Transp Porous Med (2009) 80:229–252DOI 10.1007/s11242-009-9353-x

Impact of Capillary-Driven Liquid Films on SaltCrystallization

Duc Le · Hai Hoang · Jagannathan Mahadevan

Received: 13 June 2008 / Accepted: 3 February 2009 / Published online: 30 May 2009© Springer Science+Business Media B.V. 2009

Abstract Flow-through drying of ionic liquids in porous media can lead to super saturationand hence crystallization of salts. A model for the evolution of solid and liquid concentrationsof salt, in porous media, due to evaporation by gas flow is presented. The model takes intoaccount the impact of capillary-driven liquid film flow on the evaporation rates as well asthe rate of transport of salt through those films. It is shown that at high capillary wickingnumbers and high dimensionless pressure drops, supersaturation of brine takes place in thehigher drying rate regions in the porous medium. This leads to solid salt crystallization andaccumulation in the higher drying rate region. In the absence of wicking, there is no transportand accumulation of solid salt. Results from experiments of flow-through drying in rockcores are compared with model prediction of salt crystallization and accumulation.

Keywords Evaporation · Capillary wicking · Porous media · Gas flow · Crystallization

1 Introduction

Salt crystallization is important in many applications such as in building materials,geothermal wells, and natural gas storage reservoirs. Salt crystallization in porous mediais generally a consequence of evaporation-driven supersaturation of ionic liquids. When thesolubility limit of the ionic salt is reached, crystallization can occur both within and outsidethe porous media. The phenomenon of crystallization within the porous medium is calledsubflorescence while that occurring outside the porous medium is called efflorescence. Thedistinction between these two is that one (efflorescence) occurs as a limiting case of thecrystallization problem within the pores. These phenomena are the subject of active researchwithin the cement and concrete research communities (Puyate and Lawrence 1998, 2000;Puyate et al. 1998; Scherer 1999).

D. Le · H. Hoang · J. Mahadevan (B)Department of Petroleum Engineering, The University of Tulsa, Tulsa, OK 74104, USAe-mail: jmahadevan@utulsa.edu

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230 D. Le et al.

In addition to the reaction kinetics of salt crystallization, transport of solute is an importantfactor in the evolution of crystallized salt in porous media. Puyate et al. (Puyate and Lawrence2000) showed that the rate of flow of the ionic solution in comparison to the rate of evapora-tion controls the occurrence of efflorescence in porous media. When the rate of liquid supplyis low and the rate of evaporation is high, salt crystallization occurs deeper within the porousmedium. The study by Puyate et al. (1998) characterized this limit as the lower Peclet numberlimit wherein the transport of salt through the wick action is a less dominant mechanism.Past theoretical studies on evaporation of saline water in porous cement concrete (Puyate andLawrence 1998, 2000; Puyate et al. 1998; Scherer 1999) and geothermal reservoirs (Tsypkin2003, 2005; Tsypkin and Caloreb 2003) show that the rate of movement of liquid throughcapillary wick action, at the evaporation or boiling front, impacts the rate of evolution ofsolid salt saturation. The studies show that a large difference in evaporation rates at the liquidfront and the rate of liquid supply through Darcian flow can lead to super saturation of brinewhich eventually can cause crystallization of salts. Therefore, a proper evaluation of the saltcrystallization problem requires a greater understanding of the evapo-transport processes andtheir driving forces in the porous medium.

The driving force for evaporation from porous media can be due to mass transfer causedby non-isothermal conditions or simply diffusive transport of volatile species through thepores to a flowing gas stream also called pass-over drying. A large body of work (Prat 1995,2002, 2007; Prat and Bouleux 1999; Yortsos and Stubos 2001; Yiotis et al. 2003, 2004) existson the problem of evaporation from porous media driven by diffusion and non-isothermalconditions. Early studies considered evaporation as equivalent to an invasion percolationproblem and subsequently introduced pore network models to understand the evolution ofwater saturation during pass over drying conditions. The pore network models were thenrefined (Yortsos and Stubos 2001; Yiotis et al. 2003, 2004) with physics of thick liquid filmflow along the corners to explain experimental observations of drying rates in porous mediasuch as a pack of silica spheres which showed faster drying than model predictions for asystem outside a porous medium (Shaw 1987). In a recent study, Sghaier et al. (2007) com-bined the modeling of pass-over drying on porous media with solute transport, during drying,using continuum modeling. The results of the study show that the concentration of the ionscan reach high values in the inner regions of the porous medium due to the receding dryingfront that forms during slow drying. Although the study does not discuss the location ofcrystallization, it may be inferred from their study that the salt will crystallize in the regionsof high drying rate and hence high salt concentration. Past studies, however, do not considerevaporation in a flow-through context which includes convection as a dominant transportmechanism.

In a recent study,(Mahadevan et al. 2006) it was shown that flow-through drying processoccurs, as opposed to pass-over drying, when a gas flows through the porous medium that ispartially saturated with liquids. This drying process is different from the processes consid-ered before in the way that the drying rates are not anymore controlled by diffusion of gas inporous media. Instead, the controlling factor is the compressibility of gas and the transportprocesses that affect the movement of liquids. It was shown that the drying rate is controlledprimarily by the compressibility of gas and additionally it is enhanced by the wicking actionof wetting liquids in the porous medium. A model including the effect of thick liquid filmswas developed and validated with experimental data obtained through X-ray CT imagingof evaporation from cylindrical rock cores. One of the main objectives of this study is tocombine the advances in the understanding of flow-through evaporation from porous mediawith solute transport theory and develop a better understanding of evolution of crystallizedsalt saturation in rocks.

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Impact of Capillary-Driven Liquid Films on Salt Crystallization 231

We hypothesize that capillary-driven films can lead to augmented solute transport andhence an accumulation of crystallized salt during flow-through evaporation. We rest ourhypothesis on the fact that capillary-driven films are a significant mechanism that enhancesthe rate of evaporation by supplying the solvent. In this paper, we report the results of model-ing and experimental study that includes the impact of liquid films on crystallized salt massevolution. The following sections describe the development of a continuum model for theevolution of concentration of salt in liquid phase and crystallized solid salt. The liquid satu-ration evolution model is adapted from previous studies and is reproduced in the appendices.The salt concentration equations are then solved in conjunction with the liquid saturationevolution in space and time using numerical and analytical methods for selected cases. Themass of solid salt crystallized is calculated by applying solubility limits on the liquid con-centrations for the case of fast crystallization kinetics and capillary-dominated conditions.A comparison between experimental data and model prediction of mass of solid salt in therock core is finally made. The experiments were conducted under similar conditions to thepreviously reported studies (Mahadevan et al. 2006, 2007) to ensure identical effects ofcapillary-driven liquid films during flow-through evaporation.

