Theory

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.

Unit Conversion System

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

Supported Unit Categories

CategoryExample Units
Pressurepsi, psia, psig, bar, kPa, MPa, atm
Lengthft, m, in, cm, mm
Volumebbl, stb, scf, m³, L, gal
Flow Ratestb/d, bbl/d, m³/d, scf/d, Mscf/d
TemperaturedegF, degC, degR, K
ViscositycP, mPa·s, Pa·s
Permeabilitymd, D, m²
Compressibility1/psi, 1/bar, 1/kPa

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

FunctionDescription
UnitConverterConvert value between any compatible units

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
γAPI=141.5γo−131.5\gamma_{API} = \frac{141.5}{\gamma_o} - 131.5 γo=141.5γAPI+131.5\gamma_o = \frac{141.5}{\gamma_{API} + 131.5}

Where:

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

Classification by API Gravity

ClassificationAPI GravitySpecific Gravity
Light crude> 31.1°< 0.87
Medium crude22.3° – 31.1°0.87 – 0.92
Heavy crude10° – 22.3°0.92 – 1.00
Extra-heavy< 10°> 1.00

Functions

FunctionConversionFormula
SG2APISG → APIγAPI=141.5γo−131.5\gamma_{API} = \frac{141.5}{\gamma_o} - 131.5
API2SGAPI → SGγo=141.5γAPI+131.5\gamma_o = \frac{141.5}{\gamma_{API} + 131.5}

Examples

InputFunctionOutput
γo=0.85\gamma_o = 0.85SG2API34.97° API
γAPI=35°\gamma_{API} = 35°API2SG0.850
γo=1.00\gamma_o = 1.00 (water)SG2API10° API
γAPI=10°\gamma_{API} = 10°API2SG1.00

Exponential Integral Function

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

Definition

Ei(x)=∫−∞xett dt(x>0)\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):

−Ei(−x)=∫x∞e−tt dt(x>0)-\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:

pD(rD,tD)=−12Ei(−rD24tD)p_D(r_D, t_D) = -\frac{1}{2} \text{Ei}\left( -\frac{r_D^2}{4 t_D} \right)

At the wellbore (rD=1r_D = 1):

pD(tD)=−12Ei(−14tD)p_D(t_D) = -\frac{1}{2} \text{Ei}\left( -\frac{1}{4 t_D} \right)

Logarithmic Approximation

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

−Ei(−x)≈−ln⁡(x)−γfor x<0.01-\text{Ei}(-x) \approx -\ln(x) - \gamma \quad \text{for } x < 0.01

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

This gives the familiar semi-log approximation:

pD≈12[ln⁡(tD)+0.80907]p_D \approx \frac{1}{2} \left[ \ln(t_D) + 0.80907 \right]

Function

FunctionDescription
ExpIntegralEiEvaluate Ei(x)\text{Ei}(x) for any real xx

Numerical Implementation

The function uses different algorithms depending on the argument magnitude:

Argument RangeMethod
$x
$6 \lex
$x

Functions Covered

See each function page for detailed parameter definitions, Excel syntax, and usage examples.

FunctionPurposeInputOutput
UnitConverterGeneral unit conversionValue, FromUnit, ToUnitConverted value
SG2APISpecific gravity to APIγo\gamma_oγAPI\gamma_{API}
API2SGAPI to specific gravityγAPI\gamma_{API}γo\gamma_o
ExpIntegralEiExponential integralxxEi(x)\text{Ei}(x)

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.

Utilities
utilitiesunit conversionAPI gravityspecific gravityexponential integralspecial functions

Related Excel Functions

4 items

This website uses cookies to enhance your experience and analyze site usage. By clicking "Accept", you consent to the use of cookies for analytics purposes. Read our privacy policy