Given. An undersaturated-to-saturated reservoir producing under solution-gas drive plus a small ("pot" / unsteady, single-tank) aquifer.
PVT and production history
$P$ (psia)
$B_o$
$R_s$
$B_g$
$N_p$ (MMSTB)
$G_p$ (MMscf)
3,000 ($p_i$)
1.316
650
–
0
0
2,500 ($p_b$)
1.324
650
0.00082
0.092
59.8
1,500
1.252
510
0.00135
0.850
490.0
1,300 (current)
1.231
450
0.00160
1.100
970.0
Find. OOIP ($N$) and cumulative water influx ($W_e$) at the current pressure (1,300 psi).
Approach. Cast the material balance in Havlena–Odeh straight-line form, $F=N\,E_o+W_e$, and combine it with the Pot (small, instantaneous-equilibrium) aquifer model $W_e=k\,\Delta p$ to get a straight line in $F/E_o$ vs. $\Delta p/E_o$, whose intercept is $N$ and slope is the aquifer constant $k$.
Underlying-drive functions.
$$F=N_pB_o+(G_p-N_pR_s)B_g,\qquad E_o=(B_o-B_{oi})+(R_{si}-R_s)B_g$$
with $B_{oi}=1.316$, $R_{si}=650$ (both at $p_i=3{,}000$ psi, above $p_b$).
Evaluate $F$ and $E_o$ at each pressure.
Havlena–Odeh terms
$P$
$E_o$ (rb/STB)
$F$ (res bbl)
$\Delta p=p_i-p$
$F/E_o$
$\Delta p/E_o$
2,500
0.0080
121,808
500
15,226,000
62,500
1,500
0.1250
1,140,475
1,500
9,123,800
12,000
1,300
0.2350
2,114,100
1,700
8,996,170
7,234
The 2,500 psi (bubble-point) row is excluded from the regression: $E_o$ there is only 0.008, so any small data error is amplified enormously in $F/E_o$ (its own value, 15.2 million, is wildly out of line with the other two rows) – the standard, well-documented Havlena–Odeh instability near $p_b$.
Fit the straight line through the 1,500 and 1,300 psi points. With the Pot aquifer model $W_e=k\,\Delta p$, the material balance becomes $F/E_o=N+k(\Delta p/E_o)$, a straight line in $(\Delta p/E_o,\,F/E_o)$:
$$k=\frac{8{,}996{,}170-9{,}123{,}800}{7{,}234-12{,}000}\approx 26.8\ \text{res bbl/psi}$$
$$N=\left(\frac{F}{E_o}\right)_{1{,}300}-k\left(\frac{\Delta p}{E_o}\right)_{1{,}300}=8{,}996{,}170-26.8(7{,}234)$$
$$\boxed{N\approx 8.80\text{ MMSTB (OOIP)}}$$
Cumulative water influx at the current pressure.
$$W_e(1{,}300\text{ psi})=k\,\Delta p=26.8\times1{,}700$$
$$\boxed{W_e\approx 45{,}525\text{ res bbl}}$$