2 Model Development and Solutions

Flow-through drying or compressibility-driven drying of porous media that is partially sat-urated with salt solution leads to super saturation of the salt solution within the porousmedium. Therefore, the first step is to understand the evaporation effects due to flowing gas,which is saturated at the inlet, on the liquid saturation profiles which are controlled by thedimensionless wicking number and the dimensionless pressure drop.

2.1 Evaporation Model

When gas flows through cylindrical porous sandstone initially saturated with water, part ofthe water is removed by displacement or physical expulsion. More than 50% of the water isstill trapped in the pore spaces (Mahadevan et al. 2006). This water is only slowly removed byevaporation which is caused due to the expansion of the gas as it flows through the core. Theslow removal of water in turn affects the recovery of gas flow rates which are initially verylow due to the large resistance offered by the water and slowly increases as the water evap-orates (Mahadevan et al. 2006, 2007). In this study, we concern ourselves to the case wherethe viscous displacement of the fluids is completed and evaporation is the dominant modeof water removal. The model for water saturation profile during evaporation is reproducedbelow from a previous study (Mahadevan et al. 2006; variables explained in the Appendix):

∂Sw

∂tD= NWi

∂

∂xD

(D(Sw)

∂Sw

∂xD

)− 1(∫ 1

0dξ

krg(Sp)

)2krg(Sp)

⎛⎝1 − C

∫ xD0

dξkrg(Sp)∫ 1

0dξ

krg(Sp)

⎞⎠

−3/2

. (1)

The relative permeability of gas is defined as a function of the total saturation (sum of solidsalt saturation and that of liquid water saturation). Included in the governing Eq. 1 are twoconstitutive relations, the relative permeability to gas and the capillary diffusivity, both ofwhich depend on the wetting phase saturation. A number of expressions, such as the Corey(1977), or the van Genuchten (1980) equations, exist in the literature. During the displacementperiod, power laws of the Corey form are typically assumed for the relative permeabilities,

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232 D. Le et al.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Wat

er S

atu

rati

on

, Sw

Dimensionless Distance, xD

tD = 0

tD = 0.1

tD = 0.2

tD = 0.3

tD = 0.4

tD = 0.43

Fig. 1 Water saturation profiles at different dimensionless times obtained by numerically solving (Mahadevanet al. 2006) Eq. 1 for the case where the capillarity is controlling (Nwi = 100 and C = 0.5)

krg = k0rg

(1 − Sw

)ng if Sw > Swr, (2a)

krg = 1 − (1 − k0rg)

Sp

Swrif Sw ≤ Swr, (2b)

where k0rg and k0

rg are prefactors, ng, nw are positive exponents, Sw = Sw−Swr1−Swr−Sgr

is the reducedsaturation, and Sp is the total saturation of residual water and crystallized salt (Sw + Ss), andSs is the solid salt saturation. When pure water is considered, Ss is equal to zero.

Equation 1 is subject to an initial condition supplied by the profile at the end of the dis-placement regime and to a no-liquid-flux boundary condition at the outlet end,

D (Sw)∂Sw

∂x

∣∣∣∣L

= 0. (3a)

The boundary condition at the inlet end varies, depending on whether the process iswet-gas or dry-gas injection. In the former case, the boundary condition is also no-liquidflux,

D (Sw)∂Sw

∂x

∣∣∣∣0

= 0. (3b)

In the above case, the governing equation and boundary conditions are solved usingnumerical methods. Also, in the case of pure water, the crystallization of salt is absent andhence the solid salt crystallization goes to zero. Figure 1 shows the model predicted purewater saturation profiles at the end of the displacement regime and during the injection of awet gas at long times. The saturation profiles are nearly uniform owing to capillary wickingeffect which causes the water to be transported through liquid films along the corners of anon-circular capillary tube. The vapor pressure of water in the gas phase is assumed to be3,166 kPa.

In a previous study, experimental data, obtained through X-ray radiography techniques,were compared with the above model solutions and this is reproduced in Fig. 2 (Mahadevanet al. 2006). The comparison is reasonably good with the exception of end effects and local

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 233

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Length, m

Wat

er S

atu

rati

on

, Sw

0.5 hrs

13 hrs

17 hrs

23 hrs

38 hrs

Fig. 2 Comparison of water saturation profiles between experimental data obtained from X-ray scanning,shown in empty circles, and the model, shown in solid line for flow-through drying and injection of a wet gas.Gas flow is from left to right. The indicated times are experimental values while the model times correspondto the experiments. These data are reproduced from Mahadevan et al. (2006) to illustrate the impact of highcapillary wicking (Nwi ∼ 120) on saturation evolution

oscillations of the saturation profiles. The first profile shows the saturation at the end ofdisplacement which is then followed by saturation profiles during the evaporation regime.Several other experimental results for saturation evolution, during flow-through drying, inlimestones and sandstones were compared satisfactorily with model results and were pub-lished in another report (Mahadevan et al. 2006). The main conclusion from all these resultsand the study is that capillary wicking leads to more uniform profiles which has importantimplications for the transport of solute under flow-through drying conditions.

2.2 Salt Conservation Equations

Flow through wetting films in the porous medium is an important factor that determines theliquid convection rates. It will therefore be necessary to account for this mechanism in mod-eling the transport of salt which is dissolved in the liquid, during drying process. The factorsaffecting salt crystallization include solubility product of salt, salinity of brine, temperatureand pressure.

Species conservation equations for salt ions are shown below (developed in the Appen-dix). The rate of change of both dissolved salt and the crystallized solid salt may be obtainedthrough a solution of the salt model equations along with the saturation evolution equations(Eqs. 1–3). In the solution for water saturation evolution, we also consider the effect of solidsalt by incorporating the solid saturation in the constitutive relationship (relative permeabilitycurve).

2.2.1 Flowing Phase Salt Equation

The governing dimensionless salt conservation equation becomes (see Appendix forderivation)

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234 D. Le et al.

Sw∂θ

∂tD=

⎧⎪⎪⎨⎪⎪⎩(

Nwi D (Sw) ∂Sw∂xD

)∂θ∂xD

+ θ 1

krg

(∫ 10

dξkrg

)2

(1 − C

∫ xD0

dξkrg∫ 1

0dξkrg

)− 32

−Np(θ − θeq)Sw H(θ − θeq

)

⎫⎪⎪⎬⎪⎪⎭

, (4)

where Np is a new dimensionless number that relates the rate of crystallization of salt to therate of evaporation of water. In the above, gas relative permeability is a function of both watersaturation (Sw) and solid salt saturation (Ss).

