2083:
relations. Advanced computational and software solutions are required here to over-come this obstacle. The method has also been criticized for over-emphasizing the role of capillarity, and for being in some ways 'overly simplistic' In one dimensional simulations of rainfall infiltration into dry soils, fine spatial discretization less than one cm is required near the land surface, which is due to the small size of the
1292:
698:
2082:
The numerical solution of the
Richards equation is one of the most challenging problems in earth science. Richards' equation has been criticized for being computationally expensive and unpredictable because there is no guarantee that a solver will converge for a particular set of soil constitutive
947:
1149:
1420:
1134:
1645:
566:
582:
1876:
The
Richards equation in any of its forms involves soil hydraulic properties, which is a set of five parameters representing soil type. The soil hydraulic properties typically consist of water retention curve parameters by van Genuchten:
335:
416:
1808:
1728:
1482:
125:
855:
996:
1533:
1936:
1864:
is the residual (minimal) water content a successful numerical solution is restricted just for ranges of water content satisfactory below the full saturation (the saturation should be even lower than
1287:{\displaystyle {\frac {\Delta \theta }{\Delta t}}\approx C(h){\frac {\Delta h}{\Delta t}},\quad {\mbox{and so}}\quad {\frac {\Delta \theta }{\Delta t}}-C(h){\frac {\Delta h}{\Delta t}}=\varepsilon .}
720:
between the atmosphere and the aquifer. It also appears in pure mathematical journals because it has non-trivial solutions. The above-given mixed formulation involves two unknown variables:
1334:
1036:
1567:
2035:
1575:
2064:
484:
249:
1314:
1862:
1835:
787:
693:{\displaystyle {\frac {\partial \theta }{\partial t}}={\frac {\partial }{\partial z}}\left(\mathbf {K} (\theta )\left({\frac {\partial h}{\partial z}}+1\right)\right)-S}
160:
274:
1956:
835:
738:
186:
472:
2074:
of second order) should also be provided. Identification of these parameters is often non-trivial and was a subject of numerous publications over several decades.
1028:
1996:
1976:
815:
758:
443:
212:
2730:
Tocci, M. D., C. T. Kelley, and C. T. Miller (1997), Accurate and economical solution of the pressure-head form of
Richards' equation by the method of lines,
2796:
279:
351:
1733:
1653:
1428:
2826:
2717:
Short, D., W.R. Dawes, and I. White, (1995). The practicability of using
Richards' equation for general purpose soil-water dynamics models.
57:
51:. The equation is based on Darcy-Buckingham law representing flow in porous media under variably saturated conditions, which is stated as
2523:
Angulo-Jaramillo, Rafael; Vandervaere, Jean-Pierre; Roulier, Stéphanie; Thony, Jean-Louis; Gaudet, Jean-Paul; Vauclin, Michel (May 2000).
1316:
that affects the mass conservation of the numerical solution, and so special strategies for temporal derivatives treatment are necessary.
942:{\displaystyle {\frac {\partial \theta (h)}{\partial t}}={\frac {{\textrm {d}}\theta }{{\textrm {d}}h}}{\frac {\partial h}{\partial t}}}
1140:
The head-based
Richards equation is prone to the following computational issue: the discretized temporal derivative using the implicit
2087:
for multiphase flow in porous media. In three-dimensional applications the numerical solution of the
Richards equation is subject to
956:
2556:"Inverse Dual-Permeability Modeling of Preferential Water Flow in a Soil Column and Implications for Field-Scale Solute Transport"
2084:
1487:
2704:
Farthing, Matthew W., and Fred L. Ogden, (2017). Numerical solution of
Richards’ Equation: a review of advances and challenges.
2801:
1880:
2148:"Clean two- and three-dimensional analytical solutions of Richards' equation for testing numerical solvers: TECHNICAL NOTE"
2660:"Automated calibration methodology to avoid convergence issues during inverse identification of soil hydraulic properties"
32:
2283:"The contributions of Lewis Fry Richardson to drainage theory, soil physics, and the soil-plant-atmosphere continuum"
2091:
constraints where the ratio of horizontal to vertical resolution in the solution domain should be less than about 7.
