4338:
157:
2385:, specifically how the hot water percolates through the coffee grinds under pressure, starting with a 2001 paper by Varlamov and Balestrino, and continuing with a 2007 paper by Gianino, a 2008 paper by Navarini et al., and a 2008 paper by W. King. The papers will either take the coffee permeability to be constant as a simplification or will measure change through the brewing process.
2819:
3744:. Hence Darcy's law is not always valid in such sediments. For flow through commercial circular pipes, the flow is laminar when the Reynolds number is less than 2000 and turbulent when it is more than 4000, but in some sediments, it has been found that flow is laminar when the value of the Reynolds number is less than 1.
2626:
Some carbonate reservoirs have many fractures, and Darcy's equation for multiphase flow is generalized in order to govern both flow in fractures and flow in the matrix (i.e. the traditional porous rock). The irregular surface of the fracture walls and high flow rate in the fractures may justify the
2622:
The flow in the middle of a sandstone reservoir is so slow that
Forchheimer's equation is usually not needed, but the gas flow into a gas production well may be high enough to justify use of Forchheimer's equation. In this case, the inflow performance calculations for the well, not the grid cell of
1654:
less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that flow regimes with
Reynolds numbers up to 10 may still be Darcian, as in the case of groundwater flow. The Reynolds number (a dimensionless parameter) for porous media flow is typically
139:
first refined Darcy's equation for a single-phase flow by including viscosity in the single (fluid) phase equation of Darcy. It can be understood that viscous fluids have more difficulty permeating through a porous medium than less viscous fluids. This change made it suitable for researchers in the
3135:
2357:
simultaneous flow and immiscible mixing of all fluid phases in the oil zone. The operator of the oil field may also inject water (and/or gas) in order to improve oil production. The petroleum industry is therefore using a generalized Darcy equation for multiphase flow that was developed by
2356:
Almost all oil reservoirs have a water zone below the oil leg, and some have also a gas cap above the oil leg. When the reservoir pressure drops due to oil production, water flows into the oil zone from below, and gas flows into the oil zone from above (if the gas cap exists), and we get a
3251:
2972:
2150:
1405:. A number of these parameters are used in alternative definitions below. A negative sign is used in the definition of the flux following the standard physics convention that fluids flow from regions of high pressure to regions of low pressure. Note that the
2654:
160:
Diagram showing definitions and directions for Darcy's law. A is the cross sectional area (m) of the cylinder. Q is the flow rate (m/s) of the fluid flowing through the area A. The flux of fluid through A is q = Q/A. L is the length of the cylinder. Δp =
2635:
For gas flow in small characteristic dimensions (e.g., very fine sand, nanoporous structures etc.), the particle-wall interactions become more frequent, giving rise to additional wall friction (Knudsen friction). For a flow in this region, where both
3433:
2855:
is the effective
Knudsen diffusivity of the porous media. The model can also be derived from the first-principle-based binary friction model (BFM). The differential equation of transition flow in porous media based on BFM is given as
3332:
2999:
2328:
1967:
2577:
4078:
3601:
3510:
3707:
140:
petroleum industry. Based on experimental results by his colleagues
Wyckoff and Botset, Muskat and Meres also generalized Darcy's law to cover a multiphase flow of water, oil and gas in the porous medium of a
2231:
1622:
the discharge rate of fluid will often be different — through different formation materials (or even through the same material, in a different direction) — even if the same pressure gradient exists in both
1592:
3146:
798:
1849:
521:
884:
4160:
Jin, Y.; Uth, M.-F.; Kuznetsov, A. V.; Herwig, H. (2 February 2015). "Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study".
2862:
1997:
1703:
1519:
3531:
For very short time scales, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of
703:
403:
2814:{\displaystyle N=-\left({\frac {k}{\mu }}{\frac {p_{a}+p_{b}}{2}}+D_{\mathrm {K} }^{\mathrm {eff} }\right){\frac {1}{R_{\mathrm {g} }T}}{\frac {p_{\mathrm {b} }-p_{\mathrm {a} }}{L}}\,,}
617:
3519:
is known as the
Klinkenberg parameter, which depends on the gas and the porous medium structure. This is quite evident if we compare the above formulations. The Klinkenberg parameter
2436:
1111:
450:
3638:
Another extension to the traditional form of Darcy's law is the
Brinkman term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1949),
1348:
1206:
1616:
if there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient — hence the negative sign in Darcy's law),
1403:
1051:
1154:
996:
930:
661:
2365:
or generalized Darcy equation (or law) or simply Darcy's equation (or law) or simply flow equation if the context says that the text is discussing the multiphase equation of
1237:
2617:
1285:
890:
3343:
2472:
in electrostatics. The flux vector is analogous to the current density, head is analogous to voltage, and hydraulic conductivity is analogous to electrical conductivity.)
2349:
to determine the flow through permeable media — the most simple of which is for a one-dimensional, homogeneous rock formation with a single fluid phase and constant fluid
960:
824:
574:
2648:
presented a semi-empirical model for flow in transition regime based on his experiments on small capillaries. For a porous medium, the
Knudsen equation can be given as
1464:
3720:
term. This correction term accounts for flow through medium where the grains of the media are porous themselves, but is difficult to use, and is typically neglected.
548:
263:
198:
4050:
Pant, L. M.; Mitra, S. K.; Secanell, M. (2012). "Absolute permeability and
Knudsen diffusivity measurements in PEMFC gas diffusion layers and micro porous layers".
1074:
314:
1368:
1305:
1257:
1016:
354:
334:
290:
237:
3262:
1409:
must be taken into account if the inlet and outlet are at different elevations. If the change in pressure is negative, then the flow will be in the positive
2623:
the 3D model, is based on the
Forchheimer equation. The effect of this is that an additional rate-dependent skin appears in the inflow performance formula.
3964:"Experimental investigation of steam pressure coffee extraction in a stove-top coffee maker" L. Navarini, E. Nobile, F. Pinto, A. Scheri, F. Suggi-Liverani
3130:{\displaystyle {\frac {\partial p}{\partial x}}=-R_{\mathrm {g} }T\left({\frac {kp}{\mu }}+D_{\mathrm {K} }\right)^{-1}{\dfrac {p}{R_{\mathrm {g} }T}}q\,.}
3934:
2276:, and the common form is obtained as below, which enables the determination of the liquid flow velocity by solving a set of equations in a given region.
2282:
1879:
3630:
groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use.
2506:
3740:
are small; thus, the flow is laminar. Coarse-grained sediments also behave similarly, but in very coarse-grained sediments, the flow may be
3541:
3444:
3644:
1469:
4018:"Knudsen diffusivity and permeability of pemfc microporous coated gas diffusion layers for different polytetrafluoroethylene loadings"
2361:
et alios. Because Darcy's name is so widespread and strongly associated with flow in porous media, the multiphase equation is denoted
2165:
3246:{\displaystyle q=-{\frac {k}{\mu }}\left(1+{\frac {D_{\mathrm {K} }\mu }{k}}{\frac {1}{p}}\right){\frac {\partial p}{\partial x}}\,.}
826:, often referred to as the Darcy flux or Darcy velocity, is not the velocity at which the fluid is travelling through the pores. The
1535:
359:
737:
4235:
589:
1788:
3919:
3610:
is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (>
482:
2967:{\displaystyle {\frac {\partial p}{\partial x}}=-R_{\mathrm {g} }T\left({\frac {kp}{\mu }}+D_{\mathrm {K} }\right)^{-1}N\,.}
2145:{\displaystyle k_{ni}\left(k^{-1}\right)_{ij}u_{j}=\delta _{nj}u_{j}=u_{n}=-{\frac {k_{ni}}{\varphi \mu }}\partial _{i}p\,,}
2369:
et alios. Multiphase flow in oil and gas reservoirs is a comprehensive topic, and one of many articles about this topic is
412:
852:
1661:
4123:
Brinkman, H. C. (1949). "A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles".
