« back to the Online Library

Modeling Scale Inhibitor Dosages for Oilfield Operations

Robert J. Ferguson
French Creek Software, Inc.
2208 Kimberton Rd, Box 684 Kimberton
Pennsylvania 19442 U.S.A.

Donald J. Weintritt WeintrittDonald J. Weintritt
Weintritt Consulting Services
305 Andrew Guidry Road
Lafayette, Louisiana 70503 U.S.A.

Mathematical models for predicting the minimum effective dosage for scale inhibitors have been used successfully to optimize treatment levels in field applications. The models were developed in the format of minimum effective dosage as a function of water chemistry, temperature and time. This paper describes the use of laboratory and field data to develop the models.

Keywords: scale, scale prediction, inhibitors, treatment, modeling


Feeding the minimum effective inhibitor dosage can reduce operating costs for chemical treatment, minimize treatment chemical discharge to the environment, and in some cases, prevent under-feed of a scale inhibitor. Common sense indicates that the same scale inhibitor dosage is not required for all waters and conditions. One size does not fit all.

Cooling water treatment chemists have capitalized on this concept since the development of the first computerized water chemistry evaluation and treatment recommendation systems in the 70’s.(1,2) The transfer of this technology to the oil field has not been a rapid process, due to the increased number of parameters and more stressed conditions encountered in oil field operations. Cooling systems operate at atmospheric pressure, temperatures close to ambient, and comparatively low ionic strength. The high ionic strength, high temperatures and high pressures encountered in oil field applications increases the difficulty of calculating reproducible indices for scale potential for use in developing general models.(3) The problem is compounded by the difficulty of measuring low level inhibitor residuals in oil field operations.(4)

The development of ion association models for evaluating the free ion potential of high ionic strength waters has allowed the application of this technology to oil field scale control problems by providing a transportable index for use as a driving force for scale formation by the models.(5) The ion association models have been further refined and the range of applicability for the models expanded by incorporating solubility products derived from improved measurements of the solubility of common scale forming species versus temperature and pressure.(6,7) Refinement of the pressure impact upon the dissociation of common ion pairs, has further assisted in developing driving forces which can be used to model scale inhibitor requirements with practical accuracy.

This paper discusses the parameters critical to developing an effective dosage modulation model for scale inhibitors from laboratory data, field data, or a combination of both. The paper draws upon the concept of induction time as a basis for the mathematical models used to develop predictive models from actual data. The models are based upon the concept that threshold effect inhibitors do not prevent scale formation, they only delay the inevitable. The models are in agreement with current theories and treat scale inhibitors as agents which extend the induction time before stable clusters form which can lead to crystal formation and/or growth on existing active sites occurs in the case of calcium carbonate.

The models predict the dosage required to inhibit deposition until the treated water has passed through a perturbed state. A water might be stable, for example, until it undergoes a change in temperature, pressure, or both. The models discussed in this paper have been used to accurately predict a delay varying from 3 to 15 seconds in rapid turn-over systems, to days.

Thermodynamic driving forces and system operating conditions are used by the models to describe the kinetics of scale formation, growth, and the impact of inhibitors upon induction time.

Similar models to those discussed have been used successfully to optimize scale inhibitor treatments in once through utility cooling systems and open recirculating cooling systems since the late 70’s. Recently, a comparable approach has been taken to model the impact of inhibitors upon scale formation in oil field applications using a simplified driving force for calcium carbonate scale.(4)


The original models described in this paper were developed from a combination of field observations, common sense, and laboratory data. Model development began in the early 1970’s. Evaluation of data from field and laboratory dosage optimization studies revealed that several parameters were critical to dosage: time, temperature, and the degree of supersaturation.

The method outlined in this paper has been used to develop models of minimum effective inhibitor dosages from laboratory data, field data, and combinations of both. The inhibition of scales ranging from calcium carbonate to calcium phosphate have been modeled using the method outlined. The models provide a natural path for bringing research data into the practical arena of the operating engineer or water chemist.

Induction Time: The Key To The Models

Reactions do not occur instantaneously. A time delay occurs once all of the reactants have been added together. They must come together in the reaction media to allow the reaction to happen. The time required before a reaction begins is termed the induction time.

Thermodynamic evaluations of a water’s scale potential predict what will happen if it is allowed to sit undisturbed under the same conditions for an infinite period of time. Even simplified indices of scale potential such as the Langelier saturation index can be interpreted in terms of the kinetics of scale formation. For example, calcium carbonate scale formation would not be expected in an operating system when the Langelier saturation index for the system is 0.1 to 0.2 . The driving force for scale formation is too low forscale formation to occur in finite, practical residence times. Scale would be expected under the same conditions if the same system had a scale potential driving force as indicated by a Langelier saturation index of 2.8 .