1415:{\displaystyle \mathbf {K} (h)\nabla h=\mathbf {K} (h){\frac {{\textrm {d}}h}{{\textrm {d}}\theta }}\nabla \theta ,}
716:
The
Richards equation appears in many articles in the environmental literature because it describes the flow in the
2447:"Simultaneous estimation of soil hydraulic and solute transport parameters from transient infiltration experiments"
2743:
Germann, P. (2010), Comment on “Theory for source-responsive and free-surface film modeling of unsaturated flow”,
1129:{\displaystyle C(h){\frac {\partial h}{\partial t}}=\nabla \cdot \left(\mathbf {K} (h)\nabla h+\nabla z\right)-S}
478:
Then substituting the fluxes by the Darcy-Buckingham law the following mixed-form
Richards equation is obtained:
2786:
999:
1536:
1541:
1640:{\displaystyle {\frac {\partial \theta }{\partial t}}=\nabla \cdot \mathbf {D} (\theta )\nabla \theta -S.}
2816:
2791:
2001:
1865:
1650:
The saturation-based
Richards equation is prone to the following computational issues. Since the limits
790:
561:{\displaystyle {\frac {\partial \theta }{\partial t}}=\nabla \cdot \mathbf {K} (h)(\nabla h+\nabla z)-S}
29:
2659:
2360:
2040:
2622:
2567:
2497:
2458:
2419:
2372:
2333:
2282:
2247:
2120:
705:
223:
35:; its analytical solution is often limited to specific initial and boundary conditions. Proof of the
2361:"An adaptive time discretization of the classical and the dual porosity model of Richards' equation"
2306:
25:
1299:
2821:
2687:
2591:
2524:
2485:
2446:
2216:
2169:
1840:
1813:
704:
Although attributed to L. A. Richards, the equation was originally introduced 9 years earlier by
40:
2609:
Younes, Anis; Mara, Thierry; Fahs, Marwan; Grunberger, Olivier; Ackerer, Philippe (3 May 2017).
2484:
Fodor, Nándor; Sándor, Renáta; Orfanus, Tomas; Lichner, Lubomir; Rajkai, Kálmán (October 2011).
763:
136:
2407:
2679:
2640:
2583:
2388:
2263:
2208:
342:
256:
36:
2756:
Gray, W. G., and S. Hassanizadeh (1991), Paradoxes and realities in unsaturated flow theory,
1941:
820:
723:
171:
2671:
2630:
2575:
2536:
2505:
2466:
2431:
2427:
2380:
2341:
2255:
2200:
2159:
2128:
48:
44:
2555:
448:
1004:
163:
2300:
2626:
2571:
2525:"Field measurement of soil surface hydraulic properties by disc and ring infiltrometers"
2501:
2462:
2423:
2376:
2337:
2251:
2124:
2408:"A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils"
1981:
1961:
800:
743:
428:
197:
2540:
2470:
2235:
2810:
2691:
2675:
2610:
2259:
2220:
2173:
215:
189:
2611:"Hydraulic and transport parameter assessment using column infiltration experiments"
2658:
Kuraz, Michal; Jačka, Lukáš; Ruth Blöcher, Johanna; Lepš, Matěj (1 November 2022).
2509:
2236:"Model for simulating soil-water content considering evapotranspiration — Comments"
2088:
330:{\displaystyle \nabla z=\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)}
2595:
2322:"A general mass-conservative numerical solution for the unsaturated flow equation"
411:{\displaystyle {\frac {\partial \theta }{\partial t}}+\nabla \cdot {\vec {q}}+S=0}
2111:
Richards, L.A. (1931). "Capillary conduction of liquids through porous mediums".
1141:
717:
298:
21:
2321:
1803:{\displaystyle \lim _{\theta \to \theta _{r}}||\mathbf {D} (\theta )||=\infty }
1723:{\displaystyle \lim _{\theta \to \theta _{s}}||\mathbf {D} (\theta )||=\infty }
2635:
2384:
1325:
846:
794:
573:
345:
for an incompressible porous medium and constant liquid density, expressed as
2683:
2644:
2587:
2392:
2267:
2212:
1477:{\displaystyle \mathbf {K} (h){\frac {{\textrm {d}}h}{{\textrm {d}}\theta }}}
2345:
2486:"Evaluation method dependency of measured saturated hydraulic conductivity"
2320:
Celia, Michael A.; Bouloutas, Efthimios T.; Zarba, Rebecca L. (July 1990).
2554:
Köhne, J. Maximilian; Mohanty, Binayak P.; Šimůnek, Jirka (January 2006).
2579:
2359:
Kuráž, Michal; Mayer, Petr; Lepš, Matěj; Trpkošová, Dagmar (2010-04-15).
2164:
2147:
2067:
2204:
2132:
2188:
2071:
120:{\displaystyle {\vec {q}}=-\mathbf {K} (\theta )(\nabla h+\nabla z),}
2367:. Finite Element Methods in Engineering and Science (FEMTEC 2009).
760:. This can be easily resolved by considering constitutive relation
2445:
Inoue, M.; Ĺ imĹŻnek, J.; Shiozawa, S.; Hopmans, J.W. (June 2000).
991:{\displaystyle {\frac {{\textrm {d}}\theta }{{\textrm {d}}h}}}
1868:) as well as satisfactory above the residual water content.
1528:{\displaystyle {\frac {\mathbf {K} (\theta )}{C(\theta )}}}
2187:
Wilhelm Alt, Hans; Luckhaus, Stephan (1 September 1983).
