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CHAPTER V: VERIFICATION PROBLEMS IN THE PORFLOW MANUAL

Shortcuts:

CASE V.01 - Transient One-Dimensional Diffusion

CASE V.02 - One-Dimensional Heat Transport by Unidirectional Flow

CASE V.03 - Theis Solution for Transient Drawdown

CASE V.04 - Finite Cylinder with Heat Source

CASE V.05 - Coupled Flow and Heat Transfer in Regional Flow

CASE V.06 - Three-Dimensional Transport of a Contaminant

CASE V.07 - Philip's Solution for Horizontal Unsaturated Flow

CASE V.08 - Philip's Solution for a Vertical Unsaturated Column

CASE V.09 - Infiltration from a Line Source to a Water Table

CASE V.10 - Transient Free-Surface Boussinesq Flow - Recharge

CASE V.11 - Transient Free-Surface Boussinesq Flow - Seepage

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************************************************************************
TITLe CASE V.02 - One-Dimensional Heat Transport by Unidirectional Flow
************************************************************************
//// Carslaw, H.S. and J.C. Jaeger, 1959. Conduction of Heat in Solids
//// Oxford Press 2nd Ed., pp. 387-389
************************************************************************
/
GRID NODEs 182 nodes in the X direction
COORdinate X minimum = 0.0 max = 4500.0
/
DENSity = 1000.0
FLUId: SPECific heat = 4185.0
FLUId: thermal CONDuctivity = 2.05E+07
/
ROCK density = 2780
ROCK porosity = 0.001
THERmal Ce = 850.0 Ke = 5.0E+07 L.D. = 2.0 T.D. = 0.0
/
SET U = 0.90 everywhere
SET T = 10.0
/
BOUNdary conditions for T: index X-, value = 20.0
BOUNdary conditions for T: index X+, value = 10.0
/
MATRix sweep X direction for T; method = ADI
/
DIAGnostic node (20,1) every 500 steps
DEBUG GEOMETRY OFF
SELEct window from (1,1) to (181,1) interval (2,1)
OUTPut for SELEcted window
SAVE T on 'V2.ARC'
/
SOLVe for 500 years in steps of 0.25 years
SAVE NOW
/
SOLVe for 500 more years in steps of 0.25 years
SAVE NOW
/
SOLVe for 1000 more years in steps of 0.25 years
/
END
/

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************************************************************************
TITLe CASE V.03 - Theis Solution for Transient Drawdown
************************************************************************
//// Theis, C.V., 1935. The Relation Between the Lowering of the
//// Piezometric Surface and the Rate and Duration of Discharge of a
//// Well Using Groundwater Storage, Trans. Amer. Geophys. Union, 2,
//// p. 519-524.
************************************************************************
/
GRID NODEs 1 by 69
COORdinate R NODES located at
0.25 .5 1.0 2.0 3.5 5.0 7.5
10.0 15.0 20.0 25.0 30.0 35.0 40.0
45.0 50.0 55.0 60.0 65.0 70.0 75.0
80.0 85.0 90.0 95.0 100.0 110.0 120.0
130.0 140.0 150.0 160.0 170.0 180.0 190.0
200.0 220.0 240.0 260.0 280.0 300.0 320.0
340.0 360.0 380.0 400.0 440.0 480.0 520.0
560.0 600.0 640.0 680.0 720.0 760.0 800.0
850.0 900.0 950.0 1000.0 1100.0 1200.0 1300.0
1400.0 1500.0 1600.0 1700.0 1800.0 2000.0
/
HYDRaulic properties S = 0.002, kx = 300 m/day, Ky = 300 m/day
/
SET P = 100.0 everywhere initially
/
BOUNdary conditions for P: index Y-, FLUX = -1273.24
/
MATRix sweep Y direction
/
DIAGnostic node (2,26) every 100 steps
DEBUG GEOMETRY OFF
SELEct (2,1) to (2,999) interval (2,2)
OUTPut V, H for SELEcted region
SAVE on 'V3.ARC' H only
/
SOLVe for 0.5 days in steps of 0.001 days
/
END
/

