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 2-D analytical model is developed to study the transport-controlled 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 2-D finite volume model was developed to validate the extended Graetz solution. The time evolution of the transport-controlled and the reaction-controlled dissolution models for a single tube with time-invariant flow rate are compared. This comparison shows that for time-invariant flow rate, the reaction-controlled dissolution model produces a positive feedback between the tube enlargement and dissolution, while the transport-controlled dissolution does not.

Key Points

• A 2-D analytical model, the extended Graetz solution, is developed to study the transport-controlled 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 reaction-controlled 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 reaction-controlled and transport-controlled, respectively. Mixed reaction/transport-controlled 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 well-mixed laboratory systems that are designed to eliminate mass transport limitations. Such rates and associated laws may not be directly applicable to model the large-scale 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 transport-controlled (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.

For transport-controlled dissolution in a tube, the dissolution mass flux q_w at the solid-liquid interface can be formulated as:

(1)

where k_t is the transport-controlled dissolution rate coefficient; C_s is the equilibrium concentration of the solid; C_b is the bulk concentration of the fluid (refer to Appendix Appendix A for a detailed definition).

The transport-controlled 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).

The transport-controlled dissolution in a cylindrical tube is analogous to the heat transfer problem in which a fluid with its initial temperature enters a cylindrical tube with constant wall temperature. Graetz (1883) derived the analytical solution to solve this heat transfer problem. The governing equations are solved in dimensionless form using the infinite series (the detailed solution is presented in Appendix Appendix A). The Sherwood number is introduced as a dimensionless quantity representing the ratio of mass transfer rate to the diffusion rate, which is the dimensionless form of the dissolution rate coefficient. The transport-controlled dissolution rate coefficient k_t can thus be calculated as:

(2)

where R is the radius of the cylindrical tube; D is the diffusion coefficient; and S_h is the Sherwood number.

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 transport-controlled 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 transport-controlled 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 2-D 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 2-D 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 2-D finite volume model is developed to validate the extended Graetz solution. A comparison is made between the transport-controlled dissolution and reaction-controlled 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 R₁. The exact profile is not prescribed for now, since it does not affect the derivation, as will be discussed in section 2.1.2.

Figure 1

Tapered tube and velocities (v_z and v_r) in the cylindrical coordinates r – z.

2.1 Analytical Solution for Transport-Controlled Dissolution in a Tapered Tube

2.1.1 Flow Velocity Field

Laminar flow in a tapered tube with gradually varying radii has been studied by Oka and Takami (1967) and Zerkle and Sunderland (1968). The fully developed velocity profile (v_z, v_r) in the tapered tube can be expressed in cylindrical coordinates (r–z):

(3)

where is the average velocity at the inlet, , and is the local gradient of the radius. Equations 3 show that the axial flow velocity v_z has the parabolic profile in each cross section and that the radial flow velocity v_r is proportional to the local gradient of radius .

The length of the hydrodynamic entrance region is: , while the length of the mass transfer entrance region is , with R_e and P_e being the Reynolds number ( ) and the Péclet number ( ), respectively. The relation between the two dimensionless quantities is: , where S_c is the Schmidt number. The Schmidt number is the ratio of momentum diffusivity (kinematic viscosity, ) to mass diffusivity (D).

(4)

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 10³. 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

Using the quasi steady state approximation, the mass conservation equation for dissolution in a tapered tube can be expressed as:

(5)

with the boundary conditions being:

(6)

Similar to the simplification made in the Graetz solution, this solution considers the case when the Péclet number and the axial diffusion term can be neglected. Upon defining the dimensionless quantities:

(7)

and the transformed dimensionless radius:

(8)

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.

The derivatives with regard to z−r can be expressed in the transformed coordinates as:

(9)

The mass conservation equation (equations 5) can be rewritten in the transformed dimensionless coordinates :

(10)

With the velocity profile equations 3, the transport equation can be rewritten as:

(11)

The underlined radial advection term in equation 11 is canceled by the first term. Then equation 11 can be simplified into the form (equation 12) with its boundary conditions (equations 13):

(12)

(13)

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.

