### 2.1 Introduction

In this chapter, I describe how to characterize reservoir pore pressures that are under capillary and gravity equilibrium (e.g., Fig. 2.1). This is approximately the state of a geological reservoir prior to production. Pore pressure is commonly described with a pressure versus depth plot (Fig. 2.1b). The reservoir pressure ( ${u}_{g}$ and ${u}_{w}$ , Fig. 2.1) commonly, but not always, lies between the hydrostatic pressure ( ${u}_{h}$ ) and the lithostatic stress ( ${\sigma}_{v}$ ) (Fig. 2.1). Pore pressure is also characterized with overpressure ( ${u}^{*}$ ), and average equivalent density (mud weight) plots (Fig. 2.1c, d).

I illustrate the approach with an example from the Bullwinkle field in the Gulf of Mexico. Initially, I neglect the impact of capillary migration pressure (the minimum pressure necessary for the non-wetting phase to migrate through the rock). Later, I describe how to incorporate this effect. I close with a conceptual discussion of the saturation and pressure distribution within reservoirs at gravitational and capillary equilibrium.

### 2.2 Pore Pressure

Hydrostatic pressure $\left({u}_{h}\right)$ (Fig. 2.1b) is the pressure caused by a static column of water. It is commonly expressed by assuming a constant pore water density ( ${\rho}_{pw}$ ) referenced from the sea surface:$TV{D}_{ss}$ is the vertical depth from the sea surface. The hydrostatic pressure is often described with a seawater density ( ${\rho}_{sw}$ ) that is distinct from the pore water density ( ${\rho}_{pw}$ ):

${Z}_{wd}$
is the water depth, and
$g$
is acceleration of gravity. Seawater density is approximately equal to 1.023 g/cm^{3} (Table 2.1) and is generally assumed constant. However, the pore water can vary regionally and with depth.

Material | Density ( $kg/{m}^{3}$ ) |
Pressure Gradient $(MPa/km$ ) |
Pressure Gradient ( $psi/ft$ ) | Mud weight ( $PPG)$ |
---|---|---|---|---|

Seawater | 1 023 | 10.0 | 0.444 | 8.5 |

Typical overburden | 2 110 | 20.7 | 0.915 | 17.6 |

Typical brine | 1 072 | 10.5 | 0.465 | 8.9 |

Typical gas | 300 | 2.9 | 0.130 | 2.5 |

Typical oil | 718 | 7.0 | 0.311 | 6.0 |

Water | 1 000 | 9.8 | 0.434 | 8.3 |

In a static aquifer, the water pressure follows the hydrostatic gradient. For the case where the pore water and seawater density are equal, the water pressure parallels the hydrostatic gradient (Fig. 2.1b) and the overpressure ( ${u}^{*}$ ) is constant (Fig. 2.1c).

Reservoirs, prior to production, are commonly described as in gravitational equilibrium. In this case, water pressures and hydrocarbon pressures follow their static pressure gradients (Fig. 2.1). Typical static gradients for fluids are shown in Table 2.1. The distribution of reservoir pore pressures is estimated by extrapolating along the static pressure gradient from measured values. For example, the pore pressure ( ${u}_{{z}_{2}}$ ) at depth ${z}_{2}$ , given a pressure measurement at depth ${z}_{1}$ , is:Here, $\rho $ is the fluid density, $g$ is gravitational acceleration, and consequently $\rho g$ is the fluid pressure gradient.

In Figure 2.1b, both a water phase pressure (blue line) and a gas phase pressure (red line) are shown and they are assumed to be equal at the gas-water contact. We explore this assumption later in this chapter.

