702:
63:
The groundwater flow equation is often derived for a small representative elemental volume (REV), where the properties of the medium are assumed to be effectively constant. A mass balance is done on the water flowing in and out of this small volume, the flux terms in the relationship being expressed
256:
1190:), located at the origin. Both this equation and the Cartesian version above are the fundamental equation in groundwater flow, but to arrive at this point requires considerable simplification. Some of the main assumptions which went into both these equations are:
482:
2089:
1988:
1882:
1796:
1702:
362:
684:
579:
108:. It is simply a statement of accounting, that for a given control volume, aside from sources or sinks, mass cannot be created or destroyed. The conservation of mass states that, for a given increment of time (
1347:
boundary condition: in addition to solving for the spatial distribution of heads, the location of this surface is also an unknown. This is a non-linear problem, even though the governing equation is linear.
1176:
899:
118:
2112:
in the unconfined case is non-linear, whereas it is linear in the confined case. For unconfined steady-state flow, this non-linearity may be removed by expressing the PDE in terms of the head squared:
1301:, and has many analogs in other fields. The Laplace equation can be solved using techniques, using similar assumptions stated above, but with the additional requirements of a steady-state flow field.
908:
3-D form of the governing groundwater flow equation. However, it has an option to run in a "quasi-3D" mode if the user wishes to do so; in this case the model deals with the vertically averaged
2240:
2173:
1248:
If the aquifer has recharging boundary conditions a steady-state may be reached (or it may be used as an approximation in many cases), and the diffusion equation (above) simplifies to the
1240:
Despite these large assumptions, the groundwater flow equation does a good job of representing the distribution of heads in aquifers due to a transient distribution of sources and sinks.
1544:
1584:
1486:
1393:
1292:
2264:
1422:
112:), the difference between the mass flowing in across the boundaries, the mass flowing out across the boundaries, and the sources within the volume, is the change in storage.
614:) on the right hand side. The hydraulic diffusivity is proportional to the speed at which a finite pressure pulse will propagate through the system (large values of
2245:
This formulation allows us to apply standard methods for solving linear PDEs in the case of unconfined flow. For heterogeneous aquifers with no recharge,
401:
2011:
1917:
1811:
1708:
1614:
290:
to turn the flux across the boundary into a flux over the entire volume, the final form of the groundwater flow equation (in differential form) is:
2698:
2429:
2397:
296:
1899:(length per time), represents the addition of water in the vertical direction (e.g., recharge). By incorporating the correct definitions for
624:
509:
2461:
391:). This equation has both head and flux as unknowns, but Darcy's law relates flux to hydraulic heads, so substituting it in for the flux (
2410:
961:
734:
251:{\displaystyle {\frac {\Delta M_{stor}}{\Delta t}}={\frac {M_{in}}{\Delta t}}-{\frac {M_{out}}{\Delta t}}-{\frac {M_{gen}}{\Delta t}}}
2271:
1352:
2647:
2311:
Corona, Oliver López; Padilla, Pablo; Escolero, Oscar; González, Tomas; Morales-Casique, Eric; Osorio-Olvera, Luis (2014-10-16).
693:, now has the same units but is divided by the appropriate storage term (as defined by the hydraulic diffusivity substitution).
1327:
Steady-state flow to a pumping well (which never truly occurs, but is sometimes a useful approximation) is commonly called the
2688:
2652:
2109:
372:
2703:
2184:
278:(density does not depend on pressure). The mass fluxes across the boundaries then become volume fluxes (as are found in
2530:
2520:
2118:
2657:
2281:
2535:
1502:
1488:, and the aquifer base is at the zero datum, then the unconfined saturated thickness is equal to the head, i.e.,
100:, to arrive at the transient groundwater flow equation. This balance is analogous to the energy balance used in
2454:
2515:
1549:
1499:
and the horizontal components of flow are uniform along the entire saturated thickness of the aquifer (i.e.,
1451:
1358:
2606:
2374:
2258:
936:
1258:
2626:
2591:
1496:
488:
1900:
1445:
1425:
2571:
2550:
2499:
1591:
721:
65:
1805:
expression, we obtain the general 2D governing equation for incompressible saturated groundwater flow:
927:. In the quasi-3D mode, flow is calculated between 2D horizontal layers using the concept of leakage.