Induction time has been modeled for economically important crystals such as sucrose. Models follow a formula similar to Equation (1):

Induction Time = 1 / k [Saturation Level - 1]P-1 (1)


  • Induction Time is the time before crystal formation and growth occurs;
  • k is a temperature dependent constant;
  • Saturation Level is the degree of super-saturation;
  • P is the critical number of molecules in a cluster prior to phase change.

Gill and his associates demonstrated that commercially available scale inhibitors extend the induction time for calcium carbonate scale(8). Their paper points out several critical parameters which impact the induction time prior to crystal growth:

  • The degree of supersaturation.
  • The temperature.
  • The presence of active sites upon which growth can occur.
  • The inhibitor level.

Gill’s study used ion association model saturation level as the thermodynamic driving force for scale growth. Saturation level calculations performed using a computerized ion pairing (ion association) method eliminate most of the assumptions inherent in simplified indices (5). They account for common ion effects which can increase the apparent solubility of a scale forming specie such as calcium carbonate. Driving forces for scale formation calculated using the ion pairing method are transportable between systems because they base their calculations upon free ion concentrations rather than the total analytical values. This is the heart of the ion pairing, or ion association method, which subtracts ion pairs (e.g. CaSO4, MgSO4, CaHCO3-) from the total analytical value to estimate the free ion present and available to react in forming seed crystals, or in driving growth on existing substrates. Table 1 summarizes ion pairs which can be of importance in estimating the free ion concentrations for reactants.

The remainder of this paper uses ion association model saturation levels for the driving force for scale formation. Table 2 provides a working definition of the term saturation level for calcium carbonate, calcium sulfate, barium sulfate and other scale forming species encountered in oil field applications.

Critical Parameters 

The parameters contributing to Equation (1) are included in the basic relationships used for inhibitor dosage modeling. Major data values required include the time period during which scale formation must be prevented, the degree of supersaturation which is the driving force which must be overcome, the temperature at which the inhibitor must function, and the pH of the cooling water. The surface area of active sites also impacts the dosage requirement.

These parameters have the following impacts upon dosage:

Time. The time selected is the residence time the inhibited water will be in a perturbed state. The inhibitor must prevent scale formation or growth until the water has passed through the system and been discharged. Figure 1 profiles the impact of induction time upon dosage with all other parameters held constant.

Degree of Supersaturation. An ion association model saturation level is the driving force for the model outlined in this paper, although other, similar driving forces have been used. Calculation of driving force requires a complete water analysis, and the temperature at which the driving force should be calculated. Figure 2 profiles the impact of saturation level upon dosage, all other parameters being constant.

Temperature. Temperature affects the rate constant for the induction time relationship. As in any kinetic formula, the temperature has a great impact upon the collision frequency of the reactants. This temperature effect is independent of the effect of temperature upon saturation level calculations. Figure 3 profiles the impact of temperature upon dosage with other critical parameters held constant.

pH. pH affects the saturation level calculations, but it also may affect the dissociation state and stereochemistry of the inhibitors(9). Inhibitor effectiveness can be a function of pH due to its impact upon the charge and shape of an inhibitor molecule. This effect may not always be significant in the pH range of interest (e.g. 6.0 to 9.5).

Active sites. It is easier to keep a clean system clean than it is to keep a dirty system from getting dirtier. This rule of thumb may well be related to the number of active sites for growth in a system. When active sites are available, scale forming species can skip the crystal formation stage and proceed directly to crystal growth.

Other factors can impact dosage such as suspended solids in the water. Suspended solids can act as sources of active sites, and can reduce the effective inhibitor concentration in a water by adsorption of the inhibitor. These other factors are not taken into account in the models in this paper. Table 2 summarizes the factors critical to dosage modeling, and their impact upon dosage.

Data Base

The dosage models used as examples in this paper were developed from data collected in field studies,(2,10) laboratory studies, published data, or a combination of these sources.

 Examples in this paper include data from side stream evaluation of the minimum effective dosages.(10,11) In these studies, two parallel fouling probes were used to develop estimates of the minimum effective dosages for the phosphonates amino-tris-methylene phosphonic acid (AMP), 1,1-hydroxy ethylidene diphosphonic acid (HEDP), and polyacrylic acid (PAA). One probe was over-treated at a level where no calcium carbonate deposition would be anticipated. The parallel probe was not treated, and the time required for a measurable deposit to form determined. This was deemed the minimum period between dosage adjustments for the test. (Note: A minimum test duration of twice the time required for fouling was allowed to pass between dosage adjustments). Dosages were decreased until failure, as indicated by a measurable deposit formation.

