24-Pet-B5 Reservoir Mechanics · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
EGBC National Exam — Petroleum Engineering, 2016-May. 3 hours, closed book, non-communicating calculator. This paper's own cover page reads “98-Pet-B5, Well Testing,” not Reservoir Mechanics — every question below is pressure-transient/well-test analysis. NOTES items 4/5 state that five (5) questions constitute a complete exam and only the first five as they appear are marked; all six questions on the paper are solved in full below. Three of the six questions (Q3, Q4, Q6) are chart-reading questions built around semilog/log-log plots with no printed data table for Q3/Q4; every value read from those charts is flagged check where it feeds a boxed result. Q6 ships a short printed data table for its early-time linear-flow fit; its flow-regime identification uses the full log-log pressure-change/derivative plot.
Reference texts: Lee, J., Well Testing, SPE Textbook Series Vol. 1 (diffusivity equation, radial flow, wellbore storage, superposition); Earlougher, R.C., Advances in Well Test Analysis, SPE Monograph Vol. 5 (Horner analysis, interference/pulse tests, reservoir-limit tests); Bourdet, D., Well Test Analysis: The Use of Advanced Interpretation Models, Elsevier (derivative diagnostic plot, flow-regime identification); Cinco-Ley, H. & Samaniego, F., “Transient Pressure Analysis for Fractured Wells,” JPT, 1981 (infinite-conductivity vertical fracture linear flow).
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.
(a) Wellbore storage. After a well is opened or shut in, the sandface flow rate lags the surface-controlled rate for a period because the wellbore fluid itself must first fill or deplete (or a gas/liquid interface must move) to accommodate the change — described by the wellbore storage coefficient $C=\Delta V/\Delta p$ (bbl/psi). On a log-log plot of $\Delta p$ vs. $t$ it produces the characteristic early-time unit-slope (45°) line before formation flow can be diagnosed.
(b) Pulse test. A multi-well interference test in which the active well is alternately produced and shut in in a series of short, regular pulses; the small pressure response at an observation well, together with the time lag and amplitude of each pulse, is matched to type curves to estimate interwell permeability-thickness and porosity-compressibility (i.e., reservoir continuity/communication) between the two wells.
(c) Diffusivity equation. The governing PDE for slightly-compressible, single-phase radial flow in porous media, combining Darcy's law, mass conservation and fluid/rock compressibility: $$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial p}{\partial r}\right)=\frac{\phi\mu c_t}{0.0002637k}\frac{\partial p}{\partial t}$$ Its solutions (line-source $E_i$, log approximation, bounded-reservoir forms) underlie every drawdown/buildup/interference analysis method used elsewhere in this exam.
(d) Back pressure test. A multi-rate deliverability test (chiefly for gas wells) in which the well is produced at several successive stabilized flow rates and the corresponding stabilized $p_{wf}$ is recorded at each; the results are fit to the empirical deliverability equation $q_g=C(p_R^2-p_{wf}^2)^n$ to establish the well's absolute open-flow (AOF) potential and the C, n deliverability constants.
(e) Non-Darcy flow. At high velocities near the wellbore (common in high-rate gas wells) the pressure drop no longer varies linearly with rate; an inertial (turbulent) term must be added to Darcy's law, giving the rate-dependent-skin form $\Delta p\propto q+Dq^2$, where $D$ is the non-Darcy flow coefficient — it manifests as an apparent skin that increases with flow rate.
(f) Fall off test. The injection-well analogue of a pressure buildup test: an injector is shut in after a period of fluid injection and the pressure “falls off” toward reservoir pressure; analyzed with a Horner-type plot (injection time in place of production time) to obtain injectivity-zone permeability, skin, and reservoir pressure, and to detect a moving injection-front radius.
(g) Line source approximation. Treats the wellbore as a line of zero radius so that the exact radial-diffusivity solution reduces to the exponential-integral form $p_D=0.5[-Ei(-1/4t_D)]$, valid at any $t_D$; for $t_D>100$ this simplifies further to the familiar logarithmic form $p_D=0.5[\ln t_D+0.809]$, the basis of standard semilog (MDH/Horner) straight-line analysis.
(h) Superposition principle. Because the diffusivity equation is linear, the pressure response to a sequence of rate changes (or to multiple wells) can be built up by adding the individual constant-rate line-source solutions, each started at its own time and scaled by its own rate change — the basis for variable-rate drawdown analysis, multi-well interference, and the Horner buildup superposition-in-time construction.
(i) Pseudo radial flow. The late-time flow regime in a well with a finite-conductivity feature (fracture, horizontal section, multi-layer completion) once the pressure transient has moved far enough into the reservoir that the near-well geometry no longer matters and the flow lines again converge radially toward an effective wellbore — the log-log pressure derivative flattens to the same 0.5 constant used for ordinary (unfractured) infinite-acting radial flow.
(j) Formation damage. A reduction in near-wellbore permeability (drilling-fluid invasion, fines migration, scale, wettability alteration) relative to the undisturbed formation, quantified by a positive skin factor $S$ and an additional pressure drop $\Delta p_{skin}=141.2q\mu B_o S/(kh)$; it is diagnosed from a well test (semilog/Horner skin equation) and commonly treated by acid or solvent stimulation.
