Question 7 of 7: Capillary Pressure, Rock Typing, and Multiphase Pore Occupancy

Question 7: Capillary Pressure, Rock Typing, and Multiphase Pore Occupancy (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. Four sketches: (i) three $P_c$–$S_w$ curves labelled i, ii, iii; (ii) a dipping four-layer reservoir ($k_1$–$k_4$) each showing its own oil/WOC/water split relative to a common free-water level (FWL); (iii) a pore-size (diameter) frequency distribution for a water-wet rock split into three regions i, ii, iii; (iv) a hysteresis loop between drainage and imbibition $P_c$ curves.

Find. (a1) permeability ranking from the $P_c$ curves; (a2) permeability ranking from the WOC positions; (b) which phase occupies which pore-size region; (c) label the four features of the drainage/imbibition loop.

Approach. All four parts use the same governing idea — capillary pressure at a given saturation is inversely related to pore-throat size (and hence to permeability): larger, better-connected pores need less capillary pressure to displace the wetting phase, giving a curve shifted toward the $P_c$-axis with a lower entry pressure and lower irreducible wetting-phase saturation.

Part (a1) — ranking curves i, ii, iii. Curve iii sits furthest left (lowest capillary entry pressure and lowest irreducible water saturation), curve i furthest right (highest entry pressure, highest irreducible $S_w$), with ii in between.

Since capillary entry pressure and irreducible water saturation both fall as pore throats get larger (i.e. as permeability rises), the ranking from higher to lower permeability is $\boxed{\text{iii}>\text{ii}>\text{i}}$.

Part (a2) — ranking $k_1$–$k_4$ from WOC position. In a stack of layers sharing one free-water level, the apparent water-oil contact (the depth where $S_w$ is judged 50%) sits ABOVE the FWL by a height set by the layer's own capillary entry pressure via $P_c=(\rho_w-\rho_o)g h$: a high-permeability layer has a low entry pressure, so its WOC lies very close to the FWL (a thin, sharp transition zone); a low-permeability layer needs a much taller oil column before capillary forces are overcome, so its WOC (and the transition zone above it) sits noticeably higher, structurally, above the FWL. Reading the sketch this way — the lower/deeper layers ($k_4$, then $k_3$) show their WOC/water label appearing almost immediately below the oil zone, close to the FWL, while the shallowest layer ($k_1$) shows no WOC/water at all within the same span, i.e. its transition zone is thickest and its WOC lies comparatively far from FWL — the ranking from higher to lower permeability is $\boxed{k_4>k_3>k_2>k_1}$.

Part (b) — three-phase pore occupancy (water-wet rock). In a water-wet rock, water is the most strongly wetting phase and gas is the most strongly non-wetting, with oil of intermediate wettability. The wetting phase preferentially occupies the smallest pores/throats (held there by the strongest capillary forces and as a film on grain surfaces); the most non-wetting phase occupies the largest, most easily entered pores. Mapping onto the distribution (i = smallest pore diameters, ii = the mid-size majority around the peak, iii = largest pore diameters): $\boxed{\text{i}=\text{water},\ \text{ii}=\text{oil},\ \text{iii}=\text{gas}}$.

Part (c) — drainage/imbibition hysteresis loop. The upper branch, running from $(S_w=1,P_c=0)$ up to a sharp rise, is the (1) drainage curve (non-wetting phase displacing water, $S_w$ decreasing); its steep initial rise marks the (3) threshold (critical/entry) pressure, the minimum $P_c$ needed for the non-wetting phase to first enter the largest connected pores. The lower branch, traced as water re-invades and $S_w$ increases back toward 1, is the (2) imbibition curve; it does NOT return all the way to $S_w=1$ at $P_c=0$ but stops short, at the gap marked (4) residual non-wetting-phase (oil) saturation — the oil saturation trapped and immobilized in pore throats that water could not displace during spontaneous/forced imbibition.