Proposal for analysis of base fluid 10-day and field drilling mud 4-day
*Leptocheirus* *plumulosus* sediment toxicity testing results -- A new
interval arithmetic approach to demonstrate compliance

M.L. Cano ^1^ and J.T. Louallen ^2^

^1^ Shell Chemical Company LP, ^2^ Shell Global Solutions

##### Prepared for the Offshore Operators Committee

# Summary

This document contains a proposed approach for handling the uncertainty
that is present in the computation used to determine compliance with
USEPA Gulf of Mexico (GOM) guidelines for sediment toxicity of base
fluids used for drilling muds and sediment toxicity of field drilling
muds. The proposed approach is based on **interval arithmetic**. The 95%
confidence intervals of the test and reference (base fluid or field
drilling mud) sediment toxicity test results are used to develop a
"compliance" interval for the ratio of the test results. Fluids and muds
might be assessed as "passing" their respective sediment toxicity tests
if the lower bound of this compliance interval is ≤ 1.0. Example
calculations are shown for some representative base fluid sediment
toxicity tests and some field drilling mud sediment toxicity tests.

# 

# Introduction

The current USEPA Gulf of Mexico (GOM) guidelines (USEPA, 2001) and
Region 6 General Permit (USEPA, 2002) for demonstration of base fluid
sediment toxicity and field drilling mud sediment toxicity compliance
involve the use of a K-factor to account for uncertainty in the toxicity
tests. The compliance tests use a ratio of the toxicity of a reference
fluid or reference mud (C1618 internal olefin (IO) as base fluid or as
basis for reference mud) to the toxicity of the tested fluid or mud. A
fluid or mud is shown to be compliant if the ratio is ≤ 1.0. It is
currently believed that when the new GOM General Permit is issued, usage
of the K-factor will be eliminated. This will result in no specific
mechanism to address uncertainty in the sediment toxicity tests. Typical
results for the tests include a reported LC~50~ value for toxicity to
*Leptocheirus* *plumulosus* with a corresponding statistical confidence
interval (usually 95%).

Some level of uncertainty in experimental measurements is always present
(including the LC~50~ results of the 10-day base fluid sediment toxicity
test and the 4-day field mud sediment toxicity test). There are many
potential sources of uncertainty, which can include limitations of
instrumentation and variability in test samples and procedures. When
computations are carried out with measured numbers that have associated
uncertainty, there are statistical techniques that are used to recognize
this uncertainty and reflect it in the computed result. This is
typically known as propagation of errors. Standard mathematical formulas
are available to determine error propagation (Taylor, 1982). However,
these techniques usually only have applications in situations when
errors are normally distributed (negative and positive errors are
equal). When errors are not normally distributed, the traditional
propagation of error formulas do not apply.

For non-normally distributed errors, an alternative way to account for
uncertainty is available. This approach is based on the concept of
**interval arithmetic** (Hayes, 2003). This document contains a proposal
for an alternative mechanism to recognize and account for the
uncertainty in sediment toxicity testing results (using the concept of
interval arithmetic) when computing a ratio that is to be used for
compliance. This concept and its application are discussed below.

# Interval arithmetic to address uncertainty

One approach to dealing with numerical errors is a technique called
interval arithmetic. It does nothing directly to improve the accuracy of
calculations, but for every number it provides a certificate of
accuracy---or the lack of it. The result of an ordinary (non-interval)
computation is a single number, a point on the real number line, which
lies at some unknown distance from the true answer. An interval
computation yields a pair of numbers, an upper and a lower bound, which
are guaranteed to enclose the exact answer. While the resulting interval
does not pinpoint the truth, it does point out how much is actually
known. Work in the area of computer programming uses interval arithmetic
to manage and mitigate the effects of round-off error in computational
algorithms.

A single formula extends the definition of the four standard arithmetic
operations to intervals. If ° represents any of the operations +, --, ×
or ÷, then the corresponding interval operation is defined as:

\[x~1~, x~2~\] ° \[y~1~, y~2~\] = \[min(x~1~ ° y~1~, x~1~ ° y~2~, x~2~ °
y~1~, x~2~ ° y~2~), max(x~1~ ° y~1~, x~1~ ° y~2~, x~2~ ° y~1~, x~2~ °
y~2~)\] (1)

Thus, one can compute the four possible combinations and choose
whichever of these results is the smallest as the new lower bound, and
likewise take the largest as the new upper bound. This interval
represents the combined uncertainty in the computation. If both
intervals have the same uncertainty, the computed interval will have the
exact same uncertainty.

For the example of a ratio, the interval calculation is as follows:

$\frac{\lbrack x_{1},x_{2}\rbrack}{\lbrack y_{1},y_{2}\rbrack} = \left\lbrack \text{min}\left( \frac{x_{1}}{y_{1}},\frac{x_{1}}{y_{2}},\frac{x_{2}}{y_{1}},\frac{x_{2}}{y_{2}} \right)\text{,max}\left( \frac{x_{1}}{y_{1}},\frac{x_{1}}{y_{2}},\frac{x_{2}}{y_{1}},\frac{x_{2}}{y_{2}} \right) \right\rbrack$
(2)

In the case of the 10-day *Leptocheirus* sediment toxicity testing
results that are compared via a ratio (toxicity of C1618 reference
fluid/toxicity of test fluid), if both results have a 95% confidence
level, the computed ratio interval will be based on these 95% confidence
levels. This calculated interval would represent the "compliance"
interval for the ratio of the two toxicity test results. A similar
approach could be used to evaluate the results from the 4-day field mud
sediment toxicity tests. This computed interval could be used to assess
compliance.

