03197 a Potential Attenuation Equation for Design and Analysis of Pipeline Cathodic Protection Systems With Displaced an (51300-03197-Sg)

Paper No.

03197

2003

CORROSION

A POTENTIAL ATTENUATION EQUATION FOR DESIGN AND ANALYSIS OF PIPELINE CATHODIC PROTECTION SYSTEMS WITH DISPLACED ANODES

Diane K. Lysogorski Center for Marine Materials Florida Atlantic University – Sea Tech Campus 101 North Beach Road Dania Beach, FL 33004 William H. Hartt Center for Marine Materials Florida Atlantic University - Sea Tech Campus 101 North Beach Road Dania Beach, FL 33004 ABSTRACT A recently proposed, first-principles based potential attenuation equation for pipelines and risers with multiple, equally spaced, identical superimposed spherical (bracelet) galvanic anodes that incorporates all relevant resistance terms (anode, coating, polarization, and metallic return path) has been modified for situations where anodes are displaced. The equation is solved numerically using the Coordinate Mapping Based Finite Difference Method, and potential versus distance plots are provided for several examples with accuracy being proven by independent calculations. The solutions are compared with those of the classical equation of Uhlig, and it is concluded that the latter is overly conservative in situations where the pipeline or a portion thereof lies in the potential field of the anode. It is demonstrated further how the equation can be employed for pipelines polarized by impressed current anodes. Key words: Cathodic protection, pipelines, inclusive equation, attenuation, offset anodes. INTRODUCTION Safe, reliable operation of pipelines, both onshore and off, is critical to the positive functionality of a modern society. An important aspect of assuring and maintaining pipeline performance is adequate control of corrosion, internal as affected by product and external in conjunction with exposure to soils and waters. While such resistance can be achieved by selection of a corrosion resistant alloy pipe material of construction, economic considerations usually dictate for situations of relatively large pipe diameter and product transport distances that some grade of structural steel be used, in which case additional corrosion control measures are required. Thus, internal corrosion is normally managed by product pretreatment with inhibitors and external by a combination of

Copyright 2003 by NACE International. Requests for permission to publish this manuscript in any form, in part or in whole must be in writing to NACE International, Publications Division, 1440 South Creek Drive, Houston, Texas 77084-4906. The material presented and the views expressed in this paper are solely those of the author(s) and not necessarily endorsed by the Association. Printed in U.S.A.

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coatings and cathodic protection (cp). Nonetheless, a recently completed study has placed the annual direct cost of pipeline corrosion in the United States alone at approximately seven billion dollars (1). With consideration of indirect costs (for example, loss of product, environmental and property damage, and loss of life) this figure becomes much higher. A dual approach for control of external pipeline corrosion is necessary because coatings invariably exhibit defects; and even if this barrier layer can be certified as defect-free at the time of construction, deterioration with time leads to localized exposure of the substrate steel. While cp could, in theory, function as a stand-alone corrosion control method, it is only practical when employed in conjunction with a coating. In effect, the cp need only provide protection at coating defects. Consequently, the pipe current demand is low compared to the bare metal case; and so anode or anode ground bed spacing can be relatively large. For example, cp systems for marine pipelines are normally designed assuming several percent coating bare area and utilizing galvanic bracelet anodes spaced about 150-250 m apart. Here, limitations on the size of bracelet anodes that can be deployed from a lay barge and cp design life are controlling. For the on-shore buried counterpart, on the other hand, higher coating quality in conjunction with impressed current (ic) cp, which is the type normally employed here, is such that metallic path ground return resistance is controlling and anode ground bed spacings can be as great as 50-100 km. Figure 1 schematically shows the potential profile that results in each of these two cases. Thus, for marine pipelines and risers with galvanic bracelet anodes (Figure 1a) potential is constant except within the field of the anode with the magnitude of polarization being determined by electrolyte resistivity, anode dimensions, and pipe current demand. Local positive potential excursions may occur, however, at coating defects. Buried pipelines with iccp and large anode/anode bed spacings, on the other hand, exhibit continued polarization decay with increasing distance beyond the field of the anode (Figure 1b). These features reflect four cp circuit components as the resistance associated with 1) anode, 2) coating, 3) pipe polarization, and 4) metallic path return. A critical design aspect in either case (offshore or buried onshore) is projection of pipe current demand. For buried pipelines with iccp, anode bed design and spacing are also important. For 1) marine pipeline cp retrofits, 2) marine pipelines deployed by reeling with subsequent anode sled placement, and 3) buried onshore pipelines with iccp systems, anode spacing is designed to be as large as feasible and, consequently, metallic path resistance may be significant, if not controlling. For this circumstance, potential attenuation is commonly projected using the classical equations of Morgan (2) and Uhlig (3). Accordingly, for pipelines polarized by identical, equally spaced anodes,