1931:{\displaystyle \alpha ,\,n,\,m,\,\theta _{s},\theta _{r}}
2189:"Quasilinear elliptic-parabolic differential equations"
1213:
797:, the Richards equation may be reformulated as either
2043:
2004:
1984:
1964:
1944:
1883:
1843:
1816:
1736:
1656:
1578:
1544:
1490:
1431:
1337:
1302:
1152:
1039:
1007:
959:
858:
823:
803:
766:
746:
726:
585:
487:
451:
431:
354:
282:
259:
226:
200:
174:
139:
60:
276:is the geodetic head gradient, which is assumed as
2058:
2029:
1990:
1970:
1950:
1930:
1856:
1829:
1802:
1722:
1639:
1561:
1527:
1476:
1414:
1308:
1286:
1128:
1022:
990:
941:
829:
809:
781:
752:
732:
692:
572:For modeling of one-dimensional infiltration this
560:
466:
437:
410:
329:
268:
243:
206:
180:
154:
119:
218:, which is negative for unsaturated porous media;
2365:Journal of Computational and Applied Mathematics
1738:
1658:
2037:. Further the saturated hydraulic conductivity
2769:Downer, Charles W., and Fred L. Ogden (2003),
1978:is the pore size distribution parameter , and
1837:is the saturated (maximal) water content and
8:
2797:Finite water-content vadose zone flow method
2305:. Cambridge, The University press. pp.
837:-form (saturation based) Richards equation.
28:who published the equation in 1931. It is a
1144:method yields the following approximation:
251:is the unsaturated hydraulic conductivity;
2634:
2234:Feddes, R. A.; Zaradny, H. (1 May 1978).
2163:
2050:
2045:
2042:
2017:
2003:
1983:
1963:
1943:
1922:
1909:
1904:
1897:
1890:
1882:
1848:
1842:
1821:
1815:
1789:
1784:
1770:
1765:
1760:
1752:
1741:
1735:
1709:
1704:
1690:
1685:
1680:
1672:
1661:
1655:
1608:
1579:
1577:
1545:
1543:
1494:
1491:
1489:
1462:
1461:
1450:
1449:
1446:
1432:
1430:
1391:
1390:
1379:
1378:
1375:
1361:
1338:
1336:
1301:
1255:
1220:
1212:
1188:
1153:
1151:
1086:
1052:
1038:
1006:
976:
975:
964:
963:
960:
958:
919:
907:
906:
895:
894:
891:
859:
857:
822:
802:
765:
745:
725:
648:
629:
609:
586:
584:
517:
488:
486:
455:
450:
430:
385:
384:
355:
353:
296:
281:
258:
227:
225:
199:
173:
141:
140:
138:
79:
62:
61:
59:
2406:van Genuchten, M. Th. (September 1980).
2706:Soil Science Society of America Journal
2432:10.2136/sssaj1980.03615995004400050002x
2412:Soil Science Society of America Journal
2302:Weather prediction by numerical process
2100:
1484:, which could be further formulated as
43:of solution was given only in 1983 by
2773:,18, pp. 1-22. DOI:10.1002/hyp.1306.
1958:is the inverse of air entry value ,
1562:{\displaystyle \mathbf {D} (\theta )}
1296:This approximation produces an error
7:
2106:
2104:
20:represents the movement of water in
2615:Hydrology and Earth System Sciences
1328:on the spatial derivative leads to
2030:{\displaystyle m=1-{\frac {1}{n}}}
1797:
1717:
1622:
1602:
1590:
1582:
1403:
1352:
1266:
1258:
1231:
1223:
1199:
1191:
1164:
1156:
1109:
1100:
1075:
1063:
1055:
930:
922:
879:
862:
659:
651:
615:
611:
597:
589:
543:
534:
511:
499:
491:
378:
366:
358:
283:
260:
105:
96:
14:
1569:. The equation is then stated as
1030:. The equation is then stated as
849:on temporal derivative leads to
297:
2676:10.1016/j.advengsoft.2022.103278
2664:Advances in Engineering Software
2085:representative elementary volume
2059:{\displaystyle \mathbf {K} _{s}}
2046:
1771:
1691:
1609:
1546:
1495:
1433:
1362:
1339:
1087:
630:
518:
228:
80:
2802:Soil Moisture Velocity Equation
1219:
1211:
474:], typically root water uptake.
337:for three-dimensional problems.
244:{\displaystyle \mathbf {K} (h)}
2827:Partial differential equations
2510:10.1016/j.geoderma.2011.07.004
2299:Richardson, Lewis Fry (1922).
