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24-Pet-B2 Oil and Gas Evaluation and Economics · December 2015

Question 5 of 7: Harmonic Decline — Economic-Limit Production Life and Reserves

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams December 2015, 98-Pet-B2, Natural Gas Engineering — 3 hours, closed book (non-communicating calculator permitted), 7 questions of 20 marks each (only the first five as they appear in the answer book are officially marked). All 7 questions are solved, not just the first five.

Reference texts: Katz et al., Handbook of Natural Gas Engineering; Lee & Wattenbarger, Gas Reservoir Engineering (SPE Textbook Series Vol. 5); Ahmed, Reservoir Engineering Handbook, 5th ed.; Mohitpour et al., Pipeline Design and Construction, 3rd ed. (ASME Press); McCain, The Properties of Petroleum Fluids, 3rd ed.

Question 5: Harmonic Decline — Economic-Limit Production Life and Reserves (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Seven ($q$, $G_p$) pairs digitized from the field chart (q in MSCFD, log scale; $G_p$ in MMSCF): (10, 78), (19, 55), (28, 42), (38, 32), (49, 25), (63, 19), (77, 14). Economic limit $q_{econ}=5$ MSCFD.

Find. Production life $t$ (days) and cumulative production $G_p$ at the economic limit.

Approach. A straight-line relationship between $G_p$ and $\ln q$ is the diagnostic signature of harmonic decline ($q=q_i/(1+Dt)$ gives $G_p=(q_i/D)\ln(q_i/q)$, from the formula sheet) — a least-squares fit of the seven points against $\ln q$ recovers both $q_i$ and $D$ directly, after which the harmonic-decline time equation and the same fitted line both evaluate at $q=5$ MSCFD.

  1. Linear fit vs. $\ln q$. Least-squares regression of $G_p$ (in MSCF) on $\ln q$ across the seven points gives $G_p=A-B\ln q$ with $B=31{,}434$ MSCF and $A=148{,}670$ MSCF (equivalently $B/q_i$ is the fitted slope’s reciprocal relationship below).
  2. Recover $q_i$ and $D$. Since $B=q_i/D$ and $A=B\ln q_i$: $q_i=e^{A/B}$: $\boxed{q_i=112.0\ \text{MSCFD}}$. Then $D=q_i/B=112.0/31{,}434$: $\boxed{D=3.564\times10^{-3}\ \text{day}^{-1}}$ (the two fit parameters, not a separate rate-vs-time chart, are enough to pin down both harmonic-decline constants).
  3. Production life at the economic limit. From $q=q_i/(1+Dt)$: $t=(q_i/q_{econ}-1)/D=(112.0/5-1)/3.564\times10^{-3}$: $\boxed{t=6004\ \text{days}}$ ($\approx16.4$ years).
  4. Cumulative production at the economic limit. Reading the same fitted line at $q=5$ MSCFD: $G_p=(A-B\ln5)/1000$: $\boxed{G_p=97.7\ \text{MMSCF}}$.
1 10 100 0 20 40 60 80 100 Gas production rate, q (MSCFD) [log scale] Cumulative gas production, Gp (MMSCF) Field data Gp=97.7 MMSCF @ q=5 MSCFD
Fig. 2 — Cumulative gas production vs. rate (log scale), with the harmonic-decline fit $G_p=(q_i/D)\ln(q_i/q)$ extrapolated to the 5 MSCFD economic limit.
QuantityValue
Fitted initial rate, $q_i$112.0 MSCFD
Decline rate, $D$$3.564\times10^{-3}$ day$^{-1}$
Production life at $q_{econ}=5$ MSCFD6004 days (≈16.4 yr)
Cumulative production at economic limit97.7 MMSCF
Check: the seven ($q$, $G_p$) pairs were digitized by eye off the source’s semilog scatter chart (log-$q$ axis from 1 to >100 MSCFD, linear $G_p$ axis 0–100+ MMSCF) — readings are estimates to the nearest whole MSCFD/MMSCF grid division, consistent with an "estimate" exam question. The fitted $q_i=112$ MSCFD is an extrapolated curve-fit parameter (the actual field data only extends down to $q\approx10$ MSCFD), not a claim that the field was ever produced at 112 MSCFD.