### 2.3 Vertical Stress and Vertical Effective Stress

The most common way to estimate the total vertical stress ( ${\sigma}_{v}$ ) is to integrate the weight of the overlying material:where ${\rho}_{b}$ is the bulk density of the sediment, which varies with depth. In the marine environment, Equation 2.5 is expanded to include the weight of the water column:

I discuss in Chapter 8 different approaches to characterize the overburden stress ( ${\sigma}_{v}$ ). However, a common approach is to integrate log-derived bulk density measurements. The compressibility of water is so small that it is commonly assumed to be constant. However, the bulk density ( ${\rho}_{b}$ ) varies greatly and cannot be assumed constant.

When a porous solid is externally loaded, the internal opposing forces are partitioned between water (or other fluids) and the porous solid (Fig. 2.2). Fluid pressure acts in opposition to external stresses over the fraction of the porous medium’s boundary represented by the porosity (Fig. 2.2). Reference TerzaghiTerzaghi (1923) first described how these forces are partitioned through the effective stress concept. Effective stress represents the average stress felt by the soil skeleton and was meant to describe the fraction of the stress that was ‘effective’ in deforming the rock. The effective vertical stress ( ${\sigma}_{v}^{\text{'}}$ ) is the total vertical stress less some portion of the pore pressure:

$\alpha $ is a pore pressure coefficient, which varies between 0 and 1 as described by Reference Ingebritsen, Sanford and NeuzilIngebritsen et al. (2006). Equation 2.7 applies for both differential changes in stress and, as written, for absolute values of stress and pressure. Equation 2.7 is written for vertical stress, but can apply to any stress orientation. Reference TerzaghiTerzaghi (1923) originally suggested $\alpha $ was equal to porosity. Subsequently, Reference Nur and ByerleeNur and Byerlee (1971) showed that it was proportional to the ratio of the bulk compressibility $\left({c}_{b}\right)$ to solid compressibility $\left({c}_{s}\right):$

Under most conditions in sedimentary basins, the bulk compressibility is orders of magnitude greater than the solid compressibility and therefore $\alpha $ is equal to 1. This assumption breaks down in very stiff, and very low porosity material. Generally, I will assume $\alpha $ = 1.

### 2.4 Overpressure, and Pressure Gradient Plots

Subsurface pressures and stresses are often presented as overpressure plots (Fig. 2.1c) or pressure gradient (mud weight) plots (Fig. 2.1d). In an overpressure plot, the hydrostatic pressure ( ${u}_{h}$ ) is subtracted from the total pressures or stresses (Eq. 2.3). In this perspective, aquifers that are at a hydrostatic gradient have a constant overpressure (blue line, Fig. 2.1c). When the overpressure equals the reduced total vertical stress ( ${\sigma}_{v}-{u}_{h}$ ), the pore pressure equals the total vertical stress ( $u={\sigma}_{v}$ ).

Finally, average density plots, or mud weight plots, express the pore pressure as the average density of a fluid measured from a datum that will balance the pressure at a particular depth (Fig. 2.1d). In this example, the sea surface is assumed to be the datum. However, generally the datum is the elevation of the rig floor. Average density as measured from the sea surface is calculated from pore pressure as follows:This density can easily be converted to equivalent mud weight or a pressure gradient (e.g., Table 2.1). For example, equivalent mud weight in units of pounds per gallon (EMW) can be expressed as a function of pressure in units of pounds per square inch and depth in feet as:

### 2.5 Example: The Bullwinkle J3 Sand

The Bullwinkle Oil Field is located on the western flank of a circular salt-withdrawal minibasin on the slope of the Gulf of Mexico, approximately 150 miles southwest of New Orleans, Louisiana in 1 350 ft water depth (412 m) (Figures 2.3 and 2.4). The field has been described extensively (Reference ComiskyComisky, 2002; Reference Flemings, Comisky, Liu and LupaFlemings et al., 2001; Reference Holman and RobertsonHolman & Robertson, 1994; Reference O’Connell, Kohli and AmosO’Connell et al., 1993; Reference Swanston, Flemings, Comisky and BestSwanston et al., 2003).

