Underground Withdrawal and Expansion Terms

Overview

The material balance equation requires accurate calculation of two fundamental quantities:

1. Underground withdrawal ($F$) — the total volume of fluids removed from the reservoir, measured at reservoir conditions
2. Expansion terms ($E_o$, $E_g$, $E_{fw}$) — the volume increase of each fluid/rock component as pressure declines

These calculations convert surface production volumes into reservoir voidage and quantify the driving forces behind production.

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Underground Withdrawal (F)

Oil Reservoir

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

The term $(R_p - R_s)$ represents free gas produced above what was dissolved in oil.

Gas Reservoir

$$F = G_p B_g + W_p B_w$$

For dry gas reservoirs without condensate or water production:

$$F = G_p B_g$$

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Expansion Terms

Oil and Dissolved Gas Expansion ($E_o$)

$$E_o = (B_o - B_{oi}) + (R_{si} - R_s)B_g$$

Or equivalently, using the two-phase formation volume factor:

$$E_o = B_t - B_{ti}$$

Where:

$$B_t = B_o + (R_{si} - R_s)B_g$$

Gas Cap Expansion ($E_g$)

$$E_g = B_{oi}\left(\frac{B_g}{B_{gi}} - 1\right)$$

This term applies only when a gas cap is present ($m > 0$). As pressure declines, the gas cap expands and provides energy for oil displacement.

For gas reservoirs:

$$E_g = B_g - B_{gi}$$

Connate Water and Rock Expansion ($E_

$$E_{fw} = B_{oi}\frac{c_w S_{wi} + c_f}{1 - S_{wi}}\Delta p$$

Where $\Delta p = p_i - p$.

This term is usually small compared to $E_o$ and $E_g$ except in:

• Undersaturated reservoirs ($P > P_b$) where $E_o$ is small
• Geopressured reservoirs where $c_f$ is large
• High water saturation rocks where $c_w S_{wi}$ contribution is significant

Total Expansion ($E_t$)

$$E_t = E_o + mE_g + E_{fw}$$

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Effective Compressibility

Undersaturated Oil

Above bubble point, oil expansion is described by effective compressibility:

$$c_e = \frac{c_o S_o + c_w S_w + c_f}{1 - S_w}$$

Gas Reservoir

$$c_e = \frac{S_g c_g + S_w c_w + c_f}{S_g}$$

Where $c_g = 1/p - (1/z)(dz/dp)$ for real gas.

Geopressured Systems

In geopressured reservoirs, the pore volume compressibility $c_f$ can be 10-50 times larger than in normally pressured formations:

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Two-Phase Formation Volume Factor ($B_t$)

Definition

$$B_t = B_o + (R_{si} - R_s)B_g$$

$B_t$ represents the total volume occupied by one stock-tank barrel of oil and its originally dissolved gas, at current reservoir pressure.

Behavior

$B_t$ always increases with decreasing pressure below $P_b$, unlike $B_o$ which decreases below $P_b$.

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Calculation Tips

Unit Consistency

Common error: Using $B_g$ in RB/Mscf instead of RB/scf. Check units carefully — this affects $E_o$, $E_g$, and $F$ by a factor of 1000.

Undersaturated vs. Saturated

For pressures above $P_b$:

• $R_s = R_{si}$ (constant)
• $B_g$ is not meaningful (no free gas)
• Use $c_o$ formulation instead of $E_o$

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Related Topics

• MBE Overview — How F and E terms fit into material balance
• Oil Reservoirs — Havlena-Odeh using these terms
• Gas Reservoirs — Simplified gas material balance
• PVT Overview — Bo, Rs, Bg correlations needed

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References

1. Dake, L.P. (1978). Fundamentals of Reservoir Engineering. Elsevier.

2. Ahmed, T. (2019). Reservoir Engineering Handbook, 5th Edition. Gulf Professional Publishing.

3. Craft, B.C. and Hawkins, M.F. (1991). Applied Petroleum Reservoir Engineering, 2nd Edition. Prentice Hall.

4. 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.

5. Walsh, M.P. and Lake, L.W. (2003). A Generalized Approach to Primary Hydrocarbon Recovery. Elsevier.