 2πrp ⋅ Rm 1 / 2   ⋅ (z − L ) or E c (z ) = E b ⋅ cosh   k ⋅ ζ     2πr ⋅ R p m E a = E b ⋅ cosh −    k ⋅ζ 

   

1/ 2

 ⋅ L ,  

(1

where Ea, Eb, and Ec(z) are the magnitudes of polarization at the drainage point, the mid-anode spacing, and distance z from the drainage point, rp is the pipe radius, Rm is the pipe resistance per unit length, k ⋅ ζ is the current density demand, and

L is the half anode spacing. A difficulty, however, is that anode resistance does not appear explicitly in these expressions. Consequently, if the pipe or a portion thereof lies within the anode potential field, an inaccurate projection of the potential-distance relationship is likely to result. Numerical methods such as Boundary Element Modeling (BEM) accommodate

2

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anode resistance and the fact that pipe potential, φc, is a function of z; however, they do not take into account metallic path resistance. Recently, a first principles based equation that projects potential attenuation along a pipeline or riser protected by multiple, equally spaced, identical superimposed (bracelet) anodes and which incorporates all four resistance terms (anode (electrolyte), coating, polarization, and metallic path) was derived (4,5). This has as its governing equation the expression,

E c (z ) = U m'' (z ) − U e'' (z ) , ''

(2

where, Um(z) and Ue(z) are potentials in the metallic pipe and electrolyte, respectively, at z. The Ec(z) term was addressed by assuming a linear relation between the polarized pipe potential, φc and cathode current density, ic, according to,

E c ( z ) = α ⋅ γ ⋅ ic ( z ) ,

(3

where,

α is the polarization resistance, and γ is the total-to-bare pipe surface area ratio (reciprocal of the coating breakdown factor). Substitution of this and a modified version of the differential equation that yielded Equation 1 (2,3), U m'' ( z ) =

Rm ⋅ 2 ⋅ π ⋅ rp

αγ

⋅ Ec ( z ) ,

(4

into Equation 2 yielded what has been termed an inclusive attenuation equation for polarization of a unidimensional system,

2⋅H L H  ( ) B E z + ⋅ =  ∫ E c (t ) ⋅ dt , c 2 z3 z z 

E c (z ) +  ''

(5

where, H =

ρ e ⋅ rp αγ

and B =

− Rm ⋅ 2πr p

αγ

with

ρe = electrolyte resistivity. The equation is unique compared to preexisting alternatives in that, as noted above, it incorporates all relevant resistance terms (coating, polarization, and metallic path return resistances are included explicitly and anode (electrolyte) resistance implicitly as the integral), whereas other expressions (2,3) preclude the anode resistance and numerical modeling the metallic path term. Accuracy and utility of the equation for cp modeling and for design of marine pipelines with bracelet anodes has been demonstrated, and example solutions have been provided (4,5). The purpose of the present paper is to extend Equation 2 to the more general case where equally spaced anodes are displaced from the pipeline.

3


DISPLACED ANODE POTENTIAL ATTENUATION EQUATION

Consider a pipeline that is cathodically polarized by identical, equally spaced, equally displaced anodes and two points thereon that are separated by a distance ∆z, as shown in Figure 2. The potential difference between the two points is,

∆U e = I e ( z ) ⋅ ∆Re ,

(6

where ∆Re is the resistance between the two points and the current remaining in the electrolyte, I e ( z ) , is, L

I e ( z ) ≡ 2 ⋅ ∫ 2 ⋅ π ⋅ rp ⋅ ic (t ) ⋅ dt = z

4 ⋅ π ⋅ rp

α ⋅γ

L

⋅ ∫ E (t ) ⋅ dt .