2370:
2362:
667:
3523:
is dependent on permeability, Knudsen diffusivity and viscosity (i.e., both gas and porous medium properties).
1437:
4277:
3955:
Gianino, Concetto. Experimental analysis of the
Italian coffee pot "moka". American Journal of Physics (2007)
2404:
628:
In the (less general) integral form, the volumetric flux and the pressure gradient correspond to the ratios:
270:
1619:
the greater the pressure gradient (through the same formation material), the greater the discharge rate, and
1079:
4387:
3946:
A. Varlamov and G. Balestrino, “La fisica di un buon caffè,” Il Nuovo Saggiatore 173-4, 59–66 2001.
87:
3626:
at constant head boundaries at very small times. This form is more mathematically rigorous but leads to a
1310:
1162:
455:
2981:
as well as porous media. The terminology of the Knudsen effect and Knudsen diffusivity is more common in
83:
1597:
Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that
1373:
1021:
4367:
4318:
4228:
1422:
1418:
1120:
216:
and in a homogeneously permeable medium, is given by a simple proportionality relationship between the
144:. The generalized multiphase flow equations by Muskat and others provide the analytical foundation for
965:
899:
633:
4308:
3765:
3615:
3428:{\displaystyle k^{\mathrm {eff} }=k\left(1+{\frac {D_{\mathrm {K} }\mu }{k}}{\frac {1}{p}}\right)\,.}
1628:
1211:
126:
4137:
2594:
1262:
3863:
3623:
4079:"A modified Maxwell–Stefan model for transport through inert membranes: The binary friction model"
943:
807:
557:
4372:
1873:
is the pressure. Assuming the viscous resisting force is linear with the velocity we may write:
4267:
4132:
3973:
King, Warren. "The physics of a stove-top espresso machine". American Journal of Physics (2008)
2990:
2982:
2630:
2458:
1526:
889:
The Darcy's constitutive equation, for single phase (fluid) flow, is the defining equation for
622:
71:
59:
530:
245:
180:
82:
Darcy's law was first determined experimentally by Darcy, but has since been derived from the
4377:
4221:
2346:
1414:
937:
721:
466:
145:
1650:
flow; however, most groundwater flow cases fall in this category. Typically any flow with a
4169:
3828:
2986:
581:
217:
122:
1059:
299:
63:
8:
3775:
3627:
3256:
Comparing this equation with conventional Darcy's law, a new formulation can be given as
2493:
1713:
1522:
141:
103:
4173:
3832:
3819:
Whitaker, S. (1986). "Flow in porous media I: A theoretical derivation of Darcy's law".
3327:{\displaystyle q=-{\frac {k^{\mathrm {eff} }}{\mu }}{\frac {\partial p}{\partial x}}\,,}
4382:
4185:
3844:
3770:
3619:
2334:
1353:
1290:
1242:
1114:
1001:
551:
339:
319:
275:
222:
201:
4094:
3438:
This is equivalent to the effective permeability formulation proposed by Klinkenberg:
4323:
4189:
3915:
3883:
3848:
3737:
2641:
1728:
is a representative grain diameter for the porous media (the standard choice is math|
1054:
4108:
Klinkenberg, L. J. (1941). "The permeability of porous media to liquids and gases".
1724:
is the specific discharge (not the pore velocity — with units of length per time),
4272:
4177:
4142:
4090:
4063:
4059:
4029:
3907:
3875:
3836:
3532:
2465:
91:
2381:
A number of papers have utilized Darcy's law to model the physics of brewing in a
2481:
2323:{\displaystyle {\boldsymbol {q}}=-{\frac {k}{\mu }}\,{\boldsymbol {\nabla }}p\,.}
1651:
713:
577:
95:
58:
difference (which is often just proportional to the pressure difference) via the
4362:
4342:
4262:
3785:
2645:
2450:
1406:
55:
1962:{\displaystyle -\left(k^{-1}\right)_{ij}\mu \varphi u_{j}-\partial _{i}p=0\,,}
1631:(based on Darcy's law and the conservation of mass) is in the construction of
54:
in electrostatics, linearly relating the volume flow rate of the fluid to the
4356:
4287:
3887:
3780:
2993:. Using the definition of molar flux, the above equation can be rewritten as
2366:
2358:
2240:
896:
With reference to the diagram to the right, the flow velocity is in SI units
827:
266:
240:
209:
136:
107:
67:
47:
3901:
2631:
Correction for gases in fine media (Knudsen diffusion or Klinkenberg effect)
2445:
is the volume flux vector of the fluid at a particular point in the medium,
1609:
if there is no pressure gradient over a distance, no flow occurs (these are
4313:
4244:
3760:
3754:
3729:
2989:. In geological and petrochemical engineering, this effect is known as the
2572:{\displaystyle \nabla p=-{\frac {\mu }{k}}q-{\frac {\rho }{k_{1}}}q^{2}\,,}
2469:
2464:, at that point. The hydraulic conductivity can often be approximated as a
1610:
1429:
729:
708:
In case of an anisotropic porous media, the permeability is a second order
474:
130:
117:
One application of Darcy's law is in the analysis of water flow through an
99:
51:
43:
3879:
1779:
1636:
1598:
31:
4033:
1987:
is the second order permeability tensor. This gives the velocity in the
4181:
4146:
3840:
3611:
2497:
1736:
726:
336:
over which the pressure drop is computed, and the cross-sectional area
4337:
3741:
3717:
3596:{\displaystyle \tau {\frac {\partial q}{\partial t}}+q=-k\nabla h\,,}
2978:
2350:
293:
111:
4083:
Chemical Engineering Journal and the Biochemical Engineering Journal
3505:{\displaystyle k^{\mathrm {eff} }=k\left(1+{\frac {b}{p}}\right)\,.}
4282:
4017:
3911:
3733:
3702:{\displaystyle -\beta \nabla ^{2}q+q=-{\frac {k}{\mu }}\nabla p\,,}
2489:
2485:
2382:
2237:
1977:
1417:
for absolute permeability, and the most famous one is probably the
839:
471:
4253:
2637:
1647:
1632:
1602:
213:
118:
2226:{\displaystyle q_{n}=-{\frac {k_{ni}}{\mu }}\,\partial _{i}p\,.}
156:
4213:
2461:
2243:
the off-diagonal elements in the permeability tensor are zero,
2155:
which gives Darcy's law for the volumetric flux density in the
709:
27:
16:
Equation describing the flow of a fluid through a porous medium
4016:
Carrigy, N.; Pant, L. M.; Mitra, S. K.; Secanell, M. (2013).
1717:
1587:{\displaystyle K={\frac {k\rho g}{\mu }}={\frac {kg}{\nu }}.}
35:
23:
3140:
This equation can be rearranged into the following equation
3937:
Journal of Colloid and Interface Science 539 (2019) 379–387
793:{\displaystyle q_{i}=-{\frac {k_{ij}}{\mu }}\partial _{j}p}
39:
2644:
friction are present, a new formulation needs to be used.
