Some Frequently Overlooked Aspects of Reactive Flow through

Ind. Eng. Chem. Res. 2000, 39, 2717-2724

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Some Frequently Overlooked Aspects of Reactive Flow through Permeable Media Aura Araque-Martinez and Larry W. Lake* Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, Texas 78712

Solid dissolution and precipitation are major reactions that occur in reactive flow through permeable media. Precipitated reaction products, in particular, can clog pore space and cause flow impairment, which is one of the reasons their simulation is important. Many calculations assume local equilibrium, ideal solutions, no aqueous reactions, and no supersaturation, but depending on the flow conditions, these assumptions can lead to a misrepresentation of the effects of the reactive flow. This paper evaluates the effects of these simplifying assumptions. We applied a method of characteristics model to run generic reactive flow cases consisting of a single mineral initially present and one possible precipitate. Results show that, under the conditions studied, the amount of precipitation predicted is less than that calculated by assuming equilibrium conditions; however, this precipitate can, nevertheless, cause flow impairment. The same conclusionsreduced precipitations appears to be true for the assumptions of no supersaturation, ideal solutions, and negligible flowing-phase reactions. Including these effects leads to no precipitation at all in some cases. It seems clear that including these effects is important for accurate geochemical modeling. This work is one of the many tangible consequences of Dr. R. S. Schechter’s distinguished work in geochemical flow modeling. This work has ranged from laboratory experiment to fieldwork to theoretical analysis to numerical simulation. The second author on this paper attributes much of his success to this diversity and to Bob’s unwavering devotion to scientific principles. We are pleased to be able to continue the pursuit of knowledge begun by Dr. Schechter and to honor him and his career at the same time. Introduction Simulation of reactive flow through permeable media can become complicated when accounting for all possible conditions that might affect the results. As a consequence, assumptions are often made to simplify the simulation. These assumptions include all or some of the following: (1) the medium and the flowing aqueous phase are in local thermodynamic equilibrium, (2) the flowing phase is an ideal solution, (3) no chemical reactions occur in the flowing phase, and (4) there is no solid supersaturation. However, there are always questions about how these assumptions affect the results. We use a simplified method of characteristics (MOC)based solution1,2 that has proven to be as accurate and much faster than a complete numerical solution with little loss of generality. The results of these simulations will study the effects of the above assumptions on precipitation and dissolution reactions to determine the error involved when invoking them. Flow Regions under Nonequilibrium Conditions We use a time-distance diagram8 in which the position of the fronts is shown as a function of dimensionless distance xD and dimensionless time tD. The xDtD diagram is the flow domain in the following. The conventional dimensionless variables are

xD )

x qt and tD ) L AφL

(1)

* To whom correspondence should be addressed. Phone: (512)471-8233. Fax: (512)471-9605. E-mail: larry•lake@ pe.utexas.edu.

Figure 1. Sketch of the mineral dissolution zonation under LEA conditions.

for constant q in one-dimensional, linear flow. xD is the fractional position between the inlet (xD ) 0) and the outlet (xD ) 1) of a linear medium. tD is the cumulative fluid injected normalized by the pore volume of the medium. See the Nomenclature section for other definitions. Before illustrating the MOC solution to describe the flow regions under nonequilibrium conditions (NLEA), we will briefly discuss the equilibrium case (LEA). [LEA, the local equilibrium assumption, means that the solid and flowing phases are in equilibrium with each other at a given xD and all tD. NLEA, the nonlocal equilibrium assumption, means the solid and flowing phases are only in equilibrium initially.] Figure 1 sketches a simple dissolution problem for LEA flow that follows the reaction AX f A+ + X-. AX is a generic solid initially present at uniform concentration in equilibrium with cation A+ and anion X-. A solid initially present is a primary solid. The injected solution is undersaturated with respect to AX, which dissolves as flow progress.

10.1021/ie990881m CCC: $19.00 © 2000 American Chemical Society Published on Web 06/29/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 8, 2000

Five different regions now occur along the flow domain. Region I is bounded by the tD ) 0 axis and the salinity wave as before. In this region, the solid and solution concentrations remain at their initial values. In region II AX remains undissolved within the medium at all times but not necessarily at its initial concentration. This region, the sparingly soluble region, is bounded by the salinity wave and the vertical line tD ) tDc, where tDc is the time at which AX is completely dissolved at the inlet (xD ) 0). tDc is also the time at which the flowing concentrations take on their injected values at the inlet. These characteristics follow from the solidconcentration integration in region II1,2 Figure 2. Sketch of the mineral dissolution and precipitation zonation under LEA conditions.