According to the definition of the dimensionless quantities (equations 7), the transformed coordinates will not affect the calculation of the bulk concentration (C_b) and Sherwood number. With the infinite series solution in the transformed coordinates, the bulk concentration C_b and Sherwood number S_h can be expressed as:

(14)

(15)

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 transport-controlled 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 transport-controlled 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

Based on the mass conservation of the solvent, the governing equation for the bulk concentration is:

(16)

The Sherwood number S_h in equations 16 is a function of the dimensionless axial coordinate Z. As discussed in section 2.1.2, the dimensionless axial coordinate Z can be expressed as:

(17)

Where D is the mass diffusion coefficient and Q is the volumetric flow rate in the tube. Given the conservation of the fluid in the tube, Q is a constant along the tube, which simplifies the calculation of the dimensionless axial coordinate Z. Equation 14 or a fitted approximation (Lienhard, 2013) can be used to obtain the Sherwood number. For example:

(18)

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 transport-controlled 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 time-invariant flow rate, the bulk concentration will also be time-invariant despite the evolving geometry.

2.2.2 Rate of Tube Enlargement

With the Sherwood number and bulk concentration calculated, the rate of tube enlargement can be expressed as:

(19)

Equation 19 shows that the rate of tube enlargement at time t is dependent on the tube profile . The integration of tube enlargement rate with time produces the tube radius profile:

(20)

where R₀ is the initial tube radius. For time-invariant-flow rate Q, as discussed in section 2.2.1, the bulk concentration C_b is also time-invariant despite the evolution of the tube geometry. By integrating equation 19 with the initial condition , the tube radius profile at time t can be expressed as:

(21)

As shown in equation 21, when the flow rate is a time-invariant, 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 transport-controlled 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 time-invariant even when the tube is enlarging due to dissolution. The assumption of time-invariant 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, time-invariant flow rate is often used to study the dissolution of rocks (James & Lupton, 1978; Smith et al., 2013).

Figure 2

Sketch of how groundwater flows into preexisting gypsum tubes.

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 reaction-controlled 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 transport-controlled. The parameters are summarized in Table 1.

Table 1. Parameters Used in the Example

	0.5	Length of the tube
	2.0	Initial tube radius
		Flow rate
	2.5	Solubility (equilibrium concentration)
		Solute diffusivity
		Reaction-controlled 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 time-invariant flow rate, the radius profiles for can be calculated using equation 21 and are plotted in Figure 3b.

Figure 3

Application of the extended Graetz solution to model gypsum dissolution in a preexisting tube. (a) Normalized bulk concentration along the tube. The quasi steady state concentration profiles for the initial tube and tapered tubes at are the same. (b) Radius profiles along the tube. The nonuniform dissolution along the tube causes more material to dissolve near the inlet, forming a tapered tube.

Since the flow rate is time-invariant, the dimensionless axial coordinate Z and the concentration profile along the tube are also time-invariant 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 example in section 3.1 shows how the extended Graetz solution models the evolution of a preexisting tube in gypsum. The example concludes that the quasi steady state bulk concentration and Sherwood number along the tube are not functions of the tube geometry, but functions of flow rate. A 2-D axisymmetric finite volume model was developed to validate the extended Graetz solution. The model solves the following transient mass conservation equation:

(22)

with the boundary conditions being the same as equations 6 and the initial condition being: .

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 2-D 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).

Figure 4

The 2-D numerical model. (a) The structured nonorthogonal finite volume grid. (b) Streamlines of the flow described in equation 3.

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 solid-liquid 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 2-D 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.

Figure 5

Numerical simulation of dissolution in a tapered tube. (a) The normalized bulk concentration and Sherwood number (S_h) along the tube. C_b and S_h calculated from the numerical simulation matches those produced by the extended Graetz solution; thus, the extended Graetz solution is validated. (b) Normalized concentration profile in the tube. Flow with zero concentration (blue) enters the tube from the top, while the solute diffuses from the solid-liquid interface to the center of the flow, similar to the case of a cylindrical tube.

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 transport-controlled 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 contact-area; 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 transport-controlled dissolution, so the Sherwood number and bulk concentration do not change with the tube geometry.

3.3 Comparison to Reaction-Controlled Dissolution in a Tube

The above example in sections 3.1 and 3.2 shows how a preexisting tube evolves due to transport-controlled dissolution: for a time-invariant flow rate, the concentration profile is also time-invariant, while the radius increases nonlinearly with time. This behavior is different from the case of reaction-controlled dissolution. It is necessary to make a comparison to differentiate the two with regard to time evolution.