1790:
1785:
1781:
1775:
1766:
1761:
1745:
1710:
1705:
1701:
1695:
1686:
1681:
1665:
1619:
1613:
1556:
1550:
1519:
1513:
1505:
1499:
1443:
1437:
1372:
1366:
1349:
1343:
1252:
1246:
1185:
1179:
1097:
1091:
1049:
1043:
1017:
1011:
874:
868:
776:
770:
640:
634:
549:
531:
528:
522:
390:
238:
232:
146:
111:
93:
90:
84:
67:
1:
2541:10.1016/S0167-1987(00)00098-2
2471:10.1016/S0309-1708(00)00011-7
33:partial differential equation
2288:. EGU General Assembly 2016.
2281:Knight, John; Raats, Peter.
2260:10.1016/0022-1694(78)90030-6
2146:Tracy, F. T. (August 2006).
1309:{\displaystyle \varepsilon }
24:soils, and is attributed to
2451:Advances in Water Resources
1857:{\displaystyle \theta _{r}}
1830:{\displaystyle \theta _{s}}
2843:
782:{\displaystyle \theta (h)}
155:{\displaystyle {\vec {q}}}
2636:10.5194/hess-21-2263-2017
2529:Soil and Tillage Research
2385:10.1016/j.cam.2009.11.056
2193:Mathematische Zeitschrift
2787:Infiltration (hydrology)
2326:Water Resources Research
2152:Water Resources Research
1000:retention water capacity
789:, which is known as the
269:{\displaystyle \nabla z}
190:volumetric water content
2346:10.1029/WR026i007p01483
1998:is usually assumed as
1951:{\displaystyle \alpha }
830:{\displaystyle \theta }
733:{\displaystyle \theta }
341:Considering the law of
181:{\displaystyle \theta }
2708:, 81(6), pp.1257-1269.
2060:
2031:
1992:
1972:
1952:
1932:
1858:
1831:
1804:
1724:
1641:
1563:
1537:soil water diffusivity
1529:
1478:
1416:
1310:
1288:
1130:
1024:
992:
943:
831:
817:-form (head based) or
811:
783:
754:
734:
694:
562:
468:
439:
412:
331:
270:
245:
208:
182:
156:
121:
2792:Water retention curve
2061:
2032:
1993:
1973:
1953:
1933:
1859:
1832:
1805:
1725:
1642:
1564:
1530:
1479:
1417:
1311:
1289:
1131:
1025:
993:
944:
832:
812:
791:water retention curve
784:
755:
735:
695:
563:
469:
467:{\displaystyle ^{-1}}
440:
413:
332:
271:
246:
209:
183:
157:
122:
2580:10.2136/vzj2005.0008
2240:Journal of Hydrology
2165:10.1029/2005WR004638
2041:
2002:
1982:
1962:
1942:
1881:
1841:
1814:
1734:
1654:
1576:
1542:
1488:
1429:
1335:
1300:
1150:
1037:
1023:{\displaystyle C(h)}
1005:
957:
856:
821:
801:
764:
744:
724:
706:Lewis Fry Richardson
583:
485:
449:
429:
352:
280:
257:
224:
198:
172:
137:
58:
2760:, 27(8), 1847-1854.
2627:2017HESS...21.2263Y
2572:2006VZJ.....5...59K
2560:Vadose Zone Journal
2502:2011Geode.165...60F
2463:2000AdWR...23..677I
2424:1980SSASJ..44..892V
2377:2010JCoAM.233.3167K
2338:1990WRR....26.1483C
2252:1978JHyd...37..393F
2125:1931Physi...1..318R
445:is the sink term [T
26:Lorenzo A. Richards
2758:Water Resour. Res.
2205:10.1007/BF01176474
2066:(which is for non
2056:
2027:
1988:
1968:
1948:
1928:
1854:
1827:
1800:
1759:
1720:
1679:
1637:
1559:
1535:, is known as the
1525:
1474:
1412:
1306:
1284:
1217:
1126:
1020:
988:
939:
827:
807:
779:
750:
730:
690:
558:
464:
435:
408:
327:
321:
320:
266:
241:
204:
178:
152:
117:
2732:Adv. Wat. Resour.
2371:(12): 3167–3177.
2133:10.1063/1.1745010
2025:
1991:{\displaystyle m}
1971:{\displaystyle n}
1737:
1657:
1597:
1523:
1472:
1465:
1453:
1401:
1394:
1382:
1273:
1238:
1216:
1206:
1171:
1070:
986:
979:
967:
937:
917:
910:
898:
886:
810:{\displaystyle h}
753:{\displaystyle h}
666:
622:
604:
506:
438:{\displaystyle S}
393:
373:
343:mass conservation
207:{\displaystyle h}
149:
70:
18:Richards equation
2834:
2774:
2767:
2761:
2754:
2748:
2747:9(4), 1000-1101.
2741:
2735:
2728:
2722:
2721:. 21(5):723-730.
2715:
2709:
2702:
2696:
2695:
2655:
2649:
2648:
2638:
2621:(5): 2263–2275.
