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Haneberg, bill@haneberg.com, ", ButtonBox["www.haneberg.com", ButtonData:>{ URL[ "http://www.haneberg.com"], None}, ButtonStyle->"Hyperlink"] }], "Subtitle", ShowCellBracket->False, CellMargins->{{Inherited, Inherited}, {5, 0}}, TextAlignment->Left, TextJustification->0, Background->RGBColor[0.996109, 0.996109, 0.500008]], Cell["Introduction", "Section", Editable->False, CellFrame->{{0, 0}, {0, 0.5}}, ShowCellBracket->False, FontSize->24, FontColor->RGBColor[0.500008, 0.500008, 0.996109]], Cell[TextData[{ "This example uses analytical solutions from Haneberg (\"Steady-State \ Groundwater Flow across Idealized Faults,\" ", StyleBox["Water Resources Research", FontSlant->"Italic"], " 31, 1815-1820, 1995) to calculate the influence of an idealized vertical \ fault on steady-state horizontal groundwater flow. The solutions are derived \ by specifying general solutions to the 1D steady-state flow equation for each \ of three subdomains (right-hand aquifer, fault, and left-hand aquifer). \ Constants of integration were eliminated by specifying head at the left and \ right boundaries, and then requiring continuity of head and flux across each \ of the two internal boundaries. Solution derivations are included at the end \ of this notebook. \n\nAll variables are dimensionless. The horizontal axis \ shows distance from the fault (normalized to the problem domain width), and \ the vertical axis shows normalized hydraulic head changed (normalized to the \ total head drop across the problem domain). " }], "Text", CellFrame->{{0, 0}, {0.5, 0}}, ShowCellBracket->False, FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ "Evaluate the ", StyleBox["FaultFlowPlot", FontFamily->"Courier"], " Function" }], "Section", CellDingbat->"\[FilledSquare]"], Cell[CellGroupData[{ Cell[TextData[{ "Syntax is ", StyleBox["FaultFlowPlot[TL, TR, Lf]", FontFamily->"Courier"], ", where ", StyleBox["TL", FontFamily->"Courier"], " is the transmissivity of the left-hand aquifer relative to that of the \ fault, ", StyleBox["TR", FontFamily->"Courier"], " is the transmissivity of the right-hand aquifer relative to that of the \ fault, and ", StyleBox["Lf", FontFamily->"Courier"], " is the width of the fault relative to the total width of the problem \ domain." }], "Text", CellDingbat->None, FontWeight->"Normal", Background->RGBColor[0.996109, 0.996109, 0.500008]], Cell[BoxData[{ \(HL[TL_, TR_, Lf_] := \(\(-TL\) + Lf\ TL + TR + Lf\ TR - 2\ Lf\ TL\ TR + 4\ TR\ x\)\ \/\(2\ \((TL - Lf\ TL + TR - Lf\ TR + 2\ Lf\ TL\ TR)\)\)\), "\n", \(HF[TL_, TR_, Lf_] := \(\(-TL\) + Lf\ TL + TR - Lf\ TR + 4\ TL\ TR\ x\)\/\(2\ \((TL \ - Lf\ TL + TR - Lf\ TR + 2\ Lf\ TL\ TR)\)\)\), "\n", \(HR[TL_, TR_, Lf_] := \(\(-TL\) - Lf\ TL + TR - Lf\ TR + 2\ Lf\ TL\ TR + 4\ TL\ x\)\ \/\(2\ \((TL - Lf\ TL + TR - Lf\ TR + 2\ Lf\ TL\ TR)\)\)\), "\n", \(FaultFlowPlot[TL_, TR_, LF_] := Block[{p1, p2, p3, x}, p1 = Plot[HL[TL, TR, LF], {x, \(-0.5\), \(-\(LF\/2\)\)}, DisplayFunction \[Rule] Identity]; p2 = Plot[HF[TL, TR, LF], {x, \(-\(LF\/2\)\), LF\/2}, DisplayFunction \[Rule] Identity]; p3 = Plot[HR[TL, TR, LF], {x, LF\/2, 0.5}, DisplayFunction \[Rule] Identity]; Show[Graphics[{GrayLevel[0. ], Dashing[{0.02}], Line[{{\(-0.5\), \(-0.5\)}, {0.5, 0.5}}]}], p1, p2, p3, AspectRatio \[Rule] 1, DefaultFont \[Rule] {"\", 12}, Frame \[Rule] True, FrameTicks \[Rule] {{\(-0.5\), \(-0.25\), 0, 0.25, 0.5}, {\(-0.5\), \(-0.25\), 0, 0.25, 0.5}, None, None}, PlotRange \[Rule] {{\(-0.51\), 0.51}, {\(-0.51\), 0.51}}, DisplayFunction \[Rule] $DisplayFunction]]\)}], "Input"], Cell[" Create a Plot", "Section", CellDingbat->"\[FilledSquare]"], Cell[CellGroupData[{ Cell["\<\ In this case, the left-hand aquifer transmissivity is 10 times that \ of the fault and the right-hand aquifer transmissivity is 50 times that of \ the fault. The width of the fault relative to the problem domain is 0.05. The \ solid line shows the calculated hydraulic head profile across the fault, and \ the dashed line shows the unperturbed hydraulic gradient that would have \ existed without the fault. As discussed in Haneberg (1995), this very simple \ model accounts exceptionally well for water-level changes observed across \ faults in the Albuquerque basin.\ \>", "Text", CellDingbat->None, FontWeight->"Normal", Background->RGBColor[0.996109, 0.996109, 0.500008]], Cell[CellGroupData[{ Cell[BoxData[ \(FaultPlot = FaultFlowPlot[10, 50, 0.05]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Helvetica %%IncludeFont: Helvetica /Helvetica findfont 12 scalefont setfont % Scaling calculations 0.5 0.980392 0.5 0.980392 [ [.0098 -0.0125 -14.5 -15.4375 ] [.0098 -0.0125 14.5 0 ] [.2549 -0.0125 -17.8438 -15.4375 ] [.2549 -0.0125 17.8438 0 ] [.5 -0.0125 -5.34375 -15.4375 ] [.5 -0.0125 5.34375 0 ] [.7451 -0.0125 -13.6875 -15.4375 ] [.7451 -0.0125 13.6875 0 ] [.9902 -0.0125 -10.3438 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StyleBox["c", FontSlant->"Italic"], "6", StyleBox[" ", FontSlant->"Italic"], "are the constants of integration. The governing equation for steady-state \ 1D flow in each of the three subdomains is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox[ StyleBox["d", FontSlant->"Plain"], "2"], RowBox[{ StyleBox["h", FontSlant->"Italic"], "/", SuperscriptBox[ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["x", FontSlant->"Italic"]}]], "2"]}]}], " ", "=", " ", "0"}], TraditionalForm]]], ". ", StyleBox["DSolve", FontFamily->"Courier"], " can also be used to solve the entire system of differential equations and \ boundary conditions, and it is included as an alternative at the end of this \ notebook." }], "Text", Background->RGBColor[0.996109, 0.996109, 0.500008]], Cell[CellGroupData[{ Cell[BoxData[ \(HL = c1 + c2\ x\)], "Input"], Cell[BoxData[ \(c1 + c2\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(HF = c3 + c4\ x\)], "Input"], Cell[BoxData[ \(c3 + c4\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(HR = c5 + c6\ x\)], "Input"], Cell[BoxData[ \(c5 + c6\ x\)], "Output"] }, Open ]] }, Open ]], Cell["Evaluate the Boundary Conditions", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "At the left and right boundaries ", StyleBox["x", FontSlant->"Italic"], " = +/- 1/2. Head is fixed at +/- one half of the total head drop, \ \[CapitalDelta], across the system." }], "Text", Background->RGBColor[0.996109, 0.996109, 0.500008]], Cell[CellGroupData[{ Cell[BoxData[ \(bc1 = HL == \[CapitalDelta]\/2 /. \[InvisibleSpace]x \[Rule] \ \(-\(1\/2\)\)\)], "Input"], Cell[BoxData[ \(c1 - c2\/2 == \[CapitalDelta]\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bc2 = HR == \(-\(\[CapitalDelta]\/2\)\) /. \[InvisibleSpace]x \[Rule] 1\/2\)], "Input"], Cell[BoxData[ \(c5 + c6\/2 == \(-\(\[CapitalDelta]\/2\)\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Evaluate the internal boundary conditions that link the aquifers to the \ fault by requiring continuity of head and flux across the boundaries (", StyleBox["x = +/-Lf/2", FontFamily->"Courier", FontSlant->"Italic"], "). 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