Volumetrics and uncertainty

By week 5 the team has a gross rock volume, pool-average rock properties and a formation volume factor. Multiplying them gives one number for the oil in place. Ethan's response to the first draft is a question: “How sure are you?” The sponsor will not decide on a single number. It needs to know the range of outcomes, what drives the range, and how the range changes with the development option. This lesson turns the static model into a probabilistic estimate of oil in place and recoverable oil, identifies the inputs that matter most, and classifies the volumes under the resources framework.

The deterministic estimate

The volumetric oil in place is the pore volume filled with oil, divided by the formation volume factor to convert it to stock-tank volume:

where the gross rock volume is in cubic metres, giving N in stock-tank cubic metres. With the best estimates from Lessons 3 and 4, a gross rock volume of 50.4 million m³, net-to-gross of 0.75, porosity of 8.3 percent, water saturation of 14 percent and Boi of 1.446, the deterministic best estimate is 1.866 million m³, about 11.7 million barrels.

A deterministic estimate can be made for low, best and high cases by choosing low, best and high values of each input. That approach is common and quick, but it has a weakness: multiplying all the low values together gives a case that is far less likely than the low end of any one input, because it requires every input to be low at once. The probabilistic method avoids that by treating each input as a distribution and combining them by simulation.

Choosing input distributions

Each input distribution should express what the team actually knows, with its shape and range justified from the data. The team uses the following.

Two points from the table deserve emphasis. First, the porosity distribution describes the uncertainty in the pool average, which is much narrower than the spread of individual log samples. Using the sample spread would exaggerate the range of oil in place enormously, because a pool average over millions of cubic metres of rock cannot be as variable as one sample. Second, the inputs are treated as independent. In reality some are correlated: in this dolomite, porosity and water saturation are negatively correlated, because better rock holds less water. Ignoring a negative correlation widens the range slightly, which is conservative; ignoring a positive correlation would understate it. The team tests the effect and finds it small here.

Monte Carlo simulation

Monte Carlo simulation draws one value of each input at random from its distribution, calculates the oil in place, and repeats many thousands of times. The resulting values form the distribution of oil in place. By the industry convention, P90 is the value that has a 90 percent probability of being equalled or exceeded, P50 the median, and P10 the value with only a 10 percent probability of being exceeded. For recoverable volumes the team multiplies each trial's oil in place by a recovery factor drawn from a distribution for each recovery mechanism: natural depletion with an uncertain aquifer, between 12 and 34 percent with 22 percent most likely, and waterflooding from the start of production, between 30 and 45 percent with 38 percent most likely. These ranges come from analogous Nisku and Red River pools and will be refined by the reservoir engineering in Lessons 6 to 8.

# Deterministic and Monte Carlo oil in place for Spruce Ridge, and recoverable volumes for two recovery mechanisms.
import random
SI = True      # True: Canadian case (m3), False: US case (STB)
random.seed(499)
tri = random.triangular                          # triangular(low, high, mode)
if SI:
    c, unit = 1.0, "million m3"                  # GRV in m3
    grv_b, ntg_b, phi_b, sw_b, bo_b = 50.4e6, 0.75, 0.083, 0.14, 1.446
    draw_grv = lambda: tri(40e6, 64e6, 49e6)
    draw_ntg = lambda: tri(0.62, 0.86, 0.75)
    draw_phi = lambda: max(0.05, random.gauss(0.083, 0.007))
    draw_sw = lambda: tri(0.11, 0.21, 0.14)
    draw_bo = lambda: random.uniform(1.43, 1.46)
else:
    c, unit = 7758.0, "million STB"              # GRV in acre-ft; 7,758 bbl per acre-ft
    grv_b, ntg_b, phi_b, sw_b, bo_b = 28500.0, 0.62, 0.115, 0.22, 1.477
    draw_grv = lambda: tri(22000, 36000, 28000)
    draw_ntg = lambda: tri(0.50, 0.72, 0.62)
    draw_phi = lambda: max(0.06, random.gauss(0.115, 0.009))
    draw_sw = lambda: tri(0.17, 0.30, 0.22)
    draw_bo = lambda: random.uniform(1.46, 1.49)
ooip = lambda grv, ntg, phi, sw, bo: c * grv * ntg * phi * (1 - sw) / bo
det = ooip(grv_b, ntg_b, phi_b, sw_b, bo_b)
print(f"deterministic best estimate: {det / 1e6:.3f} {unit}")