The J sands are of early Nebraskan (3.35 Ma) age and host the majority of the reserves at Bullwinkle. The five J sands (J0–J4) are bowl-shaped interconnected channel and sheet turbidite sands that are interbedded with debris flow deposits and shales, and overlain by a thick section (500 ft) of bathyal shales. Pressure drawdown at each well followed the same depletion curve, indicating that the sands are in pressure communication. The J3 sand is very fine to fine-grained and has a blocky log character (Fig. 2.4, inset). The lithology and the grain size are relatively homogenous across the field. It is interpreted to be a ponded, internally amalgamated sheet sand. There is a small oil and gas pool at its crest, while the majority of it is brine saturated (Fig. 2.5).

The parameters necessary to characterize pressure in the J3 sand are shown in Table 2.2. Three critical measurements were made: (1) pressure was measured to be 56.7 MPa within the gas leg of the J3 sand at 3 493 m true vertical depth below sea surface ( $TV{D}_{SS}$ ) with Schlumberger’s Repeat Formation Tester (RFT); (2) the gas-oil contact was observed from logging data (neutron-density cross over) at 3 504 m $TV{D}_{SS}$ ; and (3) the oil-water contact was imaged from seismic data at 3 613 m $TV{D}_{SS}$ . With these data and additional information about fluid densities (Table 2.2), bulk densities, and water depth, I characterize the initial reservoir pressure (Fig. 2.6).

Parameter | Field Units | Metric Units |
---|---|---|

Seafloor Depth ( $TV{D}_{ss}$ ) | 1 350 ft | 411.6 m |

Structural Crest ( $TV{D}_{ss}$ ) | 11 100 ft | 3 384.1 m |

Pressure Measurement ( $TV{D}_{ss}$ ) | 11 460 ft | 3 493.9 m |

Gas-oil Contact ( $TV{D}_{ss}$ ) | 11 493 ft | 3 504.0 m |

OWC ( $TV{D}_{ss}$ ) | 11 850 ft | 3 612.8 m |

Structural Base ( $TV{D}_{ss}$ ) | 13 000 ft | 3 963.4 m |

Gas Density ( ${\rho}_{g}$ ) | 0.13 psi/ft | 300 kg/m^{3} |

Oil Density ( ${\rho}_{o}$ ) | 0.31 psi/ft | 714 kg/m^{3} |

Pore Water Density ( ${\rho}_{pw}$ ) | 0.465 psi/ft | 1 071 kg/m^{3} |

Seawater Density ( ${\rho}_{sw}$ ) | 0.44 psi/ft | 1 013 kg/m^{3} |

Local Overburden Density ( ${\rho}_{b}$ ) | 0.915 psi/ft | 2 108 kg/m^{3} |

Vertical Stress at Crest ( ${\sigma}_{v}$ ) | 9 683 psi | 66.76 MPa |

Pressure at Measurement ( ${u}_{g}$ ) | 8 224 psi | 56.70 MPa |

The measured pore pressure is shown as the solid dot (Fig. 2.6). From this point, the gas pressure is extended along its static gradient (Eq. 2.4) upward to the crest and downward to the gas-oil contact. Next, the oil pressure is assumed to equal the gas pressure at the gas-oil contact. We explore this assumption in the following section. The oil pressure is then extended along its static gradient upward from the gas-oil contact to the crest and downward to the oil-water contact. Finally, the water pressure is assumed to equal the gas pressure at the oil-water contact and from this, the water pressure is calculated by extending the water pressure along its hydrostatic gradient upward to the crest of the structure and down to the base of the structure.

At the crest of the structure, the water pressure is 55.1 MPa, the oil pressure is 55.9 MPa, and the gas pressure is 60.4 MPa (Table 2.3). The difference in pressure between adjacent, immiscible, phases is the capillary pressure ( ${u}_{c}$ ). Thus, at the crest, the gas-oil capillary pressure ( ${u}_{cgo}$ ) is 0.49 MPa and the oil-water capillary pressure is 0.8 MPa. In the overpressure plot (Fig. 2.6b), the water phase overpressure (blue line, Fig. 2.6b) is constant and equal to 19.5 MPa. Thus, the J3 sand has an overpressure ( ${u}^{*}$ ) equal to 19.5 MPa.

