Pressure-Depth Relationships

Equation 2.1 is a separable, first-order differential equation. It can be separated and integrated as follows  

However, to perform the integration, it is necessary to have some relation between p, g, and z. In situations on the surface of the earth, g is practically constant (see Sec. 2.6), so we may take it outside the integral sign. Several possible kinds of behavior of p in relation to z lead to simple integrations of the equation, as shown below. 

No real substances have constant density the density of every substance increases as the pressure increases. However, for most liquids at temperatures far below their critical temperatures, the effect of pressure on density is very small. For example, raising the pressure of water at 100 F from 1 to 1000 Ibf/ in causes the density to increase by 0.3 percent. In most engineering calculations, we can neglect such small changes in density. Then we take p outside the integral sign in Eq. 2.8 and find that the pressure is 

Example 2.2. When the submarine Thresher sank in the Atlantic, it was estimated in the newspapers that the accident had occurred at a depth of 1000 ft (304.9 m). What is the pressure of the sea at that depth  

Seawater may be considered incompressible, with density 63.91bm/ft The pressure at the surface is atmospheric pressure, which is approximately 14.71bf/in. The acceleration of gravity is 32.2 ft/s Therefore, 

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Gas density at reservoir conditions is useful for calculation the pressure gradient of the gas when constructing pressure-depth relationships (see Section 5.2.8). 

The density of the oil at reservoir conditions is useful in calculating the gradient of oil and constructing a pressure - depth relationship in the reservoir (see section 5.2.8). 

In Section 5.2.8 we shall look at pressure-depth relationships, and will see that the relationship is a linear function of the density of the fluid. Since water is the one fluid which is always associated with a petroleum reservoir, an understanding of what controls formation water density is required. Additionally, reservoir engineers need to know the fluid properties of the formation water to predict its expansion and movement, which can contribute significantly to the drive mechanism in a reservoir, especially if the volume of water surrounding the hydrocarbon accumulation is large. 

In abnormally pressured reservoirs, the continuous pressure-depth relationship is interrupted by a sealing layer, below which the pressure changes. If the pressure below the seal is higher than the normal (or hydrostatic) pressure the reservoir is termed overpressured. Extrapolation of the fluid gradient in the overpressured reservoir back to the surface datum would show a pressure greater than one atmosphere. The actual value by which the extrapolated pressure exceeds one atmosphere defines the level of overpressure in the reservoir. Similarly, an underpressured reservoir shows an pressure less than one atmosphere when extrapolated back to the surface datum. 

The term fluidization is applied to processes in which a loose, porous bed of solids is converted to a fluid system, having the properties of surface leveling, flow, and pressure-depth relationships, by passing the fluid up through the bed. 

For simple fluids at rest, the pressure-depth relationship is given by the basic equation of fluid statics dPIdz -pg. This equation is found by considering the weight of a small element of fluid and the pressure change with depth necessary to support that weight. 

Fig. 12. GEA pressure/depth plot showing the relationship between aquifer overpressure, fracture pressures, crestal reservoir pressures and closure style. Aquifer pressures are grouped into a terrace domain and a deep graben domain.

Adsorption for gas purification comes under the category of dynamic adsorption. Where a high separation efficiency is required, the adsorption would be stopped when the breakthrough point is reached. The relationship between adsorbate concentration in the gas stream and the solid may be determined experimentally and plotted in the form of isotherms. These are usually determined under static equilibrium conditions but dynamic adsorption conditions operating in gas purification bear little relationship to these results. Isotherms indicate the affinity of the adsorbent for the adsorbate but do not relate the contact time or the amount of adsorbent required to reduce the adsorbate from one concentration to another. Factors which influence the service time of an adsorbent bed include the grain size of the adsorbent depth of adsorbent bed gas velocity temperature of gas and adsorbent pressure of the gas stream concentration of the adsorbates concentration of other gas constituents which may be adsorbed at the same time moisture content of the gas and adsorbent concentration of substances which may polymerize or react with the adsorbent adsorptive capacity of the adsorbent for the adsorbate over the concentration range applicable over the filter or carbon bed efficiency of adsorbate removal required. 

There are two effects from the adsorbent bed depth on mass transfer. First, it is important that the bed be deeper than the length of the transfer zone which is unsaturated. The second is that any multiplication of the minimum bed depth gives more than a proportionally increased capacity. Generally, it is advantageous to size the adsorbent bed to the maximum length allowed by pressure-drop considerations. The determination of the depth of the MTZ or unsaturated depth may be determined experimentally, and applying the following relationship ... 

Pressure is the force per unit area exerted by or on a fluid. In a static fluid the pressure increases with depth, but according to Pascal s principle it is the same in all directions at any given depth. Pressure may be specified as either absolute, or gauge, the relationship between the two being ... 

Subsurface Rock Fracture Pressure (Fracture Pressure Gradient). The subsurface rock fracture pressure can be approximated by utilizing the known pore pressure at the same depth. The relationship between rock fracture pressure p (psi) and pore pressure p (psi) is [34]... 

FIGURE 3.24 Groundwater conditions near the ground surface. Saturated and saturated zones (a), profile of moisture content vs. depth (b), pressure head and hydraulic head relationships insets = water retention under pressure heads less than (top) and greater than (bottom) atmospheric (c), profile of pressure head vs. depth (d), and profile of hydraulic head vs. depth (e). (After Freeze and Cherry, 1979.)... 

Lowenstam, H. A. Biogeochemistry of hard tissues, their depth and possible pressure relationships. In Barobiology and the experimental biology of the deep sea, pp. 19. Brauer,... 

Although spherical vessels have a limited process application, the majority of pressure vessels are made with cylindrical shells. The heads may be flat if they are suitably buttressed, but preferably they are some curved shape. The more common types of heads are illustrated on Figure 18.16. Formulas for wall thicknesses are in Table 18.3. Other data relating to heads and shells are collected in Table 18.5. Included are the full volume V0 and surface S as well as the volume fraction V/V0 corresponding to a fractional depth H/D in a horizontal vessel. Figure 18.17 graphs this last relationship. For ellipsoidal and dished heads the formulas for V/V0 are not exact but are within 2% over the whole range. 

Figure 1. Attenuation of cosmogenic production rates with atmospheric pressure (elevation) and depth in rock. A) Log-linear plot of depth as a function of normalized production rate as per equation (5), assuming a rock density of 2.7 g cm-3. Slope is rock attenuation coefficient, A. B) Log-linear plot of atmospheric depth as a function of normalized production rate as per equation (3) (solid line) and as per the scaling function presented in Stone (2000) for 40° latitude (dashed line). The offset between the two lines indicates the importance of the elevation scaling relationships when reconstructing high paleoaltitudes.

Two major types of variability in the relationship between overlying water chemistry and carbonate accumulation in deep sea sediments occur. The first is the previously discussed relation of the saturation state of the water to the R0, FL and CCD. The second is the relative separation of these different sedimentary features. In some areas of the ocean these relations can be influenced by transitions in water masses having different chemical and hydrographic characteristics (e.g., Thunell, 1982), but in many areas of the ocean the only major variable influencing the saturation state over wide areas is pressure, which leads to a nearly uniform gradient in saturation state with respect to depth. 

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