C h AX C h IAX

(

) 1 - NDA 1 -

)

CA+CXKeff sp

(tD - xD)

tDc is given as follows:

tDc )

(

1-

1 CXJ -

CAJ +

Keff sp

)

(2)

NDa

where NDa is the Damkholer number and Keff sp is the effective solubility product. They are defined as

NDa )

Figure 3. Sketch of the mineral dissolution zonation in the flow domain under NLEA conditions. Compare to Figure 1.

In this case, the mineral zonation is defined based on physical conditions as described in refs 3, 9, and 10. See ref 1 for derivation of the solutions we will be discussing. In LEA flow three different regions develop within the flow domain, each of which represents a region of constant concentration separated from the adjacent regions by waves or changes in concentration. In the absence of dispersion, which we assume to be negligible here, these waves are sharp or pistonlike (shocks). The first wave, known as the salinity wave,4 separates the initial region I from a region II carrying the reaction products from upstream reactions. The dissolution wave (the AX front) represents the boundary between regions II and J. In regions I and J, all concentrations are at their initial and injected values, respectively. The dissolution is occurring at the AX front. When secondary solid precipitation is possible, A+ + Y- f AY, the flow regions are as shown in Figure 2. The term secondary solid identifies a possible precipitate, which forms as a consequence of another solid dissolving within the flow domain. Under these conditions, there will be a wave in addition to the two shown in Figure 1. This additional wave corresponds to the precipitation wave for AY or the AY front, which separates region III from region J. Here the dissolution wave for AX represents the boundary between regions II and III. Solid AY disappears downstream of the AX front, because solid concentrations do not change across the salinity wave. In other words, only the solids initially present exist across the salinity wave. Let us now illustrate the NLEA solution for the cases in Figures 1 and 2. Figure 3 sketches the flow regions for a dissolutiononly problem under nonequilibrium conditions (NLEA), which again takes place by the following reaction: AX f A+ + X-.

krdφL C h IAXu

(3)

and

Keff sp )

Ksp n

(4)

γi ∏ i)1 where n is the number of components comprising solid k. n ) 2 for our cases. See the Nomenclature section for definitions of additional terms. Local equilibrium applies when NDa becomes large h IAX or u is small). (i.e., krd or L approaches infinity or C The critical time will obviously be zero under local equilibrium conditions, and the AX front will start at xD ) tD ) 0 (Figure 1). We will adjust the critical time below as a means to deviate from equilibrium. For tD > tDc, the solid has been dissolved at the inlet of the medium and a dissolution front appears in the flow domain. Here we use the term front to describe the position at which the AX is first encountered in the downstream direction for tD > tDc. Note that there is a distinction between a front and a wave in NLEA flow. A wave is simply a propagating change in concentration; a front is a line that has different solids on either side of it. Waves and fronts are synonymous under LEA conditions. Two different regions can be differentiated for tD > tDc: region III, in which the dissolution front is moving at a variable specific velocity, and region IV, in which an asymptotic behavior is closely approached and the dissolution front moves at constant specific velocity. The specific velocity is defined as the ratio between the dissolution front velocity and the Darcy velocity of the injected fluid. The boundary between regions III and IV is the time tD ) tDcv at which the specific dissolution front velocity becomes constant at

Ind. Eng. Chem. Res., Vol. 39, No. 8, 2000 2719

ve )

1

[

1+

C h JAX - C h IAX CAJ + - CAeq+

1 ∆C h AX 1+ ∆CA

][ )

]

(5)

This constant specific velocity is the same as the wave velocity under LEA flow1 for tD > tDcv. The last region J to appear in the flow domain is bounded by the dissolution front and the inlet of the medium (xD ) 0). It also appears at tD > tDc and is defined as the region where the solution concentrations are at their injected values and the AX concentration is zero. The AX concentration in region IV approaches the LEA value at tD f ∞. The solution for the sparingly soluble region has been discussed in a previous publication;2 here we will only discuss briefly the solution for regions III and IV. The flow conditions that prevail in region III (i.e., the region with the variable specific velocity upstream) do not allow for any simplification of the system of partial differential equations (pde’s). To solve for the concentration profiles in this region, we must solve the complete system of pde’s. This type of solution results in simulations that are computing time and memory consuming. Under most real conditions, the specific dissolution front velocity, ve, is small or the time affected by region III is very short (i.e., tDcv - tDc is small). As a consequence, region III can be omitted from the calculations without making much difference in the final results. Unlike region III, in region IV the dissolution front is moving at the equilibrium specific velocity. In this case we solve the conservation equations using a new position coordinate zD ) xD - ve(tD - tDc), where ve is the constant specific velocity defined in eq 5. Details of the derivations are discussed in ref 1. The final expressions for solution and solid concentration are