1466:
one can decline the integral form also into the equation:
2345:
Another derivation of Darcy's law is used extensively in
1640:
4015:
1844:{\displaystyle \mu \nabla ^{2}u_{i}-\partial _{i}p=0\,,}
1627:
A graphical illustration of the use of the steady-state
1428:
By considering the relation for static fluid pressure (
962:, or discharge per unit area, is also defined in units
516:{\displaystyle \mathbf {q} =-{\frac {k}{\mu }}\nabla p}
3935:
Tailoring Porous Media For Controllable Capillary Flow
3614:). The main reason for doing this is that the regular
2591:
is known as inertial permeability, in units of length
4159:
3647:
3544:
3447:
3346:
3265:
3149:
3095:
3002:
2865:
2657:
2597:
2509:
2407:
2285:
2168:
2000:
1882:
1791:
1664:
1538:
1472:
1440:
1376:
1356:
1313:
1293:
1265:
1245:
1214:
1165:
1123:
1082:
1062:
1024:
1004:
968:
946:
902:
855:
810:
740:
670:
636:
592:
560:
533:
485:
415:
362:
342:
322:
302:
278:
248:
225:
183:
3526:
1747:
For stationary, creeping, incompressible flow, i.e.
879:{\displaystyle \mathbf {q} =\varphi \,\mathbf {u} .}
461:
The Darcy's law can be generalised to a local form:
62:. In fact, the Darcy's law is a special case of the
1698:{\displaystyle \mathrm {Re} ={\frac {qd}{\nu }}\,,}
1413:direction. There have been several proposals for a
3997:
3701:
3595:
3504:
3427:
3326:
3245:
3129:
2966:
2813:
2611:
2571:
2500:behavior of the pressure difference vs flow data.
2488:effects can also become significant. Sometimes an
2430:
2322:
2225:
2144:
1961:
1843:
1697:
1586:
1514:{\displaystyle Q={\frac {kAg}{\nu L}}\,{\Delta h}}
1513:
1458:
1397:
1362:
1342:
1299:
1279:
1251:
1231:
1200:
1148:
1105:
1068:
1045:
1010:
990:
954:
924:
878:
818:
792:
697:
655:
611:
568:
542:
515:
444:
397:
348:
328:
308:
284:
257:
231:
192:
4112:. American Petroleum Institute. pp. 200–213.
4049:
4011:
4009:
208:In the integral form, Darcy's law, as refined by
4354:
2492:term is added to the Darcy's equation, known as
2398:Darcy's law can be expressed very generally as:
2337:for single-phase fluid flow in a porous medium.
269:. The proportionality constant is linked to the
3736:. In fine-grained sediments, the dimensions of
3633:
2340:
1778:, the Navier–Stokes equation simplifies to the
1739:analysis using sieves — with units of length).
34:based on results of experiments on the flow of
4045:
4043:
4006:
3998:Cunningham, R. E.; Williams, R. J. J. (1980).
3903:Read "Memorial Tributes: Volume 14" at NAP.edu
1159:In the integral form, the total pressure drop
698:{\displaystyle \nabla p={\frac {\Delta p}{L}}}
4229:
398:{\displaystyle Q={\frac {kA}{\mu L}}\Delta p}
3806:Les fontaines publiques de la ville de Dijon
22:is an equation that describes the flow of a
4107:
4040:
2496:term. This term is able to account for the
4236:
4222:
3723:
2393:
4205:Soil Mechanics and Foundation Engineering
4136:
3695:
3589:
3498:
3421:
3320:
3239:
3123:
2960:
2807:
2565:
2316:
2307:
2263:and the diagonal elements are identical,
2219:
2205:
2138:
1955:
1837:
1691:
1502:
1307:, or discharge, is also defined in units
867:
612:{\displaystyle \sigma ={\frac {k}{\mu }}}
121:; Darcy's law along with the equation of
4122:
3818:
2376:
1782:, which by neglecting the bulk term is:
155:
4076:
2431:{\displaystyle \mathbf {q} =-K\nabla h}
2287:
1735:, which is the 30% passing size from a
4355:
4022:Journal of the Electrochemical Society
1106:{\displaystyle \mathrm {(Pa\cdot s)} }
4217:
4202:
3982:
3864:"Henry Darcy and the making of a law"
3861:
3803:
1259:is the length of the sample in units
445:{\displaystyle R={\frac {\mu L}{kA}}}
70:, in turn deriving from the momentum
1646:Darcy's law is only valid for slow,
1343:{\displaystyle \mathrm {(m^{3}/s)} }
1201:{\displaystyle \Delta p=p_{b}-p_{a}}
716:one can write the more general law:
129:, one of the basic relationships of
4000:Diffusion in Gases and Porous Media
2388:
1287:, the Darcy's volumetric flow rate
204:applied between the points a and b.
13:
3689:
3655:
3583:
3559:
3551:
3460:
3457:
3454:
3391:
3359:
3356:
3353:
3311:
3303:
3289:
3286:
3283:
3230:
3222:
3189:
3107:
3074:
3035:
3014:
3006:
2937:
2898:
2877:
2869:
2795:
2780:
2759:
2736:
2733:
2730:
2723:
2602:
2510:
2422:
2207:
2126:
1937:
1819:
1796:
1669:
1666:
1504:
1398:{\displaystyle \mathrm {(m^{2})} }
1382:
1333:
1319:
1270:
1222:
1219:
1166:
1139:
1131:
1128:
1096:
1090:
1087:
1046:{\displaystyle \mathrm {(m^{2})} }
1030:
981:
973:
915:
907:
778:
683:
671:
621:can be thought as the Darcy's law
534:
507:
454:can be defined as the Darcy's law
389:
249:
184:
30:medium. The law was formulated by
14:
4399:
3527:Darcy's law for short time scales
1149:{\displaystyle \mathrm {(Pa/m)} }
4336:
4243:
4110:Drilling and Production Practice
2475:
2409:
2309:
991:{\displaystyle \mathrm {(m/s)} }
948:
925:{\displaystyle \mathrm {(m/s)} }
869:
857:
812:
656:{\displaystyle q={\frac {Q}{A}}}
562:
487:
4196:
4153:
4116:
4101:
4070:
3991:
3976:
2627:use of Forchheimer's equation.
2480:For flows in porous media with
2371:Darcy's law for multiphase flow
2363:Darcy's law for multiphase flow
1232:{\displaystyle \mathrm {(Pa)} }
846:) with the following equation:
4064:10.1016/j.jpowsour.2012.01.099
3967:
3958:
3949:
3940:
3928:
3894:
3855:
3812:
3797:
3757:, a unit of fluid permeability
2612:{\displaystyle \mathrm {(m)} }
2605:
2599:
1391:
1378:
1336:
1315:
1280:{\displaystyle \mathrm {(m)} }
1273:
1267:
1225:
1216:
1142:
1125:
1099:
1084:
1039:
1026:
984:
970:
918:
904:
151:
1:
4095:10.1016/S0923-0467(96)03134-X
3791:
1742:
1350:and the cross-sectional area
893:(single phase permeability).
77:
3634:Brinkman form of Darcy's law
2484:greater than about 1 to 10,
2341:Use in petroleum engineering
1635:, to quantify the amount of
955:{\displaystyle \mathbf {q} }
819:{\displaystyle \mathbf {q} }
569:{\displaystyle \mathbf {q} }
90:methods. It is analogous to
7:
4125:Applied Scientific Research
3747:
2977:This equation is valid for
292:of the medium, the dynamic
10:
4404:
4162:Journal of Fluid Mechanics
2582:where the additional term
834:) is related to the flux (
580:which here is called also
4332:
4309:Groundwater flow equation
4296:
4251:
4002:. New York: Plenum Press.
3821:Transport in Porous Media
3766:Groundwater flow equation
3728:Darcy's law is valid for
3616:groundwater flow equation
1629:groundwater flow equation
1459:{\displaystyle p=\rho gh}
932:, and since the porosity
804:Notice that the quantity
148:that exists to this day.
127:groundwater flow equation
4052:Journal of Power Sources
3987:. John Wiley & Sons.