2606:
2600:
2599:
2551:
2545:
2544:
2520:
2514:
2513:
2481:
2475:
2474:
2442:
2436:
2435:
2403:
2397:
2396:
2356:
2350:
2349:
2332:(7): 1483–1496.
2317:
2311:
2310:
2296:
2290:
2289:
2287:
2278:
2272:
2271:
2231:
2225:
2224:
2184:
2178:
2177:
2167:
2143:
2137:
2136:
2108:
2065:
2063:
2062:
2057:
2055:
2054:
2049:
2036:
2034:
2033:
2028:
2026:
2018:
1997:
1995:
1994:
1989:
1977:
1975:
1974:
1969:
1957:
1955:
1954:
1949:
1937:
1935:
1934:
1929:
1927:
1926:
1914:
1913:
1863:
1861:
1860:
1855:
1853:
1852:
1836:
1834:
1833:
1828:
1826:
1825:
1809:
1807:
1806:
1801:
1793:
1788:
1774:
1769:
1764:
1758:
1757:
1756:
1729:
1727:
1726:
1721:
1713:
1708:
1694:
1689:
1684:
1678:
1677:
1676:
1646:
1644:
1643:
1638:
1612:
1598:
1596:
1588:
1580:
1568:
1566:
1565:
1560:
1549:
1534:
1532:
1531:
1526:
1524:
1522:
1508:
1498:
1492:
1483:
1481:
1480:
1475:
1473:
1471:
1467:
1466:
1463:
1459:
1455:
1454:
1451:
1447:
1436:
1421:
1419:
1418:
1413:
1402:
1400:
1396:
1395:
1392:
1388:
1384:
1383:
1380:
1376:
1365:
1342:
1324:By applying the
1320:Saturation-based
1315:
1313:
1312:
1307:
1293:
1291:
1290:
1285:
1274:
1272:
1264:
1256:
1239:
1237:
1229:
1221:
1218:
1214:
1207:
1205:
1197:
1189:
1172:
1170:
1162:
1154:
1135:
1133:
1132:
1127:
1119:
1115:
1090:
1071:
1069:
1061:
1053:
1029:
1027:
1026:
1021:
998:is known as the
997:
995:
994:
989:
987:
985:
981:
980:
977:
973:
969:
968:
965:
961:
948:
946:
945:
940:
938:
936:
928:
920:
918:
916:
912:
911:
908:
904:
900:
899:
896:
892:
887:
885:
877:
860:
845:By applying the
836:
834:
833:
828:
816:
814:
813:
808:
788:
786:
785:
780:
759:
757:
756:
751:
739:
737:
736:
731:
699:
697:
696:
691:
683:
679:
678:
674:
667:
665:
657:
649:
633:
623:
621:
610:
605:
603:
595:
587:
576:form reduces to
567:
565:
564:
559:
521:
507:
505:
497:
489:
473:
471:
470:
465:
463:
462:
444:
442:
441:
436:
417:
415:
414:
409:
395:
394:
386:
374:
372:
364:
356:
336:
334:
333:
328:
326:
322:
275:
273:
272:
267:
250:
248:
247:
242:
231:
213:
211:
210:
205:
187:
185:
184:
179:
161:
159:
158:
153:
151:
150:
142:
126:
124:
123:
118:
83:
72:
71:
63:
2842:
2841:
2837:
2836:
2835:
2833:
2832:
2831:
2807:
2806:
2783:
2778:
2777:
2768:
2764:
2755:
2751:
2742:
2738:
2729:
2725:
2716:
2712:
2703:
2699:
2657:
2656:
2652:
2608:
2607:
2603:
2553:
2552:
2548:
2522:
2521:
2517:
2483:
2482:
2478:
2444:
2443:
2439:
2405:
2404:
2400:
2358:
2357:
2353:
2319:
2318:
2314:
2298:
2297:
2293:
2285:
2280:
2279:
2275:
2233:
2232:
2228:
2186:
2185:
2181:
2145:
2144:
2140:
2110:
2109:
2102:
2097:
2080:
2044:
2039:
2038:
2000:
1999:
1980:
1979:
1960:
1959:
1940:
1939:
1918:
1905:
1879:
1878:
1874:
1872:Parametrization
1866:air entry value
1844:
1839:
1838:
1817:
1812:
1811:
1748:
1732:
1731:
1668:
1652:
1651:
1589:
1581:
1574:
1573:
1540:
1539:
1509:
1493:
1486:
1485:
1460:
1448:
1427:
1426:
1389:
1377:
1333:
1332:
1322:
1298:
1297:
1265:
1257:
1230:
1222:
1198:
1190:
1163:
1155:
1148:
1147:
1085:
1081:
1062:
1054:
1035:
1034:
1003:
1002:
974:
962:
955:
954:
929:
921:
905:
893:
878:
861:
854:
853:
843:
819:
818:
799:
798:
793:. Applying the
762:
761:
742:
741:
722:
721:
714:
658:
650:
647:
643:
628:
624:
614:
596:
588:
581:
580:
498:
490:
483:
482:
452:
447:
446:
427:
426:
365:
357:
350:
349:
319:
318:
312:
311:
305:
304:
292:
278:
277:
255:
254:
222:
221:
196:
195:
170:
169:
164:volumetric flux
135:
134:
56:
55:
12:
11:
5:
2840:
2838:
2830:
2829:
2824:
2819:
2809:
2808:
2805:
2804:
2799:
2794:
2789:
2782:
2779:
2776:
2775:
2762:
2749:
2745:Vadose Zone J.
