Gradient pressure

Dieterici s equation A modification of van der Waals equation, in which account is taken of the pressure gradient at the boundary of the gas. It is written... 

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). 

Normal pressure regimes follow a hydrostatic fluid gradient from surface, and are approximately linear. Abnormal pressure regimes include overpressured and underpressured fluid pressures, and represent a discontinuity in the normal pressure gradient. Drilling through abnormal pressure regimes requires special care. 

In a normal pressure regime the pressure in a hydrocarbon accumulation is determined by the pressure gradient of the overlying water (dP / dD), which ranges from 0.435 psi/ ft (10 kPa/m) for fresh water to around 0.5 psi/ft (11.5 kPa/m) for salt saturated brine. At any depth (D), the water pressure (PJ can be determined from the following equation, assuming that the pressure at the surface datum is 14.7 psia (1 bara) ... 

The water pressure gradient is related to the water density (p, kg/m ) by the following equation ... 

Flence it can be seen that from the density of a fluid, the pressure gradient may be caloulated. Furthermore, the densities of water, oil and gas are so significantly different, that they will show quite different gradients on a pressure-depth plot. 

In some cases when drilling fluids invade a very low permeability zone, pressure equalisation in the formation can take a considerable time. The pressure recorded by the tool will then be close to the pressure of the mud and much higher than the true formation pressure. This is known as supercharging. Supercharging pressures indicate tight formation, but are not useful in establishing the true fluid pressure gradient. 

For a single fluid flowing through a section of reservoir rock, Darcy showed that the superficial velocity of the fluid (u) is proportional to the pressure drop applied (the hydrodynamic pressure gradient), and inversely proportional to the viscosity of the fluid. The constant of proportionality is called the absolute permeability which is a rock property, and is dependent upon the pore size distribution. The superficial velocity is the average flowrate... 

The field unit for permeability is the Darcy (D) or millidarcy (mD). For clastic oil reservoirs, a good permeability would be greater than 0.1 D (100 mD), while a poor permeability would be less than 0.01 D (10 mD). For practical purposes, the millidarcy is commonly used (1 mD = 10" m ). For gas reservoirs 1 mD would be a reasonable permeability because the viscosity of gas is much lower than that of oil, this permeability would yield an acceptable flowrate for the same pressure gradient. Typical fluid velocities in the reservoir are less than one metre per day. 

Capillary pressure gradients and Marongoni flow induce flow in porous media comprising glass beads or sand particles [40-42], Wetting and spreading processes are an important consideration in the development of inkjet inks and paper or transparency media [43] see the article by Marmur [44] for analysis of capillary penetration in this context. 

A simple law, known as Darcy s law (1936), states that the volume flow rate per unit area is proportional to the pressure gradient if applied to the case of viscous flow through a porous medium treated as a bundle of capillaries,... 

Clearly the general situation is very complicated, since all three mechanisms operate simultaneously and might be expected to interact in a complex manner. Indeed, this problem has never been solved rigorously, and the momentum transfer arguments we shall describe circumvent the difficulty by first considering three simple situations in which each of the three separate mechanisms in turn operates alone. In these circumstances Che relations between fluxes and composition and/or pressure gradients can be found without too much difficulty. Rules of combination, which are essea-... 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

Now the force per unit volume exerted on the porous medium by the pressure gradient in the gas is -grad p, where p, as distinct from is the physical pressure of the gaseous mixture. This is the force which must be balanced in our model by the external forces acting on the dust particles, so... 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

There is a further simplification which is often justifiable, but not by consideration of the flux equations above. The nature of many problems is such that, when the permeability becomes large, pressure gradients become very small ialuci uidiii iiux.es oecoming very large. in catalyst pellets, tor example, reaction rates limit Che attainable values of the fluxes, and it then follows from equation (5,19) that grad p - 0 as . But then the... 

This simplification must be used with caution, of course, making sure that the specification of the problem does not determine the magnitude of the pressure gradient, but it is very useful in the important case of a coarsely porous catalyst pellet. 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

In the first, motion is generated by composition gradients in the absence of any temperature or pressure gradient, in the second, motion is a consequence of pressure gradients alone, and in the third,both pressure and composition gradients are present but a constraint of zero net molar flux is imposed. These will be discussed in turn. 

To appreciate the questions raised by Knudsen s results, consider first the relation between molar flow and pressure gradient for a pure gas flowing through a porous plug, rather than a capillary. The form predicted by the dusty gas model can be obtained by setting = 1, grad = 0 in equation... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

Let us first consider experiments without composition gradients. These are permeability measurements, in which flow is induced by a pressure gradient Consider first the flow of pure substance 1, setting x = 1, = 0 and... 

It must be emphasized that equation (10.14) is based on the assumption of strictly isobaric diffusion. In practice, we may ask, how closely must the pressures on the two sides of the diffusion cell be balanced if undue error is not to be associated with the use of (10.14) The answer, of course, is that pressure gradients must be small enough that the term in grad p on the right hand side of equation (10.4) is negligible coranared with the term in grad Since the viscous term dominates the coeffi-... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

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