CA + ) -CAeq+

[

CAJ + + CXeq-

CAJ + - CAeq+

1-

[

(

)]

(CAeq+ + CXeq-)C h IAX exp zD (1 - ve)KAX sp

(

-

(CAeq+ + CXeq-)C h IAX exp zD (1 - ve)KAX sp

CAJ + + CXeq-

CAJ + - CAeq+

CXeq-

)]

CX- ) CXJ - - CAJ + + CA+ h IAX C h AX ) C

(6)

(7)

(1 - ve)(CAeq+ + CXeq-) 1 ve 1-f

(8)

[

CAJ + + CXeq-

CAJ + - CAeq+

exp

(

(CAeq+ - CXeq-)C h IAX zD (1 - ve)KAX sp

)]

h JAX (eq 8). These values for while C h AX approaches zero, C zD correspond to the boundary conditions for this problem. Where precipitation of secondary solid AY is possible, A+ + Y- f AY, the flow regions are as shown in Figure 4. Under these conditions, there will now be a precipitation front superimposed on all of the regions illustrated in Figure 3. The location of this additional front is determined by the critical saturation index (SI*) of AY. The saturation index of a solid k (AY in this case) is the ratio between the ionic activity product of the component(s) comprising the solid in a solution (IAP) and what is allowed by equilibrium (Ksp). SI* corresponds to a critical point equal to or greater than the saturation (SI ) 1) point above which a secondary solid starts nucleating (forming new crystals).11,12 As a consequence, secondary solids can precipitate at any xD if SI > SI*. When they precipitate, their precipitation rate is proportional to the difference between the IAP and Ksp at that xD as follows:

{

precipitation rate ) h AY ) 0 krp(IAP - Ksp) for SI > SI* and C or 1 < SI and C h AY > 0 for 1 < SI < SI* and C h AY ) 0 0 (metastable)

}

(10)

where krp is the precipitation rate constant. Unlike the LEA problem (Figures 1 and 2), the regions shown in Figures 3 and 4 are not regions of constant concentration; in these NLEA cases the waves formed are spreading waves.4 Sensitivity Runs

where f is defined as

f)

Figure 4. Sketch of the mineral dissolution and precipitation zonation in the flow domain under NLEA conditions. Compare to Figure 2.

(9)

in the absence of aqueous speciation. See the Nomenclature section for the definition of other terms. In the limits zD f ∞ and f f ∞, then CA+ and CXapproach their equilibrium concentrations, CAeq+ and h AX CXeq- (CAeq+f . CXeq- and f . 1, eqs 6 and 7), while C approaches its initial value, C h IAX (eq 8). At zD ) 0 and f ) (CAJ + + CXeq-)/(CAJ + - CAeq+), CA+ and CX- approach their injected concentrations, CAJ + and CXJ - (eqs 6 and 7),

We use the same generic case previously described to illustrate results. The injected solution is rich in cation A+ and a second anion Y- that is undersaturated with respect to AX. Because of the dissolution of AX, the solution can become supersaturated with respect to AY and precipitate it as a secondary solid. Fluid and rock compositions as well as equilibrium constants used in these simulations are in Table 1. As stated before, reactive flow simulation is often simplified by assuming local equilibrium, spontaneous solid precipitation (no supersaturation), ideal solutions, and no aqueous reactions. Here we consider different runs that vary the critical time, the amount of aqueous speciation, the ionic strength, and critical supersaturation to evaluate their effects on fluid-rock interactions.

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Table 1. Basic Input Data for All Simulations solubility concn, mol/LPV rate constant, mol/s‚LPV product initial injected at 25 °C dissolution precipitation AX(s) AY(s) Aa Xa Ya

10.0 0.0 1.0 1.0 0.0

0.0 0.0 0.6 0.0 3.0

1.0 2.5

0.01 0.025

0.01 0.01

a Total aqueous concentration ) free ion concentration + aqueous species concentration.

