A B C D
DALL TUBE DALTON'S LAW DALTON'S LAW OF PARTIAL PRESSURES DAMAGE INTERFACE DAMPING, OF HEAT EXCHANGER TUBES DARCY EQUATION DARCY FREE CONVECTION DARCY NUMBER DARCY'S LAW DARCY-PRANDTL NUMBER Data for molecules of practical interest DATING OF ARCHAEOLOGICAL SAMPLES DDT, DEFLAGRATION TO DETONATION TRANSITION DE LAVAL NOZZLES DE-ICING DEBORAH NUMBER DEBYE TEMPERATURE DECANTATION DECANTERS DECELERATING DROPS DECONVOLUTION, OPTICS DEEP SHAFT DEFINITE INTEGRALS DEFLAGRATION DEFLAGRATION TO DETONATION TRANSITION, DDT DEGENERACY DEGREES OF FREEDOM DEHYDRATION DELIQUESCENCE DELTA FUNCTION DENDRITE LAYER DENDRITIC CRYSTALS DENSITY GAS MODEL DENSITY MAXIMUM OF WATER DENSITY MEASUREMENT DENSITY OF GASES DENSITY OF LIQUIDS DENSITY, HOMOGENEOUS DENSITY, OF THE ATMOSPHERE DENSITY-WAVE OSCILLATIONS DEPARTMENT OF THE ENVIRONMENT, DoE DEPARTURE FROM FILM BOILING DEPARTURE FROM NUCLEATE BOILING, DNB DEPHLEGMATOR DEPOSITION DEPOSITION OF HOMO- AND HETERO-EPITAXIAL SILICON THICK FILMS BY MESO-PLASMA CVD DEPOSITION OF PARTICLES DEPOSITION RATE OF DROPLETS IN ANNULAR FLOW DERIAZ TURBINES Derivative (derived) unit of measurement DESALINATION DESALINATION OF OIL DESALINATION, FLASH EVAPORATION FOR DESICCANTS DESICCATION DESIGN BASIS ACCIDENT DESTRUCTION OF SURFACES DETECTION OF CHEMICAL AND BIOLOGICAL AEROSOLIZED POLLUTANTS DETERGENTS DETERMINANTS Determination of material properties: optical tomography applications DETERMINISTIC CHAOS DETONATION DETONATION WAVE DEUTERIUM DEUTERIUM OXIDE DEUTSCH-ANDERSON EQUATION DEVIATORIC STRESS DEVOLATIZATION OF COAL PARTICLES DEW POINT DEWAR DEWATERING DIAMETER, HYDRAULIC DIAMOND-SHAPED CYLINDER BUNDLE DIAPHRAGM GAUGE DIE DIE-CASTING DIELECTRIC HEATING DIELECTROPHORETIC FORCES DIESEL CONDITIONS DIESEL ENGINES DIESEL FUEL DIESEL JET DESTRUCTION DIESEL SPRAY DIESEL-EMITTED PARTICLES Differential approximations DIFFERENTIAL CONDENSATION CURVE DIFFERENTIAL EQUATIONS DIFFERENTIAL PRESSURE FLOWMETERS DIFFERENTIAL PRESSURE TRANSDUCERS DIFFRACTION DIFFUSER DIFFUSION DIFFUSION APPROXIMATION IN MULTIDIMENSIONAL RADIATIVE TRANSFER PROBLEMS DIFFUSION COEFFICIENT DIFFUSION COEFFICIENT OF GASES DIFFUSION EQUATIONS DIFFUSION FLAMES DIFFUSION IN ELECTROLYTE SOLUTION DIFFUSION LAW DIFFUSION PUMP DIFFUSIVE CONVECTION DILATANCY DILATANT FLUIDS DILATION OF GRANULAR MATERIAL DILUTANT FLUIDS DILUTE SUSPENSION Dimension (of a secondary quantity with respect to a given primary quantity) Dimensional Analysis Dimensional Analysis and Similarity Dimensional equation DIMENSIONAL MATRIX Dimensional quantity (variable) DIMENSIONAL STABLE ANODES, DSAS DIMENSIONALLY HOMOGENEOUS EQUATIONS DIMENSIONLESS GROUPS Dimensionless Parameters Dimensionless quantity (Dimensionless variable) DIMERS