Parameter | Value | Unit |
---|---|---|

Overpressure ( ${u}^{*}$ ) | 19.5 | MPa |

Crestal Water Pressure ( ${u}_{w}$ ) | 55.1 | MPa |

Gas-oil Capillary Pressure @ Crest ( ${u}_{cgo}$ ) | 0.5 | MPa |

Oil-water Capillary Pressure @ Crest ( ${u}_{cow}$ ) | 0.8 | MPa |

Basal Water Pressure ( ${u}_{w}$ ) | 61.2 | MPa |

Effective Stress at Crest ( ${\sigma}_{v}\text{'}$ ) | 11.7 | MPa |

Effective Stress at Base ( ${\sigma}_{v}\text{'}$ ) | 17.6 | MPa |

Overpressure Ratio at Crest ( ${\lambda}^{*}$ ) | 0.63 | - |

${\lambda}^{*}$ describes the relative position of the pressure between the hydrostatic pressure and the lithostatic stress. At Bullwinkle, ${\lambda}^{*}$ = 0.63 at the crest and 0.53 at the base of the J3 reservoir.

### 2.6 Capillary Pressure

#### 2.6.1 Surface Tension

Capillary pressure was introduced by describing the gas-oil capillary pressure ( ${u}_{cgo}$ ) as the difference between the gas pressure and the oil pressure, and the oil-water capillary pressure ( ${u}_{cow}$ ) as the difference between the oil and water pressures (Fig. 2.6, Table 2.3). I now explore capillary behavior more deeply because it controls reservoir pore pressures, trap integrity, and hydrocarbon migration.

Reference Dake and DakeDake (1978b) gives an elegant overview of applications of capillary pressure in petroleum engineering. Reference SmithSmith (1966), Reference BergBerg (1975), and Reference SchowalterSchowalter (1979) apply these concepts to understand reservoir pressures, trap integrity, and secondary migration. Reference de Gennes, Brochard-Wyart and Quéréde Gennes et al. (2004) and Reference BluntBlunt (2017) provide wonderful descriptions that I rely heavily on in the ensuing section.

In a liquid, the molecules are attracted to each other. A molecule at the surface of a gas-liquid interface loses half of its cohesive interactions. If the cohesion energy per molecule is $U$ inside the liquid, a molecule at the surface finds itself short by roughly $\frac{U}{2}$ . The surface tension ( $\gamma $ ) is a direct measure of this energy shortfall per unit surface area. If the molecule is of size $a$ and ${a}^{2}$ is the exposed area on the surface, the surface tension is of order $\gamma \cong \frac{U}{2{a}^{2}}$ . For most oils, for which the interactions are of the van der Waals type, at 25 degrees, $\gamma =20\frac{mJ}{{m}^{2}}$ . Water involves hydrogen bonds and its cohesive energy is larger ( $\gamma =70\frac{mJ}{{m}^{2}}$ ). Mercury, a strongly cohesive liquid metal, has $\gamma \cong 500\frac{mJ}{{m}^{2}}$ . The surface energy between two immiscible liquids A and B also has an interfacial tension, ${\gamma}_{AB}$ , which is approximately the difference in the surface tensions of the individual components. For example, the oil-water interfacial tension is ~50 mJ/m.