3985:Convection Heat Transfer
3868:Water Resources Research
2333:The above equation is a
543:{\displaystyle \nabla p}
258:{\displaystyle \Delta p}
193:{\displaystyle \nabla p}
3724:Validity of Darcy's law
2468:. (Note the analogy to
2394:Differential expression
1865:is the velocity in the
584:. Note that the ratio:
84:Navier–Stokes equations
42:, forming the basis of
4268:hydraulic conductivity
4207:. Standard Publishers.
3703:
3597:
3506:
3429:
3328:
3247:
3131:
2968:
2815:
2613:
2573:
2459:hydraulic conductivity
2432:
2324:
2227:
2146:
1963:
1845:
1699:
1588:
1527:hydraulic conductivity
1515:
1460:
1423:Kozeny–Carman equation
1399:
1364:
1344:
1301:
1281:
1253:
1233:
1202:
1150:
1107:
1070:
1047:
1012:
992:
956:
926:
880:
820:
794:
699:
657:
623:hydraulic conductivity
613:
570:
544:
517:
446:
399:
350:
330:
310:
286:
259:
233:
205:
194:
72:Navier-Stokes equation
60:hydraulic conductivity
4203:Arora, K. R. (1989).
3862:Brown, G. O. (2002).
3704:
3598:
3507:
3430:
3329:
3248:
3132:
2969:
2837:is the gas constant,
2816:
2614:
2574:
2433:
2377:Use in coffee brewing
2347:petroleum engineering
2325:
2228:
2147:
1964:
1846:
1700:
1605:exhibits, including:
1589:
1516:
1461:
1415:constitutive equation
1400:
1365:
1345:
1302:
1282:
1254:
1234:
1203:
1151:
1108:
1071:
1048:
1013:
993:
957:
938:nondimensional number
927:
891:absolute permeability
881:
821:
795:
722:constitutive equation
700:
658:
614:
571:
545:
518:
467:constitutive equation
447:
407:Note that the ratio:
400:
351:
331:
316:, the given distance
311:
287:
260:
234:
195:
159:
146:reservoir engineering
50:. It is analogous to
4077:Kerkhof, P. (1996).
3880:10.1029/2001WR000727
3645:
3542:
3445:
3344:
3263:
3147:
3000:
2987:chemical engineering
2863:
2841:is the temperature,
2655:
2595:
2507:
2405:
2283:
2166:
1998:
1880:
1789:
1662:
1536:
1525:. The corresponding
1470:
1438:
1374:
1354:
1311:
1291:
1263:
1243:
1212:
1163:
1121:
1080:
1069:{\displaystyle \mu }
1060:
1022:
1002:
966:
944:
900:
853:
808:
738:
668:
634:
590:
582:superficial velocity
558:
531:
483:
456:hydraulic resistance
413:
360:
340:
320:
309:{\displaystyle \mu }
300:
276:
246:
223:
218:volumetric flow rate
214:gravitational forces
212:, in the absence of
181:
123:conservation of mass
4388:Transport phenomena
4297:Governing equations
4174:2015JFM...766...76J
4034:10.1149/2.036302jes
3833:1986TPMed...1....3W
3776:Black-oil equations
2828:is the molar flux,
2741:
1714:kinematic viscosity
1523:kinematic viscosity
998:; the permeability
142:petroleum reservoir
104:electrical networks
4343:Geology portal
4324:Hooghoudt equation
4182:10.1017/jfm.2015.9
4147:10.1007/BF02120313
3983:Bejan, A. (1984).
3841:10.1007/BF01036523
3804:Darcy, H. (1856).
3771:Mathematical model
3699:
3620:diffusion equation
3593:
3502:
3425:
3324:
3243:
3127:
3118:
2991:Klinkenberg effect
2964:
2811:
2717:
2609:
2569:
2428:
2335:governing equation
2320:
2223:
2142:
1959:
1858:is the viscosity,
1841:
1695:
1584:
1511:
1456:
1395:
1360:
1340:
1297:
1277:
1249:
1229:
1198:
1146:
1115:hydraulic gradient
1103:
1066:
1043:
1008:
988:
952:
922:
876:
816:
790:
695:
653:
609:
566:
552:hydraulic gradient
540:
513:
442:
395:
346:
326:
306:
282:
255:
229:
206:
202:hydraulic gradient
190:
125:simplifies to the
4368:Civil engineering
4350:
4349:
3921:978-0-309-15218-1
3808:. Paris: Dalmont.
3687:
3566:
3491:
3414:
3404:
3318:
3298:
3237:
3212:
3202:
3167:
3117:
3063:
3021:
2926:
2884:
2805:
2769:
2712:
2680:
2553:
2530:
2305:
2203:
2123:
1689:
1579:
1561:
1500:
1363:{\displaystyle A}
1300:{\displaystyle Q}
1252:{\displaystyle L}
1055:dynamic viscosity
1011:{\displaystyle k}
940:, the Darcy flux
775:
693:
651:
607:
505:
440:
387:
349:{\displaystyle A}
329:{\displaystyle L}
285:{\displaystyle k}
232:{\displaystyle Q}
4395:
4341:
4340:
4238:
4231:
4224:
4215:
4214:
4209:
4208:
4200:
4194:
4193:
4157:
4151:
4150:
4140:
4120:
4114:
4113:
4105:
4099:
4098:
4074:
4068:
4067:
4047:
4038:
4037:
4013:
4004:
4003:
3995:
3989:
3988:
3980:
3974:
3971:
3965:
3962:
3956:
3953:
3947:
3944:
3938:
3932:
3926:
3925:
3898:
3892:
3891:
3859:
3853:
3852:
3816:
3810:
3809:
3801:
3716:is an effective
3715:
3708:
3706:
3705:
3700:
3688:
3680:
3663:
3662:
3609:
3602:
3600:
3599:
3594:
3567:
3565:
3557:
3549:
3522:
3518:
3511:
3509:
3508:
3503:
3497:
3493:
3492:
3484:
3465:
3464:
3463:
3434:
3432:
3431:
3426:
3420:
3416:
3415:
3407:
3405:
3400:
3396:
3395:
3394:
3383:
3364:
3363:
3362:
3333:
3331:
3330:
3325:
3319:
3317:
3309:
3301:
3299:
3294:
3293:
3292:
3276:
3252:
3250:
3249:
3244:
3238:
3236:
3228:
3220:
3218:
3214:
3213:
3205:
3203:
3198:
3194:
3193:
3192:
3181:
3168:
3160:
3136:
3134:
3133:
3128:
3119:
3116:
3112:
3111:
3110:
3096:
3093:
3092:
3084:
3080:
3079:
3078:
3077:
3064:
3059:
3051:
3040:
3039:
3038:
3022:
3020:
3012:
3004:
2973:
2971:
2970:
2965:
2956:
2955:
2947:
2943:
2942:
2941:
2940:
2927:
2922:
2914:
2903:
2902:
2901:
2885:
2883:
2875:
2867:
2854:
2853:
2852:
2840:
2836:
2827:
2820:
2818:
2817:
2812:
2806:
2801:
2800:
2799:
2798:
2785:
2784:
2783:
2772:
2770:
2768:
2764:
2763:
2762:
2748:
2746:
2742:
2740:
2739:
2727:
2726:
2713:
2708:
2707:
2706:
2694:
2693:
2683:
2681:
2673:
2618:
2616:
2615:
2610:
2608:
2590:
2578:
2576:
2575:
2570:
2564:
2563:
2554:
2552:
2551:
2539:
2531:
2523:
2482:Reynolds numbers
2437:
2435:
2434:
2429:
2412:
2389:Additional forms
2329:
2327:
2326:
2321:
2312:
2306:
2298:
2290:
2275:
2262:
2252:
2232:
2230:
2229:
2224:
2215:
2214:
2204:
2199:
2198:
2186:
2178:
2177:
2158:
2151:
2149:
2148:
2143:
2134:
2133:
2124:
2122:
2114:
2113:
2101:
2093:
2092:
2080:
2079:
2070:
2069:
2054:
2053:
2044:
2043:
2035:
2031:
2030:
2013:
2012:
1990:
1986:
1975:
1968:
1966:
1965:
1960:
1945:
1944:
1932:
1931:
1916:
1915:
1907:
1903:
1902:
1872:
1868:
1864:
1857:
1850:
1848:
1847:
1842:
1827:
1826:
1814:
1813:
1804:
1803:
1777:
1775:
1773:
1772:
1767:
1764:
1727:
1723:
1711:
1704:
1702:
1701:
1696:
1690:
1685:
1677:
1672:
1639:flowing under a
1593:
1591:
1590:
1585:
1580:
1575:
1567:
1562:
1557:
1546:
1520:
1518:
1517:
1512:
1510:
1501:
1499:
1491:
1480:
1465:
1463:
1462:
1457:
1412:
1404:
1402:
1401:
1396:
1394:
1390:
1389:
1369:
1367:
1366:
1361:
1349:
1347:
1346:
1341:
1339:
1332:
1327:
1326:
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1276:
1258:
1256:
1255:
1250:
1238:
1236:
1235:
1230:
1228:
1207:
1205:
1204:
1199:
1197:
1196:
1184:
1183:
1155:
1153:
1152:
1147:
1145:
1138:
1112:
1110:
1109:
1104:
1102:
1075:
1073:
1072:
1067:
1052:
1050:
1049:
1044:
1042:
1038:
1037:
1017:
1015:
1014:
1009:
997:
995:
994:
989:
987:
980:
961:
959:
958:
953:
951:
935:
931:
929:
928:
923:
921:
914:
885:
883:
882:
877:
872:
860:
845:
837:
833:
825:
823:
822:
817:
815:
799:
797:
796:
791:
786:
785:
776:
771:
770:
758:
750:
749:
704:
702:
701:
696:
694:
689:
681:
662:
660:
659:
654:
652:
644:
618:
616:
615:
610:
608:
600:
575:
573:
572:
567:
565:
549:
547:
546:
541:
522:
520:
519:
514:
506:
498:
490:
451:
449:
448:
443:
441:
439:
431:
423:
404:
402:
401:
396:
388:
386:
378:
370:
355:
353:
352:
347:
335:
333:
332:
327:
315:
313:
312:
307:
291:
289:
288:
283:
264:
262:
261:
256:
238:
236:
235:
230:
199:
197:
196:
191:
102:in the field of
94:in the field of
38:through beds of
4403:
4402:
4398:
4397:
4396:
4394:
4393:
4392:
4353:
4352:
4351:
4346:
4335:
4328:
4314:Theis equation
4292:
4247:
4242:
4212:
4201:
4197:
4158:
4154:
4138:10.1.1.454.3769
4121:
4117:
4106:
4102:
4075:
4071:
4048:
4041:
4014:
4007:
3996:
3992:
3981:
3977:
3972:
3968:
3963:
3959:
3954:
3950:
3945:
3941:
3933:
3929:
3922:
3900:
3899:
3895:
3860:
3856:
3817:
3813:
3802:
3798:
3794:
3750:
3726:
3713:
3679:
3658:
3654:
3646:
3643:
3642:
3636:
3607:
3558:
3550:
3548:
3543:
3540:
3539:
3529:
3520:
3516:
3483:
3476:
3472:
3453:
3452:
3448:
3446:
3443:
3442:
3406:
3390:
3389:
3385:
3384:
3382:
3375:
3371:
3352:
3351:
3347:
3345:
3342:
3341:
3310:
3302:
3300:
3282:
3281:
3277:
3275:
3264:
3261:
3260:
3229:
3221:
3219:
3204:
3188:
3187:
3183:
3182:
3180:
3173:
3169:
3159:
3148:
3145:
3144:
3106:
3105:
3101:
3100:
3094:
3085:
3073:
3072:
3068:
3052:
3050:
3049:
3045:
3044:
3034:
3033:
3029:
3013:
3005:
3003:
3001:
2998:
2997:
2948:
2936:
2935:
2931:
2915:
2913:
2912:
2908:
2907:
2897:
2896:
2892:
2876:
2868:
2866:
2864:
2861:
2860:
2851:
2848:
2847:
2846:
2842:
2838:
2835:
2829:
2825:
2794:
2793:
2789:
2779:
2778:
2774:
2773:
2771:
2758:
2757:
2753:
2752:
2747:
2729:
2728:
2722:
2721:
2702:
2698:
2689:
2685:
2684:
2682:
2672:
2671:
2667:
2656:
2653:
2652:
2633:
2598:
2596:
2593:
2592:
2589:
2583:
2559:
2555:
2547:
2543:
2538:
2522:
2508:
2505:
2504:
2478:
2408:
2406:
2403:
2402:
2396:
2391:
2379:
2343:
2308:
2297:
2286:
2284:
2281:
2280:
2269:
2264:
2254:
2249:
2244:
2210:
2206:
2191:
2187:
2185:
2173:
2169:
2167:
2164:
2163:
2156:
2129:
2125:
2115:
2106:
2102:
2100:
2088:
2084:
2075:
2071:
2062:
2058:
2049:
2045:
2036:
2023:
2019:
2015:
2014:
2005:
2001:
1999:
1996:
1995:
1988:
1985:
1981:
1973:
1940:
1936:
1927:
1923:
1908:
1895:
1891:
1887:
1886:
1881:
1878:
1877:
1870:
1869:direction, and
1866:
1863:
1859:
1855:
1822:
1818:
1809:
1805:
1799:
1795:
1790:
1787:
1786:
1780:Stokes equation
1768:
1765:
1761:
1752:
1751:
1749:
1748:
1745:
1734:
1725:
1721:
1709:
1678:
1676:
1665:
1663:
1660:
1659:
1652:Reynolds number
1568:
1566:
1547:
1545:
1537:
1534:
1533:
1521:where ν is the
1503:
1492:
1481:
1479:
1471:
1468:
1467:
1439:
1436:
1435:
1419:Kozeny equation
1410:
1385:
1381:
1377:
1375:
1372:
1371:
1355:
1352:
1351:
1328:
1322:
1318:
1314:
1312:
1309:
1308:
1292:
1289:
1288:
1266:
1264:
1261:
1260:
1244:
1241:
1240:
1215:
1213:
1210:
1209:
1192:
1188:
1179:
1175:
1164:
1161:
1160:
1134:
1124:
1122:
1119:
1118:
1083:
1081:
1078:
1077:
1061:
1058:
1057:
1033:
1029:
1025:
1023:
1020:
1019:
1003:
1000:
999:
976:
969:
967:
964:
963:
947:
945:
942:
941:
933:
910:
903:
901:
898:
897:
868:
856:
854:
851:
850:
843:
835:
831:
811:
809:
806:
805:
802:
781:
777:
763:
759:
757:
745:
741:
739:
736:
735:
714:tensor notation
682:
680:
669:
666:
665:
643:
635:
632:
631:
599:
591:
588:
587:
578:volumetric flux
561:
559:
556:
555:
532:
529:
528:
525:
497:
486:
484:
481:
480:
432:
424:
422:
414:
411:
410:
379:
371:
369:
361:
358:
357:
356:, in the form:
341:
338:
337:
321:
318:
317:
301:
298:
297:
277:
274:
273:
247:
244:
243:
224:
221:
220:
182:
179:
178:
176:
172:
168:
164:
154:
96:heat conduction
80:
64:Stokes equation
17:
12:
11:
5:
4401:
4391:
4390:
4385:
4380:
4375:
4373:Soil mechanics
4370:
4365:
4348:
4347:
4333:
4330:
4329:
4327:
4326:
4321:
4319:Thiem equation
4316:
4311:
4306:
4300:
4298:
4294:
4293:
4291:
4290:
4285:
4280:
4275:
4270:
4265:
4263:hydraulic head
4259:
4257:
4249:
4248:
4241:
4240:
4233:
4226:
4218:
4211:
4210:
4195:
4152:
4115:
4100:
4089:(3): 319–343.