1398:
1209:
1911:, we can transform this into two unique governing equations for confined and unconfined conditions:
2683:
2566:
2447:
2394:
286:
to represent the in and out flux terms across the boundaries of the control volume, and using the
2693:
2596:
2586:
2433:
947:
axis — causing converging radial flow). Under these conditions the above equation becomes (
368:
287:
41:
1424:
extending from the aquifer base to the unsaturated surface. This distance is referred to as the
1343:, the solution to the 3D form of the equation is complicated by the presence of a free surface
2678:
2581:
2540:
2406:
2352:
2334:
2291:
1305:
1298:
77:
1448:
is defined as the vertical distance between the water table surface and the aquifer base. If
2342:
2324:
1904:
1433:
1351:
An alternative formulation of the groundwater flow equation may be obtained by invoking the
1321:
1249:
1233:
585:
53:
2379:
An excellent beginner's read for groundwater modeling. Covers all the basic concepts, with
2611:
2601:
2576:
2544:
2489:
2401:
1219:
49:
37:
1339:
The above groundwater flow equations are valid for three dimensional flow. In unconfined
2631:
2616:
2347:
2312:
2246:
2102:
1908:
1587:
1309:
1195:
376:
279:
275:
97:
81:
69:
57:
2672:
2621:
283:
105:
101:
45:
2423:
1215:
for the 1D radial problem the pumping well is fully penetrating a non-leaky aquifer,
1802:
1328:
1317:
477:{\displaystyle S_{s}{\frac {\partial h}{\partial t}}=-\nabla \cdot (-K\nabla h)-G.}
73:
17:
2375:
Introduction to
Groundwater Modeling: Finite Difference and Finite Element Methods
2084:{\displaystyle S_{y}{\frac {\partial h}{\partial t}}=\nabla \cdot (Kh\nabla h)+N.}
618:
lead to fast propagation of signals). The groundwater flow equation then becomes
2494:
2002:
1344:
29:
25:
2484:
2470:
2384:
1983:{\displaystyle S{\frac {\partial h}{\partial t}}=\nabla \cdot (Kb\nabla h)+N.}
1877:{\displaystyle {\frac {\partial nb}{\partial t}}=\nabla \cdot (Kb\nabla h)+N.}
1791:{\displaystyle Q_{y}=\int _{0}^{b}q_{y}dz=-Kb{\frac {\partial h}{\partial y}}}
1697:{\displaystyle Q_{x}=\int _{0}^{b}q_{x}dz=-Kb{\frac {\partial h}{\partial x}}}
1355:, where it is assumed that heads do not vary in the vertical direction (i.e.,
1205:
940:
905:
701:
380:
2338:
1395:). A horizontal water balance is applied to a long vertical column with area
1230:
725:
500:
2356:
499:), it can be taken out of the spatial derivative, simplifying them to the
2276:
A simplification of the groundwater flow equation regarding vertical flow
1892:
1340:
1198:(no change in matrix due to changes in pressure — aka subsidence),
357:{\displaystyle S_{s}{\frac {\partial h}{\partial t}}=-\nabla \cdot q-G.}
1441:
1313:
713:
679:{\displaystyle {\frac {\partial h}{\partial t}}=\alpha \nabla ^{2}h-G.}
267:
33:
2329:
1436:, the saturated thickness is determined by the height of the aquifer,
574:{\displaystyle S_{s}{\frac {\partial h}{\partial t}}=K\nabla ^{2}h-G.}
52:). The steady-state flow of groundwater is described by a form of the
496:
271:
1186:
This equation represents flow to a pumping well (a sink of strength
700:
85:
1440:, and the pressure head is non-zero everywhere. In an unconfined
709:
Especially when using rectangular grid finite-difference models (
1171:{\displaystyle {\frac {\partial h}{\partial t}}=\alpha \left-G.}
894:{\displaystyle {\frac {\partial h}{\partial t}}=\alpha \left-G.}
717:
375:(PDE). This mathematical statement indicates that the change in
2443:
943:
is a line source located at the origin — parallel to the
2439:
2313:"Complex groundwater flow systems as traveling agent models"
2249:
methods may be applied for mixed confined/unconfined cases.
728:
operator becomes (for three-dimensional flow) specifically
1324:, allowing complex geometries to be solved approximately.
