24-Pet-A3 Fundamental Reservoir Engineering · December 2018
Question 7 of 7: Decline-Curve Analysis — Harmonic Decline from a Rate-vs-Cumulative Plot
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
17-Pet-A3 — Fundamental Reservoir Engineering · National Exams, December 2018 · 3 hours, closed book, approved Casio/Sharp calculator only · five (5) questions constitute a complete exam paper (the first five as they appear in the answer book are marked), all questions equal value, all parts of a multipart question equal weight.
Reference texts: Craft, B.C. & Hawkins, M.F., Applied Petroleum Reservoir Engineering, 3rd ed. (material balance, well testing, relative permeability, Darcy flow); Ahmed, T., Reservoir Engineering Handbook, 5th ed. (material balance, transient well testing, gas PVT, decline-curve analysis); Lyons, W.C. (ed.), Standard Handbook of Petroleum and Natural Gas Engineering, 3rd ed.; McCain, W.D., The Properties of Petroleum Fluids, 3rd ed. (PVT properties, Z-factor correlations).
Question 7: Decline-Curve Analysis — Harmonic Decline from a Rate-vs-Cumulative Plot (20 marks)
Given. Rate $q_t$ (MMSCFD) vs. cumulative production $G_p$ (MMMSCF), plotted with $q_t$ on a log axis and $G_p$ on a linear axis; eight points: (2.5, 9.8), (4.0, 8.0), (5.0, 7.0), (6.0, 6.0), (7.0, 5.2), (8.0, 4.5), (9.0, 3.8), (10.0, 3.0).
Find. The gas production rate after 8 years.
Approach. A straight line on a semilog plot of $q$ vs. $G_p$ (not vs. time) is the diagnostic signature of harmonic decline ($b=1$), since integrating $q=q_i/(1+Dt)$ gives $\ln q=\ln q_i-(D/q_i)G_p$. Least-squares fit $\ln q_t$ vs. $G_p$ to get $q_i$ and $D/q_i$, recover $D$ in 1/day using the unit link between $G_p$ (MMMSCF) and $q\,dt$ (MMSCFD$\times$day = MMSCF), then evaluate $q(t=8\ \text{yr})$ directly from the harmonic decline equation.
Fig. 1: $q_t$ vs. $G_p$ on a semilog scale. The 8 points fall on a straight line ($R^2=0.993$), confirming harmonic decline; the fitted line is extrapolated to $t=8$ yr.
Least-squares fit of $\ln q_t$ vs. $G_p$. Regressing the 8 points gives $\boxed{\ln q_i=2.700\ (q_i=14.88\ \text{MMSCFD})}$ and slope $-D'=-0.1536\ \text{MMMSCF}^{-1}$ ($R^2=0.993$), i.e. $q=q_i\exp(-D'G_p)$ with $D'=0.1536\ \text{MMMSCF}^{-1}$.
Recover the time-based decline constant $D$. Since $G_p=\int q\,dt$, with $q$ in MMSCFD and $t$ in days, $G_p$ in MMSCF $=1000\times$ MMMSCF; matching the harmonic-decline integral $G_p=(q_i/D)\ln(q_i/q)$ to the fitted $G_p=\ln(q_i/q)/D'$ gives $D=q_iD'/1000=14.88(0.1536)/1000$, so $\boxed{D=2.286\times10^{-3}\ \text{day}^{-1}}$.
Rate at $t=8$ years. $t=8(365)=2920$ days. Harmonic decline: $q=\dfrac{q_i}{1+Dt}=\dfrac{14.88}{1+2.286\times10^{-3}(2920)}$, so $\boxed{q(t=8\ \text{yr})=1.94\ \text{MMSCFD}}$.
Cross-check. At $t=2920$ days, $G_p=(q_i/D)\ln(q_i/q)=13.26$ MMMSCF, well within the extrapolation range of the fitted line; substituting back into $q=q_i\exp(-D'G_p)$ reproduces the same $q=1.94$ MMSCFD.
Check: the source chart gives only 8 discrete $(G_p,q_t)$ pairs with no explicit time axis, so the decline TYPE (harmonic, $b=1$) is inferred from the straight-line fit on this semilog $q$-vs-$G_p$ plot, and the absolute time scale is recovered purely from unit consistency between $G_p$ (MMMSCF) and $q\,dt$ (MMSCFD$\times$days) — a genuinely digitized source chart with a time axis would let this be checked directly.