2736:
2734:, 20(1), 1–14.
2723:
2710:
2697:
2650:
2601:
2546:
2515:
2476:
2457:(7): 677–688.
2437:
2418:(5): 892–898.
2398:
2351:
2312:
2291:
2273:
2246:(3): 393–397.
2226:
2199:(3): 311–341.
2179:
2138:
2119:(5): 318–333.
2099:
2098:
2096:
2093:
2079:
2076:
2070:environment a
2053:
2048:
2024:
2021:
2016:
2013:
2010:
2007:
1987:
1967:
1947:
1925:
1921:
1917:
1912:
1908:
1903:
1900:
1896:
1893:
1889:
1886:
1873:
1870:
1851:
1847:
1824:
1820:
1799:
1796:
1792:
1787:
1783:
1780:
1777:
1773:
1768:
1763:
1755:
1751:
1747:
1744:
1740:
1719:
1716:
1712:
1707:
1703:
1700:
1697:
1693:
1688:
1683:
1675:
1671:
1667:
1664:
1660:
1648:
1647:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1611:
1607:
1604:
1601:
1595:
1592:
1587:
1584:
1558:
1555:
1552:
1548:
1521:
1518:
1515:
1512:
1507:
1504:
1501:
1497:
1470:
1458:
1445:
1442:
1439:
1435:
1423:
1422:
1411:
1408:
1405:
1399:
1387:
1374:
1371:
1368:
1364:
1360:
1357:
1354:
1351:
1348:
1345:
1341:
1321:
1318:
1305:
1283:
1280:
1277:
1271:
1268:
1263:
1260:
1254:
1251:
1248:
1245:
1242:
1236:
1233:
1228:
1225:
1210:
1204:
1201:
1196:
1193:
1187:
1184:
1181:
1178:
1175:
1169:
1166:
1161:
1158:
1138:
1137:
1125:
1122:
1118:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1089:
1084:
1080:
1077:
1074:
1068:
1065:
1060:
1057:
1051:
1048:
1045:
1042:
1019:
1016:
1013:
1010:
984:
972:
951:
950:
935:
932:
927:
924:
915:
903:
890:
884:
881:
876:
873:
870:
867:
864:
842:
839:
826:
806:
778:
775:
772:
769:
749:
729:
713:
710:
702:
701:
689:
686:
682:
677:
673:
670:
664:
661:
656:
653:
646:
642:
639:
636:
632:
627:
620:
617:
613:
608:
602:
599:
594:
591:
570:
569:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
520:
516:
513:
510:
504:
501:
496:
493:
476:
475:
461:
458:
454:
434:
420:
419:
407:
404:
401:
398:
392:
389:
383:
380:
377:
371:
368:
363:
360:
339:
338:
325:
317:
314:
313:
310:
307:
306:
303:
300:
299:
295:
291:
288:
285:
265:
262:
252:
240:
237:
234:
230:
219:
214:is the liquid
203:
193:
177:
167:
148:
145:
128:
127:
116:
113:
110:
107:
104:
101:
98:
95:
92:
89:
86:
82:
78:
75:
69:
66:
13:
10:
9:
6:
4:
3:
2:
2839:
2828:
2825:
2823:
2820:
2818:
2815:
2814:
2812:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2784:
2780:
2772:
2771:Hydrol. Proc.
2766:
2763:
2759:
2753:
2750:
2746:
2740:
2737:
2733:
2727:
2724:
2720:
2714:
2711:
2707:
2701:
2698:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2654:
2651:
2646:
2642:
2637:
2632:
2628:
2624:
2620:
2616:
2612:
2605:
2602:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2550:
2547:
2542:
2538:
2535:(1–2): 1–29.