705:
Three-dimensional finite difference grid used in MODFLOW
2426:— free groundwater modeling software like MODFLOW
2286:
Groundwater flow equations based on the energy balance
96:
A mass balance must be performed, and used along with
2187:
2121:
2014:
1920:
1814:
1711:
1617:
1552:
1505:
1454:
1401:
1361:
1261:
964:
737:
627:
512:
404:
299:
274:, and under most conditions, water can be considered
121:
495:) is spatially uniform and isotropic (rather than a
2640:
2559:
2508:
2477:
28:relationship which is used to describe the flow of
2235:{\displaystyle \nabla ^{2}h^{2}=-{\frac {2N}{K}}.}
2234:
2167:
2083:
1982:
1876:
1790:
1696:
1578:
1538:
1480:
1416:
1387:
1304:A common method for solution of this equations in
1286:
1201:the water is of constant density (incompressible),
1170:
893:
678:
573:
476:
356:
250:
40:flow of groundwater is described by a form of the
2168:{\displaystyle \nabla \cdot (K\nabla h^{2})=-2N.}
1320:of hydraulic head and the stream function make a
379:with time (left hand side) equals the negative
1297:This equation states that hydraulic head is a
2455:
1312:is to use the graphical technique of drawing
8:
1539:{\displaystyle \partial q_{x}/\partial z=0}
2462:
2448:
2440:
2392:Freeze, R. Allan; Cherry, John A. (1979).
2265:solution of partial differential equations
904:MODFLOW code discretizes and simulates an
2346:
2328:
2214:
2202:
2192:
2186:
2141:
2120:
2025:
2019:
2013:
1924:
1919:
1815:
1813:
1768:
1744:
1734:
1729:
1716:
1710:
1674:
1650:
1640:
1635:
1622:
1616:
1559:
1551:
1519:
1513:
1504:
1461:
1453:
1400:
1368:
1360:
1275:
1260:
1204:any external loads on the aquifer (e.g.,
1145:
1127:
1120:
1108:
1090:
1083:
1075:
1066:
1043:
1033:
1021:
1003:
996:
965:
963:
868:
850:
843:
831:
813:
806:
794:
776:
769:
738:
736:
658:
628:
626:
553:
523:
517:
511:
415:
409:
403:
310:
304:
298:
226:
220:
195:
189:
167:
161:
132:
122:
120:
48:to describe the flow of heat in a solid (
2303:
1579:{\displaystyle \partial K/\partial z=0}
1481:{\displaystyle \partial h/\partial z=0}
1388:{\displaystyle \partial h/\partial z=0}
935:Another useful coordinate system is 3D
1287:{\displaystyle 0=\alpha \nabla ^{2}h}
367:This is known in other fields as the
88:or fractured rocks (i.e. volcanic)
7:
1244:Laplace equation (steady-state flow)
371:or heat equation, it is a parabolic
60:and has analogs in numerous fields.
1218:the groundwater is flowing slowly (
724:. In these coordinates the general
262:Diffusion equation (transient flow)
2189:
2134:
2122:
2063:
2048:
2036:
2028:
1962:
1947:
1935:
1927:
1856:
1841:
1829:
1818:
1779:
1771:
1685:
1677:
1564:
1553:
1524:
1506:
1466:
1455:
1373:
1362:
1272:
1138:
1124:
1101:
1087:
1054:
1046:
1014:
1000:
976:
968:
861:
847:
824:
810:
787:
773:
749:
741:
655:
639:
631:
550:
534:
526:
456:
441:
426:
418:
336:
321:
313:
239:
208:
177:
149:
125:
72:, which requires that the flow is
14:
697:Rectangular cartesian coordinates
2263:A numerical method used for the
1417:{\displaystyle \delta x\delta y}
1335:Two-dimensional groundwater flow
931:Circular cylindrical coordinates
595:), puts hydraulic diffusivity (
2699:Partial differential equations
2178:Or, for homogeneous aquifers,
2147:
2128:
2069:
2054:
1968:
1953:
1862:
1847:
462:
447:
80:to incorporate the effect of
76:Other approaches are based on
64:in terms of head by using the
1:
2373:H. F. Wang and M.P. Anderson
2272:Dupuit–Forchheimer assumption
2110:partial differential equation
1353:Dupuit–Forchheimer assumption
373:partial differential equation
1225:the hydraulic conductivity (
689:Where the sink/source term,
939:(typically where a pumping
266:Mass can be represented as
2720:
2282:Groundwater energy balance
951:being radial distance and
503:, this makes the equation
44:, similar to that used in
2536:Hydrological optimization
2526:Groundwater flow equation
2424:USGS groundwater software
1801:Inserting these into our
22:groundwater flow equation
1194:the aquifer material is
584:Dividing through by the
387:) and the source terms (
2531:Hazen–Williams equation
2521:Darcy–Weisbach equation
2259:Analytic element method
1590:in terms of integrated
937:cylindrical coordinates
2236:
2169:
2085:
1984:
1878:
1792:
1698:
1592:groundwater discharges
1580:
1540:
1497:hydraulic conductivity
1482:
1418:
1389:
1288:
1172:
895:
706:
680:
575:
489:hydraulic conductivity
478:
358:
252:
2689:Hydraulic engineering
2551:Pipe network analysis
2516:Bernoulli's principle
2500:Hydraulic engineering
2430:Groundwater Hydrology
2237:
2170:
2086:
1985:
1879:
1793:
1699:
1581:
1541:
1483:
1419:
1390:
1289:
1222:less than unity), and
1173:
896:
722:Cartesian coordinates
704:
681:
576:
479:
359:
253:
66:constitutive equation
56:, which is a form of
2185:
2119:
2094:(unconfined), where
2012:
1918:
1812:
1709:
1615:
1550:
1503:
1452:
1399:
1359:
1259:
1210:atmospheric pressure
962:
735:
625:
510:
402:
297:
119:
2704:Transport phenomena
1901:saturated thickness
1895:. The source term,
1739:
1645:
1446:saturated thickness
1426:saturated thickness
2434:MIT OpenCourseware
2400:2020-04-06 at the
2232:
2165:
2081:
1993:(confined), where
1980:
1874:
1788:
1725:
1694:
1631:
1586:), we can express
1576:
1536:
1495:Assuming both the
1478:
1414:
1385:
1284:
1168:
891:
707:
676:
571:
474:
369:diffusion equation
354:
288:divergence theorem
248:
78:Agent Based Models
42:diffusion equation
2666:
2665:
2541:Open-channel flow
2405:. Prentice Hall.
2330:10.7717/peerj.557
2292:Richards equation
2227:
2043:
1942:
1836:
1786:
1692:
1306:civil engineering
1299:harmonic function
1152:
1115:
1081:
1061:
1041:
1028:
983:
875:
838:
801:
756:
646:
606:or equivalently,
541:
433:
328:
246:
215:
184:
156:
104:to arrive at the
84:aquifers such as
2711:
2464:
2457:
2450:
2441:
2361:
2360:
2350:
2332:
2308:
2241:
2239:
2238:
2233:
2228:
2223:
2215:
2207:
2206:
2197:
2196:
2174:
2172:
2171:
2166:
2146:
2145:
2105:of the aquifer.
2090:
2088:
2087:
2082:
2044:
2042:
2034:
2026:
2024:
2023:
1989:
1987:
1986:
1981:
1943:
1941:
1933:
1925:
1905:specific storage
1883:
1881:
1880:
1875:
1837:
1835:
1827:
1816:
1797:
1795:
1794:
1789:
1787:
1785:
1777:
1769:
1749:
1748:
1738:
1733:
1721:
1720:
1703:
1701:
1700:
1695:
1693:
1691:
1683:
1675:
1655:
1654:
1644:
1639:
1627:
1626:
1585:
1583:
1582:
1577:
1563:
1545:
1543:
1542:
1537:
1523:
1518:
1517:
1487:
1485:
1484:
1479:
1465:
1434:confined aquifer
1423:
1421:
1420:
1415:
1394:
1392:
1391:
1386:
1372:
1322:curvilinear grid
1293:
1291:
1290:
1285:
1280:
1279:
1250:Laplace equation
1177:
1175:
1174:
1169:
1158:
1154:
1153:
1151:
1150:
1149:
1136:
1132:
1131:
1121:
1116:
1114:
1113:
1112:
1099:
1095:
1094:
1084:
1082:
1080:
1079:
1067:
1062:
1060:
1052:
1044:
1042:
1034:
1029:
1027:
1026:
1025:
1012:
1008:
1007:
997:
984:
982:
974:
966:
900:
898:
897:
892:
881:
877:
876:
874:
873:
872:
859:
855:
854:
844:
839:
837:
836:
835:
822:
818:
817:
807:
802:
800:
799:
798:
785:
781:
780:
770:
757:
755:
747:
739:
720:), we deal with
685:
683:
682:
677:
663:
662:
647:
645:
637:
629:
586:specific storage
580:
578:
577:
572:
558:
557:
542:
540:
532:
524:
522:
521:
483:
481:
480:
475:
434:
432:
424:
416:
414:
413:
363:
361:
360:
355:
329:
327:
319:
311:
309:
308:
257:
255:
254:
249:
247:
245:
237:
236:
221:
216:
214:
206:
205:
190:
185:
183:
175:
174:
162:
157:
155:
147:
146:
145:
123:
54:Laplace equation
2719:
2718:
2714:
2713:
2712:
2710:
2709:
2708:
2669:
2668:
2667:
2662:
2641:Public networks
2636:
2555:
2545:Manning formula
2504:
2490:Hydraulic fluid
2473:
2468:
2420:
2402:Wayback Machine
2370:
2368:Further reading
2365:
2364:
2310:
2309:
2305:
2300:
2255:
2216:
2198:
2188:
2183:
2182:
2137:
2117:
2116:
2099:
2035:
2027:
2015:
2010:
2009:
2001:is the aquifer
1998:
1934:
1926:
1916:
1915:
1891:is the aquifer
1828:
1817:
1810:
1809:
1778:
1770:
1740:
1712:
1707:
1706:
1684:
1676:
1646:
1618:
1613:
1612:
1606:
1599:
1548:
1547:
1509:
1501:
1500:
1450:
1449:
1397:
1396:
1357:
1356:
1337:
1271:
1257:
1256:
1246:
1220:Reynolds number
1212:) are constant,
1184:
1141:
1137:
1123:
1122:
1104:
1100:
1086:
1085:
1071:
1053:
1045:
1017:
1013:
999:
998:
995:
991:
975:
967:
960:
959:
933:
925:
864:
860:
846:
845:
827:
823:
809:
808:
790:
786:
772:
771:
768:
764:
748:
740:
733:
732:
699:
654:
638:
630:
623:
622:
604:
593:
549:
533:
525:
513:
508:
507:
425:
417:
405:
400:
399:
320:
312:
300:
295:
294:
264:
238:
222:
207:
191:
176:
163:
148:
128:
124:
117:
116:
94:
50:heat conduction
12:
11:
5:
2717:
2715:
2707:
2706:
2701:
2696:
2691:
2686:
2681:
2671:
2670:
2664:
2663:
2661:
2660:
2655:
2650:
2644:
2642:
2638:
2637:
2635:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2563:
2561:
2557:
2556:
2554:
2553:
2548:
2538:
2533:
2528:
2523:
2518:
2512:
2510:
2506:
2505:
2503:
2502:
2497:
2492:
2487:
2481:
2479:
2475:
2474:
2469:
2467:
2466:
2459:
2452:
2444:
2438:
2437:
2427:
2419:
2418:External links
2416:
2415:
2414:
2411:978-0133653120
2390:
2389:
2388:
2369:
2366:
2363:
2362:
2302:
2301:
2299:
2296:
2295:
2294:
2289:
2288:
2287:
2279:
2278:
2277:
2269:
2268:
2267:
2254:
2251:
2247:Potential flow
2243:
2242:
2231:
2226:
2222:
2219:
2213:
2210:
2205:
2201:
2195:
2191:
2176:
2175:
2164:
2161:
2158:
2155:
2152:
2149:
2144:
2140:
2136:
2133:
2130:
2127:
2124:
2108:Note that the
2103:specific yield
2097:
2092:
2091:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2041:
2038:
2033:
2030:
2022:
2018:
1996:
1991:
1990:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1940:
1937:
1932:
1929:
1923:
1909:specific yield
1885:
1884:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1834:
1831:
1826:
1823:
1820:
1799:
1798:
1784:
1781:
1776:
1773:
1767:
1764:
1761:
1758:
1755:
1752:
1747:
1743:
1737:
1732:
1728:
1724:
1719:
1715:
1704:
1690:
1687:
1682:
1679:
1673:
1670:
1667:
1664:
1661:
1658:
1653:
1649:
1643:
1638:
1634:
1630:
1625:
1621:
1604:
1597:
1575:
1572:
1569:
1566:
1562:
1558:
1555:
1535:
1532:
1529:
1526:
1522:
1516:
1512:
1508:
1477:
1474:
1471:
1468:
1464:
1460:
1457:
1413:
1410:
1407:
1404:
1384:
1381:
1378:
1375:
1371:
1367:
1364:
1336:
1333:
1329:Thiem solution
1310:soil mechanics
1295:
1294:
1283:
1278:
1274:
1270:
1267:
1264:
1245:
1242:
1238:
1237:
1223:
1216:
1213:
1202:
1199:
1196:incompressible
1183:
1180:
1179:
1178:
1167:
1164:
1161:
1157:
1148:
1144:
1140:
1135:
1130:
1126:
1119:
1111:
1107:
1103:
1098:
1093:
1089:
1078:
1074:
1070:
1065:
1059:
1056:
1051:
1048:
1040:
1037:
1032:
1024:
1020:
1016:
1011:
1006:
1002:
994:
990:
987:
981:
978:
973:
970:
955:being angle),
932:
929:
923:
916:, rather than
902:
901:
890:
887:
884:
880:
871:
867:
863:
858:
853:
849:
842:
834:
830:
826:
821:
816:
812:
805:
797:
793:
789:
784:
779:
775:
767:
763:
760:
754:
751:
746:
743:
716:, made by the
698:
695:
687:
686:
675:
672:
669:
666:
661:
657:
653:
650:
644:
641:
636:
633:
602:
591:
582:
581:
570:
567:
564:
561:
556:
552:
548:
545:
539:
536:
531:
528:
520:
516:
485:
484:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
431:
428:
423:
420:
412:
408:
377:hydraulic head
365:
364:
353:
350:
347:
344:
341:
338:
335:
332:
326:
323:
318:
315:
307:
303:
276:incompressible
263:
260:
259:
258:
244:
241:
235:
232:
229:
225:
219:
213:
210:
204:
201:
198:
194:
188:
182:
179:
173:
170:
166:
160:
154:
151:
144:
141:
138:
135:
131:
127:
93:
90:
58:potential flow
13:
10:
9:
6:
4:
3:
2:
2716:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2676:
2674:
2659:
2656:
2654:
2651:
2649:
2646:
2645:
2643:
2639:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2607:Power network
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2564:
2562:
2558:
2552:
2549:
2546:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2513:
2511:
2507:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2482:
2480:
2476:
2472:
2465:
2460:
2458:
2453:
2451:
2446:
2445:
2442:
2435:
2431:
2428:
2425:
2422:
2421:
2417:
2412:
2408:
2404:
2403:
2399:
2396:
2391:
2386:
2382:
2378:
2377:
2376:
2372:
2371:
2367:
2358:
2354:
2349:
2344:
2340:
2336:
2331:
2326:
2322:
2318:
2314:
2307:
2304:
2297:
2293:
2290:
2285:
2284:
2283:
2280:
2275:
2274:
2273:
2270:
2266:
2262:
2261:
2260:
2257:
2256:
2252:
2250:
2248:
2229:
2224:
2220:
2217:
2211:
2208:
2203:
2199:
2193:
2181:
2180:
2179:
2162:
2159:
2156:
2153:
2150:
2142:
2138:
2131:
2125:
2115:
2114:
2113:
2111:
2106:
2104:
2100:
2078:
2075:
2072:
2066:
2060:
2057:
2051:
2045:
2039:
2031:
2020:
2016:
2008:
2007:
2006:
2004:
2000:
1977:
1974:
1971:
1965:
1959:
1956:
1950:
1944:
1938:
1930:
1921:
1914:
1913:
1912:
1910:
1906:
1902:
1898:
1894:
1890:
1871:
1868:
1865:
1859:
1853:
1850:
1844:
1838:
1832:
1824:
1821:
1808:
1807:
1806:
1804:
1782:
1774:
1765:
1762:
1759:
1756:
1753:
1750:
1745:
1741:
1735:
1730:
1726:
1722:
1717:
1713:
1705:
1688:
1680:
1671:
1668:
1665:
1662:
1659:
1656:
1651:
1647:
1641:
1636:
1632:
1628:
1623:
1619:
1611:
1610:
1609:
1607:
1600:
1593:
1589:
1573:
1570:
1567:
1560:
1556:
1533:
1530:
1527:
1520:
1514:
1510:
1498:
1493:
1491:
1475:
1472:
1469:
1462:
1458:
1447:
1443:
1439:
1435:
1431:
1427:
1411:
1408:
1405:
1402:
1382:
1379:
1376:
1369:
1365:
1354:
1349:
1346:
1342:
1334:
1332:
1330:
1325:
1323:
1319:
1318:contour lines
1315:
1311:
1307:
1302:
1300:
1281:
1276:
1268:
1265:
1262:
1255:
1254:
1253:
1251:
1243:
1241:
1235:
1232:
1228:
1224:
1221:
1217:
1214:
1211:
1207:
1203:
1200:
1197:
1193:
1192:
1191:
1189:
1181:
1165:
1162:
1159:
1155:
1146:
1142:
1133:
1128:
1117:
1109:
1105:
1096:
1091:
1076:
1072:
1068:
1063:
1057:
1049:
1038:
1035:
1030:
1022:
1018:
1009:
1004:
992:
988:
985:
979:
971:
958:
957:
956:
954:
950:
946:
942:
938:
930:
928:
926:
919:
915:
911:
907:
888:
885:
882:
878:
869:
865:
856:
851:
840:
832:
828:
819:
814:
803:
795:
791:
782:
777:
765:
761:
758:
752:
744:
731:
730:
729:
727:
723:
719:
715:
712:
703:
696:
694:
692:
673:
670:
667:
664:
659:
651:
648:
642:
634:
621:
620:
619:
617:
613:
609:
605:
598:
594:
587:
568:
565:
562:
559:
554:
546:
543:
537:
529:
518:
514:
506:
505:
504:
502:
498:
494:
490:
471:
468:
465:
459:
453:
450:
444:
438:
435:
429:
421:
410:
406:
398:
397:
396:
394:
390:
386:
383:of the flux (
382:
378:
374:
370:
351:
348:
345:
342:
339:
333:
330:
324:
316:
305:
301:
293:
292:
291:
289:
285:
284:Taylor series
281:
277:
273:
269:
261:
242:
233:
230:
227:
223:
217:
211:
202:
199:
196:
192:
186:
180:
171:
168:
164:
158:
152:
142:
139:
136:
133:
129:
115:
114:
113:
111:
107:
106:heat equation
103:
102:heat transfer
99:
91:
89:
87:
83:
79:
75:
71:
67:
61:
59:
55:
51:
47:
46:heat transfer
43:
39:
35:
31:
27:
23:
19:
2627:Rescue tools
2592:Drive system
2560:Technologies
2525:
2393:
2383:examples in
2380:
2320:
2316:
2306:
2244:
2177:
2107:
2095:
2093:
1994:
1992:
1896:
1888:
1886:
1803:mass balance
1800:
1602:
1595:
1494:
1489:
1437:
1429:
1350:
1338:
1326:
1303:
1296:
1247:
1239:
1226:
1187:
1185:
952:
948:
944:
934:
921:
917:
913:
909:
903:
710:
708:
690:
688:
615:
611:
607:
600:
596:
589:
583:
492:
486:
392:
388:
384:
366:
265:
109:
95:
92:Mass balance
62:
26:mathematical
21:
18:hydrogeology
15:
2572:Accumulator
2495:Fluid power
2395:Groundwater
2003:storativity
1588:Darcy's law
1345:water table
1182:Assumptions
395:) leads to
280:Darcy's law
98:Darcy's law
70:Darcy's law
32:through an
30:groundwater
2684:Hydraulics
2673:Categories
2658:Manchester
2485:Hydraulics
2471:Hydraulics
2385:FORTRAN 77
2298:References
1206:overburden
906:orthogonal
381:divergence
2694:Hydrology
2648:Liverpool
2567:Machinery
2339:2167-8359
2212:−
2190:∇
2154:−
2135:∇
2126:⋅
2123:∇
2064:∇
2052:⋅
2049:∇
2037:∂
2029:∂
1963:∇
1951:⋅
1948:∇
1936:∂
1928:∂
1857:∇
1845:⋅
1842:∇
1830:∂
1819:∂
1780:∂
1772:∂
1760:−
1727:∫
1686:∂
1678:∂
1666:−
1633:∫
1565:∂
1554:∂
1525:∂
1507:∂
1467:∂
1456:∂
1409:δ
1403:δ
1374:∂
1363:∂
1273:∇
1269:α
1231:isotropic
1160:−
1139:∂
1125:∂
1106:θ
1102:∂
1088:∂
1055:∂
1047:∂
1015:∂
1001:∂
989:α
977:∂
969:∂
883:−
862:∂
848:∂
825:∂
811:∂
788:∂
774:∂
762:α
750:∂
742:∂
726:Laplacian
668:−
656:∇
652:α
640:∂
632:∂
563:−
551:∇
535:∂
527:∂
501:Laplacian
466:−
457:∇
451:−
445:⋅
442:∇
439:−
427:∂
419:∂
346:−
340:⋅
337:∇
334:−
322:∂
314:∂
282:). Using
240:Δ
218:−
209:Δ
187:−
178:Δ
150:Δ
126:Δ
38:transient
2679:Aquifers
2597:Manifold
2587:Cylinder
2509:Modeling
2478:Concepts
2398:Archived
2357:25337455
2323:: e557.
