Kalkulus dalam Decline Curve Analysis

Pengenalan Decline Curve Analysis (DCA)

Decline Curve Analysis adalah metode untuk memprediksi future production berdasarkan historical production data. Metode ini heavily relies on calculus concepts.

Mengapa Penting?

• Reserve estimation: Menghitung recoverable reserves
• Economic evaluation: NPV, IRR calculations
• Production planning: Facility sizing, pipeline capacity
• Well performance: Identify underperforming wells

Konsep Kalkulus yang Digunakan

1. Turunan (Derivative)

Decline rate adalah turunan dari production rate terhadap waktu:

$D = -\frac{1}{q}\frac{dq}{dt}$

Di mana:

• D = Decline rate (1/time)
• q = Production rate (bbl/day)
• t = Time

Interpretasi:

• D = 0.1/year → production turun 10% per tahun
• Semakin besar D, semakin cepat decline

2. Integral

Cumulative production adalah integral dari rate terhadap waktu:

$N_p = \int_0^t q(t) \, dt$

Ultimate recovery (EUR) adalah integral hingga economic limit:

$EUR = \int_0^{t_{econ}} q(t) \, dt$

Tiga Jenis Decline

1. Exponential Decline

Karakteristik:

• Decline rate konstan (D = constant)
• Paling umum untuk mature wells
• Paling konservatif untuk forecasting

Persamaan Rate:

$q(t) = q_i e^{-Dt}$

Persamaan Cumulative:

$N_p = \frac{q_i - q}{D}$

Contoh:

• q_i = 500 bbl/day
• D = 0.15/year
• Setelah 2 tahun: q = 500 × e^(-0.15×2) = 370 bbl/day

2. Harmonic Decline

Karakteristik:

• Decline rate menurun seiring waktu
• Lebih optimistic dari exponential
• Cocok untuk naturally fractured reservoirs

Persamaan Rate:

$q(t) = \frac{q_i}{1 + bDt}$

dengan b = 1 (harmonic)

Persamaan Cumulative:

$N_p = \frac{q_i}{bD} \ln\left(\frac{q_i}{q}\right)$

3. Hyperbolic Decline

Karakteristik:

• Paling general (exponential dan harmonic adalah special cases)
• Parameter b menentukan curvature (0 < b < 1)
• Paling flexible untuk curve fitting

Persamaan Rate:

$q(t) = \frac{q_i}{(1 + bDt)^{1/b}}$

Persamaan Cumulative:

$N_p = \frac{q_i^b}{D(1-b)}\left[q_i^{1-b} - q^{1-b}\right]$

Derivasi Persamaan (Calculus in Action)

Exponential Decline Derivation

Starting from definition:

$D = -\frac{1}{q}\frac{dq}{dt}$

Rearrange:

$\frac{dq}{q} = -D \, dt$

Integrate both sides:

$\int_{q_i}^q \frac{dq}{q} = -D \int_0^t dt$

$\ln(q) - \ln(q_i) = -Dt$

$\ln\left(\frac{q}{q_i}\right) = -Dt$

$q = q_i e^{-Dt}$

Cumulative Production

$N_p = \int_0^t q(t) \, dt = \int_0^t q_i e^{-Dt} \, dt$

$N_p = q_i \left[-\frac{1}{D}e^{-Dt}\right]_0^t$

$N_p = \frac{q_i}{D}(1 - e^{-Dt})$

Atau dalam terms of q:

$N_p = \frac{q_i - q}{D}$

Aplikasi Praktis

Case Study: Sumur X

Data:

• Initial rate (q_i) = 800 bbl/day
• Current rate (q) = 400 bbl/day
• Time elapsed = 3 years
• Economic limit = 50 bbl/day

Step 1: Determine decline type

Plot log(q) vs time:

• Linear → Exponential
• Curved → Hyperbolic

Assume exponential decline.

Step 2: Calculate decline rate

$D = \frac{1}{t}\ln\left(\frac{q_i}{q}\right) = \frac{1}{3}\ln\left(\frac{800}{400}\right) = 0.231/year$

Step 3: Forecast future production

At t = 5 years:

$q(5) = 800 \times e^{-0.231 \times 5} = 252 \, \text{bbl/day}$

Step 4: Calculate remaining reserves

Time to economic limit:

$t_{econ} = \frac{1}{D}\ln\left(\frac{q_i}{q_{econ}}\right) = \frac{1}{0.231}\ln\left(\frac{800}{50}\right) = 12.1 \, \text{years}$

Remaining reserves (from current):

$N_p = \frac{400 - 50}{0.231} = 1,515 \, \text{Mbbl}$

Advanced Topics

Type Curve Matching

Menggunakan Fetkovich type curves untuk:

• Identify decline type
• Estimate reserves
• Predict future performance

Dimensionless variables:

$q_D = \frac{q}{q_i}$

$t_D = Dt$

Arps Equation Limitations

Cautions:

1. Tidak cocok untuk transient flow (early time)
2. Overpredicts reserves untuk shale/tight reservoirs
3. Ignores operational changes (workovers, stimulation)

Modern approaches:

• Stretched exponential (SEPD)
• Power law exponential (PLE)
• Duong model untuk unconventional

Tips Praktis

1. Use semi-log plot: Exponential decline akan linear di semi-log
2. Check for trend changes: Workover, artificial lift changes
3. Conservative forecasting: Use exponential untuk reserves booking
4. Validate with analogs: Compare dengan offset wells
5. Update regularly: Re-forecast dengan new data

Kesimpulan

Decline curve analysis adalah perfect example bagaimana kalkulus diaplikasikan dalam petroleum engineering. Memahami derivation dan mathematical foundation akan membantu engineer membuat better forecasts, understand limitations, and communicate results with confidence.