2534:
2530:
2526:
2519:
2516:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2480:
2477:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2441:
2438:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2402:
2399:
2394:
2390:
2386:
2382:
2378:
2374:
2370:
2366:
2362:
2355:
2352:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2316:
2313:
2308:
2304:
2303:
2295:
2292:
2284:
2277:
2274:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2230:
2227:
2222:
2218:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2183:
2180:
2175:
2171:
2166:
2161:
2157:
2153:
2149:
2142:
2139:
2134:
2130:
2126:
2122:
2118:
2114:
2107:
2105:
2101:
2094:
2092:
2090:
2086:
2077:
2075:
2073:
2069:
2051:
2022:
2019:
2014:
2011:
2008:
2005:
1985:
1965:
1945:
1923:
1919:
1915:
1910:
1906:
1901:
1898:
1894:
1891:
1887:
1884:
1871:
1869:
1867:
1849:
1845:
1822:
1818:
1794:
1778:
1753:
1749:
1742:
1714:
1698:
1673:
1669:
1662:
1634:
1631:
1628:
1625:
1616:
1605:
1599:
1593:
1585:
1572:
1571:
1570:
1553:
1538:
1516:
1510:
1502:
1468:
1456:
1440:
1409:
1406:
1397:
1385:
1369:
1358:
1355:
1346:
1331:
1330:
1329:
1327:
1319:
1317:
1303:
1294:
1281:
1278:
1275:
1269:
1261:
1249:
1243:
1240:
1234:
1226:
1208:
1202:
1194:
1182:
1176:
1173:
1167:
1159:
1145:
1143:
1123:
1120:
1116:
1112:
1106:
1103:
1094:
1082:
1078:
1072:
1066:
1058:
1046:
1040:
1033:
1032:
1031:
1014:
1008:
1001:
982:
970:
933:
925:
913:
901:
888:
882:
871:
865:
852:
851:
850:
848:
840:
838:
824:
804:
796:
792:
773:
767:
747:
727:
719:
711:
709:
707:
687:
684:
680:
675:
671:
668:
662:
654:
644:
637:
625:
618:
606:
600:
592:
579:
578:
577:
575:
555:
552:
546:
540:
537:
525:
514:
508:
502:
494:
481:
480:
479:
459:
456:
453:
432:
425:
424:
423:
405:
402:
399:
396:
387:
381:
375:
369:
361:
348:
347:
346:
344:
323:
315:
308:
301:
293:
289:
286:
263:
253:
235:
220:
217:
216:pressure head
201:
194:
191:
175:
168:
165:
143:
133:
132:
131:
114:
108:
102:
99:
87:
76:
73:
64:
54:
53:
52:
50:
46:
42:
38:
34:
31:
27:
23:
19:
2817:Soil physics
2770:
2765:
2757:
2752:
2744:
2739:
2731:
2726:
2719:Envir. Int'l
2718:
2713:
2705:
2700:
2667:
2663:
2653:
2618:
2614:
2604:
2566:(1): 59–76.
2563:
2559:
2549:
2532:
2528:
2518:
2496:(1): 60–68.
2493:
2489:
2479:
2454:
2450:
2440:
2415:
2411:
2401:
2368:
2364:
2354:
2329:
2325:
2315:
2301:
2294:
2276:
2243:
2239:
2229:
2196:
2192:
2182:
2155:
2151:
2141:
2116:
2112:
2089:aspect ratio
2081:
1875:
1649:
1424:
1323:
1295:
1146:
1139:
952:
844:
715:
712:Formulations
703:
571:
477:
421:
340:
129:
17:
15:
2078:Limitations
718:vadose zone
30:quasilinear
22:unsaturated
2811:Categories
2670:: 103278.
2095:References
1326:chain rule
847:chain rule
841:Head-based
795:chain rule
574:divergence
41:uniqueness
2822:Hydrology
2692:252508220
2684:0965-9978
2645:1607-7938
2588:1539-1663
2393:0377-0427
2268:0022-1694
2221:120607569
2213:1432-1823
2174:119938184
2068:isotropic
2015:−
1946:α
1938:), where
1920:θ
1907:θ
1885:α
1846:θ
1819:θ
1798:∞
1779:θ
1750:θ
1746:→
1743:θ
1718:∞
1699:θ
1670:θ
1666:→
1663:θ
1629:−
1626:θ
1623:∇
1617:θ
1606:⋅
1603:∇
1591:∂
1586:θ
1583:∂
1554:θ
1517:θ
1503:θ
1469:θ
1407:θ
1404:∇
1398:θ
1353:∇
1304:ε
1279:ε
1267:Δ
1259:Δ
1241:−
1232:Δ
1227:θ
1224:Δ
1200:Δ
1192:Δ
1174:≈
1165:Δ
1160:θ
1157:Δ
1121:−
1110:∇
1101:∇
1079:⋅
1076:∇
1064:∂
1056:∂
971:θ
931:∂
923:∂
902:θ
880:∂
866:θ
863:∂
825:θ
768:θ
728:θ
708:in 1922.