For dimensionless salt concentration at equilibrium, we have

θeq = ceq

ci, (5)

where ceq is the equilibrium concentration of salt in the water and ci is the concentration ofsalt initially.

2.2.2 Solid Phase Produced by Crystallization

We define the dimensionless solid phase concentration by

θ = c

ci, (6)

where c is the concentration of solid salt in the pore space. This solid concentration isexpressed as the number of moles of salt per liter of pore space. The solid salt saturation, Ss,can then be derived as

Ss = θci MKCl

ρKCl. (7)

In the above, MKCl is the molecular weight of potassium chloride and ρKCl is the density ofsolid potassium chloride salt. The solid salt, or crystallized salt concentration evolution istherefore

∂θ

∂tD= Np(θ − θeq)Sw H

(θ − θeq

). (8)

In the above equations we define Heaviside’s unit step function as follows:

H(θ − θeq

) ={

0 if θ ≤ θeq

1 if θ ≤ θeq. (9)

2.2.3 Boundary and Initial Conditions

The saturation boundary conditions are as follows. At the inlet, no water enters; therefore,

D(Sw)∂Sw

∂xD

∣∣∣∣0

= 0. (10a)

At the outlet, no water leaves; hence,

D(Sw)∂Sw

∂xD

∣∣∣∣1

= 0. (10b)

The concentration initial condition is given as

θ (xD, tD = 0) = 1. (10c)

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 235

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Dim

ensi

on

less

Liq

uid

Sal

t C

on

cetr

atio

n

Dimensionless Distance,xD

tD =0.4

tD =0.1

tD =0.3

tD =0.2

tD =0.35

Fig. 3 Dimensionless liquid phase salt concentration profiles when capillary is non-negligible (Nwi = 100,C = 0.5, Np = 10,000). The profiles are obtained by solving Eq. 4 numerically. Initial water saturation wasassumed to be given by the profile at zero dimensionless time as indicated in Fig. 1

The concentration boundary condition is written as

∂θ

∂xD

∣∣∣∣(xD=0,tD)

= 0. (10d)

The above boundary condition arises due to the fact that evaporation losses at the inlet arenegligible (due very small pressure gradients). For the dimensionless crystallized solid saltconcentration, the initial condition is given as

θ (xD, tD = 0) = 0. (10e)

The salt conservation equations were solved along with the water saturation in the coreby developing a finite difference model. An explicit method was used to compute both sat-uration and salt concentration in time and space for the one-dimensional linear system. Thesolutions to Eq. 4 and the corresponding boundary conditions (Eqs. 10a–d) are given, wereverified for accuracy by conducting a sensitivity study in which the spatial and the temporalgrid block sizes varied. The impact of variation of the grid blocks on the error in total molesof salt (material balance error) is calculated. The results showed that for the dimensionlessgrid block sizes of 0.001 for spatial step and 0.0001 for the temporal step the material balanceerrors were 0.03 and 0.18%, respectively.

The solubility limit of potassium chloride in water is approximately 4 mol/l and we alsoassume that the vapor pressure of water is lowered from 3,166 kPa in proportion to the molefraction of water in the liquid phase. In making this assumption, we neglect the impact ofnon-ideal solutions in lowering of vapor pressure. We begin with an initial dimensionless con-centration of 1 (which corresponds to 1 mol/l initial concentration). Figure 3 shows the resultof numerical computation of the liquid phase salt concentration for the case of high wickingnumber (NWi = 100) and fast crystallization kinetics (Np =10,000). The profiles show thatthe liquid phase concentrations are greater near the exiting end of the porous medium thanthe inlet. This is due to the fact that the evaporation rates are greater near the exiting end. Inaddition, the capillary wicking transports water from the inlet end region to the outlet end

123

236 D. Le et al.

as evaporation takes place. The model predictions obtained here reflects the conclusions ofPuyate et al. (1998; Puyate and Lawrence 1998; Scherer 1999) which show that liquid phaseconcentration of dissolved sodium chloride in cement increase due to evaporation combinedwith wick effect. Our result for the case of higher wicking number is equivalent to the resultsfor higher “Peclet” number in the work by Puyate et al. (1998) The definition of Peclet num-ber in their paper corresponds to the relative rate of transport of salt ions through the wickaction and the rate of diffusive transport of ions through the porous concrete. The drivingforce for the evaporation in their study, however, is diffusion of vapor through the pores ofthe rock. We validate the result of numerical solutions in the next section where an analyticalexpression for liquid salt concentration is developed based on the limiting conditions of highwicking number.

3 Validation of Numerical Solution at Capillary-Dominated Conditionsand Fast Crystallization Kinetics

3.1 Liquid Phase Salt Concentration

When capillary effects dominate, the liquid saturation profile is flat and the gas relative per-meability is approximately a constant along the length. Under these conditions, the governingequation (Eq. 1), for water saturation becomes, upon integration with distance (Mahadevanet al. 2006),

1

krg〈Sw〉d〈Sw〉dtD

= − 2

C

((1 − C)−1/2 − 1

), (11a)

where

krg = 1 − (1 − k0rg)

〈Sw〉Swr

if Sw ≤ Swr. (11b)

In the above, we have neglected the impact of solid salt saturation although we haveincluded this effect for the numerical solution. In addition, the assumption of flat saturationprofiles due to high wicking number leads to the disappearance of the second term on theright-hand side of Eq. 1 and a constant gas relative permeability term. Details of this deriva-tion are provided in the Appendix of Mahadevan et al. (2006). The water saturation profilecan be obtained by integration of Eq. 11, after substituting for the relative permeability term,to give the following result:

〈Sw〉 =

(1 −

(1 − (1−k0

rg)

Swr〈Sw〉|tD=0

)exp

((1−k0

rg)

Swr

2C

((1 − C)−1/2 − 1

)tD

))Swr

(1 − k0rg)

(12)

The governing equation (Eq. 4) for describing salt concentration distribution then becomes,with the assumption of constant spatial liquid saturation, 〈Sw〉, the relative permeability togas flow also becomes a constant,

〈Sw〉 ∂θ

∂tD=(

Nwi D (〈Sw〉) ∂ 〈Sw〉∂xD

)∂θ

∂xD+ θkrg

(1 − CxD)3/2

−Np(θ − θeq

) 〈Sw〉 H(θ − θeq

). (13)