4069:
4039:
4005:
3990:
3975:
3966:
3957:
3948:
3939:
3927:
3920:
3912:10.17226/12884
3893:
3854:
3811:
3795:
3793:
3790:
3789:
3788:
3786:Ergun equation
3783:
3778:
3773:
3768:
3763:
3758:
3749:
3746:
3725:
3722:
3710:
3709:
3698:
3694:
3691:
3686:
3683:
3678:
3675:
3672:
3669:
3666:
3661:
3657:
3653:
3650:
3635:
3632:
3604:
3603:
3592:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3564:
3561:
3556:
3553:
3547:
3528:
3525:
3513:
3512:
3501:
3496:
3490:
3487:
3482:
3479:
3475:
3471:
3468:
3462:
3459:
3456:
3451:
3436:
3435:
3424:
3419:
3413:
3410:
3403:
3399:
3393:
3388:
3381:
3378:
3374:
3370:
3367:
3361:
3358:
3355:
3350:
3335:
3334:
3323:
3316:
3313:
3308:
3305:
3297:
3291:
3288:
3285:
3280:
3274:
3271:
3268:
3254:
3253:
3242:
3235:
3232:
3227:
3224:
3217:
3211:
3208:
3201:
3197:
3191:
3186:
3179:
3176:
3172:
3166:
3163:
3158:
3155:
3152:
3138:
3137:
3126:
3122:
3115:
3109:
3104:
3099:
3091:
3088:
3083:
3076:
3071:
3067:
3062:
3058:
3055:
3048:
3043:
3037:
3032:
3028:
3025:
3019:
3016:
3011:
3008:
2975:
2974:
2963:
2959:
2954:
2951:
2946:
2939:
2934:
2930:
2925:
2921:
2918:
2911:
2906:
2900:
2895:
2891:
2888:
2882:
2879:
2874:
2871:
2849:
2833:
2822:
2821:
2810:
2804:
2797:
2792:
2788:
2782:
2777:
2767:
2761:
2756:
2751:
2745:
2738:
2735:
2732:
2725:
2720:
2716:
2711:
2705:
2701:
2697:
2692:
2688:
2679:
2676:
2670:
2666:
2663:
2660:
2632:
2629:
2607:
2604:
2601:
2587:
2580:
2579:
2568:
2562:
2558:
2550:
2546:
2542:
2537:
2534:
2529:
2526:
2521:
2518:
2515:
2512:
2477:
2474:
2451:hydraulic head
2439:
2438:
2427:
2424:
2421:
2418:
2415:
2411:
2395:
2392:
2390:
2387:
2378:
2375:
2342:
2339:
2331:
2330:
2319:
2315:
2311:
2304:
2301:
2296:
2293:
2289:
2267:
2247:
2234:
2233:
2222:
2218:
2213:
2209:
2202:
2197:
2194:
2190:
2184:
2181:
2176:
2172:
2153:
2152:
2141:
2137:
2132:
2128:
2121:
2118:
2112:
2109:
2105:
2099:
2096:
2091:
2087:
2083:
2078:
2074:
2068:
2065:
2061:
2057:
2052:
2048:
2042:
2039:
2034:
2029:
2026:
2022:
2018:
2011:
2008:
2004:
1983:
1970:
1969:
1958:
1954:
1951:
1948:
1943:
1939:
1935:
1930:
1926:
1922:
1919:
1914:
1911:
1906:
1901:
1898:
1894:
1890:
1885:
1861:
1852:
1851:
1840:
1836:
1833:
1830:
1825:
1821:
1817:
1812:
1808:
1802:
1798:
1794:
1759:
1744:
1741:
1732:
1706:
1705:
1694:
1688:
1684:
1681:
1675:
1671:
1668:
1625:
1624:
1620:
1617:
1614:
1595:
1594:
1583:
1578:
1574:
1571:
1565:
1560:
1556:
1553:
1550:
1544:
1541:
1529:is therefore:
1509:
1506:
1498:
1495:
1490:
1487:
1484:
1478:
1475:
1455:
1452:
1449:
1446:
1443:
1407:elevation head
1393:
1388:
1384:
1380:
1359:
1338:
1335:
1331:
1325:
1321:
1317:
1296:
1275:
1272:
1269:
1248:
1227:
1224:
1221:
1218:
1195:
1191:
1187:
1182:
1178:
1174:
1171:
1168:
1144:
1141:
1137:
1133:
1130:
1127:
1101:
1098:
1095:
1092:
1089:
1086:
1065:
1041:
1036:
1032:
1028:
1007:
986:
983:
979:
975:
972:
950:
920:
917:
913:
909:
906:
887:
886:
875:
871:
866:
863:
859:
814:
801:
800:
789:
784:
780:
774:
769:
766:
762:
756:
753:
748:
744:
718:
692:
688:
685:
679:
676:
673:
650:
647:
642:
639:
606:
603:
598:
595:
564:
539:
536:
524:
523:
512:
509:
504:
501:
496:
493:
489:
463:
438:
435:
430:
427:
421:
418:
394:
391:
385:
382:
377:
374:
368:
365:
345:
325:
305:
281:
254:
251:
228:
189:
186:
174:
170:
166:
162:
153:
150:
88:homogenization
79:
76:
56:hydraulic head
48:earth sciences
46:, a branch of
15:
9:
6:
4:
3:
2:
4400:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4358:
4345:
4344:
4339:
4331:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4301:
4299:
4295:
4289:
4288:water content
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4260:
4258:
4255:
4250:
4246:
4239:
4234:
4232:
4227:
4225:
4220:
4219:
4216:
4206:
4199:
4191:
4187:
4183:
4179:
4175:
4171:
4167:
4163:
4156:
4148:
4144:
4139:
4134:
4130:
4126:
4119:
4111:
4104:
4096:
4092:
4088:
4084:
4080:
4073:
4065:
4061:
4057:
4053:
4046:
4044:
4035:
4031:
4028:(2): F81–89.
