Statistika untuk Quantifying Uncertainty dalam Reserve Estimation

Mengapa Statistika Penting dalam Migas?

Industri migas penuh dengan uncertainty:

• Reservoir properties (porosity, permeability, saturation)
• Volumetric calculations (area, thickness)
• Recovery factors
• Economic parameters (oil price, costs)

Statistika membantu kita:

• Quantify uncertainty
• Make probabilistic forecasts
• Communicate risk
• Support decision-making

Konsep Dasar Statistika

1. Probability Distribution

Normal Distribution: Cocok untuk parameters yang symmetric around mean:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Lognormal Distribution: Cocok untuk parameters yang always positive (porosity, permeability):

$f(x) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln x-\mu)^2}{2\sigma^2}}$

Triangular Distribution: Simple distribution dengan min, most likely, max values.

2. Measures of Central Tendency

• Mean (μ): Average value
• Median: Middle value (50th percentile)
• Mode: Most frequent value

3. Measures of Dispersion

• Standard Deviation (σ): Spread of data
• Variance (σ²): Square of standard deviation
• Coefficient of Variation (CV): σ/μ (normalized measure)

Reserve Estimation dengan Probabilistic Method

Deterministic vs Probabilistic

Deterministic:

• Single value estimate
• Uses "best estimate" parameters
• No uncertainty quantification

Probabilistic:

• Range of possible values
• Uses probability distributions
• Quantifies uncertainty with P10, P50, P90

Tiga Metode Estimasi

P90 (Proven/1P):

• 90% probability reserves ≥ this value
• Conservative estimate
• Low risk

P50 (Proven + Probable/2P):

• 50% probability reserves ≥ this value
• Best estimate
• Medium risk

P10 (Proven + Probable + Possible/3P):

• 10% probability reserves ≥ this value
• Optimistic estimate
• High risk

Volumetric Equation

Original Oil in Place (OOIP):

$OOIP = \frac{A \times h \times \phi \times (1-S_w)}{B_o} \times 7758$

Recoverable Reserves:

$Reserves = OOIP \times RF$

Di mana:

• A = Area (acres)
• h = Net pay thickness (ft)
• φ = Porosity (fraction)
• S_w = Water saturation (fraction)
• B_o = Formation volume factor (rb/stb)
• RF = Recovery factor (fraction)

Monte Carlo Simulation

Konsep

Monte Carlo adalah metode untuk:

1. Define probability distributions untuk input parameters
2. Random sampling dari distributions
3. Calculate output (reserves) untuk each sample
4. Analyze distribution of results

Step-by-Step Process

Step 1: Define Input Distributions

Example untuk reservoir X:

Parameter	Min	Most Likely	Max	Distribution
Area (acres)	800	1000	1200	Triangular
Thickness (ft)	25	35	50	Lognormal
Porosity	0.15	0.20	0.25	Normal
Sw	0.20	0.25	0.35	Triangular
Bo	1.15	1.20	1.25	Normal
RF	0.25	0.35	0.45	Triangular

Step 2: Run Simulation

Untuk 10,000 iterations:

1. Sample random value dari each distribution
2. Calculate OOIP dan Reserves
3. Store result

Step 3: Analyze Results

From output distribution:

• P90 = 4.2 MMstb
• P50 = 6.8 MMstb
• P10 = 10.5 MMstb
• Mean = 7.1 MMstb

Interpretation

Key insights:

• Range of uncertainty: 4.2 - 10.5 MMstb
• Expected value: 7.1 MMstb (mean)
• Risk assessment: P90 untuk conservative planning

Correlation dan Dependency

Independent vs Dependent Variables

Independent:

• Area dan porosity (usually)
• Thickness dan recovery factor

Dependent (Correlated):

• Porosity dan permeability (positive correlation)
• Water saturation dan porosity (negative correlation)

Handling Correlation

Correlation coefficient (ρ):

• ρ = +1: Perfect positive correlation
• ρ = 0: No correlation
• ρ = -1: Perfect negative correlation

Impact on uncertainty:

• Positive correlation → increases variance
• Negative correlation → decreases variance

Sensitivity Analysis

Tornado Diagram

Identify which parameters have biggest impact on reserves:

1. Vary each parameter ±20%
2. Calculate impact on reserves
3. Rank by magnitude of impact

Typical ranking untuk reserves:

1. Recovery factor (highest impact)
2. Area
3. Net pay thickness
4. Porosity
5. Water saturation
6. Bo (lowest impact)

Spider Plot

Shows how reserves change with each parameter variation.

Practical Application

Case Study: Field Development Decision

Scenario: Evaluating field development dengan uncertainty in reserves.

Data:

• P90 = 15 MMstb
• P50 = 25 MMstb
• P10 = 40 MMstb
• Development cost = $200 MM
• Oil price = $70/bbl
• Operating cost = $15/bbl

Analysis:

P90 Case (Conservative):

• Revenue = 15 × 70 = $1,050 MM
• Opex = 15 × 15 = $225 MM
• Capex = $200 MM
• Net = $625 MM
• NPV@10% ≈ $350 MM

P50 Case (Base):

• Net = $1,175 MM
• NPV@10% ≈ $650 MM

Decision: Even at P90, project is economic → Proceed with development.

Best Practices

1. Use appropriate distributions: Lognormal untuk permeability, triangular untuk simple estimates
2. Validate with analogs: Compare dengan offset fields
3. Consider correlations: Don't assume independence
4. Run enough iterations: Minimum 10,000 untuk stable results
5. Sensitivity analysis: Understand key drivers
6. Document assumptions: Transparency is critical

Common Pitfalls

1. Garbage in, garbage out: Bad input distributions → meaningless results
2. Ignoring correlations: Can significantly underestimate uncertainty
3. Over-confidence: P50 is not guaranteed
4. Misuse of mean: Use P50 (median) untuk reserves booking, not mean

Kesimpulan

Statistika adalah essential tool untuk petroleum engineers dalam managing uncertainty. Probabilistic methods memberikan more realistic view of reserves dan support better decision-making. Memahami konsep probability distributions, Monte Carlo simulation, dan sensitivity analysis akan significantly improve quality of reserve estimates dan economic evaluations.