In the development of **Equation 4.58d**, we assumed that the reservoir was volumetric (the pore-volume occupied by the oil was constant, and oil production was due to oil expansion only). If we remove this restriction and allow the pore-volume and rock to expand, then the volume of oil displaced to the wells (in bbl) becomes:

We have already seen that expansion of the original oil-in-place (in bbl) was described by **Equation 4.58d**:

The expansion of the water and rock can be determined by the definitions of compressibility. These definitions were given by **Equation 3.21** and **Equation 3.32**. If we assume the initial pressure is the reference pressure, then:

Now, the expansion of water (in bbl) becomes:

Note that the use of this equation implies that the water volume is becoming larger (expanding) as the average pressure, $\overline{p}$ decreases.

The change in pore-volume (in bbl) becomes:

Note that the use of this equation implies that the pore-volume is becoming smaller (contracting) as the average pressure, $\overline{p}$, decreases.

Both the increase in the water volume and the decrease in the pore-volume cause oil to be expelled from the reservoir, resulting in an increase in oil production (in bbl):

or,

Where:

- $\Delta \widehat{N}$, $\Delta {\widehat{N}}_{o}$, $\Delta {\widehat{N}}_{w}$, and $\Delta {\widehat{N}}_{PV}$ are the volume changes, bbl
- $N$ is the stock tank oil originally oil-in-place, STB
- $\overline{{B}_{oi}}$ is the initial oil formation volume factor averaged over the reservoir, bbl/STB
- $\overline{{B}_{o}}$ is the oil formation volume at a future time averaged over the reservoir, bbl/STB
- ${V}_{b}$ is the net rock volume, bbl
- ${V}_{PV}$ is the pore-volume, bbl
- ${V}_{PV\text{}i}$ is the pore-volume, bbl
- $\overline{{\varphi}_{i}}$ is the initial porosity averaged over the reservoir, fraction
- $\overline{\varphi}$ is the current porosity averaged over the reservoir, fraction
- $\overline{{S}_{wir}}$ is the irreducible water saturation averaged over the reservoir, fraction
- $\overline{{S}_{w}}$ is the current water saturation averaged over the reservoir, fraction
- ${c}_{w}$ is the water compressibility, 1/psi
- ${c}_{PV}$ is the rock, pore-volume compressibility, 1/psi
- $\overline{p}$ is the current pressure averaged over the reservoir, psi
- ${p}_{i}$ is the current pressure averaged over the reservoir, psi
- ${N}_{p}$ is the oil production, STB

**Equation 4.65b** is the Material Balance Equation for * Non-Volumetric*,

*that remain above the bubble-point pressure. This equation will become more complicated as we add additional expansion terms. We can simplify the material balance equation using standard Society of Petroleum Engineering symbols as:*

**Undersaturated Reservoirs**Where:

- $F$ is the sum of all
**F**low (production) from the reservoir, bbl - $N$ is the stock tank oil originally oil-in-place, STB
- ${E}_{t}$ is the total
**E**xpansion of the system, bbl/STB

Again, we can use this equation for estimating the original oil-in-place and making future reservoir forecasts. To estimate the original oil-in-place, we divide **Equation 4.66** by ${E}_{t}$ and plot $\frac{F}{{E}_{t}}$ (y-axis) versus ${N}_{p}$ (x-axis), (similar to how we generated **Figure 4.08**).

**Equation 4.65b** is valid for reservoirs with oil production caused by the expansion of the rock and fluids. It will now be instructive to go back to our discussion on the * Drive Mechanisms for Oil Reservoirs*. In that discussion, we listed all of the drive mechanisms associated with oil production and their approximate recovery factors (see

**Table 4.01**). These drive mechanisms are:

- rock and fluid expansion
- solution gas drive
- gas cap drive
- gravity drainage
- natural aquifer drive (or water encroachment)

Without going into the details of the of each drive mechanism, **Table 4.03** lists the expansion terms and production associated with each.

Reservoir Drive Mechanisms in Terms of Standard Rock and Fluid Properties (from Lesson 3) | ||||
---|---|---|---|---|

Drive Mechanism | Expansion | Quantified | Production | Maximum Recoveries |

Rock and Fluid Expansion | Oil Expansion | $$N\left(\overline{{B}_{o}}-\overline{{B}_{oi}}\right)$$ | $$---$$ | Up to 5 percent |

Water Expansion (Interstitial Water) |
$$N\frac{\overline{{S}_{wir}}{c}_{w}\left({p}_{i}-\overline{p}\right)}{\left(1.0-\overline{{S}_{wir}}\right)}$$ | $$---$$ | Up to 5 percent | |

Rock Expansion | $$N\frac{{c}_{PV}\left({p}_{i}-\overline{p}\right)}{\left(1.0-\overline{{S}_{wir}}\right)}$$ | Up to 5 percent | ||