Figure 7. Effect of aqueous reactions on the AX concentration profile at tD ) 2. Table 2. Equilibrium Constants for Aqueous Reactions at 25 °C

Figure 5. AX concentration profile for different critical times at tD ) 2. Small tDc means flow more nearly in LEA.

Figure 6. AY concentration profile for different critical times at tD ) 2. Small tDc means flow more nearly in LEA.

Only the condition being studied is varied in each simulation. Critical Time. We evaluated four different runs in which NDa was varied. According to eq 2, increasing NDa causes tDc to decrease. NDa f ∞ means tDc f 0, and the flows approach LEA conditions. Figures 5 and 6 show the results for AX and AY solid concentrations as a function of dimensionless position at tD ) 2, when tDc is varied from 0.56 to 2.5. All runs in Figures 6 and 7 assumed ideal solutions, no aqueous reactions, and no supersaturation (SI* ) 1). Results show that the AX concentration change becomes more spread as tDc increases. In other words, the more out of equilibrium the flow is, the larger the region between the injected and the initial AX concentrations will be. Another observation from Figure 5 is that all AX concentration profiles cross at a common xD. This point is defined by the location of the dissolution wave under equilibrium conditions xD ) vetD, which means that all profiles spread around the LEA wave position. The AX equilibrium velocity ve remains the same in all simulations, because we are only varying NDa, which has no effect on ve (eq 9). The independence of ve from NDa is one of the bases of the MOC solution under NLEA conditions.1

product

equilibrium constant

AX(aq) AY(aq) HX-

0.5 0.05 1000

product

equilibrium constant

H2X H2O

0.1 1 × 10-14

When AY can precipitate, its maximum concentration will decrease for larger tDc. This is mainly a consequence of the decrease in the distance over which the solid AX dissolves for large tDc (eqs 5-7), with the front position being closer to the inlet, at a given tD, for large tDc. The concentration of cation A+ first promotes the precipitation of AY, and then the spreading in the AX dissolution wave causes the AY precipitation also to spread around the same point, xD ) vetD. NLEA flow will decrease the maximum (peak) concentration for AY, will spread the AY out more, and will move it further from the inlet compared to LEA flow. When tDc is large, the dissolution of solid AX is very small; as a consequence, the solution will be unable to reach the level (SI g 1) needed to precipitate AY. Under this condition, the LEA problem will have solids AX and AY (Figure 2), but the NLEA one will have only AX (Figure 3). A very large tDc can be brought about by a large rate, small dissolution rate constant, and/or large initial solid concentration. In general, the majority of real conditions lead to NLEA fluid-rock interactions (tDc > 0), which can make LEA a poor assumption, especially near point sources such as wells. Aqueous Reactions. Another approximation in modeling reactive flow is to neglect most of flowing-phase reactions. This is mainly because of the complexity that these reactions add to the simulation.13 Here we evaluate the effect of equilibrium chemical reactions in the flowing phase on dissolution and precipitation reactions. We now include in the simulations the species AX(aq), AY(aq), HX-, and H2X as equilibrium flowing-phase products. Table 2 shows the equilibrium constants used for these reactions. All runs assumed ideal solutions, no supersaturation, and tDc ) 0.714. In this case we made two runs, one of which omitted the aqueous reactions. Figures 7 and 8 show the concentration profiles for solids AX and AY, respectively. Results show that including the speciation increases ve, which qualitatively has the same effect as was already observed when we increased tDc. In other words, including the aqueous reactions causes more spreading for the AX dissolution wave, and the region over which AY precipitates will be larger. Furthermore, the AY peak concentration will be smaller, because there will be fewer free ions (A+ and X-) in the injected solution.


Figure 8. Effect of aqueous reactions on the AY concentration profile at tD ) 2.

Figure 10. AY concentration profile for different ionic strengths at tD ) 2.

Figure 9. AX concentration profile for different ionic strengths at tD ) 2.

Figure 11. AX dissolution front velocity for different ionic strengths at tD ) 2.

