Overview
The material balance equation (MBE) is a fundamental reservoir engineering tool that accounts for all fluids produced, injected, and remaining in a reservoir. It relates cumulative production to reservoir pressure decline through fluid expansion and drive mechanisms. The Havlena-Odeh (1963) straight-line method reformulates the MBE for graphical interpretation, making it easier to identify drive mechanisms and estimate original oil in place (OOIP).
Theory
The general MBE for an oil reservoir:
N*Eo + N*m*Eg + We + Winj*Bw + Ginj*Bg = Np*(Bo + (Rp-Rs)*Bg) + Wp*Bw
Havlena-Odeh rearranges this as F = N(Eo + mEf,w) + We, where plotting F vs Eo yields a straight line with slope = N (OOIP).
Formulas
Expansion Terms
Oil expansion + dissolved gas:
Eo = (Bo - Boi) + (Rsi - Rs)*Bg
Gas cap expansion:
Eg = Boi * ((Bg/Bgi) - 1)
Connate water + pore volume compressibility:
Ef,w = Boi * ((cw*Swi + cf) / (1 - Swi)) * ΔP
Underground Withdrawal (F)
F = Np * (Bo + (Rp - Rs)*Bg) + Wp*Bw - Winj*Bw - Ginj*Bg
Havlena-Odeh Straight-Line Forms
No gas cap, no water influx (volumetric undersaturated):
F = N * Eo
Plot F vs Eo → slope = N
Gas cap, no water influx:
F = N * (Eo + m*Eg)
Plot F vs (Eo + m*Eg) → slope = N
Water influx, no gas cap:
F/Eo = N + We/Eo
Plot F/Eo vs We/Eo → y-intercept = N
Key Variables
| Symbol | Description | Units |
|---|---|---|
| N | Original oil in place | STB |
| Np | Cumulative oil produced | STB |
| Bo, Boi | Oil FVF (current, initial) | RB/STB |
| Bg, Bgi | Gas FVF (current, initial) | RB/scf |
| Rs, Rsi | Solution GOR (current, initial) | scf/STB |
| Rp | Cumulative produced GOR | scf/STB |
| m | Gas cap ratio = GBgi/(NBoi) | dimensionless |
| We | Cumulative water influx | RB |
Worked Example
Given: Volumetric undersaturated reservoir. Boi = 1.25 RB/STB, cw = 3e-6 /psi, cf = 5e-6 /psi, Swi = 0.25.
| Pressure (psia) | Np (MSTB) | Bo (RB/STB) |
|---|---|---|
| 4000 (initial) | 0 | 1.2500 |
| 3500 | 500 | 1.2480 |
| 3000 | 1200 | 1.2450 |
Step 1: Calculate Eo at 3500 psi:
Eo = Bo - Boi = 1.2480 - 1.2500 = -0.0020 RB/STB
(Above Pb, include Ef,w term)
Ef,w = 1.25 * (3e-6*0.25 + 5e-6)/(1-0.25) * 500 = 1.25 * 7.67e-6 * 500 = 0.00479
Eo_total = -0.002 + 0.00479 = 0.00279 RB/STB
Step 2: F = Np Bo = 500,000 1.248 = 624,000 RB
Step 3: N = F / Eo_total = 624,000 / 0.00279 = 223.7 MMSTB
Valid Ranges
- MBE requires accurate PVT data; small errors in Bo propagate into large N errors
- Minimum 3–5 pressure/production data points for reliable straight-line analysis
- Gas cap ratio m is often unknown and must be estimated or treated as a variable
- Water influx models (van Everdingen-Hurst, Carter-Tracy, Fetkovich) needed if We is significant
References
- Havlena, D. & Odeh, A.S. (1963). "The Material Balance as an Equation of a Straight Line." JPT, 15(8), 896–900.
- Dake, L.P. (1978). Fundamentals of Reservoir Engineering. Elsevier. Chapter 3.
- Craft, B.C. & Hawkins, M.F. (1991). Applied Petroleum Reservoir Engineering. Prentice Hall.
- PetroWiki — Material balance: https://petrowiki.spe.org/Material_balance