model produces increasing concentration profile. Although the discussion and examples are mainly about dissolution of gypsum, the extended Graetz solution is applicable to other materials if the dissolution process is transportcontrolled.
Acknowledgments
The data used in the models as examples are from the cited references. This work was funded by the Cooperative Agreement between the Masdar Institute of Science and Technology and the Massachusetts Institute of Technology. Sincere thanks to: John T. Germaine, Pierre F. J. Lermusiaux, John Lienhard, and R. Shankar Subramanian for providing help on the development of the analytical solutions. The authors also appreciate the editors' and reviewers' comments.
Appendix A: Graetz Solution
Notation

 An

 Coefficients in the Graetz solution.

 Aw

 Interfacial area .

 C(z, r)

 Solute concentration (g/L).


 Bulk concentration (g/L).

 Cs

 Equilibrium concentration (g/L).

 D

 Diffusivity ( ).

 h

 Mass transfer coefficient (m/s).

 Pe

 Péclet Number.

 Q

 Injection volumetric flow rate ( ).

 qw

 Dissolution mass flux ( ).

 R(z)

 Radius of the tube (m).

 R0

 Initial radius of the tube (m).

 R1

 Radius at the inlet (m).

 r

 Radial coordinate (m).


 Sherwood number.

 t

 Time (day).

 vr

 Radial velocity (m/s).

 vz

 Axial velocity (m/s).

 Y

 Dimensionless radial coordinate.

 Ys

 Transformed dimensionless radial coordinate.

 Z

 Dimensionless axial coordinate.

 z

 Axial coordinate (m).

 θ

 Dimensionless concentration.

 λn

 Eigenvalues in the Graetz solution.


 Eigenfunctions in the Graetz solution.

 ρr

 Density of the rock ( ).