When the aqueous reactions account for a large amount of the total component (A, X, and Y) concentrations, as would happen if the flowing-phase reaction had large equilibrium constants, ve will be large and the injected solution will have a very small concentration of free components (A+, X-, and Y-). Under this condition, AY might not precipitate at all (Figure 3). This will completely change the nature of the problem, because AX and AY were present in the runs that neglected the aqueous speciation (Figure 4). In general, including the aqueous reactions or speciation will decrease the free-ion concentrations, which will decrease the tendency to form secondary precipitates. Nonideal Solutions. Another common assumption in modeling fluid-rock interactions is to assume dilute or ideal solutions4,9,14,15 to avoid the complexity of accounting for activity coefficient corrections.13 In this case, we allow the equilibrium constant to be corrected based on the ionic strength (I) of the flowing phase, according to the Davies equation.16 We show results of four runs between I ) 0 (ideal solution) and I ) 0.47 M (the Davies equation is accurate only up to I ) 0.5 M). We adjusted I by adding a nonreacting charged component to the injected and initial solutions. All runs assumed no aqueous reactions and spontaneous precipitation (SI* ) 1). An important feature of the ionic strength correction is that the activity coefficients γi of charged components are less than 1 when 0 < I < 2.17,18 This range includes sea water and groundwater. Then, eq 4 shows that, as I increases, so does the effective solubility product of solids AX and AY. As a consequence, the initial solution concentrations for A+ and X- are greater and the initial solid concentration is less than that for I ) 0, because the change in ionic strength is caused by the addition of a nonreactive component. Under these conditions, the

Figure 12. Dimensionless critical time for different ionic strengths at tD ) 2.

results are also affected by the change in the initial solid concentration. However, as we will show later, this effect is much smaller than the one caused by the change in ionic strength. Figures 9 and 10 show AX and AY solid concentration profiles for tD ) 2. The location of the AX front advances as I increases. For I > 0.02, no AY precipitates. Increasing I causes both AX and AY to become more soluble. How this happens is discussed next. As Keff ap increases and the initial solid concentration decreases (Figure 9), the ve of the AX front will increase (eq 5 and Figure 11) and tDc will decrease (eq 2 and Figure 12) with respect to the base case. Both figures show steep changes in ve and tDc up to I ) 0.1. For I > 0.1, ve and tDc become constant, mainly because the predicted γi become nearly constant. Thus, as well as making the solids more soluble, increasing I effectively makes the runs behave more like LEA flow. Between of these two effects, the increase in ve appears to be larger than the decrease in tDc. In this case the change in tDc apparently mitigates the spreading of the dissolution front caused by the change in

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Figure 13. AX dissolution front velocity for different ionic strengths and constant initial solid concentration at tD ) 2.

Figure 15. AX concentration profile for different AY critical supersaturation indices (SI*) and a large precipitation rate at tD ) 2.

Figure 14. Schematic of a saturation state plot (modified by Stumm and Morgan16 from Zhang and Nancollas20).

velocity. Increasing I will have, qualitatively, the same effect as was discussed for aqueous reactions; after all, one way of increasing I is allowing more speciation within the flowing phase. Results for solid AY show that a small increase in I s in fact, for I g 0.02 (Figure 10)s will completely prevent secondary precipitation. This absence defines a problem different from that solved for dilute solutions (I ) 0). The same results are observed in the case of changing I by increasing the initial solution concentrations, which allows the initial solid concentration to be constant. In fact, the change in the initial concentration accounts for only 20% of the increase in the specific front velocity as shown in Figures 11 and 13. Therefore, neglecting the electrolyte effects is a poor assumption under this condition as well. Similar effects have been reported in previous work19 on thermal effects in which the coupling between the reaction and temperature can be considered also as a deviation from ideality analogous to the ionic strength correction. Critical Supersaturation. We use the term critical supersaturation index (SI*) to define the value of SI above which the secondary solid AY starts precipitating. See eq 10. In this case, we ran three different runs with SI* between 1 and 1.2. All runs assumed no aqueous reactions and ideal solutions. Figure 14 shows a schematic of the saturation states for a secondary mineral AY. The plot shows CA+ ) KspSI*/CY- in the cases of SI* ) 1 and SI* > 1. For SI* > 1, the region between the saturation line (SI ) 1) and the critical value (SI*) is metastable and no precipitation occurs as long as no previous precipitation has occurred at the same location. In other words, once AY has appeared at a specific time, it will precipitate at later times regardless of the value of SI* as long as SI > 1. Figures 15 and 16 show the calculated AX and AY profiles, respectively, at tD ) 2. Those results show that SI* > 1 causes the profiles to oscillate in space. The

Figure 16. AY concentration profile for different AY critical supersaturation indices (SI*) and a large precipitation rate at tD ) 2.