A dosage model is only as good as the data from which it is derived. The most generally applicable models include data points over the anticipated ranges for critical parameters. For example, a model developed using data in the temperature range of 30 to 40 ºC might be totally useless in predicting a dosage for a system operating at 230 ºC.

Models should be derived from data over the range of water chemistry anticipated as well as over the range of saturation level anticipated. If a calcium carbonate scale inhibitor model will be used in waters ranging from a calcium level of 40 ppm to over 1000 ppm, this range should be covered from laboratory and/or field sources. The saturation level range anticipated should also be bracketed (e.g. 1.0 to 250 saturation level for calcite).

Although field data is the source of choice, field conditions can rarely be adjusted to cover the temperature, pH, time, and water chemistry ranges desired. The use of static laboratory tests designed to elucidate the variation of dosage with any of the parameters can be used to supplement field data. Field data, although desirable, is not always necessary for the development of a preliminary correlation. Each model developed should be compared to field results to assure that a correlation exists between the test data, the model, and actual field results.


A modified version of Equation (1) provided the basis for model correlation. Dosage was added as a factor to the equation on the right side to produce Equation (2).

Induction Time= DosageM / k' [Saturation level -1]P-1

The temperature dependent rate constant k’ was found to correlate with the Arrhenius relationship shown by Equation (3).

k' = A e -Ea/RT (3)

Saturation levels were calculated from water analysis input using a computerized ion association model. The time used for the correlation was the time to failure in laboratory tests, the residence time in a heated state for utility once through cooling systems, and the holding time index in open recirculating cooling systems.

Equation (2) was rearranged to solve for dosage in the first order. Regression analysis was used to estimate the coefficients.

Typical Models

Models were developed using this method for the inhibition of calcium carbonate, calcium sulfate, and barium sulfate by commercially available inhibitors. Figure 4, Figure 5 and Figure 6 profile the minimum effective dosage for these calcium carbonate inhibitors predicted by the models for a short residence time (180 seconds). Figure 7 profiles the driving force (calcite saturation level) used to calculate the dosages. The models used in Figures 4 through 6 were developed from extensive laboratory and field data. Not all models developed using this method must be derived from dozens or hundreds of points. Table 4 summarizes the water chemistry used for the examples.

The same principals can be used to model limited data sets such as those derived from a jar test series for a particular water. Figure 8 and Figure 9 depict the correlation derived from limited testing. The models are used in these cases to summarize a series of tests and to allow a limited extrapolation of dosages from the laboratory data. The calcium sulfate inhibitor model was developed from published data.(12) The models can be useful in a field environment for limited extrapolation of results. Models derived from large data bases of laboratory and field data are recommended when the correlations are destined for use as generalized treatment recommendation tools.


Laboratory and field dosage optimization data can be converted to a mathematical model using standard statistical methods and a relationship derived from theoretical models for induction time. The models provide a practical method for collating laboratory and field data for a scale inhibitor. The correlations developed can then be used to predict the dosage based upon water chemistry and operating parameters without the necessity for laboratory or in-depth field studies to determine the minimum effective dosage. Dosages predicted by models developed in this manner are typically accurate as long as the system parameters and water chemistry data are within the range of the data used to develop the models. The examples presented in this paper are by necessity limited. The basic models described in this paper have been used successfully in systems ranging from short residence time, low scale potential systems, to high residence time, high scale potential systems for calcium carbonate control.

As with any predictive method, dosage recommendations from such models should be evaluated by an experienced water treatment chemist prior to implementation in an operational system. Predicted dosages should be used as a guideline, not as an ultimate treatment recommendation due to factors which may not be taken into account by the models.


1. C.J. Schell, “The Use of Computer Modeling in Calguard to Mathematically Simulate Cooling Water Systems and Retrieve Data,” International Water Conference, paper no.IWC-80-43, (Pittsburgh, PA: Engineers’ Society of Western Pennsylvania, 1980).

2. R.J. Ferguson, O. Codina, W. Rule, R. Baebel, “Real Time Control of Scale Inhibitor Feed Rate,” International Water Conference, paper no.IWC-88-57, (Pittsburgh, PA: Engineers’ Society of Western Pennsylvania, 1988).

3.J.E. Oddo, M.B. Tomson, “Scale Control, Prediction And Treatment Or How Companies Evaluate A Scaling Problem And What They Do Wrong,” CORROSION/92, Paper no. 34, (Houston, TX: NACE, 1992).

4. Richard G. Finley,"Field Evaluation of CaCO3 Scaling And Its Inhibition," Latin American Petroleum Engineering Conference, (Richardson, TX: SPE, 1992).

5. R.J. Ferguson, “Computerized Ion Association Model Profiles Complete Range of Cooling System Parameters,” International Water Conference, paper no.IWC-91-47, (Pittsburgh, PA: Engineers’ Society of Western Pennsylvania, 1991).