When the reaction rate is much lower than the radial diffusion rate, the dissolution in the tube is reaction-controlled. The Damköhler number of a reaction-controlled dissolution is much smaller than 1, . Take the example of the dissolution of limestone in carbon dioxide (CO₂) rich water. The reaction-controlled dissolution rate coefficient (k_r) of limestone in CO₂-rich water is: (Pokrovsky et al., 2009). The Damköhler number in this case is 0.04, so the dissolution is reaction-controlled. Assuming the dissolution is a first-order reaction, the dissolution flux can be expressed as:

(23)

Given that the dissolution problem is an inlet distribution problem (Subramanian, 1977) and that the entrance length for Taylor dispersion is (Wu & Chen, 2014), which is longer than the length of the tube L, the dispersion in the z direction can be neglected. The quasi steady state mass conservation of the solute can be expressed as:

(24)

The dissolution induced tube enlargement rate can be expressed similar to equation 19:

(25)

With the help of quasi steady state approximation, the reaction-controlled 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.

Figure 6

Example of reaction-controlled dissolution in a preexisting tube. (a) Normalized bulk concentration along the tube. Because of the increasing tube radius, the flow velocity decreases for a time-invariant flow rate and the solid-liquid interface area increases. As a result, the concentration profile increases with time. (b) Radius profiles along the tube. With the dissolution rate coefficient k_r being the same along the tube, the nonuniform dissolution is mainly caused by the concentration difference ( ), so it is less pronounced than for the case of transport-controlled dissolution.

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 transport-controlled dissolution shown in Figure 3. This is reasonable given that the reaction-controlled dissolution rate is lower than that of the transport-controlled dissolution rate. The nonuniform dissolution along the tube is less pronounced in Figure 6b than in Figure 3b. For the reaction-controlled dissolution, the nonuniform dissolution is caused by the concentration difference along the tube only, while for transport-controlled 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 time-invariant flow rate, the transport-controlled dissolution model produces a time-invariant effluent concentration as shown in blue, while the reaction-controlled dissolution model produces an effluent concentration that increases with time, as shown in red.

Figure 7

Effluent concentrations predicted by the transport-controlled (blue) and reaction-controlled (red) dissolution models. For the time-invariant flow rate condition, the transport-controlled dissolution model produces a time-invariant effluent concentration, while the reaction-controlled dissolution model produces an effluent concentration that increases with time.

The effluent concentration can be used to quantify the overall dissolution rate for the tube, which is defined as:

(26)

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 time-invariant flow rate, the overall dissolution rate for the transport-controlled dissolution case is time-invariant despite the tube enlargement. However, the tube enlargement enhances the overall dissolution rate for the reaction-controlled dissolution case, as indicated by the red curve in Figure 6. The increasing overall dissolution rate for the reaction-controlled dissolution indicates a positive feedback between the tube enlargement and dissolution even when the flow rate is time-invariant.

4 Conclusions

A 2-D axisymmetric analytical model is developed to study the transport-controlled 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 transport-controlled 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 transport-controlled and the reaction-controlled dissolution models to show their behavioral difference in time evolution during dissolution: for time-invariant flow rate, the transport-controlled dissolution model produces time-invariant concentration profile, despite the evolving geometry, while the transport-controlled 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 transport-controlled.

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

The steady-state mass conservation for transport-controlled dissolution in a cylindrical tube can be expressed as:

(A1)

where is the solute concentration and D is the diffusion coefficient of dissolved gypsum.

The boundary conditions are:

(A2)

The governing equation above can by written in dimensionless form by defining the following dimensionless quantities:

(A3)

where Z is the dimensionless axial coordinate; P_e is the Péclet number; Y is the dimensionless radial coordinate; and θ is the dimensionless concentration.

Equations A1 and A2 can be rewritten with the dimensionless quantities:

(A4)

(A5)

(A6)

(A7)

In equation A4, the Péclet number is the ratio of the mass transfer rate by advection to that by diffusion. For cases with a Péclet number , the axial diffusion, , can be safely neglected (for the effect of axial diffusion in low P_e cases, please refer to Acrivos (1980). The quasi steady state form of the transport equation A4 becomes:

(A8)

Equation A8 can be solved by separation of variables, which yields an infinite series solution:

(A9)

Where λ_n are the eigenvalues for the confluent hypergeometric function or Kummer's function (often written in the form ):

(A10)

is the eigenfunction, which can be expressed in terms of the Kummer's equation:

(A11)

And A_n is the coefficient in the solution by separation of variables,

(A12)

Given the Graetz solution of concentration (equation A9), the bulk concentration and Sherwood number can be expressed as:

(A13)

(A14)

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 ( ).