(7

z

Further, the resistance between the surface of a spherical anode and a point p on the pipeline is given by, Rra → p =

ρe 4 ⋅π

 1 1 r − p , a 

(8

where p ≡ z 2 + OF 2 with OF being the distance by which the anode is offset from the pipe. Similarly, the resistance between the surface of the anode and a point p+∆p (Figure 2) is given by, Rra → p + ∆p =

ρe 4 ⋅π

1 1   r − p + ∆p  . a 

(9

The difference between the two resistances, ∆Re, is determined by subtracting the former from the latter which, upon simplification using binomial expansion, yields, 2 3  ρe   ∆p   ∆p   ∆p  ∆R e = 1 − (1 −   +   −   K) , 4 ⋅π ⋅ p   p   p   p   

(10

and, upon substitution into Equation 7, 2 3  ρe   ∆p   ∆p   ∆p  ∆U e = I e ( z ) ⋅ 1 − (1 −   +   −   K) . 4 ⋅π ⋅ p   p   p   p   

(11

Dividing both sides by ∆p and taking the limit for ∆p approaching zero then gives, dU e dp

= I e (z ) ⋅

ρe . 4 ⋅ π ⋅ p2

(12

Further, the derivative of p with respect to z produces,

dp dz

(

= z 2 + OF 2

)

−1

2

⋅z;

(13

4


and, upon combining this with Equation 12,

U e' (z ) =

dU e dz

= I e (z ) ⋅

ρe z ⋅ 4 ⋅ π z 2 + OF 2

(

)

3

.

(14

2

Taking the derivative of Equation 14 with respect to z then yields, d 

 d ρe ⋅ z  + (I (z )) ⋅ , e 3 3 2 2 2 dz  4 ⋅ π ⋅ (z 2 + OF 2 ) 2  dz ( ) 4 ⋅ π ⋅ z + OF  

U e'' (z ) = I e (z ) ⋅

ρe ⋅ z

(15

and, upon substituting Equation 7, U e'' (z ) =

ρ e ⋅ rp ⋅ z

(

α ⋅ γ ⋅ z + OF 2

)

3

2

2

⋅ E (z ) +

ρ e ⋅ rp α ⋅γ

⋅

  L 1 3 ⋅ z2 −   ⋅ ∫z E (t ) ⋅ dt . 3 5  (z 2 + OF 2 ) 2 (z 2 + OF 2 ) 2 

(16

Lastly, upon combining Equations 2-4 and 16,  H ⋅z E c" (z ) +  B +  z 2 + OF 2 

(



where Q = 

(

1

 z 2 + OF 2

)

3

2

−

)

32

 L  ⋅ E (z ) = − H ⋅ Q ⋅ E (t ) ⋅ dt , ∫ c  z 

(17

 . 5 2 2 2  z + OF 

(

3⋅ z 2

)

Using an offset distance (OF) of zero, which is synonymous with the anode being superimposed, it can be shown that Equation 17 reduces to Equation 5. Thus Equation 17 encompasses Equation 5 and is presented as a more general expression, based on first principles, for pipelines protected by identical, evenly-spaced, equally displaced anodes. COMPARISON OF RESULTS

Solutions to Equation 17 were obtained using a Coordinate Mapping Based Finite Difference Method (CoMB-FDM) numerical procedure, similar to what was employed in conjunction with Equation 5 (5). As for the superimposed anode case, this numerical solution requires also that the drain-point potential, φdp, be known. In addition, since Equation 5 was developed for galvanic anode (ga) cp, its application to pipelines with iccp systems necessitates that potential of an equivalent galvanic anode that provides the same polarization as the ic one, φa(eq), be determined. This equivalent anode potential can be calculated from the expression (6), E = φ a (ic ) − φ a ( eq ) ,

(18

where E is the rectifier voltage and φa(ic) is the potential of the ic anode. A series of models was created assuming a rectifier voltage of 10 V, an ic anode potential of 1.5 VAg/AgCl, and the parameters listed in Table 1. The radius of a spherical galvanic anode, ra(eq), the resistance of which is equivalent to that of the ic one, was calculated using McCoy’s formula (7) such that,