6. Gordon Atkinson, K.U.G. Raju, Robert D. Howell, Microslaw Mecik, “A Comprehensive Scale Prediction Program For Oil And Gas Production,” CORROSION/93, Paper no. 276 (Houston, TX: NACE, 1993).

7. Gordon Atkinson, K.U.G. Raju, Robert D. Howell, “Thermodynamics Of ‘Scale’ Mineral Solubilities: The Effect Of Pressure,” CORROSION/92, Paper no. 31 (Houston, TX: NACE, 1992).

8. J.S. Gill, C.D. Anderson, R.G. Varsanik, “Mechanism of Scale Inhibition by Phosphonates,” International Water Conference, paper no.IWC-83-4, (Pittsburgh, PA: Engineers’ Society of Western Pennsylvania, 1983).

9. W.M. Hann, J. Natoli,"Acrylic Acid Polymers And Copolymers As Deposit Control Agents In Alkaline Cooling Water Systems," CORROSION/85, Paper no. 315 (Houston, TX: NACE, 1985).

10. R.J. Ferguson,"A Kinetic Model For Calcium Carbonate Deposition," CORROSION/84, Paper no. 120 (Houston, TX: NACE, 1984).

11. B.W. Ferguson, R.J. Ferguson, “Sidestream Evaluation of Fouling Factors in a Utility Surface Condenser,” Journal of the Cooling Tower Institute,2,(1981):p. 31-39.

12. Z. Amjad, W.F. Masler, III,"The Inhibition of Calcium Sulfate Dihydrate Crystal Growth By Polyacrylates And The Influence of Molecular Weight," CORROSION/85, Paper no. 357 (Houston, TX: NACE, 1985).


[Calcium] = [Ca+II] + [CaSO4] + [CaHCO3+I] + [CaCO3] + [Ca(OH)+I]
  + [CaHPO4] + [CaPO4-I] + [CaH2PO4+I]
[Magnesium] = [Mg+II] + [MgSO4] + [MgHCO3+I] + [MgCO3] + [Mg(OH)+I]
  + [MgHPO4] + [MgPO4-I]+[MgH2PO4+I]+[MgF+I]
 [Barium] = [Ba+II] + [BaSO4] + [BaHCO3+I] + [BaCO3] + [Ba(OH)+I]
 [Strontium] = [Sr+II] + [SrSO4] + [SrHCO3+I] + [SrCO3] + [Sr(OH)+I]
[Sodium] = [Sr+II] + [SrSO4] + [SrHCO3+I] + [SrCO3] + [Sr(OH)+I]
[Potassium] = [K+I]+[KSO4-I] + [KHPO4-I] + [KCl]
[Iron] = [Fe+II] + [Fe+III] + [Fe(OH)+I] + [Fe(OH)+II] + [Fe(OH)3-I]
  + [FeHPO4+I] + [FeHPO4] + [FeCl+II] + [FeCl2+I] + [FeCl3]
  + [FeSO4] + [FeSO4+I] + [FeH2PO4+I] + [Fe(OH)2+I] + [Fe(OH)3]
  + [Fe(OH)4-I] + [Fe(OH)2] + [FeH2PO4+II]
[Aluminum] = [Al+III] + [Al(OH)+II] + [Al(OH)2+I] + [Al(OH)4-I] + [AlF+II] + [AlF2+I]


        • Saturation level is the ratio of the Ion Activity
          Product to the Solubility Product for the scale
          forming specie.

        For calcium carbonate:
        SL = (Ca)(CO3)/Ksp'

        For barium sulfate:
        SL = (Ba)(SO4)/Ksp'

        For calcium sulfate:
        SL = (Ca)(SO4)/Ksp'

        Saturation Levels should be calculated based upon free ion activities using the solubility product for the form typical of the conditions studied (e.g. calcite for low temperature calcium carbonate, aragonite at higher temperatures.)

        Saturation levels can be interpreted as follows:

        • A water will tend to dissolve scale of
          the compound if the saturation level is less than 1.0
        • A water is at equilibrium when the Saturation Level
          is 1.0 . It will not tend to form or dissolve scale.
        • A water will tend to form scale as the Saturation
          Level increases above 1.0 .




Time Dosage increases with residence time
Degree of Supersaturation Dosage increases with saturation level
pH Dosage increases with temperature due to its impact on reaction rate (in addition to any positive or negative effects temperature may have upon saturation level).
Suspended solids Dosage requirements may increase as suspended solids increase due to adsorbtion of the inhibitor on the solids.
Active sites Dosage requirements increase if active sites for scale growth are present.

It is easier to keep a clean system clean than it is to keep a dirty system from getting dirtier.