2253:See also
1893:porosity
1341:aquifers
1316:; where
1314:flownets
1229:) is an
74:laminar.
16:Used in
2582:Circuit
2348:4203025
2101:is the
1442:aquifer
1432:. In a
714:MODFLOW
487:Now if
268:density
86:karstic
82:complex
68:called
34:aquifer
24:is the
2653:London
2409:
2381:simple
2355:
2345:
2337:
1907:, and
1887:Where
1444:, the
1234:scalar
497:tensor
272:volume
270:times
36:. The
20:, the
2612:Press
2602:Motor
2577:Brake
2317:PeerJ
2632:Seal
2617:Pump
2407:ISBN
2353:PMID
2335:ISSN
2005:and
1601:and
1546:and
1308:and
941:well
920:and
912:and
718:USGS
711:e.g.
2622:Ram
2343:PMC
2325:doi
1995:S=S
1490:b=h
612:T/S
601:K/S
2675::
2351:.
2341:.
2333:.
2319:.
2315:.
1903:,
1608::
1594:,
1492:.
1428:,
1331:.
1252:.
1208:,
610:=
599:=
110:Δt
2547:)
2543:(
2463:e
2456:t
2449:v
2436:)
2432:(
2413:.
2387:.
2359:.
2327::
2321:2
2230:.
2225:K
2221:N
2218:2
2209:=
2204:2
2200:h
2194:2
2163:.
2160:N
2157:2
2151:=
2148:)
2143:2
2139:h
2132:K
2129:(
2098:y
2096:S
2079:.
2076:N
2073:+
2070:)
2067:h
2061:h
2058:K
2055:(
2046:=
2040:t
2032:h
2021:y
2017:S
1999:b
1997:s
1978:.
1975:N
1972:+
1969:)
1966:h
1960:b
1957:K
1954:(
1945:=
1939:t
1931:h
1922:S
1897:N
1889:n
1872:.
1869:N
1866:+
1863:)
1860:h
1854:b
1851:K
1848:(
1839:=
1833:t
1825:b
1822:n
1783:y
1775:h
1766:b
1763:K
1757:=
1754:z
1751:d
1746:y
1742:q
1736:b
1731:0
1723:=
1718:y
1714:Q
1689:x
1681:h
1672:b
1669:K
1663:=
1660:z
1657:d
1652:x
1648:q
1642:b
1637:0
1629:=
1624:x
1620:Q
1605:y
1603:Q
1598:x
1596:Q
1574:0
1571:=
1568:z
1561:/
1557:K
1534:0
1531:=
1528:z
1521:/
1515:x
1511:q
1476:0
1473:=
1470:z
1463:/
1459:h
1438:H
1430:b
1412:y
1406:x
1383:0
1380:=
1377:z
1370:/
1366:h
1282:h
1277:2
1266:=
1263:0
1236:.
1227:K
1188:G
1166:.
1163:G
1156:]
1147:2
1143:z
1134:h
1129:2
1118:+
1110:2
1097:h
1092:2
1077:2
1073:r
1069:1
1064:+
1058:r
1050:h
1039:r
1036:1
1031:+
1023:2
1019:r
1010:h
1005:2
993:[
986:=
980:t
972:h
953:θ
949:r
945:z
924:s
922:S
918:k
914:S
910:T
889:.
886:G
879:]
870:2
866:z
857:h
852:2
841:+
833:2
829:y
820:h
815:2
804:+
796:2
792:x
783:h
778:2
766:[
759:=
753:t
745:h
691:G
674:.
671:G
665:h
660:2
649:=
643:t
635:h
616:α
608:α
603:s
597:α
592:s
590:S
588:(
569:.
566:G
560:h
555:2
547:K
544:=
538:t
530:h
519:s
515:S
493:K
491:(
472:.
469:G
463:)
460:h
454:K
448:(
436:=
430:t
422:h
411:s
407:S
393:q
389:G
385:q
352:.
349:G
343:q
331:=
325:t
317:h
306:s
302:S
243:t
234:n
231:e
228:g
224:M
212:t
203:t
200:u
197:o
193:M
181:t
172:n
169:i
165:M
159:=
153:t
143:r
140:o
137:t
134:s
130:M
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.