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 237

The first term in the above equation is neglected due to the assumption of flat satura-tion profiles. However, the condition of flat profiles may never be achieved in reality, andthere will always be a certain gradient, although very small. This can lead to a situationwhere the first term may not be negligible. Nevertheless, we proceed with this simplifyingassumption of flat saturation profiles, to eliminate the mathematical complexities in solvingfor the liquid phase salt concentrations. It will be shown later, using a complete finite differ-ence-based numerical solution to Eq. 4, that this is not very far from actual values of liquidconcentration. Consequently, we get

〈Sw〉 ∂θ

∂tD= θkrg

(1 − CxD)3/2 − Np(θ − θeq) 〈Sw〉 H(θ − θeq

). (14)

The salt crystallization does not begin until supersaturation of the salt is achieved. Thisis achieved once the evaporation and wicking process drives the concentrations to highervalues locally. In order to model this regime we consider the salt and water governing equa-tions without the crystallization reaction term as follows. Equation 14 becomes, for a systemwithout any supersaturation,

〈Sw〉 ∂θ

∂tD= θkrg

(1 − CxD)3/2 . (15)

However, the saturation is in turn a function of time as given by Eq. 12. Substituting theabove equation into Eq. 15 we get(

1 −(

1 − (1 − k0rg)

Swr〈Sw〉|tD=0

)exp

((1 − k0

rg)

Swr

2

C

((1 − C)−1/2 − 1

)tD

))∂θ

∂tD

= (1 − k0rg)

Swr

θkrg

(1 − CxD)3/2 . (16)

Re-writing the above equation as

∂θ

∂tD= θ F (xD) A exp (BtD)

(1 − A exp (BtD)). (17a)

In the above,

A =(

1 − (1 − k0rg)

Swr〈Sw〉|tD=0

), (17b)

B = (1 − k0rg)

Swr

2

C

((1 − C)−1/2 − 1

), (17c)

F(xD) = (1 − k0rg)

Swr

1

(1 − CxD)3/2 . (17d)

Equation 17a may be rearranged and integrated with the initial condition to yield

θ = θi

(A − 1

A exp(BtD) − 1

)F(xD)/B

. (18)

Figure 4 shows the plot of dimensionless concentration with distance at different dimen-sionless times calculated using Eq. 18. The figure shows analytical calculation, assumingcapillary-dominated conditions (high wicking numbers and flat saturation profiles) and com-plete numerical calculations at the same wicking number. The analytical and numerical

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238 D. Le et al.

1

1.5

2

2.5

3

3.5

4

4.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Dim

ensi

on

less

Liq

uid

Sal

t C

on

cetr

atio

n

Dimensionless Distance, xD

analytical numerical

tD = 0.1

tD = 0.15

tD = 0.2

tD = 0.3

tD = 0.35

tD = 0.4

tD = 0.25

Fig. 4 Comparison of analytical (Eq. 18) and numerical solution (Eq. 4) for dimensionless liquid phase saltconcentration profile when capillary is dominating (Nwi = 100, C = 0.5, Np = 10,000). Initial water saturationwas assumed to be flat at 0.58

solutions agree well at early times, but do not at later times. The difference in numericalprediction arises due to the presence of small values of the saturation gradient term in Eq. 4which is neglected in the analytical calculation, with the assumption of flat saturation profiles.With a rigorous calculation of the liquid salt from Eq. 4, using numerical methods, we findthat the liquid saturations are not necessarily flat even at high capillary wicking numbers. Thisis especially true at lower saturation levels, which occur at later times. Thus, the analyticalprofiles overestimate the liquid salt concentrations.

3.2 Evolution of Solid Salt

In this section, we model the solid salt deposition under capillary-dominated conditions. Wenote that the extent and rate of solid salt crystallization in actual systems will depend onmany other factors such as heterogeneous nucleation kinetics and surface chemistry of rockgrains. However, we assume that the crystallization reactions proceed fast and the only factorgoverning the precipitation is the threshold or equilibrium concentration at supersaturation.Once supersaturation is achieved, the crystallization reaction proceeds to deposit solids. Thisleads to accumulation of salt while the liquid phase concentration drops to values determinedby the solubility product of the salt. If a fast reaction is present then the excess salt, broughtinto the regions with liquid phase concentration of salt at θeq, through capillary wicking, willcrystallize instantaneously. Therefore, the liquid phase concentration is uniformly at θeq atall times.

In all the subsequent numerical calculations, we consider the effect of solid salt saturationon the gas flow. However, in the development of analytical model for liquid saturations, weneglect the effect of solid salt on gas flow and only consider the residual trapped liquid satu-ration. This may be reasonable considering that the initial salt concentration is a low value of1 mol/l. Equation 18 can be used to obtain the position in space and time of the regions wherethe concentrations are greater than or equal to the equilibrium value. After letting θ = θeq

and rearranging Eq. 18 we get

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 239

xD = 1

C

⎛⎜⎜⎝1 −

⎛⎝(

1 − k0rg

Swr

)ln(

A−1A exp(BtD)−1

)

B ln(

θeqθi

)⎞⎠

23

⎞⎟⎟⎠ . (19)

It is expected that the salt crystallization is likely to begin in positions where the supersat-uration is exceeded. The time required to achieve supersaturation at any position is obtainedby rearranging Eq. 19 to give

tDs = 1

Bln

(1

A+ A − 1

AθB/F(xD)eq

). (20)

When capillarity is dominating, the saturation profiles are flat (Figs. 1, 2) due to the redis-tribution of liquids which maintains additional supply of the evaporating liquid in the higherdrying rate regions. In the absence of any redistribution, the salt ions do not undergo transportand the deposition progresses only as long as the dissolved salt is available at supersaturatedconditions. Equation 14 may be rearranged and substituted in Eq. 8 to give the following:

∂θ

∂tD= θkrg

(1 − CxD)3/2 − 〈Sw〉 ∂θ

∂tD. (21)

When the reaction rate is fast the local concentration is expected to be at a constantequilibrium value and Eq. 21 becomes

∂θ

∂tD= θeq

krg

(1 − CxD)3/2 . (22)

In the above we substitute the expression for krg as a function of time to get

∂θ

∂tD= θeq

A exp (BtD)

(1 − CxD)3/2 . (23)

Separating the variables and integrating, from tDs to tD we get

θ = θeqA (exp (BtD) − exp (BtDs))

B (1 − CxD)3/2 . (24)

Figure 5 shows the solid salt concentration and solid salt saturation profile evolution withtime. The model predicts that the solid salt accumulates at regions near the outlet end due tothe transport of salt through the liquid phase by capillary wicking. Both numerical and ana-lytical methods give similar prediction for the solid salt in the porous medium but not exactlythe same. This difference arises due to the non-zero saturation gradients that the rigorousnumerical solution gives when compared to the assumption of flat saturation profiles in theanalytical solution procedure. This leads to overestimation of liquid saturations and hencethat of solid salt predicted by the analytical method as seen in Fig. 5. In the next section, themodel equations are used to predict the impact of capillary-driven liquid films and the initialliquid concentration of salt on the evolution of solid salt in the rock.

