4.5.1.2: Non-Volumetric, Undersaturated Oil Reservoirs

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
• $V_{PVi}$ 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, Undersaturated Reservoirs 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:

Where:

• $F$ is the sum of all Flow (production) from the reservoir, bbl
• $N$ is the stock tank oil originally oil-in-place, STB
• $E_t$ is the total Expansion 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.

Table 4-3 Drive Mechanisms in Terms of Rock and Fluid Properties

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.

Table 4.04 Material Balance Equation in Terms of Rock and Fluid Properties

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 $B_g$ (ft3/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
$$E_t=E_o+\frac{5.615\overline{B_{oi}}}{\overline{B_{gi}}}m{E}_g+\overline{B_{oi}}\left(1+m\right)E_{fw}$$	Total expansion in the reservoir in bbl/STB: $B_{oi}$ (bbl/STB) and $B_{gi}$ (ft3/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.