5


ra ( eq ) = 0.282 ⋅ Aa ,

(19

where Aa is surface area of the ic anode. Knowing φa(eq) and ra(eq), the equivalent spherical anode was superimposed on the pipeline; and potential at the first node was set equal to φa(eq). The total anode output current, IA, was then numerically calculated from the potential profile generated from the CoMB-FDM based FORTRAN program. This current was used to calculate φdp according to the expression,

φ dp = φ a (eq) + I A ⋅ Rr

a ( eq )→OF

= φ a ( eq ) +

I A ⋅ ρe  1

1  −  .  ra (eq ) OF 

4π

(20

Table 2 lists values of anode current output and φdp for the various combinations of parameters in Table 1, the rectifier and anode parameters listed above, and offset distances of 25, 100, and 500 m. Correspondingly, Figures 3-5 present plots of potential versus distance for αγ = 17,500 Ω m2, ρe = 100 Ω·m, and OF = 25, 100, and 500 m. For each offset, three values for the native (corrosion) potential (φcorr, see Table 2) are shown. Also, each profile determined using Equation 17 is compared with the corresponding one calculated from Equation 1. In each case, potential progressively attenuates, albeit at an ever decreasing rate, with increasing distance from the anode, as projected schematically in Figure 1b. Also, the magnitude of attenuation increases in proportion to φcorr (greater attenuation the more positive φcorr). However, while the anode potential field contributes to a majority of the attenuation in the shortest OF case (Equation 17 solution in Figure 3), this progressively moderates as OF increases and is nil for OF = 500 m. Because Equation 1 does not incorporate the anode potential field, it yielded a more negative potential-distance projection in the cases where this factor contributes meaningfully to attenuation (OF = 25 m and, to a lesser extent, 100 m). However, where this field was negligible (Figure 5, OF = 500 m), the profiles projected by both equations essentially superimpose. This, coupled with the fact that accuracy of the anode potential field projection has been documented in the case of Equation 5 (5) and is here first-principles based (Equation 6), confirms accuracy of Equation 17. In all cases, polarization at the mid-anode location (z = L) was approximately 100 mV. Figures 6-8 show potential attenuation plots for αγ = 1,750 Ω m2 (a ten-fold reduction in pipe current demand compared to Figures 3-5 but with other parameters the same). In each case, the profiles are more flat than where current demand was greater (Figures 3-5); and the distinction between the Equation 17 and 1 solutions is significant only for z ≤ 2,500 m and OF = 25 m. However, polarization at the mid-anode location is in all cases nil. Figures 9-11 provide attenuation plots for the same conditions as in Figures 3-5 but with ρe = 25 Ω·m (relatively high pipe current demand but four-fold greater electrolyte conductivity). Comparison of these results with those in Figures 3-5 indicates, first, a reduced effect of the electrolyte upon attenuation and greater influence from the metallic path and, second, closer agreement between the Equation 17 and 1 solutions in the ρe = 25 Ω·m cases, as should be expected intuitively. Polarization at the mid-anode position exceeds 300 mV in all cases. Lastly, Figures 12-14 show attenuation plots for the lower pipe current demand (αγ = 1,750 Ω m2), lower electrolyte resistivity (ρe = 25 Ω·m) case. The results here are comparable with those in Figures 6-8 except that attenuation in the anode field is less and metallic path contribution greater in Figures 12-14. Here also, however, polarization at the mid-anode position is essentially nil. The mid-anode potentials for each of the profiles in Figures 3-14 are listed for the Equation 17 solutions in Table 3 and for the Equation 1 ones in Table 4. Correspondingly, Table 5 shows the mid-anode potential difference between Equations 17 and 1, and Table 6 lists the mid anode polarization predicted by Equation 17 for each of the examples.