4 Sensitivity of Solid Salt Saturation to Capillarity and Initial Conditions

The uniformity of solid saturation profiles depends on the strength of capillarity in the porousmedium. When a strong capillary pressure gradient exists, the relative movement of liquidin comparison to the evaporation rates is high and hence the salt solution will be carried to

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240 D. Le et al.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8 1

So

lid S

alt

Sat

ura

tio

n, S

s

Dimensionless distance, xD

Analytical Numerical

tD=0.35

tD=0.40tD=0.43

Fig. 5 Comparison of analytical (using Eq. 24) and numerical solution (Eq. 4) for solid salt saturation, Ss,when capillary is dominating (Nwi = 100, C = 0.5, Np = 10,000). Initial water saturation was assumed to beflat at 0.58

Fig. 6 Solid salt saturation profiles (using numerical solution for Eq. 8) after water has completely evaporatedfor different wicking numbers (Np = 10,000, C = 0.5). Initial water saturation was assumed to be flat at 0.58

the high drying rate regions. This corresponds to the high wicking number case in this study.Figure 6 shows the effect of capillary wicking on the solid salt saturation with length. Whenthe capillary wicking effect, due to liquid films, is considered, the solid salt accumulatestowards the exiting end. This is a result of the non-uniform evaporation rates in the porousmedium, caused by the varying pressure gradients in the gas phase, and the associated liquidtransport due to capillary wicking. In the absence of any wicking effect the solid saturationis uniform which reflects the condition that the salt present in the pore space at any locationis crystallized at the same location. In the presence of increasing levels of capillary wicking

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 241

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

So

lid S

alt

Sat

ura

tio

n, S

s

Dimensionless Distance, xD

Ci = 1 M

Ci = 2 M

Ci = 3 M

Ci = 4 M

Fig. 7 Solid salt saturation profiles (using numerical solution for Eq. 8) at tD = 0.43 (when water saturationis close to zero at all positions) at different initial concentrations (Nwi = 100, C = 0.5, Np = 10,000). Initialwater saturation was assumed to be flat at 0.58

the salt is transported to the higher drying rate regions and accumulated. This is evident fromboth the liquid and solid salt saturation profiles calculated using the models.

Figure 7 shows the effect of initial salt concentration on the salt saturation profile. Whenthe initial salt concentration increases, the solid salt saturation increases to greater amountsnear the exiting end. The increase is somewhat subdued due to the slowing of gas flow ratesas the solid salt forms and blocks the flow of gas. It must be noted that the slowing of gasflow rate also lowers the rate of evaporation. In the following section, experimental study ofthe solid salt crystallization is presented along with a comparison to model prediction of thesolid salt mass in the rock core.

5 Experiments of Salt Crystallization in Cores

There is limited experimental information in the literature on the extent of crystallized solidsalt saturation, especially under flow-through drying conditions. In the experiments involvingtransport of chloride ion in concrete, by Puyate et al. (1998), the total chloride concentrationat high Peclet number (Peclet number being defined as a ratio of convection to that of diffu-sion rates of ions in the matrix) distinctly shows the formation of a supersaturated region, inthe region behind the predicted interfacial position. But the study does not report the extentof solid crystallized salt in the dry region.

In order to understand, the impact of solid salt evolution gas injection experiments wereperformed on two sets of Berea sandstone cores. The Berea cores were prepared as describedin Appendix B. A set of sandstone core plugs cut from a single long core was placed in thecore holder with axial pressure of 100 psi and radial confining pressure of 100 psi. Figure 8shows a schematic of the rock core arrangement within the core holder and the direction ofgas flow. The axial pressure is expected to generate sufficient capillary contact between theadjacent core plugs. The gravimetric measurements were done on the individual core plugs asdescribed in Appendix B before and after each experiment of gas flooding to dry conditions.

123

242 D. Le et al.

TU6 TU7 TU8 TU9

Gas Injection

Fig. 8 Arrangement of core plugs for wet gas injection experiment. The alphanumeric key represents thecore plug whose properties are shown in Table 1. Core plug “TU6” is the first core in the direction of gas flow.The core plugs are cut from a single cylindrical core of Berea sandstone of 6 inches length

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

Core Number (Gas Flow from Left to Right)

Res

idu

al S

alt,

gra

m

Experimental Result 30% Initial Salt Concentration

Experimental Result 20% Initial Salt Concentration

Fig. 9 Experimental result for crystallized salt mass along with the distance for the two cases withdifferent initial salt concentrations after flow-through drying process. The experiments were performed atwicking numbers close to 120 which reproduces the conditions for which flat saturations were observedexperimentally from a previous study (Mahadevan et al. 2006). Properties of the rock sample are shown inTables 1 and 2

An initial concentration of 30 weight % of potassium chloride salt (30 g in 100 g of water)corresponding to 3.6 M and 20 weight % corresponding to 2.6 M was used to saturate the rockbefore start of gas injection in each case. The pressure drop during gas injection through theBerea sandstone core was kept at 1 atm and the temperature was kept constant at 25◦C.

In similar humid gas injection experiments (Mahadevan et al. 2006, 2007) which weresubjected to X-ray imaging for saturation monitoring, flat saturation profiles were observed,which indicate high wicking number and rapid redistribution of water within the rock sam-ple. In this study, the experiments are conducted at similar conditions with the exception ofthe length. The length of the rock core used in this study is 6 inches which is twice that ofthe previous study. The longer core lengths provide greater initial salt mass in the systemand hence results in more accurate gravimetric analysis and mass balance verification. Thecapillary wicking number is still higher and hence it is expected that the saturation profilesof this experiment will also be flat.

