Unit Conversions and Special Functions

Overview

Petroleum engineering calculations frequently require:

• Unit conversions — between field units (psi, bbl, ft) and SI units (Pa, m³, m)
• Gravity transformations — between API gravity and specific gravity
• Special mathematical functions — particularly the exponential integral for pressure transient analysis

These utility functions provide the foundation for accurate engineering calculations.

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Unit Conversion System

The UnitConverter function provides general-purpose conversion between compatible units.

Supported Unit Categories

Usage Pattern

The function takes a value in the source unit and converts to the target unit:

Result = UnitConverter(Value, "SourceUnit", "TargetUnit")

Example: Convert 2,500 psi to bar:

UnitConverter(2500, "psi", "bar") → 172.4 bar

Function

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Gravity Transformations

Crude oil density is commonly expressed as either:

• Specific Gravity (SG) — density relative to water at 60°F
• API Gravity — American Petroleum Institute scale, inversely related to density

Relationship Between API and Specific Gravity

The API gravity scale was designed so that:

• Water has API gravity of 10°
• Lighter oils have higher API values
• Heavier oils have lower API values

$$\gamma_{API} = \frac{141.5}{\gamma_o} - 131.5$$$$\gamma_o = \frac{141.5}{\gamma_{API} + 131.5}$$

Where:

• $\gamma_{API}$ = API gravity, °API
• $\gamma_o$ = oil specific gravity (water = 1.0)

Classification by API Gravity

Functions

Examples

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Exponential Integral Function

The exponential integral $\text{Ei}(x)$ is a fundamental special function in pressure transient analysis.

Definition

$$\text{Ei}(x) = \int_{-\infty}^{x} \frac{e^t}{t} \, dt \quad (x > 0)$$

For practical computation with negative arguments (as used in well testing):

$$-\text{Ei}(-x) = \int_{x}^{\infty} \frac{e^{-t}}{t} \, dt \quad (x > 0)$$

Role in Well Testing

The line source solution for pressure drawdown in an infinite-acting reservoir is:

$$p_D(r_D, t_D) = -\frac{1}{2} \text{Ei}\left( -\frac{r_D^2}{4 t_D} \right)$$

At the wellbore ($r_D = 1$):

$$p_D(t_D) = -\frac{1}{2} \text{Ei}\left( -\frac{1}{4 t_D} \right)$$

Logarithmic Approximation

For small arguments (large $t_D$, i.e., late-time behavior):

$$-\text{Ei}(-x) \approx -\ln(x) - \gamma \quad \text{for } x < 0.01$$

Where $\gamma = 0.5772...$ is the Euler-Mascheroni constant.

This gives the familiar semi-log approximation:

$$p_D \approx \frac{1}{2} \left[ \ln(t_D) + 0.80907 \right]$$

Function

Numerical Implementation

The function uses different algorithms depending on the argument magnitude:

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Related Documentation

• Dimensionless Variables — pD, tD definitions using Ei
• Infinite Reservoir Solution — Line source with Ei function
• PVT Overview — Gravity used in correlations
• Interpolation Functions — Companion utility functions

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References

1. American Petroleum Institute. "API Gravity." API Manual of Petroleum Measurement Standards, Chapter 11.1.

2. Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions. National Bureau of Standards. Chapter 5: Exponential Integral and Related Functions.

3. Cody, W.J. and Thacher, H.C. (1968). "Rational Chebyshev Approximations for the Exponential Integral E₁(x)." Mathematics of Computation, 22(103), pp. 641-649.

4. Lee, J., Rollins, J.B., and Spivey, J.P. (2003). Pressure Transient Testing. SPE Textbook Series Vol. 9. Chapter 2.