#### 2.6.2 Capillary Pressure

Surface tension causes the pressure inside a bubble to be greater than that outside of it. As described by Reference de Gennes, Brochard-Wyart and Quéréde Gennes et al. (2004), consider a drop of oil (o) in water (w) (Fig. 2.7). The drop adopts a spherical shape of radius $R$ to minimize its surface energy. If the o/w interface is displaced by an amount $dR$ , the work done by the pressure and capillary force is:with $R\gg dR,d{V}_{o}=4\pi {R}^{2}dR=-d{V}_{w},$ and $dA=8\pi RdR$ . These are the increase in volume and surface area, respectively, of the drop. ${u}_{o}$ and ${u}_{w}$ are the pressures in the oil and the water, and ${\gamma}_{ow}$ is the oil-water interfacial tension. At mechanical equilibrium, $\delta W=0$ and Equation 2.12 reduces to

Equation 2.13 is the Laplace equation for a fluid interface of spherical geometry. It shows that the smaller the drop, the larger the differential pressure, also known as the capillary pressure ( ${\text{u}}_{\text{c}}$ ) or the Laplace pressure.

#### 2.6.3 Contact Angle ( $\theta $ )

The contact angle ( $\theta $ ) is the angle formed between two immiscible fluids that are in contact with a solid phase. It can be calculated from the interfacial tensions between the solids and the two fluids and by the interfacial tension of the fluid-fluid interface (Fig. 2.8). Treating the interfacial tensions as forces and performing a horizontal force balance yields:Equation 2.14 is rearranged to solve for the wetting angle:

By convention, the contact angle ( $\theta $ ) is measured through the denser fluid phase. Thus, in Figure 2.8, if the oil is the non-wetting phase and water is the wetting phase, the contact angle is measured through the water. The contact angle can vary from 0 to 180 degrees as a function of Equation 2.15 (Fig. 2.9). The wetting phase preferentially contacts the surface. If the contact angle is less than 90 degrees, the denser phase is wetting. If the contact is greater than 90 degrees, the denser phase is non-wetting. As described by Reference BluntBlunt (2017), while Equation 2.15 gives physical insight, it is seldom used. Instead, the contact angle is measured such as illustrated in Figure 2.9.

#### 2.6.4 Capillary Rise

The pressure distribution in a reservoir can be viewed through the concept of capillary rise. Consider the case of a glass tube placed in water (Fig. 2.10). The water inside the tube rises to a height $h$ above the water surface outside the tube (Fig. 2.11). The angle of the meniscus at the glass interface is the wetting angle ( $\theta $ ) and the meniscus within the tube is a portion of a sphere with a radius of curvature $R$ . The relationship between the tube radius ( $r$ ) and the radius of curvature ( $R$ ) is:

I consider the case where the water is overlain by oil. Equations 2.16 and 2.13 are combined to calculate the water pressure ( ${u}_{w}$ ) at point A, just beneath the meniscus:

${\text{u}}_{\text{o}}$ is the oil pressure immediately above the interface. At the level interface, far from the capillary tube, the tube radius can be considered infinite and the water and oil pressures are equal to each other and equal to ${u}_{1}$ . Inside the tube, the pressure at point A within the water is due to the weight of the water column:

Outside the tube, the oil pressure at point A (just above the interface) is:

By combining Equations 2.17, 2.18, and 2.19, the relationship between capillary radius and capillary rise is:

where,

In the example in Figure 2.11b, the water rises 6.4 cm above the interface for an oil-water system with a 1 mm capillary tube. An equivalent calculation for an air-water system results in a capillary rise of ~1.6 cm, which is more similar to what is shown in Figure 2.10. The difference is largely due to the fact that the relative density ( $\Delta \rho $ ) is much smaller in the oil-water case.

#### 2.6.5 Interpreting Mercury Injection Capillary Pressure (MICP) Curves

Imbibition occurs when the wetting phase saturation increases, and drainage occurs when it decreases. Drainage occurs as hydrocarbons fill a reservoir over geologic time through secondary migration, and it occurs during *CO _{2}* injection. We estimate reservoir water phase pressure (see below) and trap integrity (Chapter 9) assuming reservoirs formed by drainage. The drainage curve is experimentally derived by forcing a non-wetting phase into a saturated sample (Fig. 2.12). This is commonly done by injecting mercury into an air-dried sample. Prior to interpretation, these measurements must be corrected for both apparatus compressibility and sample conformance. The conformance correction is the pressure required to allow the mercury to surround or conform to the sample exterior without intruding the pores (Reference Comisky, Santiago, McCollom, Buddhala and NewshamComisky et al., 2011) .