Figure 9 shows the salt mass crystallized in each of the four sections of the rock core inthe direction of flow for the two cases. The plot shows that the salt mass is greater towards theouter end of the consolidated rock core which can result only due to solute transport throughliquid films which is the central hypothesis of this study. The first trend shows the salt massin the experiment on a consolidated Berea sandstone core whose native permeability before

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 243

Table 1 Measured rock core properties and gravimetric analysis (initial salt concentration is 30 weight %)

Core Length Diameter Permeability Weight before Weight after Measured Calculatedplug # (inches) (inches) (mD) saturation (g) drying (g) salt mass salt mass

(g) (g)

TU6 1.411±0.003 0.952±0.002 298.1937 34.4687±0.0001 35.5433±0.0001 0.1557 0.1287TU7 1.388±0.002 0.953±0.003 289.3808 33.8645±0.0001 35.0035±0.0001 0.2935 0.1929TU8 1.384±0.003 0.952±0.001 288.7057 33.5974±0.0001 34.7524±0.0001 0.3038 0.3135TU9 1.379±0.003 0.950±0.003 282.3639 34.5022±0.0001 35.8284±0.0001 0.4612 0.5765

Drying time=65 h

Table 2 Measured rock core properties and gravimetric analysis (initial salt concentration is 20 weight %)

Core Length Diameter Permeability Weight before Weight after Measured Calculatedplug # (inches) (inches) (mD) saturation (g) drying (g) salt mass salt mass

(g) (g)

TU11 1.132±0.001 1.010±0.003 165.0000 30.6302±0.0001 30.6492±0.0001 0.0190 0.0597TU12 1.132±0.003 1.011±0.004 165.0000 30.7347±0.0001 30.8077±0.0001 0.0730 0.1079TU13 1.135±0.002 1.010±0.002 165.0000 30.804±0.0001 31.0764±0.0001 0.2724 0.2045TU14 1.098±0.003 1.010±0.002 165.0000 29.5985±0.0001 30.0148±0.0001 0.4163 0.4141

Drying time=90 h

cutting is 289 mD. Table 1 shows the permeability of the individual pieces of the two rockcores. The second trend shows the salt mass measured in the experiment conducted on Bereasandstone core whose properties are shown in Table 2.

6 Discussion of Experimental Results and Model Comparison

Figure 10 shows the variation of the crystallized salt mass along the direction of flow predictedby the model in comparison to the experimental observation in Fig. 9. The theoretical saltmass, which is calculated by averaging the salt mass in the equivalent length of the core plug,predicts the general trend observed in the experiment well. The deviations of the data frommodel behavior may be attributed to variations in the local permeability and hence the poresizes of the core plug samples. These local variations can induce variable capillary wickingrates and hence solute transport characteristics. The observed profile, although affected bylocal heterogeneities, are reasonable indicators of the evolution of solid salt which seems toaccumulate in one end of the rock core. The main reason for the accumulation of the salt isdue to capillary wicking which supplies solute laden liquid to the higher drying rate regionin the rock core which occurs at the near outlet region.

The theoretical results and experimental study concerning solute transport under flow-through drying conditions presented in this study are first of its kind. The only previoustreatment, related to this study, is that by Sghaier et al. (2007). However, their theoreticalstudy did not include any film flow effects. In the experimental results of this study, thesupply of liquid, through thick films, is limited to the residual saturation in the core at theend of displacement. However, when there is a continuous supply of ionic liquid, as might bethe case in geologic systems, there will be a continuous evolution of solid salt which couldeventually plug the pores of the formation. This scenario has indeed been considered in the

123

244 D. Le et al.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2

Core Number (Gas Flow from Left to Right)

Res

idu

al S

alt,

gra

m

Model Prediction - 30% Initial Salt Concentration

Experimental Result 30% Initial Salt Concentration

Model Prediction - 20% Initial Salt Concentration

Experimental Result 20% Initial Salt Concentration

3 4 51

Fig. 10 Comparison of experiment and model prediction for the crystallized salt mass along with distanceafter flow-through drying process. The experiments were performed at wicking numbers close to 120 whichreproduces the conditions for which flat saturations were observed experimentally from a previous study(Mahadevan et al. 2006). Properties of the rock sample are shown in Tables 1 and 2

studies by Tsypkin et al. (2003, 2005), in the context of geothermal systems, albeit withoutthe effects of liquid films and the convection of gas phase.

7 Conclusions

1. We have developed a mathematical model, using mass conservation based on continuumassumption, to describe the movement of dissolved salts and crystallization caused bysuper saturation due to flow-through drying. The mathematical model includes the flowand transport of salt through capillary pressure-driven thick films.

2. The resulting mathematical model for liquid salt concentration and solid salt is solvedusing finite difference approximation, along with the evolution equation for water satura-tion, and the solutions were verified with asymptotic analytical expressions forcapillary-dominated and fast crystallization reaction conditions.

3. In the limit of fast movement of liquids through thick films, represented by a high wickingnumber, and fast crystallization, we predict accumulation of solid salt in the region wherethe drying rates are high.

4. Flow-through drying experiments, of potassium chloride in de-ionized water solution inBerea sandstone cores, show that the solute transport, enhanced by the capillary wickingprocess, leads to greater crystallized salt mass in the regions of high drying rates, thusreflecting the model predictions. These results, for the first time, clearly show the impactof capillary pressure-driven liquid films on solute transport under flow-through dryingconditions.

Acknowledgements We wish to acknowledge the financial support to work on this project by the AmericanChemical Society—Petroleum Research Fund through award number 46375-G9. We also thank Prof. YannisYortsos for useful suggestions in developing the solid saturation evolution model.

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 245

Appendix A: Model Development

In the below, we develop transient continuum models for transport of salt through liquidfilms in porous medium. The solution to the salt concentration model along with solutionto Eqs. 1–3 will give the profiles of the salinities which will then determine the crystalliza-tion spatially. We make the following assumptions to the evaporation and salt crystallizationmodel:

1. One dimensional system.2. Temperature variation caused by Joule–Thompson cooling is negligible.3. The phase behavior is described by Raoult’s law.4. The gas obeys ideal gas law.5. Local thermodynamic equilibrium exists (this means that chemical potential is equal in

all phases).6. Inter-phase mass transfer is fast.7. The maximum concentration of the salt given by the solubility product.8. Dispersion or diffusion of salt, axially, is negligible.9. All ions are transported at equal velocity and no electrochemical or adsorption effects

exist.10. Changes in mobility of gas, due to solid salt, are accounted for in the relative permeability

of gas as a function of phase saturations.11. Vapor pressure lowering of water is proportional to the mole fraction of water: ideal

solution assumption.