n = 20000
trials = []
for _ in range(n):
    N = ooip(draw_grv(), draw_ntg(), draw_phi(), draw_sw(), draw_bo())
    rf_dep = tri(0.12, 0.34, 0.22)              # natural depletion with an uncertain aquifer
    rf_wf = tri(0.30, 0.45, 0.38)               # waterflood from the start
    trials.append((N, N * rf_dep, N * rf_wf))
def pct(vals, p):                                # P90 = value exceeded with 90 percent probability
    s = sorted(vals); return s[int((1 - p / 100) * (len(s) - 1))]
for i, name in enumerate(("oil in place", "recoverable, depletion", "recoverable, waterflood")):
    v = [t[i] for t in trials]
    print(f"{name:24s} P90 {pct(v, 90) / 1e6:6.3f}  P50 {pct(v, 50) / 1e6:6.3f}  P10 {pct(v, 10) / 1e6:6.3f}  mean {sum(v) / n / 1e6:6.3f}  ({unit})")
v = [t[0] for t in trials]
print(f"P10/P90 ratio for oil in place: {pct(v, 10) / pct(v, 90):.2f}")
# Which input matters most? Swing each input from its low to high value with the others at their best estimates.
swings = {"gross rock volume": (draw_grv, 0), "net-to-gross": (draw_ntg, 1), "porosity": (draw_phi, 2), "water saturation": (draw_sw, 3), "formation volume factor": (draw_bo, 4)}
base = [grv_b, ntg_b, phi_b, sw_b, bo_b]
for name, (draw, k) in swings.items():
    s = sorted(draw() for _ in range(4000)); lo, hi = s[400], s[3600]  # 10th and 90th percentiles of the input
    a = base.copy(); a[k] = lo; b = base.copy(); b[k] = hi
    print(f"{name:24s} swing {abs(ooip(*b) - ooip(*a)) / 1e6:5.3f} {unit}")

deterministic best estimate: 1.866 million m3
oil in place             P90  1.505  P50  1.825  P10  2.211  mean  1.845  (million m3)
recoverable, depletion   P90  0.291  P50  0.410  P10  0.557  mean  0.418  (million m3)
recoverable, waterflood  P90  0.551  P50  0.686  P10  0.853  mean  0.695  (million m3)
P10/P90 ratio for oil in place: 1.47
gross rock volume        swing 0.498 million m3
net-to-gross             swing 0.325 million m3
porosity                 swing 0.401 million m3
water saturation         swing 0.121 million m3
formation volume factor  swing 0.031 million m3

The P50 oil in place is 1.825 million m³, with P90 and P10 of 1.505 and 2.211 million m³: a P10/P90 ratio of 1.47, which is typical for a discovery with two wells and good 3D seismic. The P50 is 2 percent below the deterministic best estimate of 1.866 million m³, because the skewed water saturation distribution pulls the median down. The recoverable volumes show what the choice of recovery mechanism is worth: the P50 under depletion is 0.41 million m³, under waterflood 0.69 million m³. The waterflood P90, 0.55 million m³, is almost the same as the depletion P10 of 0.56 million m³. The swings show that gross rock volume dominates the uncertainty, followed by porosity and net-to-gross; water saturation and the formation volume factor matter little.

The tornado result has a direct practical use. If the team wants to narrow the range of oil in place, the data that would help most are those that reduce the gross rock volume uncertainty: another well on the flank, or a better velocity model. More porosity logs from the crest would barely change the range. That conclusion goes into the data acquisition plan as an argument for drilling the first development well where it also tests the flank structure.

Classifying the volumes

Under COGEH and the Petroleum Resources Management System, Spruce Ridge's oil is a discovered accumulation, because wells have found it and tested it at commercial rates. But it cannot yet be booked as reserves. Reserves require a defined development project that is commercially justified, that the company intends to carry out, and that is reasonably expected to be approved and started within a reasonable time, normally five years. None of that exists yet: the development plan is what the team is writing. The recoverable volumes are therefore contingent resources, in the development pending subclass, because the contingencies, the choice of development option, sanction by the company and regulatory approval, are being actively resolved. Once Prairie Crest sanctions a plan, the volumes for that plan can move to reserves, divided into proved, probable and possible categories.

The classification is not a formality. Lenders do not lend against contingent resources, and the company's reported reserves will not include Spruce Ridge until the plan is approved. Daniel notes in the report that the choice of development option will also change the reserves that can be booked. A plan that relies on depletion would support smaller reserves, with more of the volume left as contingent resources awaiting a later waterflood decision.

The free water level as a hidden input

One input does not appear in the table of distributions, because it was fixed by the pressure data: the free water level. It is worth checking how much it matters. The reef's area at the free water level is about 131 hectares, so moving the level by 1 m changes the gross rock volume by about 1.3 million m³, or 2.6 percent. The pressure-derived level has an uncertainty of perhaps ±1.5 m, from the scatter of the gradient fits, which is small compared with the depth-conversion uncertainty at the flanks, and the team leaves it fixed. Had the supercharged test been kept, the level would have moved 19 m deeper and the gross rock volume would have grown by roughly half, far outside the range of any distribution in the table. That is the practical meaning of the data review in Lesson 2: some errors are not uncertainties to be sampled but mistakes to be removed before sampling begins.

Documenting the estimate

Graham Whitlock's first request when he reviews the volumetrics is for the documentation. He wants to see the maps and their versions, the log cutoffs and the wells used, the source and shape of every distribution and the reason for it, any dependencies included or ignored, the number of trials and the random seed, the results with the deterministic check, and the inputs that drive the range. An estimate that another engineer cannot reproduce from its documentation will not survive a reserves audit, however good the geology behind it. Connor writes a two-page volumetrics note in that form and attaches the code. It takes an afternoon, and it saves a week of questions when the sponsor's reserves group picks up the work in the define stage.

The note ends with the statement that matters most for the decision. Recoverable volume is far more sensitive to the recovery mechanism than to anything the volumetrics can resolve: the waterflood P50 is about two-thirds larger than the depletion P50, while the full P90 to P10 range of oil in place is about two-fifths of its P50 value. Whatever the geologists and petrophysicists do next, the development choice will matter more.