Solution Gas Drive | Solution Gas Expansion | $$\frac{N\overline{{B}_{g}}\left(\overline{{R}_{si}}-\overline{{R}_{s}}\right)}{5.615}$$ | $$---$$ | 20 percent |

Gas Cap Drive | Gas Cap Expansion | $$\frac{5.615mN\overline{{B}_{oi}}}{\overline{{B}_{gi}}}\left(\overline{{B}_{g}}-\overline{{B}_{gi}}\right)$$ | $$---$$ | 30 percent |

Gravity Drainage | Gravity Drainage | Not explicit in Material Balance Equation | $$---$$ | 40 percent |

Natural Aquifer Drive | Water Encroachment | $${W}_{e}\overline{{B}_{w}}$$ | $$---$$ | 45 percent |

Production | ||||

Drive Mechanism | Expansion | Quantified | Production | Maximum Recoveries |

Production | Oil Production | $$---$$ | $${N}_{p}\overline{{B}_{o}}$$ | $$---$$ |

Gas Production | $$---$$ | $${N}_{p}\frac{\left[\left({R}_{p}-\overline{{R}_{s}}\right)\overline{{B}_{g}}\right]}{5.615}$$ | $$---$$ | |

Water Production | $$---$$ | $${W}_{p}\overline{{B}_{w}}$$ | $$---$$ |

Using the definitions shown in **Table 4.03**, we can rewrite **Equation 4.66** as:

Where the entries in **Equation 4.67** are listed in **Table 4.04**.

Term | Description |
---|---|

$$F={N}_{p}\left[\overline{{B}_{o}}+\frac{\overline{{B}_{g}}\left({R}_{p}-\overline{{R}_{s}}\right)}{5.615}\right]+{W}_{p}\overline{{B}_{w}}$$ | Total volume of withdrawal (production) at reservoir conditions in bbl: ${R}_{p}$ and ${R}_{s}$ (SCF/STB) and ${B}_{g}$ (ft^{3}/SCF) |

$${N}_{p}\overline{{B}_{o}}$$ | Cumulative oil Production in bbl |

$${N}_{p}\frac{\left[\left({R}_{p}-\overline{{R}_{s}}\right)\overline{{B}_{g}}\right]}{5.615}$$ | Cumulative gas production in bbl |

$${R}_{p}=\frac{{\displaystyle {\sum}_{time}{G}_{p}}}{{\displaystyle {\sum}_{time}{N}_{p}}}$$ | Cumulative GOR (Gas-Oil Ratio) = Total gas produced over time divided by total oil produced over time. |

$${W}_{p}\overline{{B}_{w}}$$ | Cumulative water production in bbl |

$${E}_{t}={E}_{o}+\frac{5.615\text{}\overline{{B}_{oi}}}{\overline{{B}_{gi}}}m\text{}{E}_{g}+\overline{{B}_{oi}}\left(1+m\right){E}_{fw}$$ | Total expansion in the reservoir in bbl/STB: ${B}_{oi}$ (bbl/STB) and ${B}_{gi}$ (ft^{3}/SCF). |

$${E}_{o}=\left(\overline{{B}_{o}}-\overline{{B}_{oi}}\right)+\frac{\overline{{B}_{g}}}{5.615}\left(\overline{{R}_{si}}-\overline{{R}_{s}}\right)$$ | Total expansion of the oil and liberated gas dissolved in it. Expansion of the oil (above the bubble-point pressure) or shrinkage of the oil (below the bubble-point pressure due to liberation of gas) plus the expansion of the liberated gas |

$$m=\frac{G}{5.615N}\frac{\overline{{B}_{gi}}}{\overline{{B}_{oi}}}$$ | $m$ is the ratio of gas cap volume, $G$ (SCF), to original oil volume, $N$ (bbl). A gas cap also implies that the initial pressure in the oil column must be equal to the bubble-point pressure. Note: $m$ is dimensionless. |

$$E=\frac{\overline{{B}_{g}}-\overline{{B}_{gi}}}{5.615}$$ | Expansion of the original of initial free gas (gas cap). |

$${E}_{fw}=\frac{\overline{{S}_{wir}}{c}_{w}+{c}_{PV}}{\left(1.0-\overline{{S}_{wir}}\right)}\left({p}_{i}-\overline{p}\right)$$ | Even though water has low compressibility, the volume of interstitial water in the system is normally large enough to be significant. The water will expand to fill the emptying pore-volume as the reservoir depletes. As the reservoir is produced, the pressure declines and the entire reservoir pore-volume is reduced due to compaction. The change in volume expels an equal volume of fluid (as production) and is additive in the expansion terms. |

$${W}_{e}\overline{{B}_{w}}$$ | If the reservoir is connected to an active aquifer, then once the pressure drop is communicated throughout the reservoir, the water will migrate into the reservoir resulting in a net water encroachment, ${W}_{e}\overline{{B}_{w}}$ in bbl. |