Figure 17. AX concentration profile for different AY critical supersaturation indices (SI*) and a small precipitation rate at tD ) 2.

precipitation of AY occurs in bands whose widths depend on the magnitude of the precipitation rate. We ran several additional simulations to determine whether these oscillations were physical or numerical. Results1 show that the main cause of the oscillations was the discontinuity in the reaction rate for precipitation at SI ) SI*, when the dissolution of the solid initially present is not enough to compensate for the decrease in SI for AY after precipitation at SI*. Similar results have been previously reported when dispersion was included in their calculations.6,21 For cases in which the dissolution rate for AX is large and/or the precipitation rate is small, the AY profile will be continuous and smooth, as shown in Figures 17 and 18. In this additional case, we decreased the initial solid concentration, which increased ve, as we decreased the AY precipitation rate (Table 3). Under these new conditions the precipitation wave contains a discontinuity, at the location at which SI* is


Acknowledgment L.W.L. holds the W. A. (Monty) Moncrief Centennial Chair at The University of Texas at Austin. The authors thank Dr. K. Sepehrnoori for his valuable and helpful discussions on numerical dispersion. Finally, A.A.-M. thanks Petroleos de Venezuela (PDVSA) for its partial support of this work. Nomenclature

Figure 18. AY concentration profile for different AY critical supersaturation indices (SI*) and a small precipitation rate at tD ) 2. Table 3. Solid Input Data for the Case on Supersaturation Sensitivity concn, mol/LPV initial injected AX(s) AY(s)

2.0 0.0

0.0 0.0

solubility product 1.0 2.5

rate constant, mol/s‚LPV dissolution precipitation 0.01 0.009

0.01 0.0036

first attained. Downstream of this point, all runs show a similar profile. This means that increasing SI* will move the precipitation front position farther downstream, but it will not change the concentration profile downstream of this point compared to the base case (SI* ) 1). For the dissolution profile for AX, there are no drastic changes in concentration in any of the cases. This is mainly because solids initially present cannot supersaturate if they do not have components in common among themselves or if there is only one present, which is the condition here. Including the SI* will cause a banded type of secondary precipitation when the secondary precipitation rate is large and/or the primary dissolution rate is small. In other cases, it will only move the precipitation front farther downstream in the flow domain. Conclusions Based on the results of this work, the following apply: 1. All cases studied predict less secondary precipitation than that calculated under equilibrium and dilute solution conditions. However, because the dissolution wave is more spreading, compared to LEA and dilute solution conditions, the region over which secondary precipitation will occur will be larger. 2. Including aqueous speciation and ionic strength correction increases the dissolution LEA velocity, which means the primary solid will dissolve faster. 3. Increasing the critical supersaturation of the secondary precipitate moves the position of the precipitation front farther downstream, but it does not affect the dissolution profile. Under conditions of large secondary precipitation rate and/or small primary dissolution, the precipitation occurs in bands whose width depends on the precipitation/dissolution rates. These results point toward an overestimation of secondary precipitation when assuming local equilibrium, no supersaturation, no aqueous reactions, and dilute solutions. However, under most conditions, the secondary precipitates do not completely disappear; in other words, some flow impairment could still appear. In general, despite the additional complexity, the effects of the assumptions investigated in this work must be included in simulating rock-fluid interactions.

CAJ + ) injected concentration of component A in the aqueous phase, mol/LPV CA+ ) concentration of component A in the aqueous phase, mol/LPV CXJ - ) injected concentration of component X in the aqueous phase, mol/LPV CX- ) concentration of component A in the aqueous phase, mol/LPV C h IAX ) initial concentration of solid AX, mol/LPV C h AX ) concentration of solid AX, mol/LPV krd ) dissolution rate constant, mol/s‚LPV Ksp ) solubility product Keff sp ) effective solubility product L ) length of the medium, m LEA ) local equilibrium assumption LPV ) liters of pore volume M ) molarity, mol/LPV NDa ) dimensionless Damkholer number NLEA ) nonlocal equilibrium assumption n ) total number of components in mineral k q ) volumetric injection rate, m3/s SI* ) dimensionless critical supersaturation SI ) dimensionless saturation index tcD ) dimensionless critical time tcv D ) dimensionless time to attain constant velocity u ) Darcy velocity, m/s ve ) specific equilibrium velocity γi ) activity coefficient of component i, fraction φ ) porosity, fraction