Conservation of salt species in solution for a porous medium is given asRate of accumulation term for a change in time of �t :

∂ (cSwφ)

∂t(A-1)

where c is the concentration of salt in the liquid phase; Sw is the saturation of the water phase.Rate of salt transported into the element during time, �t :

∂(uT f ′

wc)

∂x(A-2)

where uT is the total velocity of fluids (Mahadevan et al. 2006)

uT = A (τ )

�= A (τ )√

�20 − 2A (τ )

∫ x0

dxλrg(Sw)

, (A-3)

and f ′w is the fractional flow of water given by,

f ′w = fw

(1 + kkrg

uTµg

∂ Pc

∂x− kkrg�ρg

uTµg

), (A-4)

where

fw =kkrwµw

kkrwµw

+ kkrgµg

. (A-5)

In the above Eqs. 6–7, k denotes permeability, kr j (Sw) is the relative permeability of phasej, a function of the liquid saturation, µ denotes viscosity, Pc(Sw) is the capillary pressure

123

246 D. Le et al.

function, �ρ is the difference in density, and g is the acceleration of gravity in the directionof displacement (which here will coincide with the downwards vertical direction). We havedefined � = Pg − Ps(T ). The transport of salt out of the element is given by an expressionsimilar to Eq. 5. The constant A(τ ) is evaluated at the outlet end of the sample (at location L),

A(τ ) = �20 − �2

L

2∫ L

01

λrg(Sw)dx

. (A-6)

Rate of salt crystallized:We assume that the salt crystallization is a kinetically controlled reaction that is described

by mass action kinetics. The rate of mass transport across the pore is not considered asa limiting factor, a reasonable assumption in micro-porous medium. Salt is crystallized ifthe degree of saturation in the liquid phase exceeds the necessary degree of supersaturationwhich is determined by a supersaturated equilibrium product, KSP. The solubility productfor a crystal whose chemical formula is of the form AaBb is

K = [aA]a [aB]b (A-7)

where, [ai] is the activity of i. The equilibrium value of the solubility product, K0, is thevalue of K when the chemical potential in the solution and crystal are equal. The solution issaid to be supersaturated if,

K > K0 (A-8)

and the crystal growth continues until the supersaturation is eliminated. However, continuoussupply of dissolved inorganic salts can sustain the crystal growth. For the case of potassiumchloride salt,

K = [aK] [aCl] (A-9)

If the activity coefficient is unity, then

K = c2 (A-10)

and,

K0 = c2eq (A-11)

where ceq is the equilibrium concentration of potassium chloride salt. It is to be noted thatunder equilibrium conditions of chemical potential in both the solid and the liquid phases,the rate of reaction is not a limiting factor and hence the concentrations in the liquid aredetermined by the solubility product of the salt. The rate of crystallization of salt in that caseis determined by the level of supersaturation.

Assuming first-order reaction kinetics, the rate of crystallization of salt, in supersaturatedregions, may then be written as

rp = kp(c − ceq), (A-12)

where

rp [=]mol

liq.vol · time(A-13)

In order to distinguish the regions of the time–space domain that satisfies the condition forcrystallization, namely, K > K0 or c > ceq, we define Heaviside’s unit step function asfollows:

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 247

H(c − ceq

) ={

01

ifif

c ≤ ceq

c > ceq(A-14)

The reaction rate may then be written as

Rp = rp H(c − ceq

)(A-15)

Salt in liquid conservation expression:The final equation is shown as below after combining Eqs. A-1,A-2 and A-15.

∂ (cSw)

∂t= − 1

φ

∂(uT f ′

wc)

∂x− kp

(c − ceq

)Sw H

(c − ceq

)(A-16)

Rate of production of solid salt by crystallization:

∂ c

∂t= kp(c − ceq)Sw H

(c − ceq

), (A-17)

where c is the moles of crystal formed per unit pore volume. The above set of differen-tial equations describe the evolution of concentrations of salt and also the solid salt crystalweight. The solution to the above set of differential equations, along with water saturationsfrom solution to Eq. 1, with the appropriate boundary conditions will provide the requiredsalt concentration profile and the solid saturation profile.

Assuming that gravity is negligible and that the viscous displacement is complete, we get

f ′w ≈ fw

(kkrg

uTµg

∂ Pc

∂x

)(A-18)

Substituting the above in the salt conservation equation we get

∂ (Swc)

∂t= − ∂

∂x

(uT fwc

kkrg

uTµg

∂ Pc

∂x

)− kp

(c − ceq

)Sw H

(c − ceq

)(A-19)

Knowing that the water velocities are very small, we assume that the first term on theright-hand side is negligible. We also assume that the gas mobilities are small relative to thatof water. Hence, upon further simplification we get

∂ (cSw)

∂t= − 1

φ

∂

∂x

(c

kkrw

µw

∂ Pc

∂x

)− kp

(c − ceq

)Sw H

(c − ceq

)(A-20)

Applying chain rule to write the above equation in terms of the derivative of the capillarypressure, we get

∂ (cSw)

∂t= − 1

φ

∂

∂x

(c

kkrw

µw

dPc

dSw

∂Sw

∂x

)− kp

(c − ceq

)Sw H

(c − ceq

)(A-21)

Writing the above equation in terms of the diffusivity term

∂ (cSw)

∂t= 1

φ

∂

∂x

(c

agmγ rc

µwD (Sw)

∂Sw

∂x

)− kp

(c − ceq

)Sw H

(c − ceq

)(A-22)

In the above, we have incorporated the expression for capillary diffusivity, which is shownbelow

− fwkkrg

µg

dPc

dSw≈ −kkrw

µw

dPc

dSw= agmγ rc

µwD(Sw). (A-23)

123

248 D. Le et al.

Combining the above expression, after applying the chain rule for differentiation, withthe expression for water saturation evolution, we get

Sw∂c

∂t= 1

φ

(agmγ rc

µwD(Sw)

∂Sw

∂x

)∂c

∂x+ c

ε

φ

∂

∂x

⎛⎝ A (t)√

�20 − 2A (t)

∫ x0

dxλrg

⎞⎠

−kp(c − ceq

)Sw H

(c − ceq

). (A-24)

In the above equation, ε, is defined as the drying scale factor and is given by

ε = α

β − α. (A-25)

The governing equation can be re-written as

Sw∂c

∂t=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1φ

(agmγ rc

µwD(Sw) ∂Sw

∂x

)∂c∂x

+ c εφ

14

(�2

0−�2L

)2

�30

kµg

1

krg

(∫ L0

dxkrg

)2

(1 −

(�2

0−�2L

)�2

0

∫ x0

dxλrg∫ L

0dxλrg

)− 32

− kp(c − ceq

)Sw H

(c − ceq

)

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(A-26)

In Eq. 2, the dimensionless space and time variables are

xD = x

L, (A-27)

tD = τ

τ ∗ , (A-28)

where

τ ∗ = 4φµg�30 L2

k(�20 − �2

L)2, (A-29)

and

τ = tα

β − α. (A-30)

