Oil Reservoir Material Balance

Overview

The Havlena-Odeh method (1963, 1964) transforms the general material balance equation into straight-line forms that enable graphical determination of original oil in place (OOIP), gas cap size, and water influx strength. This technique remains one of the most powerful tools in reservoir engineering for validating volumetric estimates and understanding reservoir drive mechanisms.

General Oil MBE

The Equation

$F = N[E_o + mE_g + E_{fw}] + W_e$

Where:

$F$ = total underground withdrawal
$N$ = original oil in place (OOIP), STB
$m$ = ratio of initial gas cap volume to initial oil volume
$E_o, E_g, E_{fw}$ = expansion terms
$W_e$ = cumulative water influx, RB

Underground Withdrawal

$F = N_p[B_o + (R_p - R_s)B_g] + W_p B_w$

This accounts for:

Oil produced (in reservoir volumes)
Free gas produced above solution GOR
Water produced

Expansion Terms

Term	Expression	Physical Meaning
$E_o$	$(B_o - B_{oi}) + (R_{si} - R_s)B_g$	Oil + dissolved gas expansion
$E_g$	$B_{oi}(B_g/B_{gi} - 1)$	Free gas cap expansion
$E_{fw}$	$B_{oi}\frac{c_w S_{wi} + c_f}{1 - S_{wi}}\Delta p$	Connate water + rock expansion
$E_t$	$E_o + mE_g + E_{fw}$	Total expansion
$B_t$	$B_o + (R_{si} - R_s)B_g$	Two-phase formation volume factor

Straight-Line Methods

Case 1: No Gas Cap, No Water Drive ( $m = 0, W_e = 0$ )

$F = N \cdot E_t$

Plot $F$ vs. $E_t$ : slope = $N$ .

Case 2: Known Gas Cap, No Water Drive ( $m$ known, $W_e = 0$ )

$F = N(E_o + mE_g + E_{fw})$

Plot $F$ vs. $(E_o + mE_g + E_{fw})$ : slope = $N$ .

Case 3: No Gas Cap, Water Drive ( $m = 0, W_e$ unknown)

$\frac{F}{E_t} = N + \frac{W_e}{E_t}$

This is the Campbell plot: plot $F/E_t$ vs. $W_e/E_t$ :

Slope = 1
Y-intercept = $N$

Case 4: Unknown Gas Cap and Water Drive ( $m$ and $W_e$ unknown)

$\frac{F - W_e}{E_o + E_{fw}} = N + Nm\frac{E_g}{E_o + E_{fw}}$

Plot $(F - W_e)/(E_o + E_{fw})$ vs. $E_g/(E_o + E_{fw})$ :

Y-intercept = $N$
Slope = $Nm$ → $m = \text{slope}/N$

Diagnostic Plots

F vs. Et (No Aquifer)

Straight line through origin = correct model
Curve upward = water influx present (We > 0)
Scatter = PVT or pressure data errors

F/Et vs. Time (Campbell Plot)

Horizontal line = correct N, no aquifer
Rising trend = water influx — N is underestimated without We
Falling trend = N overestimated or m wrong

Drive Index Plot

Plot DDI, GDI, WDI, CDI vs. time:

Shows how drive mechanisms evolve
All indices must sum to 1.0 at each point
Helps identify the dominant recovery mechanism

Practical Considerations

Data Requirements

Data	Source	Quality Concern
Average reservoir pressure	Buildup tests	Must be average, not flowing
Cumulative oil production	Production records	Usually reliable
Cumulative gas production	Gas metering	Often uncertain
Cumulative water production	Water handling records	Can be estimated
PVT properties	Lab or correlations	Consistency critical
$m$ ratio	Volumetric or log data	Often uncertain

When MBE Works Best

Significant pressure decline — at least 10-15% of initial pressure
Multiple pressure surveys — 5+ data points minimum
Reliable production data — metered oil, gas, and water
Known PVT — lab data or well-calibrated correlations

When MBE Fails

Problem	Symptom	Cause
Insufficient depletion	Scatter, no trend	Not enough pressure change
Poor pressure data	Non-straight lines	Inaccurate average pressure
Wrong PVT	Inconsistent N values	Bad fluid properties
Compartmentalization	Multiple trends	Not a single tank reservoir

MBE Overview — Context and workflow for material balance
Gas Reservoirs — p/z method for gas reservoirs
Aquifer Models — Water influx calculation methods
Underground Withdrawal — Calculating F and expansion terms

References

Havlena, D. and Odeh, A.S. (1963). "The Material Balance as an Equation of a Straight Line." Journal of Petroleum Technology, 15(8), 896-900. SPE-559-PA.
Havlena, D. and Odeh, A.S. (1964). "The Material Balance as an Equation of a Straight Line — Part II, Field Cases." Journal of Petroleum Technology, 16(7), 815-822. SPE-869-PA.
Dake, L.P. (1978). Fundamentals of Reservoir Engineering. Elsevier, Chapter 3.
Ahmed, T. (2019). Reservoir Engineering Handbook, 5th Edition. Gulf Professional Publishing.
Campbell, R.A. (1978). Mineral Property Economics, Volume 3: Petroleum Property Evaluation. Campbell Petroleum Series.