4027:
4023:
4019:
4012:
4010:
4001:
3994:
3986:
3979:
3970:
3961:
3952:
3943:
3936:
3931:
3923:
3917:
3913:
3909:
3905:
3904:
3897:
3889:
3885:
3881:
3877:
3873:
3869:
3865:
3858:
3850:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3815:
3807:
3800:
3796:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3756:
3752:
3751:
3745:
3743:
3739:
3735:
3731:
3721:
3719:
3696:
3692:
3684:
3681:
3676:
3673:
3670:
3667:
3664:
3659:
3651:
3648:
3641:
3640:
3639:
3631:
3629:
3625:
3624:singularities
3621:
3617:
3613:
3590:
3586:
3580:
3577:
3574:
3571:
3568:
3562:
3554:
3545:
3538:
3537:
3536:
3534:
3533:Fourier's law
3524:
3499:
3494:
3488:
3485:
3480:
3477:
3473:
3469:
3466:
3449:
3441:
3440:
3439:
3422:
3417:
3411:
3408:
3401:
3397:
3386:
3379:
3376:
3372:
3368:
3365:
3348:
3340:
3339:
3338:
3321:
3314:
3306:
3295:
3278:
3272:
3269:
3266:
3259:
3258:
3257:
3240:
3233:
3225:
3215:
3209:
3206:
3199:
3195:
3184:
3177:
3174:
3170:
3164:
3161:
3156:
3153:
3150:
3143:
3142:
3141:
3124:
3120:
3113:
3102:
3097:
3089:
3086:
3081:
3069:
3065:
3060:
3056:
3053:
3046:
3041:
3030:
3026:
3023:
3017:
3009:
2996:
2995:
2994:
2992:
2988:
2984:
2980:
2961:
2957:
2952:
2949:
2944:
2932:
2928:
2923:
2919:
2916:
2909:
2904:
2893:
2889:
2886:
2880:
2872:
2859:
2858:
2857:
2845:
2832:
2808:
2802:
2790:
2786:
2775:
2765:
2754:
2749:
2743:
2718:
2714:
2709:
2703:
2699:
2695:
2690:
2686:
2677:
2674:
2668:
2664:
2661:
2658:
2651:
2650:
2649:
2647:
2643:
2639:
2628:
2624:
2620:
2586:
2566:
2560:
2556:
2548:
2544:
2540:
2535:
2532:
2527:
2524:
2519:
2516:
2513:
2503:
2502:
2501:
2499:
2495:
2491:
2487:
2483:
2476:Quadratic law
2473:
2471:
2467:
2463:
2460:
2456:
2452:
2449:is the total
2448:
2444:
2425:
2419:
2416:
2413:
2401:
2400:
2399:
2386:
2384:
2374:
2372:
2368:
2364:
2360:
2354:
2352:
2348:
2338:
2336:
2317:
2313:
2302:
2299:
2294:
2291:
2279:
2278:
2277:
2274:
2270:
2261:
2257:
2250:
2242:
2239:
2220:
2216:
2211:
2200:
2195:
2192:
2188:
2182:
2179:
2174:
2170:
2162:
2161:
2160:
2139:
2135:
2130:
2119:
2116:
2110:
2107:
2103:
2097:
2094:
2089:
2085:
2081:
2076:
2072:
2066:
2063:
2059:
2055:
2050:
2046:
2040:
2037:
2032:
2027:
2024:
2020:
2016:
2009:
2006:
2002:
1994:
1993:
1992:
1979:
1956:
1952:
1949:
1946:
1941:
1933:
1928:
1924:
1920:
1917:
1912:
1909:
1904:
1899:
1896:
1892:
1888:
1883:
1876:
1875:
1874:
1838:
1834:
1831:
1828:
1823:
1815:
1810:
1806:
1800:
1792:
1785:
1784:
1783:
1781:
1771:
1762:
1755:
1740:
1738:
1731:
1719:
1715:
1692:
1686:
1682:
1679:
1673:
1658:
1657:
1656:
1655:expressed as
1653:
1649:
1644:
1642:
1638:
1634:
1630:
1621:
1618:
1615:
1612:
1608:
1607:
1606:
1604:
1600:
1581:
1576:
1572:
1569:
1563:
1558:
1554:
1551:
1548:
1542:
1539:
1532:
1531:
1530:
1528:
1524:
1507:
1496:
1493:
1488:
1485:
1482:
1476:
1473:
1453:
1450:
1447:
1444:
1441:
1433:
1431:
1426:
1424:
1421:(also called
1420:
1416:
1408:
1386:
1357:
1329:
1323:
1294:
1246:
1193:
1189:
1185:
1180:
1176:
1172:
1169:
1157:
1135:
1116:
1093:
1063:
1056:
1034:
1005:
977:
939:
911:
894:
892:
873:
864:
861:
849:
848:
847:
841:
829:
828:flow velocity
787:
782:
772:
767:
764:
760:
754:
751:
746:
742:
734:
733:
731:
728:
724:
723:
717:
715:
711:
706:
690:
686:
677:
674:
663:
648:
645:
640:
637:
629:
626:
624:
619:
604:
601:
596:
593:
585:
583:
579:
553:
537:
510:
502:
499:
494:
491:
479:
478:
476:
473:
469:
468:
462:
459:
457:
452:
436:
433:
428:
425:
419:
416:
408:
405:
392:
383:
380:
375:
372:
366:
363:
343:
323:
303:
296:of the fluid
295:
279:
272:
268:
267:porous medium
252:
242:
241:pressure drop
226:
219:
215:
211:
210:Morris Muskat
203:
187:
158:
149:
147:
143:
138:
137:Morris Muskat
134:
132:
128:
124:
120:
115:
113:
109:
105:
101:
97:
93:
92:Fourier's law
89:
85:
75:
73:
69:
68:momentum flux
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
25:
21:
4378:Soil physics
4334:
4303:
4278:permeability
4245:Hydrogeology
4204:
4198:
4165:
4161:
4155:
4128:
4124:
4118:
4109:
4103:
4086:
4082:
4072:
4055:
4051:
4025:
4021:
3999:
3993:
3984:
3978:
3969:
3960:
3951:
3942:
3930:
3902:
3896:
3871:
3867:
3857:
3824:
3820:
3814:
3805:
3799:
3761:Hydrogeology
3730:laminar flow
3727:
3711:
3637:
3605:
3530:
3514:
3437:
3336:
3255:
3139:
2976:
2843:
2830:
2823:
2634:
2625:
2621:
2584:
2581:
2479:
2454:
2446:
2442:
2440:
2397:
2380:
2355:
2344:
2332:
2272:
2265:
2259:
2255:
2245:
2241:porous media
2235:
2154:
1971:
1853:
1769:
1757:
1753:
1746:
1729:
1707:
1645:
1626:
1613:conditions),
1596:
1434:
1430:Stevin's law
1427:
1208:is in units
1158:
1117:is in units
895:
888:
803:
730:porous media
727:anisotropic
719:
707:
664:
630:
627:
620:
586:
526:
475:porous media
464:
460:
453:
409:
406:
271:permeability
207:
135:
131:hydrogeology
116:
81:
44:hydrogeology
19:
18:
4304:Darcy's law
4273:storativity
4058:: 153–160.
3738:interstices
3622:) leads to
3612:nanoseconds
2979:capillaries
2494:Forchheimer
2159:direction,
1991:direction,
1637:groundwater
1611:hydrostatic
1601:flowing in
1599:groundwater
152:Description
32:Henry Darcy
20:Darcy's law
4357:Categories
4256:properties
4168:: 76–103.
3792:References
3781:Fick's law
3628:hyperbolic
2983:mechanical
2498:non-linear
1743:Derivation
1737:grain size
472:isotropic
265:through a
239:, and the
108:Fick's law
78:Background
26:through a
4383:Hydrology
4252:Physical
4190:119946306
4133:CiteSeerX
4131:: 27–34.
3888:0043-1397
3849:121904058
3742:turbulent
3734:sediments
3718:viscosity
3690:∇
3685:μ
3677:−
3656:∇
3652:β
3649:−
3584:∇
3578:−
3560:∂
3552:∂
3546:τ
3398:μ
3312:∂
3304:∂
3296:μ
3273:−
3231:∂
3223:∂
3196:μ
3165:μ
3157:−
3087:−
3061:μ
3027:−
3015:∂
3007:∂
2950:−
2924:μ
2890:−
2878:∂
2870:∂
2787:−
2678:μ
2665:−
2541:ρ
2536:−
2525:μ
2520:−
2511:∇
2470:Ohm's law
2423:∇
2417:−
2351:viscosity
2310:∇
2303:μ
2295:−
2238:isotropic
2208:∂
2201:μ
2183:−
2127:∂
2120:μ
2117:φ
2098:−
2060:δ
2025:−
1938:∂
1934:−
1921:φ
1918:μ
1897:−
1884:−
1820:∂
1816:−
1797:∇
1793:μ
1687:ν
1577:ν
1559:μ
1552:ρ
1505:Δ
1494:ν
1448:ρ
1370:in units
1186:−
1167:Δ
1094:⋅
1076:in units
1064:μ
1018:in units
865:φ
838:) by the
779:∂
773:μ
755:−
712:, and in
684:Δ
672:∇
605:μ
594:σ
535:∇
508:∇
503:μ
495:−
426:μ
390:Δ
381:μ
304:μ
294:viscosity
250:Δ
200:= Δp/L =
185:∇
112:diffusion
100:Ohm's law
52:Ohm's law
4283:porosity
3906:. 2011.