The concentrations of water in gas phase and liquid phase are

α = yw Pg

RgT, (A-31)

β = ρw

Mw. (A-32)

The dimensionless pressure drop is given by

C = �20 − �2

L

�20

, (A-33)

and the dimensionless wicking number for the linear case is given by

NWi = 4agmγ rc�30µg

εµw(�2

0 − �2L

)2k. (A-34)

In the above the dimensionless scaling factor is given by,

ε = α

β − α. (A-35)

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 249

The subscripts 0 and L correspond to the injection and production ends, respectively. φ

is porosity, ρ is mass density, M is the molecular weight, y denotes mole fraction in the gasphase, and u is the flow velocity; k is permeability, γ is the interfacial tension, rc is the mean

pore throat size (rc =√

8kφ

, from a bundle of capillary tubes model), µ is viscosity, τ is the

drying time, and agm is a geometric constant (Mahadevan et al. 2006).We then introduce the following dimensionless variables and redefine the governing equa-

tion for salt transport. For dimensionless salt concentration, we have

θ = c

ci, (A-36)

where ci is the initial concentration of salt in the water.

Sw∂θ

∂tD=

⎧⎪⎪⎨⎪⎪⎩(

Nwi D (Sw) ∂Sw∂xD

)∂θ∂xD

+ θ 1

krg

(∫ 10

dξkrg

)2

(1 − C

∫ xD0

dξλrg∫ 1

0dξλrg

)− 32

−Np(θ − θeq

)Sw H

(θ − θeq

)

⎫⎪⎪⎬⎪⎪⎭

(A-37)

A new dimensionless number that relates the rate of crystallization of salt to the rate ofevaporation of water through the gas phase is defined below.

Np = 4�30µgφL2

(�2

0 − �2L

)2kε

kp (A-38)

The numerical error is the error arising due to the finite difference approximation to thecontinuous functions such as the salt concentrations (both liquid phase and the solid). In orderto check the accuracy of the numerical result a sensitivity study was conducted to understandthe impact of the grid block and time period sizes on the material balance. The materialbalance is defined as follows:

1∫0

θiSwidxD =1∫

0

(θ + θ Sw

)dxD (A-39)

In the above equation, the term on the left-hand side stands for the total moles of salt in thesystem initially (θi is the initial liquid phase salt concentration, while Swi is the initial watersaturation distribution along the length of the rock sample). The right-hand side is the totalmoles of salt in the system at any time t after the flow-through drying process has commenced(θ is the solid salt concentration).

It is expected that this material balance be satisfied during the numerical calculations at alltimes. However, this may not be possible considering the errors introduced by the calculationprocedure itself. We define the material balance error as follows:

error =⎡⎣

1∫0

(θ + θ Sw

)dxD

⎤⎦

model

−1∫

0

θiSwidxD (A-40)

In the above equation, the first term on the right-hand side is the total salt as calculated bythe model using numerical solution in the system. The second term is the initial moles of saltin the system. The above equation may be approximated by the following,

error =[

N∑1

(θ + θ Sw

)�xD

]

model− θi (0.58) (A-41)

123

250 D. Le et al.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 0.002 0.004 0.006 0.008 0.01 0.012

Dimensionless Space Step, xD

Dim

ensi

on

less

Mat

eria

l Bal

ance

Err

or

Fig. 11 Variation of material balance error as calculated by Eq. A-41 with respect to the dimensionless spatialgrid block size

0

0.005

0.01

0.015

0.02

0.025

0.0002 0.0004 0.0006 0.0008 0.001 0.0012

Dimensionless Time Step, D

Dim

ensi

on

less

Mat

eria

l Bal

ance

Err

or

0

t

Fig. 12 Variation of material balance error as calculated by Eq. A-41 with respect to the dimensionlesstemporal grid block size

Thus, the error in material balance may be calculated. The impact of dimensionless spa-tial and temporal grid block sizes may be obtained by calculating the error as per the aboveequation. Figure 11 shows the impact of changing the spatial grid block size on the materialbalance error as calculated by Eq. A-41. The material balance error continuously decreasesand approaches values very close to zero at small spatial grid block sizes. The value of dimen-sionless spatial grid block size used to solve Eq. A-37 is 0.001 which shows a very smallmaterial balance error (0.03%). In the above sensitivity calculation the temporal grid blocksize was fixed at 0.00001.

Figure 12 shows the impact of the dimensionless temporal grid block size on the materialbalance error, with the dimensionless spatial grid block size kept at 0.0001. The plot showsthat the variation in error is nearly linear with a positive slope. The temporal grid block sizeused in this study is 0.0001 which results in a very small error (0.18%).

123

Impact of Capillary-Driven Liquid Films on Salt Crystallization 251

Pressure regulator

Nitrogen supply

Gauge Pressure Indicator

Differential Pressure Indicator

Flow meter

To vent hood

Expelled fluid collector

Epoxy CoreHolder

N2 saturator (Demin water)

To vacuum pump

X-Ray Scan

Fig. 13 Schematic of the experimental set-up on flow-through drying

Appendix B: Core Preparation and Experimental Apparatus

A single 6 inch long cylindrical sample of Berea sandstone was cut from the same block toobtain similar specimens. The samples were then sliced into core plugs of 1.5 inches long.The Klinkenberg permeability was measured for the samples using nitrogen. The cores wereevacuated for 2 h and saturated with the required potassium chloride solution by placing thecores in a solution of potassium chloride in a beaker. A 30% by weight potassium chloridesolution in de-ionized water was used as the saturating fluid. The cores were subsequentlyplaced in a Hassler apparatus with an axial confining pressure of 100 psi and a radial confiningpressure of 100 psi. The axial confining pressure is expected to provide adequate capillarycontact between the cut rock cores.

Building instrument air supply, after passing it through a filter to remove entrained impu-rities and organic vapors, was used as the injected gas (see schematic in Fig. 13). For thewet-gas injection experiments, the gas was bubbled through a column of water filled withglass beads, to completely saturate it with water, before injection. The presence of glassbeads increases the residence time of the air bubbles, thus providing close to a 100% relativehumidity air. This was verified experimentally using a hygrometer at the maximum flowrates expected in the experiment. The experimental parameters monitored and recorded withtime are flow rate of gas at the inlet end, pressure drop (which is maintained constant at 1atmosphere), and humidity. At the conclusion of the gas injection the core plugs are removedand dried to remove any remaining water. A gravimetric analysis of the cores at the endof experiment is conducted after oven drying and compared with the initial dry weight todetermine the salt mass deposited.

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