menu_book Chapter 10ReservesHavlena-Odehvan Everdingen-Hurst

Material-Balance OOIP + Aquifer

Estimate original oil in place and spot a water drive — the #1 reserves-overstatement trap — with the Havlena-Odeh straight line.

What it does

Builds the Havlena-Odeh material-balance straight line, then shows how ignoring an aquifer overstates OOIP and how the van Everdingen-Hurst unsteady-state water-influx model corrects it back to the true value.

How it works

1Compute oil expansion (Eo) and underground withdrawal (F) from the pressure and production history.
2Model water influx with van Everdingen-Hurst dimensionless influx and superposition.
3Plot F vs. Eo — an upward curve is the tell-tale of an active aquifer.
4Regress OOIP ignoring the aquifer (overstated) and after subtracting the influx (recovered).
5See how much reserves you would have overbooked — change the aquifer strength and re-run.

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Run it live

Runs entirely in your browser — no install. Edit any value and press Run to see the result change. This is the whole project; read it top to bottom.

material-balance-ooip.pystarting Python…

import numpy as np
import matplotlib.pyplot as plt

# ── van Everdingen-Hurst dimensionless influx WeD(tD), infinite radial aquifer ──
def WeD(tD):
    tD = np.asarray(tD, float); out = np.zeros_like(tD)
    s = np.sqrt(np.clip(tD, 1e-12, None))
    small, large = tD < 0.01, tD > 100
    mid = ~small & ~large
    out[small] = 2*np.sqrt(tD[small]/np.pi)
    out[mid] = ((1.12838*s[mid] + 1.19328*tD[mid] + 0.269872*tD[mid]*s[mid]
                 + 0.00855294*tD[mid]**2) /
                (1 + 0.616599*s[mid] + 0.0413008*tD[mid]))
    out[large] = (2.02566*tD[large] - 4.29881)/np.log(tD[large])
    return out

# ── Reservoir + aquifer — tweak these ───────────────────────────────────
Pi, N_true, Boi = 3000.0, 80e6, 1.25             # psia, STB (the answer), rb/STB
k, phi, mu, ct, h, ro, f = 150., .18, 1.0, 7e-6, 30., 2000., 0.055
B = 1.119*phi*ct*ro**2*h*f                        # aquifer constant, RB/psi
tconv = 6.328e-3*k/(phi*mu*ct*ro**2)              # tD per day

# ── Pressure history, annual steps (P[0] = Pi) ──────────────────────────
t = np.arange(0, 11)*365.0                        # days
P = Pi - np.array([0,60,140,230,320,400,470,530,580,620,650], float)

# ── Cumulative water influx by van Everdingen-Hurst superposition ───────
We = np.zeros(len(P))
for n in range(1, len(P)):
    dPj = P[:n] - P[1:n+1]                         # pressure drop over interval j
    We[n] = B*np.sum(dPj*WeD(tconv*(t[n]-t[:n])))

# ── Oil expansion Eo, and the underground withdrawal F = N*Eo + We ──────
Bo = Boi*(1 + 12e-6*(Pi - P)); Eo = Bo - Boi
rng = np.random.default_rng(1)
F = (N_true*Eo + We) * (1 + rng.normal(0, 0.01, len(P)))

# ── Havlena-Odeh: regress OOIP with and without the aquifer ─────────────
m = Eo > 0
N_naive = np.polyfit(Eo[m], F[m], 1)[0]           # ignore influx
N_rec   = np.polyfit(Eo[m], (F - We)[m], 1)[0]    # subtract VEH influx
print(f"True OOIP                  : {N_true/1e6:6.1f} MMSTB")
print(f"Ignoring the aquifer       : {N_naive/1e6:6.1f} MMSTB  ({100*(N_naive/N_true-1):+.0f}%)")
print(f"With van Everdingen-Hurst  : {N_rec/1e6:6.1f} MMSTB  (recovered)")
print(f"Cumulative water influx    : {We[-1]/1e6:6.2f} MMrb")

# ── Plot influx + Havlena-Odeh straight line ────────────────────────────
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
ax[0].plot(P, We/1e6, "o-", color="navy"); ax[0].invert_xaxis(); ax[0].grid(alpha=0.3)
ax[0].set_xlabel("Reservoir pressure (psia)"); ax[0].set_ylabel("Cumulative We (MMrb)")
ax[0].set_title("Water influx (van Everdingen-Hurst)")
ax[1].scatter(Eo[m], F[m]/1e6, c="firebrick", label="F vs Eo  (curves up = aquifer)")
ax[1].scatter(Eo[m], (F - We)[m]/1e6, c="green", label="(F - We) vs Eo  ->  straight")
xx = np.linspace(0, Eo[m].max(), 20); ax[1].plot(xx, N_rec*xx/1e6, "g--", lw=1)
ax[1].set_xlabel("Eo (rb/STB)"); ax[1].set_ylabel("F (MMrb)")
ax[1].legend(fontsize=8); ax[1].grid(alpha=0.3); ax[1].set_title("Havlena-Odeh straight line")
fig.suptitle(f"Material Balance OOIP = {N_rec/1e6:.0f} MMSTB (aquifer-corrected)", fontweight="bold")
plt.tight_layout(); plt.show()

school

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