# 

# Example results for Base Fluids

The proposed interval approach has been used (Equation (2)) to assess
some specific examples of sediment toxicity tests for base fluids. These
four specific examples are shown in Table 1 below.

**Table 1. Example results from sediment toxicity tests.**

![](media/image1.wmf){width="5.99375in" height="3.4034722222222222in"}

Results are shown for two different products. Three test results are
shown for Product 1. In this example, Product 1 "passes" the compliance
test (Ratio ≤ 1) two times and fails once. Product 2 fails the test with
a Ratio of 7.8. Using interval arithmetic to compute the compliance
intervals shows the following compliance intervals for the ratios for
Product 1: \[0.1, 1.5\], \[0.5, 0.8\], and \[0.9, 2.5\]. In all cases
for Product 1, the required passing value (1.0) falls within the
compliance interval. For Product 2, the compliance interval for the
ratio is \[5.5, 11,0\]. In this case, the required passing value (1.0)
falls outside of the compliance interval.

**Example results for field muds**

Recently, the inter-industry Synthetic Based Mud (SBM) Toxicity Working
Group assembled a database of results from \~ 500 field mud sediment
toxicity tests. This database is documented in Louallen et al. (2004),
and represents results from field mud 4-day sediment toxicity testing
from 2002 and 2003 conducted in 4 different laboratories. Louallen et
al. (2004) carried out an analysis of these data to study the
appropriateness of the current K factor for these 4-day sediment
toxicity tests (0.25) and concluded that the value of 0.25 was a
statistically appropriate value based on the field mud sediment toxicity
database.

This same database was used to assess the effects of applying the
proposed interval arithmetic approach described herein. These results
are shown in Table 2.

**Table 2: Application of interval arithmetic approach to determine
compliance of field mud samples from industrial database. A total of 409
samples were considered to have adequate data for the analysis (needed
to have upper and lower confidence interval about each experimental
result). Results are presented as probabilities (≤ 1.0).**

![](media/image2.wmf){width="4.709722222222222in"
height="2.4166666666666665in"}

The data in Table 2 indicate that the interval approach has an effect
similar to the use of a K factor equal to 0.25. Overall, the use of the
interval approach resulted in a compliance rate of 96% compared to 95%
with the use of a 0.25 K factor and 85% with the use of a 0 K factor
(i.e., no K factor). This analysis shows that the interval approach can
adequately account for the variability that is (1) observed in the
sediment toxicity tests and (2) propagated in the calculation of the
sediment toxicity ratio.

**Proposed passing criterion**

Given the interval arithmetic approach discussed above, one proposed
criterion for passing the 10-day base fluid sediment toxicity test or
the 4-day field mud sediment toxicity test could be:

lower bound of compliance interval ≤ 1.0 (3)

In other words, a base fluid or field mud would pass the ecotoxicity
test if the lower bound of the compliance interval were ≤ 1.0. This
approach would provide some statistical significance to the passing
criterion. If Equation (3) is applied as a passing criterion for the
four examples shown in Table 1, Product 1 "passes" in all cases while
Product 2 "fails".

It is possible to consider other passing criteria with this approach.
These may be based on different confidence intervals. In addition, it
may also be possible to define some criteria that would help to define a
"good" test as opposed to a "bad" test. These criteria could be used to
objectively drop out (or not consider) some specific tests because of
high test variability. This would be manifested in an abnormally large
compliance interval for the ratio.

# Conclusions and recommendations

A new proposed approach for calculating toxicity ratios to determine
compliance with the 10-day base fluid sediment toxicity test or the
4-day field drilling mud sediment toxicity test has been presented. This
approach uses interval arithmetic to develop a "compliance" interval for
the ratio that is based on the 95% confidence intervals of the specific
toxicity tests used to develop the ratio. A new proposed pass/fail
criterion has also been presented. To further develop and refine this
approach, it is recommended that this approach be applied to a larger
data set of test results for base fluid sediment toxicity in order to
determine the robustness of the proposed method and to determine
additional criteria that may be used to help define the "quality" of a
set of test results. Application of the approach to a large database of
4-day mud sediment toxicity tests showed that the method could
effectively manage the variability observed in these test results.

# References

Hayes, B. 2003. A lucid interval. American Scientist. Vol. 91. No. 6,
pp. 484-488.

Louallen, J.T., D.C.L. Wong, and P.B. Dorn. 2004. Statistical analysis
of the sediment toxicity ratio for synthetic-based drilling muds. Shell
Global Solutions.

Taylor, J.R. 1982. An introduction to error analysis. The study of
uncertainties in physical measurements. University Science Books, Mill
Valley, CA.

USEPA. 2001. 40 CFR Parts 9 and 435 \[FRL-6929-8\] RIN 2040-AD14
Effluent Limitations Guidelines and New Source Performance Standards for
the Oil and Gas Extraction Point Source Category; OMB Approval Under the
Paperwork Reduction Act: Technical Amendment. 66 Federal Register 14.
January 22, 6858. Website address:
<http://www.epa.gov/waterscience/guide/sbf/final/finalsbf.pdf>.

USEPA. 2002. Final NPDES General Permit for New and Existing Sources and
New Dischargers in the Offshore Subcategory of the Oil and Gas
Extraction Category for the Western Portion of the Outer Continental
Shelf of the Gulf of Mexico
([GMG290000](http://www.epa.gov/region6/6wq/npdes/genpermt/offshore/fr_not.pdf)).