Mercury injection tests for a mudrock and a siltstone that were shallowly buried are illustrated in Figure 2.12. The drainage experiment starts at the right edge ( ${S}_{w}=1$ ). As the pressure outside the sample is raised (the mercury pressure or capillary pressure), there is initially little change in the saturation until the displacement pressure ( ${u}_{d}$ ) is reached. For these samples, as capillary pressure is increased above the displacement pressure, saturation increases rapidly. Finally, as capillary pressure is increased to very high values, the saturation does not increase as rapidly. When the water saturation does not reduce further, the residual saturation is reached where the non-wetting phase saturation cannot be reduced further regardless of the capillary pressure.

Reference PittmanPittman (1992) and Reference Comisky, Santiago, McCollom, Buddhala and NewshamComisky et al. (2011) discuss the interpretation of capillary curves. The pressure at which the non-wetting phase starts to enter the pores is commonly called the displacement pressure ( ${u}_{d}$ ). I term the pressure at which the non-wetting fluid forms a connected filament across the sample the migration pressure ( ${u}_{cmig}$ ). This is the pressure necessary for migration of the non-wetting fluid through the rock. I introduce the term migration pressure because there is a lot of ambiguity as to how to interpret the migration pressure from the capillary curve and because the migration pressure and the displacement pressure are often used interchangeably. For example, Reference SmithSmith (1966) describes the displacement pressure as the migration pressure. I define three terms that can be objectively determined from the capillary curve: the displacement pressure ( ${u}_{d}$ ), the extrapolated displacement pressure ( ${u}_{de}$ ), and the threshold pressure ( ${u}_{t}$ ). The displacement pressure ( ${u}_{d}$ ) is, approximately, the first data point where ${S}_{w}$ drops below 1 (after correction for conformance). The extrapolated displacement pressure ( ${u}_{de}$ ) is determined from a hyperbolic fit to the displacement curve and its projection to a horizontal asymptote (see Figure 1 of Reference ThomeerThomeer (1960)). In practice, this value is approximately equal to the point where the plateau of the capillary curve is extended to 100% wetting saturation. Reference ThomeerThomeer (1960) suggested this is the most accurate estimate of displacement pressure. Finally, the threshold pressure ( ${u}_{t}$ ) was defined by Reference Katz and ThompsonKatz and Thompson (1986, Reference Katz and Thompson1987) to occur when the curvature of the capillary curve changes from concave downward to concave upward ( ${u}_{t}$ , Fig. 2.12). This is the point where the maximum injected pore volume occurs for a given increase in capillary pressure (Fig. 2.12b). ${u}_{t}$ can be expressed as a threshold pore throat radius (Eq. 2.17), termed ${r}_{t}$ .

I illustrate the interpretation of ${u}_{d},{u}_{de}$ , and ${u}_{t}$ for a mudrock and a siltstone (Fig. 2.12). The extrapolated displacement pressure ( ${u}_{de}$ ) for the mudrock is approximately half the threshold pressure ( ${u}_{t}$ ) and the non-wetting phase saturation at the threshold pressure is significantly higher than at the displacement pressure (Fig. 2.12a). Given the ambiguity in the literature, I view ${u}_{t}$ as an upper bound and ${u}_{de}$ as a lower bound for the migration pressures ( ${u}_{cmig}$ ). Reference SchowalterSchowalter (1979), based on experimental results, avoided this ambiguity and suggested that the migration pressure occurs at a non-wetting phase saturation of ~10%.