DIOXINS DIPHENYL DIPOLE MOMENT DIRAC DELTA FUNCTION DIRECT CONTACT CONDENSERS DIRECT CONTACT EVAPORATORS DIRECT CONTACT HEAT EXCHANGERS DIRECT CONTACT HEAT TRANSFER DIRECT CONTACT MASS TRANSFER DIRECT INVERSION OPTICAL TECHNIQUE DIRECT NUMERICAL SIMULATIONS, DNS DIRICHLET CONDITIONS DIRICHLET'S PROBLEM DISCHARGE COEFFICIENT DISCRETE ENERGY DISCRETE ORDINATE APPROXIMATION Discrete ordinates and finite volume methods Discrete ordinates method Discrete ordinates method for one-dimensional problems DISK AND DOUGHNUT BAFFLES DISK TYPE CENTRIFUGE DISK TYPE STEAM TURBINE DISORDER Dispersed Flow DISPERSED FLOW, IN NOZZLES DISPERSED LIQUID FLOWS DISPERSION IN POROUS MEDIA DISPERSION OF PARTICLES DISPERSION RELATIONSHIPS, FOR WAVES IN FLUIDS DISPLACEMENT CHROMATOGRAPHY Displacement thickness DISPLACEMENT THICKNESS, OF BOUNDARY LAYER DISSIPATION OF HEAT FROM EARTH'S SURFACE DISSIPATIVE SYSTEMS DISSOLVED AIR FLOTATION, DAF DISSOLVED SOLIDS DISTILLATION DISTILLATION REBOILERS DISTRIBUTIONS DISTURBANCE WAVES, IN ANNULAR FLOW DISYMMETRY OF SCATTERED LIGHT DITTUS-BOELTER CORRELATION DITTUS-BOELTER EQUATION DNB, DEPARTURE FROM NUCLEATE BOILING DOE DOLINSKII, A.A. DOMESTIC WATER HEATER DONNEN EFFECT DOPING DOPPLER ANEMOMETRY DOPPLER BROADENING DOPPLER BURST DOPPLER EFFECT DOPPLER GLOBAL VELOCIMETRY DOPPLER SHIFT DOUBLE DIFFUSIVE CONVECTION IN A ROTATING POROUS LAYER DOUBLE EXPOSURE HOLOGRAPHY DOUBLE FLASH METHODS DOUBLE-DIFFUSIVE MAGNETOCONVECTION DOUBLE-PIPE EXCHANGERS DOUBLING TIME DOUBLY STRATIFIED DARCY POROUS MEDIUM DOWTHERM DRAFT TUBE MIXER DRAG Drag Coefficient DRAG FORCE DRAG FORCE ON PARTICLES DRAG INDUCED FLOW DRAG ON A PARTICLE DRAG ON PARTICLES AND SPHERES DRAG ON REACTOR DRAG ON SOLID SPHERES AND BUBBLES DRAG REDUCTION DRAINING INTENSELY EVAPORATED WAVE FILMS DREITSER, G.A. DRIFT FLUX Drift flux models DRIFT VELOCITY DROP FORMATION DROP SHAPES DROP SPLITTING DROP TOWERS DROPLET COLLISION DROPLET DEPOSITION AND ENTRAINMENT, IN ANNULAR FLOW DROPLET DETECTION DROPLET GENERATION DROPLET MEASUREMENTS DROPLET SIZE DISTRIBUTION DROPLET SPRAYS DROPLET STREAM DROPLET SURFACE TENSION DROPLET/LIQUID SEPARATION DROPLETS Drops DROPS, MASS TRANSFER TO AND FROM Dropsize measurement DROPWISE CONDENSATION DROPWISE PROMOTERS DROWNING OUT DRUM TYPE STEAM TURBINE DRY CONTAINMENTS, FOR NUCLEAR REACTORS DRY-BULB TEMPERATURE DRYERS DRYING DRYING CHAMBERS DRYOUT DUAL-PURPOSE HEAT PUMPS DUBOIS' BODY SURFACE DUCTILE FRACTURE DUCTS, NONCIRCULAR, FLOW IN DUFOUR EFFECT DUNE FLOW DUST, AS AN AIR POLLUTANT DUSTS DUSTY PLASMAS DWARF GALAXIES DYE LASERS DYNAMIC INSTABILITIES IN TWO-PHASE SYSTEMS Dynamic Pressure DYNAMIC WAVES DYNAMICAL SIMILARITY DYNAMICS
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DARCY'S LAW