685:−
660:∂
652:∂
638:θ
616:∂
612:∂
598:∂
593:θ
590:∂
553:−
544:∇
535:∇
515:⋅
512:∇
500:∂
495:θ
492:∂
457:−
391:→
382:⋅
379:∇
367:∂
362:θ
359:∂
284:∇
261:∇
176:θ
147:→
106:∇
97:∇
88:θ
77:−
68:→
37:existence
2781:See also
2490:Geoderma
1810:, where
188:is the
162:is the
49:Luckhaus
2623:Bibcode
2568:Bibcode
2498:Bibcode
2459:Bibcode
2420:Bibcode
2373:Bibcode
2334:Bibcode
2248:Bibcode
2121:Bibcode
2113:Physics
953:where
422:where
130:where
2690:
2682:
2643:
2596:781417
2594:
2586:
2391:
2266:
2219:
2211:
2172:
2072:tensor
1425:where
1215:and so
2688:S2CID
2592:S2CID
2286:(PDF)
2217:S2CID
2170:S2CID
2158:(8).
1142:Rothe
2680:ISSN
2641:ISSN
2584:ISSN
2389:ISSN
2264:ISSN
2209:ISSN
1730:and
740:and
47:and
39:and
16:The
2672:doi
2668:173
2631:doi
2576:doi
2537:doi
2506:doi
2494:165
2467:doi
2428:doi
2381:doi
2369:233
2342:doi
2307:262
2256:doi
2201:doi
2197:183
2160:doi
2129:doi
1739:lim
1659:lim
45:Alt
2813::
2686:.
2678:.
2666:.
2662:.
2639:.
2629:.
2619:21
2617:.
2613:.
2590:.
2582:.
2574:.
2562:.
2558:.
2533:55
2531:.
2527:.
2504:.
2492:.
2488:.
2465:.
2455:23
2453:.
2449:.
2426:.
2416:44
2414:.
2410:.
2387:.
2379:.
2363:.
2340:.
2330:26
2328:.
2324:.
2262:.
2254:.
2244:37
2242:.
2238:.
2215:.
2207:.
2195:.
2191:.
2168:.
2156:42
2154:.
2150:.
2127:.
2115:.
2103:^
2694:.
2674::
2647:.
2633::
2625::
2598:.
2578::
2570::
2564:5
2543:.
2539::
2512:.
2508::
2500::
2473:.
2469::
2461::
2434:.
2430::
2422::
2395:.
2383::
2375::
2348:.
2344::
2336::
2309:.
2270:.
2258::
2250::
2223:.
2203::
2176:.
2162::
2135:.
2131::
2123::
2117:1
2052:s
2047:K
2023:n
2020:1
2012:1
2009:=
2006:m
1986:m
1966:n
1924:r
1916:,
1911:s
1902:,
1899:m
1895:,
1892:n
1888:,
1877:(
1850:r
1823:s
1795:=
1791:|
1786:|
1782:)
1776:(
1772:D
1767:|
1762:|
1754:r
1715:=
1711:|
1706:|
1702:)
1696:(
1692:D
1687:|
1682:|
1674:s
1635:.
1632:S
1620:)
1614:(
1610:D
1600:=
1594:t
1557:)
1551:(
1547:D
1520:)
1514:(
1511:C
1506:)
1500:(
1496:K
1464:d
1457:h
1452:d
1444:)
1441:h
1438:(
1434:K
1410:,
1393:d
1386:h
1381:d
1373:)
1370:h
1367:(
1363:K
1359:=
1356:h
1350:)
1347:h
1344:(
1340:K
1282:.
1276:=
1270:t
1262:h
1253:)
1250:h
1247:(
1244:C
1235:t
1209:,
1203:t
1195:h
1186:)
1183:h
1180:(
1177:C
1168:t
1136:.
1124:S
1117:)
1113:z
1107:+
1104:h
1098:)
1095:h
1092:(
1088:K
1083:(
1073:=
1067:t
1059:h
1050:)
1047:h
1044:(
1041:C
1018:)
1015:h
1012:(
1009:C
983:h
978:d
966:d
949:,
934:t
926:h
914:h
909:d
897:d
889:=
883:t
875:)
872:h
869:(
805:h
777:)
774:h
771:(
748:h
700:.
688:S
681:)
676:)
672:1
669:+
663:z
655:h
645:(
641:)
635:(
631:K
626:(
619:z
607:=
601:t
568:.
556:S
550:)
547:z
541:+
538:h
532:(
529:)
526:h
523:(
519:K
509:=
503:t
460:1
433:S
418:,
406:0
403:=
400:S
397:+
388:q
376:+
370:t
324:)
316:1
309:0
302:0
294:(
290:=
287:z
264:z
239:)
236:h
233:(
229:K
202:h
192:;
166:;
144:q
115:,
112:)
109:z
103:+
100:h
94:(
91:)
85:(
81:K
74:=
65:q
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