6


CONCLUSIONS

1. A first principles based equation that incorporates resistance terms associated with the anode (electrolyte), coating, polarization, and metallic path return was derived that describes potential attenuation along a pipeline cathodically polarized by identical, evenly-spaced, equally offset spherical anodes as,  H ⋅z '' E c (z ) +  B +  z 2 + OF 2 

(

where H =

ρ e ⋅ rp αγ

, B=

)

32

− Rm ⋅ 2πr p

αγ

 L  ⋅ E (z ) = − H ⋅ Q ⋅ E (t ) ⋅ dt , ∫ c  z 



, and Q = 

1

(

 z 2 + OF 2

)

3

2

−

 . 5 z 2 + OF 2 2 

(

3⋅ z 2

)

2. Potential attenuation plots determined from Coordinate Mapping Based Finite Difference Method numerical solutions to this equation indicate that it provides a more accurate representation than the classical equations of Morgan and Uhlig for situations where the pipeline or a portion thereof lies within the potential field of the anode. ACKNOWLEDGEMENTS

The authors are indebted to member organizations of a joint industry project, including Chevron (now ChevronTexaco), ExxonMobil, Equilon Pipeline Company, Texaco (now ChevronTexaco), and the Minerals Management Service for financial sponsorship of this research and for permission to publish. Also appreciated is the technical guidance and assistance from members of the Technical Advisory Committee which included Fred Corsiglia, Russell Lewis, Mark Mateer, Steve Smith, Neill Strickland, and Robert Smith. BIBLIOGRAPHY

1. Corrosion Costs and Preventive Strategies in the United States,” Report No. FHWA-RD-01-156, Federal Highway Administration, Washington DC, March, 2002, pp. 24,25. 2. J. Morgan, Cathodic Protection, Macmillan, New York, 1960, pp. 140-143. 3. H.H. Uhlig and R.W. Revie, Corrosion and Corrosion Control, Third Ed., J Wiley and Sons, New York, 1985, pp. 421-423. 4. P. Pierson, K. Bethune, W.H. Hartt, and P. Ananthakrishnan, Corrosion, Vol. 56, 2000, p. 350. 5. D. Lysogorski, W.H. Hartt, and P. Ananthakrishnan, “A Modified Potential Attenuation Equation for Cathodically Polarized Marine Pipelines and Risers,” paper no. 077 to be presented at CORROSION/03. submitted to Corrosion. 6. W.H. Hartt, “The Slope Parameter Approach to Marine Cathodic Protection Design and Its Application to Impressed Current Systems,” in Designing Cathodic Protection Systems for Marine Structures and Vehicles, Ed. H. Hack, Special Technical Publication 1370, Am. Soc. For Testing and Materials, West Conshohocken, PA, 1999. 7. J.E. McCoy, The Institute of Marine Engineers Transactions, vol. 82, 1970, p. 210.

7


Table 1: Model parameters. Pipe Outer Radius, m Pipe Inner Radius, m Anode Spacing, m Anode Radius, m Anode Length, m Equivalent Spherical Anode Radius, m Polarization Resistance, α, Ω-m2 Pipe bare area, percent Corresponding, αγ, Ω-m2 Pipe Corrosion Potential, VAg/AgCl Electrolyte Resistivity, Ω-m Metallic Resistivity, Ω-m

0.136 0.128 100,000 0.374 2.677 0.749 17.5 0.1, 1.0 1,750, 17,500 -0.650, -0.500, -0.300 100, 25 1.70E-07

Table 2: Predicted anode current outputs and drain point potentials: (a) αγ = 17,500 Ωm2; ρe = 100 Ωm, (b) αγ = 17,500 Ωm2; ρe = 100 Ωm; (c) αγ = 1,750 Ωm2; ρe = 25 Ωm; (d) αγ = 1,750 Ωm2; ρe = 25 Ωm. Pipe φcorr, VAg/AgCl -0.650 -0.500 -0.300

Anode Current, A 0.709 0.723 0.741

Drainage Point Potential, φdp, VAg/AgCl OF = 25m OF = 100m OF = 500m -1.188 -1.019 -0.974 -1.049 -0.876 -0.830 -0.862 -0.686 -0.638 (a)

Pipe φcorr, VAg/AgCl -0.650 -0.500 -0.300

Anode Current, A 0.738 0.752 0.771

Drainage Point Potential, φdp, VAg/AgCl OF = 25m OF = 100m OF = 500m -0.893 -0.717 -0.670 -0.748 -0.569 -0.521 -0.554 -0.370 -0.321 (b)

Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300

Anode Current, A 2.568 2.617 2.682

Drainage Point Potential, φdp, VAg/AgCl OF = 25m OF = 100m OF = 500m -1.884 -1.731 -1.690 -1.758 -1.602 -1.560 -1.589 -1.429 -1.387 (c)

Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300

Anode Current, A 2.829 2.883 2.955

Drainage Point Potential, φdp, VAg/AgCl OF = 25m OF = 100m OF = 500m -1.210 -1.041 -0.996 -1.071 -0.899 -0.853 -0.885 -0.709 -0.662 (d)

8


Table 3: Mid-anode potential values predicted using Equation 17: (a) αγ = 17,500 Ωm2; ρe = 100 Ωm, (b) αγ = 17,500 Ωm2; ρe = 100 Ωm; (c) αγ = 1,750 Ωm2; ρe = 25 Ωm; (d) αγ = 1,750 Ωm2; ρe = 25 Ωm. Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300

OF = 25m OF = 100m -0.748 -0.752 -0.600 -0.604 -0.403 -0.407 (a) OF = 25m OF = 100m -0.650 -0.650 -0.500 -0.500 -0.301 -0.300 (b) OF = 25m OF = 100m -0.989 -0.991 -0.846 -0.848 -0.655 -0.656 (c) OF = 25m OF = 100m -0.640 -0.653 -0.504 -0.503 -0.304 -0.303 (d)

OF = 500m -0.754 -0.606 -0.408 OF = 500m -0.650 -0.500 -0.300 OF = 500m -0.995 -0.851 -0.660 OF = 500m -0.653 -0.503 -0.303

Table 4: Mid-anode potential values predicted using Equation 1: (a) αγ = 17,500 Ωm2; ρe = 100 Ωm, (b) αγ = 17,500 Ωm2; ρe = 100 Ωm; (c) αγ = 1,750 Ωm2; ρe = 25 Ωm; (d) αγ = 1,750 Ωm2; ρe = 25 Ωm. Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300

OF = 25m OF = 100m -0.828 -0.772 -0.682 -0.625 -0.486 -0.428 (a) OF = 25m OF = 100m -0.652 -0.650 -0.502 -0.501 -0.302 -0.301 (b) OF = 25m OF = 100m -1.059 -1.008 -0.916 -0.865 -0.727 -0.674 (c) OF = 25m OF = 100m -0.640 -0.653 -0.504 -0.503 -0.304 -0.303 (d)

OF = 500m -0.757 -0.609 -0.412 OF = 500m -0.650 -0.500 -0.300 OF = 500m -0.995 -0.851 -0.660 OF = 500m -0.653 -0.503 -0.303

9


Table 5: Mid-anode potential difference projected by Equation 17 compared to Equation 1: (a) αγ = 17,500 Ωm2; ρe = 100 Ωm, (b) αγ = 17,500 Ωm2; ρe = 100 Ωm; (c) αγ = 1,750 Ωm2; ρe = 25 Ωm; (d) αγ = 1,750 Ωm2; ρe = 25 Ωm. Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300

OF = 25m OF = 100m OF = 500m 0.080 0.020 0.003 0.082 0.021 0.003 0.083 0.021 0.004 (a) Pipe φ corr, VAg/AgCl OF = 25m OF = 100m OF = 500m -0.650 0.002 0.000 0.000 -0.500 0.002 0.001 0.000 -0.300 0.001 0.001 0.000 (b) Pipe φ corr, VAg/AgCl OF = 25m OF = 100m OF = 500m -0.650 0.070 0.017 0.000 -0.500 0.070 0.017 0.000 -0.300 0.072 0.018 0.000 (c) Pipe φ corr, VAg/AgCl OF = 25m OF = 100m OF = 500m -0.650 0.000 0.000 0.000 -0.500 0.000 0.000 0.000 -0.300 0.000 0.000 0.000 (d) Table 6: Magnitude of the mid-anode polarization predicted using Equation 17: (a) αγ = 17,500 Ωm2; ρe = 100 Ωm, (b) αγ = 17,500 Ωm2; ρe = 100 Ωm; (c) αγ = 1,750 Ωm2; ρe = 25 Ωm; (d) αγ = 1,750 Ωm2; ρe = 25 Ωm. Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300 Pipe φ corr, VAg/AgCl -0.650 -0.500 -0.300