The mercury air capillary pressure ( ${u}_{Hg-air}$ ) can be converted to describe capillary pressures in other fluid systems if the wetting angle ( $\theta $ ) and interfacial tension ( $\gamma $ ) are known. For example, to convert mercury-air capillary pressures to equivalent oil-water capillary pressures:Typical values for interfacial tension and wetting angle are shown in Table 2.4. In Figure 2.12, the equivalent gas-water capillary pressure is illustrated on the right-hand side, and in Figure 2.13, the equivalent oil-water capillary pressure is shown on the right hand side.

System | Contact Angle $\left(\theta \right)$ | Interfacial Tension $\left(\gamma \right)\left(\frac{dyn}{cm}\right)$ |
---|---|---|

Laboratory | ||

Air-water | 0 | 72 |

Oil-water | 30 | 48 |

Air-Mercury | 140 | 480 |

Air-oil | 0 | 24 |

Reservoir | ||

Water-oil | 30 | 30 |

Water-gas | 0 | 50* |

Oil-gas | 0 | 24 |

*pressure/temp dep. |

A common and simple way to conceptualize a reservoir is through a ‘bundle of tubes model.’ The reservoir is assumed to comprise a range of different cylindrical tubes of different diameters. As the capillary pressure is raised, the non-wetting phase enters smaller and smaller tubes. In this conceptual view, each capillary pressure on the capillary curve can be related to an equivalent cylindrical radius through Equation 2.17 as is illustrated on the right side of Figures 2.12 and 2.13.

### 2.7 Capillary/Gravity Equilibrium

#### 2.7.1 Two Phase Capillary/Gravity Equilibrium

I use an aquarium to illustrate the distribution of fluid phases and pore pressures present in a two-phase system at capillary and gravity equilibrium (Fig. 2.14). A fine-grained ‘seal’ composed of 0.5 mm sand overlies a coarse-grained ‘reservoir’ composed of ~5 mm gravel in an anticlinal geometry. As air is added from below, it fills the reservoir and displaces the water that was there originally (shown with black). The gas-water contact separates the zone above where gas is present (white) from the water leg below (black) (Fig. 2.14a). Pressure measurements within the gas leg are approximately constant (red line), whereas the water pressures follow the hydrostatic gradient (blue line) (Fig. 2.14b). At the crest, the gas pressure is ~930 Pa more than the water pressure. At the gas-water contact, the gas and water pressures differ by 120 Pa.

The capillary pressure at the gas-water contact is the migration pressure ( ${u}_{cmig}=120\text{}\text{P}\text{a}$ ), the differential pressure necessary for the gas to migrate downward into the water-saturated reservoir. The pore throat radius given this capillary pressure (Eq. 2.17) is 1.2 mm. This is approximately 20% of the grain diameter (5 mm), which is in the range expected for the ratio of pore throat radius to grain diameter for a well-sorted sand.

The elevation of the gas-water contact above the free water level can be calculated directly from the capillary pressure observed at the gas-water contact:It is found to be 1.2 cm as shown in Figure 2.14b.

I next describe the fluid distribution in the Bullwinkle J2 reservoir. In this oil reservoir, the depth of the crest, the depth of the oil-water contact, the fluid properties, and the pressure within the oil column were known. I combine these data with the reservoir capillary curves shown in Figure 2.13 to describe the distribution of fluid phases with depth (Fig. 2.15). The free water level is the depth where the oil and water pressures are equal and thus the capillary pressure is zero ( ${u}_{o}={u}_{w}$ ) (Fig. 2.15). Above this, there is a zone where the non-wetting phase is absent, because the capillary pressure is not large enough for the non-wetting phase to enter the rock. The oil-water contact occurs where the capillary pressure equals the migration pressure for the reservoir ( ${u}_{cmig}$ ):${h}_{owc}$ is the height of the oil-water contact above the free water level. In this example, I assume the migration pressure is the displacement pressure ( ${u}_{d}$ ), or the minimum pressure where significant saturation of the non-wetting phase occurs. This is easy to interpret in the examples given, where there is an abrupt break in slope of the capillary curve at low saturations and where the capillary curve is fairly flat above the displacement pressure (Fig. 2.12).