Literature Cited (1) Araque-Martinez, A. Modeling the Effects of Geochemistry on Well Impairment. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 2000, in progress. (2) Araque-Martinez, A.; Lake, L. W. A Simplified Approach to Geochemical Modeling and Its Effect on Well Impairment. Presented at the 1999 SPE Annual Technical Conference and Exhibition, Houston, TX, Oct 1999; Paper SPE 56678. (3) Bryant, S. L. Wave Behavior in Reactive Flow Through Permeable Media. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 1986. (4) Dria, M. A Chemical and Thermochemical Wave Behavior in Multiphase Fluid Flow Through Permeable Media: Wave-Wave Interactions. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 1988. (5) Novak, C. Metasomatic Structure Formation in Permeable Media Produced by Infiltration or Diffusion. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 1989. (6) Sevougian, S. D. Partial Local Equilibrium and the Propagation of Mineral Alteration Zones. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 1992. (7) Stohs, M. A Study of Metal Ion Migration in Soils From Drilling Mud Pit Discharges. M.S. Thesis, The University of Texas at Austin, Austin, TX, 1986. (8) Walsh, M. P. Geochemical Flow Modeling. Ph.D. Dissertation, The University of Texas at Austin, Austin, TX, 1983. (9) Novak, C.; Schechter, R. S.; Lake, L. W. Rule-Based Mineral Sequences in Geochemical Flow Processes. AIChE J. 1988, 34 (10), 1607-1614.

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(10) Walsh, M. P.; Bryant, S. L.; Schechter, R. S.; Lake, L. W. Precipitation and Dissolution of Solids Attending Flow Through Porous Media. AIChE J. 1984, 30 (2), 317-328. (11) Nielsen, A. E. Electrolyte Crystal Growth Mechanisms. J. Cryst. Growth 1984, 67, 289-310. (12) Steefel, C.; Van Cappelen, P. A New Kinetic Approach to Modeling Water-Rock Interaction: The Role of Nucleation, Precursors, and Ostwald Ripening. Geochim. Cosmochim. Acta 1990, 54, 2657-2677. (13) Liu, Y.; Watanasiri, S. Successfully Simulated Electrolyte Systems. Chem. Eng. Prog. 1999, 25-42. (14) Novak, C.; Schechter, R. S.; Lake, L. W. Diffusion and Solid Dissolution/Precipitation in Permeable Media. AIChE J. 1989, 35 (7), 1057-1072. (15) Sevougian, S. D.; Schechter, R. S.; Lake, L. W. Effect of Partial Local Equilibrium on the Propagation of Precipitation/ Dissolution Waves. Ind. Eng. Chem. Res. 1993, 32, 2281-2304. (16) Stumm, W.; Morgan, J. J. Aquatic Chemistry, 2nd ed.; John Wiley and Sons: New York, 1981.

(17) Harned, O. The Physical Chemistry of Electrolytic Solutions, 3rd ed.; Reinhold Pub. Corp.: New York, 1958. (18) Latimer, W. The Oxidation States of the Elements and Their Potentials in Aqueous Solutions, 2nd ed.; Prentice-Hall: New York, 1952. (19) Dria, M. A.; Lake, L. W. Behavior of Thermochemical Waves during Reactive Flow Through Permeable Media. Ind. Eng. Chem. 1995, 34 (8), 2889-2897. (20) Zhang, J. W.; Nancollas, G. H. Mechanism of Growth and Dissolution of Sparingly Soluble Salts. Reviews in Mineralogy; Hochella, M. F., White, A. F., Eds.; Mineralogical Society of America: Washington, DC, 1990; Vol. 23, pp 365-396. (21) Sultan, R.; Ortoleva, P.; DePasquale, F.; Tartaglia, P. Bifurcation of the Ostwald-Liesagang Supersaturation-Nucleation-Depletion Cycle. Earth-Sci. Rev. 1990, 29, 163-174.

Received for review December 8, 1999 Revised manuscript received May 3, 2000 Accepted May 4, 2000 IE990881M

Some Frequently Overlooked Aspects of Reactive Flow through

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