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************************************************************************
TITLe CASE V.04 - Finite Cylinder with Heat Source
************************************************************************
//// Carslaw, H.S. and J.C. Jaeger, 1959. Conduction of Heat in Solids
//// Oxford Press 2nd Ed., p. 223-224
************************************************************************
/
GRID NODEs 22 by 22
COORdinate R range = 1
/
BOUNdary conditions for T ib=X-, value = 0.
BOUNdary conditions for T ib=X+, value = 0.
BOUNdary conditions for T ib=Y-, GRAD=0.
BOUNdary conditions for T ib=Y+, value = 0.
/
SOURce for T: constant at 4 per unit VOLUme everywhere
/
DENSity 1
ROCK DENSity = 1
ROCK POROsity = 1
THERmal props cp = 1, kt = 1
FLUId SPECific heat = 1
FLUId thermal CONDuctivity = 1
/
DIAGnostic node at 12,2 print every 20 steps
DEBUG GEOMETRY OFF
SELEct (1,1) to (999,999) interval (2,2)
OUTPut in SELEcted region in NARRow mode
SAVE on 'V4.ARC' T only
/
SOLVE in steady mode max 200 steps min 200
/
END
/

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************************************************************************
TITLe CASE V.05 - Coupled Flow and Heat Transfer in Regional Flow
************************************************************************
//// Domenico, P.A. and V.V. Palciauskas, 1973. Theoretical Analysis
//// of Forced Convective Heat Transfer in Regional Ground-Water Flow,
//// Geological Society of America Bulletin, Vol. 84, p. 3803-3814,
//// December 1973.
************************************************************************
/
GRID NODEs is 42 by 42
/
COORdinate X RANGE 100
COORdinate Y RANGE 100
/
/**************** Rock properties
ROCK DENSity = 1
ROCK POROsity = 1
THERmal properties: specific heat = 1, thermal conductivity = 1
HYDRaulic properties: ss = 1, hydraulic conductivity = 2*0.01
/
/**************** Fluid properties
DENSity = 1
FLUId SPECific heat = 1
FLUId thermal CONDuctivity = 1
/
/**************** Boundary for pressure
BOUNdary for P index X-: GRADient 0
BOUNdary for P index X+: GRADient 0
BOUNdary for P index Y-: GRADient 0
BOUNdary index Y+: P = -11.591957 * COS ( 0.0314159 * X )
/
/**************** Boundary for temperature
BOUNdary for T index X-: GRADient 0
BOUNdary for T index X+: GRADient 0
BOUNdary for T index Y-: outward GRADient +0.0002
/
DIAGnostics at 11,11 every 100 steps
DEBUG GEOMETRY OFF
SELEct (1,1) to (999,999) interval (2,2)
OUTPut P and T in SELEcted region in NARRow mode
SAVE on file 'V5.ARC' T and H
/
SOLVe in STEAdy mode for 2000 minimum of 2000
/
END
/

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************************************************************************
TITLe CASE V.06 - Three-Dimensional Transport of a Contaminant
************************************************************************
//// Codell, R.B., T.K. Key and G. Whelan, 1982. A Collection of
//// Mathematical Models for Dispersion in Surface Water and
//// Groundwater, NUREG-0868, Division of Engineering, Office of
//// Nuclear Reactor Regulation, Washington, D.C.
************************************************************************
/
GRID NODEs 65 by 26 by 20
/
COORdinate X user specifiied values at NODEs are:
-700. -500. -370. -280. -220. -180. -150. -130. -110. -90.
-70. -50. -40. -30. -20. -10. 0. 10. 20. 30.
40. 50. 70. 90. 110. 130. 150. 170. 190. 210.
240. 270. 300. 330. 360. 400. 440. 480. 520. 560.
600. 650. 700. 750. 800. 850. 900. 950. 1000. 1075.
1150. 1225. 1300. 1375. 1450. 1525. 1600. 1700. 1800. 1900.
2050. 2200. 2400. 2700. 3000.
/
COORdinate Y user specifiied values at NODEs are:
0. 10. 30. 50. 70. 90. 110. 130. 150. 170.
190. 210. 230. 250. 275. 300. 325. 360. 400. 440.
480. 520. 570. 620. 690. 800.
/
COORdinate Z User specified values at NODEs
-50. -44. -36. -29. -22. -17. -12. -9. -6. -4.
-3. -2. -1.5 -1.0 -0.75 -0.50 -0.25 -0.15 -0.05 0.
/
ROCK POROsity = 0.1
TRANsport kd = 0., md = 0, Ld = 91.0, Td = 20.0
/
LOCATE (10,1,19) to (24,6,19) $ Represents the Contaminant Source
SOURce C is constant at 2.5E-4 VOLUmetric g/sec/cu m in SELEcted area
/
SET U to 1.4E-06
/
BOUNdary conditions for C: all boundaries at zero FLUX = 0.
BOUNdary conditions for C: X-: value = 0.0
/
CONVERGENCE criteria: C 1.E-7, iterations = 20
/
Diagnostic node (27,4,19) output every 100 steps
DEBUG GEOMETRY OFF
HISTory at (27,4,19) (27,4,2) (27,16,19)
HISTory of C only on 'V6.HIS' every 1 steps
SELEct (1,1,1) to (999,999,999) interval (6,2,3)
OUTPut C in SELEcted region in NARRow mode
SAVE only C on 'V6.ARC'
/
SOLVe C for 1.5768E+08 secs in steps of 2.E+3 increase 1.10 max 5.0E6
***> Use next SOLVe rather than one above for better analytic match
/SOLVe C for 1.5768E+08 secs in steps of 2.E+3 increase 1.01 max 5.E4
***********
/
END
/