Like the aquarium example, this example shows that if the reservoir rock has a significant migration pressure, the depth where the oil and water pressures are equal (the free water level) is significantly below the fluid contact. If the reservoir has the capillary properties of the Nankai Siltstone and the migration pressure is 300 psi (0.15 MPa) (Fig. 2.12), then the free water level is ~42 meters below the oil-water contact (OWC) (horizontal dashed lines, Fig. 2.15). In this case, the water pressure is 300 psi less than the oil pressure at the OWC (note separation between the green and the blue solid lines at the OWC in Figure 2.15b). In contrast, if the reservoir has the capillary properties of Facies 1 of the Bullwinkle sandstone with a migration pressure of only 6 psi (0.0029 MPa) (Fig. 2.13), then the free water level is only 0.8 meters below the OWC. If we compare the two examples (dashed versus solid blue line, Fig. 2.15), we see that the reservoir with a higher migration pressure results in a lower predicted water pressure (solid line) and a higher predicted capillary pressure at any depth.

As the energy industry explores and produces from increasingly fine-grained reservoirs, it is becoming important to account for reservoir capillary behavior to determine reservoir pressure. Without incorporating capillary behavior, the water pressure will not be calculated correctly.

#### 2.7.2 Three Phase Capillary/Gravity Equilibrium

I close by returning to the original Bullwinkle J3 example to consider the presence of all three phases (Fig. 2.6). I use the capillary model of the Nankai Siltstone to illustrate the behavior (Fig. 2.16). When three phases are present, the water saturation is calculated exactly as done previously from the oil-water capillary pressure ( ${u}_{cow}$ ). However, in the case of the gas phase, the sum of the water saturation and the oil saturation is equal to the wetting phase on the gas-oil capillary curve.

The results are illustrated in Figure 2.16. The water saturation is everywhere the same as in the two-phase example (compare Fig. 2.15c with Fig. 2.16b). However, above the gas-water contact, the gas saturation increases dramatically at the expense of the oil saturation. It can be envisioned that with sufficient structural height, the oil saturation would ultimately go to zero. Reference BluntBlunt (2017) describes this process analytically. At the gas-oil contact, the oil pressure is offset from the gas pressure by the gas-oil migration pressure (0.135 MPa) and at the oil-water contact, the water pressure is offset from the oil pressure by the oil-water migration pressure (0.146 MPa). The difference between the blue line and the dashed black line is the difference between water pressure predicted with and without considering the effects of the migration pressure (0.281 MPa). The example given with the Nankai Siltstone curve demonstrates the effect of a relatively high migration pressure on reservoir pressure estimation and on saturation. However, the basic physical processes will be present in reservoirs with much lower migration pressures.

Figure 2.17 emphasizes an important conceptual view of the reservoir which, although highly simplified, provides important insight. A pore scale conceptual view (Fig. 2.17a) is that the grains are surrounded by water and the gas phase is surrounded by oil. Each phase is connected along some pathway through the reservoir. At the reservoir scale, this system can be viewed as a vertical column. Water is present everywhere (Fig. 2.17b). The gas phase is the non-wetting fluid with respect to the oil phase. Pressures within each phase follow their own static pressure gradients. The pressures between the phases are different due to the interfacial tension present at the boundary between the fluid phases.

### 2.8 Summary

I have described how to characterize reservoir pore pressures that are under capillary and gravity equilibrium through reservoir examples and through a simple fish tank demonstration. Different immiscible fluid phases have distinct pressures. These pressures result from capillary behavior and through an understanding of this behavior, we can systematically describe these pressures and the associated pore fluid saturations. It is critical to account for capillary behavior to successfully recover the water phase pressure, which is one of the critical parameters we need to know in pore pressure analysis.