OF = 25m OF = 100m -0.098 -0.102 -0.100 -0.104 -0.103 -0.107 (a) OF = 25m OF = 100m 0.000 0.000 0.000 0.000 -0.001 0.000 (b) OF = 25m OF = 100m -0.339 -0.341 -0.346 -0.348 -0.355 -0.356 (c) OF = 25m OF = 100m 0.010 -0.003 -0.004 -0.003 -0.004 -0.003 (d)

OF = 500m -0.104 -0.106 -0.108 OF = 500m 0.000 0.000 0.000 OF = 500m -0.345 -0.351 -0.360 OF = 500m -0.003 -0.003 -0.003

10


Anode Potential Anode IR Drop

+ Potential -

Polarized Pipeline Potential

Pipeline Free Corrosion Potential

Pipeline

Anode (2)

+ Potential -

(a) Mid-Anode Spacing Anode IR Drop

Pipeline IR Drop

Anode Bed

Polarized Pipeline Potential

Pipeline Free Corrosion Potential

Pipeline

(b) Figure 1: Schematic illustration of potential profiles that arise from (a) galvanic cp of marine pipelines with bracelet anodes and (b) impressed current cp of buried pipelines.

11


ra

p+∆p OF

p ∆z

Z=0

z1

Drainage Point

z2 Figure 2: Illustration of offset (displaced) anode cathodic protection system.

0.00 Equation 17, OF=25m Uhlig

Potential, V (Ag/AgCl)

-0.20 -0.40 -0.60 -0.80 -1.00 -1.20 -1.40 0

10000

20000

30000

40000

50000

Distance from anode, m

Figure 3: CoMB-FDM solution to Equation 17 and solution to Equation 1, αγ = 17,500 Ω-m2, ρe = 100 Ω-m, and Offset of 25 m.

12


0.00 Equation 17, OF=100m Uhlig


-0.20 -0.40 -0.60 -0.80 -1.00 -1.20 -1.40 0

10000

20000

30000

40000

50000



0.00 Equation 17, OF=500 m Uhlig


-0.20 -0.40 -0.60 -0.80 -1.00 -1.20 -1.40 0

10000

20000

30000

40000

50000



13


0.00


-0.20 -0.40 -0.60 -0.80 Equation 17, OF=25 m Uhlig

-1.00 -1.20 -1.40 0

10000

20000

30000

40000

50000


Figure 6: CoMB-FDM solution to Equation 17 and solution to Equation 1, αγ = 1,750 Ω-m2, ρe = 100 Ω-m, and offset of 25 m.

0.00


-0.20 -0.40 -0.60 -0.80 Equation 17, OF=100 m Uhlig

-1.00 -1.20 -1.40 0

10000

20000

30000

40000

50000



14


0.00 -0.20

Potential, V

-0.40 -0.60 -0.80 -1.00

Equation 17, OF=500 m Uhlig

-1.20 -1.40 0

10000

20000

30000

40000

50000


Figure 8: CoMB-FDM solution to Equation 17 and solution to Equation 1, αγ = 1,750 Ω-m2, ρe = 100 Ω-m, and offset of 500 m. 0.00 -0.20

Equation 17, OF=25m Uhlig


-0.40 -0.60 -0.80 -1.00 -1.20 -1.40 -1.60 -1.80 -2.00 0

10000

20000

30000

40000

50000



15


0.00 -0.20



-0.40 -0.60 -0.80 -1.00 -1.20 -1.40 -1.60 -1.80 -2.00 0

10000

20000

30000

40000

50000



0.00 -0.20


-0.40


-0.60 -0.80 -1.00 -1.20 -1.40 -1.60 -1.80 -2.00 0

10000

20000

30000

40000

50000



16


0.00 -0.20


-0.40 -0.60 -0.80 -1.00 -1.20 -1.40


-1.60 -1.80 -2.00 0

10000

20000

30000

40000

50000



0.00 -0.20


-0.40 -0.60 -0.80 -1.00 -1.20 Equation 17, OF=100m Uhlig

-1.40 -1.60 -1.80 -2.00 0

10000

20000

30000

40000

50000

Distance, m


17


0.00 -0.20


-0.40 -0.60 -0.80 -1.00 -1.20 -1.40 -1.60


-1.80 -2.00 0

10000

20000

30000

40000

50000



18


03197 a Potential Attenuation Equation for Design and Analysis of Pipeline Cathodic Protection Systems With Displaced an (51300-03197-Sg)

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