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************************************************************************
TITLe CASE V.07 - Philip's Solution for Horizontal Unsaturated Flow
************************************************************************
//// Philip, J.R., 1957. Numerical Solution of Equations of the
//// Diffusion Type with Diffusivity Concentration-Dependent,
//// Transactions, Faraday Society 51:885-892.
************************************************************************
/
GRID NODEs 52 in the X direction
COORdinate X at NODEs are:
0. 0.302 0.376 0.509 0.726 0.741 1.003 1.249
1.440 1.483 1.744 1.950 2.107 2.237 2.406 2.780
2.852 3.059 3.290 3.403 3.719 3.972 4.141 4.558
4.572 4.970 5.173 5.379 5.479 5.785 6.112 6.190
7.004 7.211 7.418 7.840 8.276 8.334 8.733 9.221
9.764 10.407 11.285 12.0 13.0 14.0 15.0 16.0
17.0 18.0 19.0 20.0
/
GRAVity is 0, 0
/
ROCK POROsity = 0.45
HYDRaulic properties: effective storativity = 0.; 2*1.157e-5 cm/sec
MULTiphase CONDuctivity, 2 sets in TABLe (S1,Kr): (0.3333,0) (1,1)
MULTiphase with 2 sets in TABLe format (S1,Pcap): (0.3333,100) (1,0)
/
SET initial S to 0.444444 everywhere
/
BOUNdary for P at boundary X- head = 0. cm
/
CONVergence of P is LOCAl criteria = 1.0e-5, iterations = 100
/
PROPerties at interfaces determined by ARIThmetic mean
/
DIAGnostic P and S at (3,1) every 100 steps
DEBUG GEOMETRY OFF
/
SOLVe for 864 sec in steps of 0.1, multiplier = 1.1, max time step = 5
SELEct (1,2) to (999,2) interval (5,1)
OUTPut P and S NOW in SELEcted region in NARRow mode
SAVE S NOW on file 'V7.ARC'
/
SOLVe for 4320 sec in steps of 5.0, multiplier = 1.1, max time step = 10
OUTPut NOW in SELEcted region
SAVE NOW
/
SOLVe for 4320 sec
/
END
/

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************************************************************************
TITLe CASE V.08 - Philip's Solution for a Vertical Unsaturated Column
************************************************************************
//// Philip, J.R., 1957. Numerical Solution of Equations of the
//// Diffusion Type with Diffusivity Concentration-Dependent II,
//// Australian Journal of Physics, 10(2), p 29-42.
************************************************************************
/
GRID NODEs 1 by 201
COORdinate Y MINImum = -15.0, max = 0.0
GRAV 0., -7.57E-5 cm/hr/hr
/
ROCK POROsity = 0.371
HYDRaulic properties: SS=1.0E-7, K = 3*0.04428 cm/hr
/
SET P = -601.76 everywhere
BOUNdary conditions for P: index Y+, value = -1.
/
MULTiphase: LOGArithmic relation n=4, alf=1, c=739, sr=0.
MULTiphase CONDuctivity: INVErse law n=1.77, alf=1, c=124.6
/
PROP using GEOMetric option
/
CONVergence criteria: 1.E-5, iterations = 200
/
Diagnostic node 2,100 print every 100 steps
DEBUG GEOMETRY OFF
SELEct window (2,1) to (2,999) interval (1,5)
OUTPut for SELEcted window IN NARRow mode
/
SOLVE until time = 2.0 hrs IN steps of 0.01, factor = 1.001, max = .1
/
SAVE on 'V8.ARC' P only
/
END
/