3827:: 3–25.
3748:See also
3732:through
2490:inertial
2486:inertial
2383:moka pot
1978:porosity
1633:flownets
1603:aquifers
1113:and the
840:porosity
720:Darcy's
465:Darcy's
114:theory.
66:for the
4254:aquifer
4170:Bibcode
3829:Bibcode
2646:Knudsen
2642:Knudsen
2638:viscous
2457:is the
1976:is the
1774:
1750:
1712:is the
1648:viscous
576:is the
550:is the
119:aquifer
4188:
4135:
3918:
3886:
3847:
3712:where
3606:where
3515:where
3337:where
2824:where
2466:scalar
2462:tensor
2453:, and
2441:where
2367:Muskat
2359:Muskat
1980:, and
1972:where
1854:where
1708:where
1623:cases.
1239:, and
1053:, the
710:tensor
527:where
163:outlet
106:, and
28:porous
4363:Water
4186:S2CID
3874:(7).
3845:S2CID
3755:darcy
1718:water
936:is a
167:inlet
36:water
24:fluid
3916:ISBN
3884:ISSN
3753:The
2985:and
2640:and
2253:for
554:and
86:via
40:sand
4178:doi
4166:766
4143:doi
4091:doi
4060:doi
4056:206
4030:doi
4026:160
3908:doi
3876:doi
3837:doi
3535:),
2251:= 0
2236:In
1776:≈ 0
1716:of
1641:dam
1432:):
1425:).
1156:.
173:- p
169:= p
165:- p
110:in
4359::
4184:.
4176:.
4164:.
4141:.
4127:.
4087:64
4085:.
4081:.
4054:.
4042:^
4024:.
4020:.
4008:^
3914:.
3882:.
3872:38
3870:.
3866:.
3843:.
3835:.
3823:.
2619:.
2373:.
2353:.
2271:=
2268:ii
2258:≠
2248:ij
1984:ij
1770:Dt
1758:ρu
1733:30
1720:,
1643:.
732:)
705:.
625:.
477:)
458:.
177:.
133:.
98:,
74:.
4237:e
4230:t
4223:v
4192:.
4180::
4172::
4149:.
4145::
4129:1
4097:.
4093::
4066:.
4062::
4036:.
4032::
3924:.
3910::
3890:.
3878::
3851:.
3839::
3831::
3825:1
3714:β
3697:,
3693:p
3682:k
3674:=
3671:q
3668:+
3665:q
3660:2
3618:(
3608:τ
3591:,
3587:h
3581:k
3575:=
3572:q
3569:+
3563:t
3555:q
3521:b
3517:b
3500:.
3495:)
3489:p
3486:b
3481:+
3478:1
3474:(
3470:k
3467:=
3461:f
3458:f
3455:e
3450:k
3423:.
3418:)
3412:p
3409:1
3402:k
3392:K
3387:D
3380:+
3377:1
3373:(
3369:k
3366:=
3360:f
3357:f
3354:e
3349:k
3322:,
3315:x
3307:p
3290:f
3287:f
3284:e
3279:k
3270:=
3267:q
3241:.
3234:x
3226:p
3216:)
3210:p
3207:1
3200:k
3190:K
3185:D
3178:+
3175:1
3171:(
3162:k
3154:=
3151:q
3125:.
3121:q
3114:T
3108:g
3103:R
3098:p
3090:1
3082:)
3075:K
3070:D
3066:+
3057:p
3054:k
3047:(
3042:T
3036:g
3031:R
3024:=
3018:x
3010:p
2962:.
2958:N
2953:1
2945:)
2938:K
2933:D
2929:+
2920:p
2917:k
2910:(
2905:T
2899:g
2894:R
2887:=
2881:x
2873:p
2850:K
2844:D
2839:T
2834:g
2831:R
2826:N
2809:,
2803:L
2796:a
2791:p
2781:b
2776:p
2766:T
2760:g
2755:R
2750:1
2744:)
2737:f
2734:f
2731:e
2724:K
2719:D
2715:+
2710:2
2704:b
2700:p
2696:+
2691:a
2687:p
2675:k
2669:(
2662:=
2659:N
2606:)
2603:m
2600:(
2588:1
2585:k
2567:,
2561:2
2557:q
2549:1
2545:k
2533:q
2528:k
2517:=
2514:p
2455:K
2447:h
2443:q
2426:h
2420:K
2414:=
2410:q
2318:.
2314:p
2300:k
2292:=
2288:q
2273:k
2266:k
2260:j
2256:i
2246:k
2221:.
2217:p
2212:i
2196:i
2193:n
2189:k
2180:=
2175:n
2171:q
2157:n
2140:,
2136:p
2131:i
2111:i
2108:n
2104:k
2095:=
2090:n
2086:u
2082:=
2077:j
2073:u
2067:j
2064:n
2056:=
2051:j
2047:u
2041:j
2038:i
2033:)
2028:1
2021:k
2017:(
2010:i
2007:n
2003:k
1989:n
1982:k
1974:φ
1957:,
1953:0
1950:=
1947:p
1942:i
1929:j
1925:u
1913:j
1910:i
1905:)
1900:1
1893:k
1889:(
1871:p
1867:i
1862:i
1860:u
1856:μ
1839:,
1835:0
1832:=
1829:p
1824:i
1811:i
1807:u
1801:2
1766:/
1763:)
1760:i
1756:(
1754:D
1730:d
1726:d
1722:q
1710:ν
1693:,
1683:d
1680:q
1674:=
1670:e
1667:R
1582:.
1573:g
1570:k
1564:=
1555:g
1549:k
1543:=
1540:K
1508:h
1497:L
1489:g
1486:A
1483:k
1477:=
1474:Q
1454:h
1451:g
1445:=
1442:p
1411:x
1392:)
1387:2
1383:m
1379:(
1358:A
1337:)
1334:s
1330:/
1324:3
1320:m
1316:(
1295:Q
1274:)
1271:m
1268:(
1247:L
1226:)
1223:a
1220:P
1217:(
1194:a
1190:p
1181:b
1177:p
1173:=
1170:p
1143:)
1140:m
1136:/
1132:a
1129:P
1126:(
1100:)
1097:s
1091:a
1088:P
1085:(
1040:)
1035:2
1031:m
1027:(
1006:k
985:)
982:s
978:/
974:m
971:(
949:q
934:φ
919:)
916:s
912:/
908:m
905:(
874:.
870:u
862:=
858:q
844:φ
842:(
836:q
832:u
830:(
813:q
788:p
783:j
768:j
765:i
761:k
752:=
747:i
743:q
725:(
691:L
687:p
678:=
675:p
649:A
646:Q
641:=
638:q
602:k
597:=
563:q
538:p
511:p
500:k
492:=
488:q
470:(
437:A
434:k
429:L
420:=
417:R
393:p
384:L
376:A
373:k
367:=
364:Q
344:A
324:L
280:k
253:p
227:Q
188:p
175:a
171:b
161:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
