Darcy

For direct measurement from core samples, the samples are mounted in a holder and gas is flowed through the core. The pressure drop across the core and the flowrate are measured. Providing the gas viscosity (ji) and sample dimensions are known the permeability can be calculated using the Darcy equation shown below. 

Keywords compressibility, primary-, secondary- and enhanced oil-recovery, drive mechanisms (solution gas-, gas cap-, water-drive), secondary gas cap, first production date, build-up period, plateau period, production decline, water cut, Darcy s law, recovery factor, sweep efficiency, by-passing of oil, residual oil, relative permeability, production forecasts, offtake rate, coning, cusping, horizontal wells, reservoir simulation, material balance, rate dependent processes, pre-drilling. 

On a microscopic scale, the most important equation governing fluid flow in the reservoir is Darcy s law, which was derived from the following situation. 

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. 

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,... 

Darcy s Law and the Basic Filtration Equation. Darcy s law combines the constants in the last term of equation 1 into one factor K known as the permeabiHty of the bed, ie. 

Modem filtration theory tends to prefer the Ruth form of Darcy s law, ie,... 

The fundamental case for pressure filters may be made using equation 10 for dry cake production capacity Y (kg/m s) derived from Darcy s law when the filter medium resistance is neglected. Eor the same cycle time (same speed), if the pressure drop is increased by a factor of four, production capacity is doubled. In other words, filtration area can be halved for the same capacity but only if is constant. If increases with pressure drop, and depending how fast it increases, the increased pressure drop may not give much more capacity and may actually cause capacity reductions. 

The framework for the solution of porous media flow problems was estabUshed by the experiments of Henri Darcy in the 1800s. The relationship between fluid volumetric flow rate, hydraulic gradient, and cross-sectional area, yi, of flow is given by the Darcy formula ... 

This form of Darcy s law is appHcable only to saturated flow. As discussed earlier, there are distinctions between the state of soil water in the saturated and unsaturated regions. These distinctions lead to an alternative form of Darcy s law for the case of unsaturated flow (2,5). 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

Table 5. Correlations for Heat-Transfer and Darcy Friction Coefficients for Noncircular Laminar Duct Flow ...

Fig. 5. Moody diagram for Darcy friction factor (13) (-----), smooth flow (----), whoUy turbulent flow ( ), laminar flow.

Fanning friction factor Darcy friction factor mass flux... 

Pressure drop in static mixers depends very strongly on geometric arrangement of the inserts. It is simply defined in relation to the pressure drop AP in an empty tube given by Darcy s equation ... 

When pure water is forced through a porous ultrafiltration membrane, Darcy s law states that the flow rate is direcdy proportional to the pressure gradient ... 

For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots/as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction fac tor used by Moody Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor ... 

Creeping flow (Re <- 1) through porous media is often described in terms or the permeability k and Darcy s Law ... 

Fluid friction in conduits Darcy friction factor = 4/... 

Darcy s law has been used to derive an expression which reflects not only the effect of a change in elevation, but also provides a means for estimating changes in air rate resulting from changes in vacuum level and cake thickness (or cake weight per unit areaj. In order for this relationship to hold for changes in vacuum and cake thickness, it must be assumed that both cakes have the same specific resistance. 

This formula is another variation on the Affinity Laws. Monsieur s Darcy and VVeisbach were hydraulic civil engineers in France in the mid 1850s (some 50 years before Mr. H VV). They based their formulas on friction losses of water moving in open canals. They applied other friction coefficients from some private experimentation, and developed their formulas for friction losses in closed aqueduct tubes. Through the years, their coefficients have evolved to incorporate the concepts of laminar and turbulent flow, variations in viscosity, temperature, and even piping with non uniform (rough) internal. surface finishes. With. so many variables and coefficients, the D/W formula only became practical and popular after the invention of the electronic calculator. The D/W forntula is extensive and eomplicated, compared to the empirieal estimations of Mr. H W. 

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