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Our current ability to predict the flow of fluids through porous media, for example, in ground water movements, the recovery of oil or the design of filter beds originates from the experimental work performed by French engineer Henry Darcy (1856).

At the time, Darcy was concerned with developing the water works for the town of Dijon, and he needed information which would enable him to determine the size of the filter bed that should be installed to handle the daily throughput of water in the system. To provide this information, he performed a series of experiments in which water was passed downwards through vertical columns of sand at a controlled rate, and fluid pressures were measured at the top and bottom of the columns using mercury manometers. The results showed that there was a linear relationship between the flow rate per unit of cross-sectional area of the column, the column height, and the pressure differential when expressed as the differences in heights of equivalent water manometers above and below the sand. Historically (before the introduction of the SI system of units), the relationship was expressed in the form:

(1)

where is the volumetric flow rate, A is the cross-sectional, area 1 is the thickness of the sand, and (h1 - h2) is the water equivalent difference in manometer levels. The constant in this equation was found to be dependent upon the sizes of sand particles used in the experiment.

Other workers have subsequently shown that by introducing the concept of fluid potential and the physical properties of fluid, Darcy's original result can be extended to provide a relationship which is universally-applicable to any single-phase fluid, and for any flow direction. Expressed in SI nomenclature, it becomes:

(2)

where η is the fluid viscosity; ρ is the fluid density; dФ/dl is the potential gradient; and K is a property of the porous medium having the dimensions of (length)2 and known as permeability. This relationship has appropriately become known as Darcy's Law.

At die Darcy Centennial Hydrology Symposium in 1956, M. King Hubbert presented a paper in which he showed that Darcy's law has a theoretical foundation, and that the law can be derived from the Navier Stokes equation for a viscous fluid. He noted that Darcy's law has many analogies to Ohm's Law for the Conduction of Electricity and Fourier's expression for the conduction of heat. He pointed out, however, mat whilst the relationship given in Eq. (2) may appear to have similarities with the Poiseuille equation for laminar flow of fluids in straight cylindrical capillaries, expressed as:

(3)

(where r is the tube radius), the fluid mechanics of flow through porous media are very different from those of flow through capillary tubes. (See Fourier's Law.)

In porous media, inertia forces play an important role since each fluid particle moves through a tortuous path in which it is continuously being accelerated and decelerated; whereas in capillary flow, the fluid particles move in straight lines at a constant velocity. Thus, the similarities between Darcy's law and Poiseuille's equation are fortuitous. (See Poiseuille Flow.)

Darcy's law strictly applies to the flow of a single-phase fluid, in which case permeability is a property of the rock and is independent of the fluid flowing through the pores. When two or more phases flow through the rock at the same time however, each phase must flow through part of the total cross-section of pore space, thus reducing the flow area available to the other phases. As a result, the effective permeability of the individual phases is reduced. To accommodate this situation, Darcy's law has been modified by introducing a multiplication factor, known as relative permeability, ki, for each phase.

(4)

The value of ki will clearly depend upon the fraction of the pore flow area occupied by the other phases, and hence upon the saturation of the other phases. Typical relative permeability curves for two-phase oil and water flow through a sandstone rock are shown in Figure 1.

Example of water/oil relative permeability curves.

Figure 1. Example of water/oil relative permeability curves.

The relative permeability of one phase decreases as the saturation of the other phase increases, and vice versa. Note that the relative permeability of each phase becomes zero at a nonzero value of the phase saturation. This is because of the fluid pore structure, which prevents phase saturations below a certain level. Thus for two-phase flow, individual phase permeabilities depend on the nature of the other fluids as well as the structure of the rock matrix.

REFERENCES

Darcy, H. (1856) Les Fontaines Publiques de la Ville de Dijon., Victor Dalmont, Paris.

King, Hubbert M. (1956) Darcy's Law and the Field Equations of the Flow of Underground Fluids., Trans. AIME, 207: 222-239.

References

  1. Darcy, H. (1856) Les Fontaines Publiques de la Ville de Dijon., Victor Dalmont, Paris.
  2. King, Hubbert M. (1956) Darcy's Law and the Field Equations of the Flow of Underground Fluids., Trans. AIME, 207: 222-239.

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