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************************************************************************
TITLe CASE V.09 - Infiltration from a Line Source to a Water Table
************************************************************************
//// Warrick, A.W. and D.O. Lomen, 1977. Flow from a Line Source above
//// a Shallow Water Table, Soil Sci. Soc. Am. J., 41, p. 849-852.
************************************************************************
/
GRID NODEs 63 by 63
COORdinate X: range 61
COORdinate Y: range 122
/
GRAVity 0, -1 normalized values
/
PROPerty mode for P GEOMetric
HYDRaulic properties ss = 0., kx = 0.00112 ky = 0.00112 cm_per_sec
/
MULTiphase: EXPOnential n=1, alpha=0.1258
MULTiphase CONDuctivity; EXPOnential n=1, alpha=0.1258
/
SET P as a LINEar function: 0 -0.5 Y
BOUNdary for P ib = X- FLUX = 0 $ No flow at left boundary
BOUNdary for P ib = X+ FLUX = 0 $ No flow at right boundary
BOUNdary for P ib = Y- value = 0 $ water table
BOUNdary for P ib = Y+ FLUX = 0.
LOCAte (2,999) source at the top
SOURce for P = 0.000525 in SELEcted zone
DIAGnostic node at (20,62) every 100 steps
DEBUG GEOMETRY OFF
/
SOLVe in STEAdy mode maximum steps 800 minimum 800
SELEct (1,1) to (999,999) interval (3,3)
OUTPut U, V, P in SELEcted region in NARRow mode
SAVE P on 'V9.ARC'
/
END
/

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************************************************************************
TITLe CASE V.10 - Transient Free-Surface Boussinesq Flow - Recharge
************************************************************************
//// Polubarinova-Kochina, P.Ya., 1954. Theory of Groundwater Movement,
//// Translated from Russian to English by J.M. Roger de Weist, 1962,
//// Princeton University Press, N.J.
************************************************************************
/
PROB WITH FREE SURFACE
/
GRID NODEs 44 BY 23
COOR X: MIN=0 MAX=200 ratio=1.1
COOR Y values at NODEs are:
0. 2.0 4.0 5.5 7.0 8.0 8.5 9.0 9.25 9.5
9.7 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7
10.8 10.9 11.0
/
ROCK POROsity = 0.25
HYDRAULIC S=0., Kx=0.1, Ky=1.0
/
SET H = 10 everywhere
BOUNdary X- for H = 11
BOUNDARY FOR P AT Y- FLUX = 0 $ No-flow bottom boundary
BOUNDARY FOR P AT Y+ FLUX = 0 $ No-flow top boundary
/
CONVergence for FLOW as a reference: 1.E-10, 25 iter
/
DIAGNOSITC NODE AT (2,6) every 100 steps
DEBUG GEOMETRY OFF
SAVE H on file 'V10.ARC'
SELEct (1,1) to (999,999) interval (2,2)
OUTPut U, V, P H, and S in SELEcted region in NARRow mode
/
SOLVE for 9 days dt_iniital=0.02, increase_fac=1.01 dt_max = 1.0
SAVE NOW
SOLVe for 27 days
SAVE NOW
SOLVe for 45 days
SAVE NOW
SOLVe for 63 days
SAVE NOW
SOLVe for 81 days
SAVE NOW
SOLVe for 99 days
/
END
/

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************************************************************************
TITLe CASE V.11 - Transient Free-Surface Boussinesq Flow - Seepage
************************************************************************
//// Polubarinova-Kochina, P.Ya., 1954. Theory of Groundwater Movement,
//// Translated from Russian to English by J.M. Roger de Weist, 1962,
//// Princeton University Press, N.J.
************************************************************************
/
PROBlem with FREE SURFace
/
GRID NODEs 44 BY 23
COOR X: MIN=0 MAX=200 ratio=1.1
COOR Y values at NODEs are:
0. 2.0 4.0 5.5 7.0 7.7 8.2 8.5 8.7 8.9
9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
10.0 10.1 10.2
/
ROCK POROsity = 0.25
HYDRAULIC S=0., Kx=0.1, Ky=1.0
/
SET H = 10 everywhere
BOUNdary X- for H = 9
/
BOUNDARY FOR P AT Y- FLUX = 0 $ No-flow bottom boundary
/
CONVergence for FLOW as a reference: 1.E-10, 25 iter
/
DIAGNOSITC NODE AT (2,6) every 100 steps
DEBUG GEOMETRY OFF
SAVE H on file 'V11.ARC'
SELEct (1,1) to (999,999) interval (2,2)
OUTPut U, V, P H, and S in SELEcted region in NARRow mode
/
SOLVE for 9 days dt_iniital=0.02, increase_fac=1.01 dt_max = 1.0
SAVE NOW
SOLVe for 27 days
SAVE NOW
SOLVe for 45 days
SAVE NOW
SOLVe for 63 days
SAVE NOW
SOLVe for 81 days
SAVE NOW
SOLVe for 99 days
/
END
/
/

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