Yhd 12 3105 subsurface hydrology
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Yhd-12.3105 Subsurface Hydrology. Transient Flow. Teemu Kokkonen. Email : [email protected] Tel. 09-470 23838 Room :272 (Tietotie 1 E). Water Engineering Department of Civil and Environmental Engineering Aalto University School of Engineering. Transient flow.

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Yhd 12 3105 subsurface hydrology

Yhd-12.3105 SubsurfaceHydrology

TransientFlow

Teemu Kokkonen

Email: [email protected]

Tel. 09-470 23838

Room:272 (Tietotie 1 E)

Water Engineering

Department of Civil and Environmental Engineering

Aalto UniversitySchool of Engineering


Transient flow

Transientflow

t = 30 min

t = 10 min

t = 5 min

  • In a transient flow problem the hydraulic head values change with time

t = 30 s

t = 10 s

Pumping well

Q

t = 1 d

t = 0

t = 1 h

t = 2 h

t = 12 h


Storage

Storage

t = 30 min

t = 10 min

t = 5 min

  • To be able to describe transient flow we need the concept of storage to account for changes in the amount of water in a control volume

t = 30 s

t = 10 s

Pumping well

Q

t = 1 d

t = 0

t = 1 h

t = 2 h

t = 12 h


Groundwater flow equation transient 3d

Step 1.Write the equation for the specificstorativity.

GroundwaterFlowEquationTransient 3D

Specificstorativity S0

volume of wateradded to storage, per unitvolume and per unitrise in hydraulichead

Step 2.Usespecificstorativity to write the change in the volume of water per unitvolume and per unittime.

Step 3.Complete the equationshownbelow (i.e. replace the questionmarkwithsomethingelse).


Groundwater flow equation transient 2d

GroundwaterFlowEquationTransient 2D

Storagecoefficient S

volume of wateradded to storage, per unitarea and per unitrise in hydraulichead


Groundwater flow equation transient 2d1

GroundwaterFlowEquationTransient 2D

How is the aboveequationsimplifiedwhen the aquifer is homogeneous and isotropic?

CanTchangewithtime? When?


Storage1

Storage

  • Unconfinedaquifer

    • The ratiobetween air and water in the pore spacechanges (i.e. air is displacewithwater, orviceversa)

  • Confinedaquifer

    • Mineralgrains of the soilmatrixarereorganisedaffecting the porosity of the aquifer

      • When the aquifercompressesmoremineralgrainsaresqueezed into a controlvolume and somewater is displaced out of it

    • Water is to someextentcompressible

  • An unconfinedaquiferhas a muchgreatercapacity to store and release waterthan a confinedaquifer


Numerical solution transient flow

t-1

t

t+1

NumericalSolution – TransientFlow

time

  • Thisfarwehavenumericallyestimatedspatialderivativesappearing in the groundwaterequations – nowwealsoneed to dealwith a temporalderivative

Forwarddifference

Superscripts (t and t +1) refer to the serialnumber of the time-step


Numerical solution transient flow1

t-1

t

t+1

NumericalSolution – TransientFlow

time

”old” time-stepused in computingspatialderivatives

x

y

Write the numericalapproximation for the aboveequation.


Numerical solution transient flow2

NumericalSolution – TransientFlow

  • Expressing the spatialderivativesusingonly ”old” hydraulicheadvaluesallows for leaving just oneunknown ”new” hydraulicheadvalue on the left-hand side of the equation and solvingitsvaluebased on known ”old” hydraulicheadvalues=> explicitsolution

    • Easy to constructbutcanlead to numericalproblems (instability)

  • Expressing the spatialderivativesusing ”new” hydraulicheadvaluesleads to an implicitsolution

    • Numericallymorestablethen the explicitsolutionbutmoretedious to solve (requiresitaration)

    • Often a weightedaverage of derivativeswith ”old” and ”new” headvalues is used


Forward problem vs inverse problem

ForwardProblem vs. InverseProblem

  • In a forwardproblem the hydraulicparameters (e.g. transimssivity) of an aquiferareknown and the problem is to simulate the hydraulicheadvalues

  • In an inverseproblemsomehydraulicheadvaluesareknown and the problem is to estimatevalues of hydraulicparameters

    • Scale problem: hydraulic properties estimated from small-sized soil samples may represent poorly hydraulic properties at a larger scale


Well posed problem vs ill posed problem

Well-PosedProblem vs. Ill-posedProblem

  • In a well-posedproblem the followingcriteriaaremet

    • Existence: a solutionexists

    • Uniqueness: there is onlyonesolution

    • Stability: a change in the solution is smallif a change in hydraulicparameters is small

  • As opposed to the well-posedproblem, in an ill-posedproblem the abovecriteriaarenotmet


Inverse problem

InverseProblem

  • Existence of solution

    • A mathematicalmodel is alwaysonly an approximation of the naturalsystemitdescribes

    • Measuredhydraulicheadvaluesarenoterror-free

    • For thesereasons the inverseproblemdoesnothave an exactsolution

    • The lack of exactsolution is notreally a problem

  • Uniqueness of the solution

    • k * 0 = 0 => k = ?

    • 2a + 3b = 10 => a = ? b = ?

    • Differenthydrogeologicalconditionscanresult in similarhydraulicheadvalues => the inversegroundwaterproblemoftendoesnothave a uniquesolution

  • Stability

    • The inversegroundwaterproblemoftensuffersfrominstabilityproblems


Pumping tests

Pumpingtests

  • Pumpingtestsareused ”stimulate” the aquifer to getdrawdown data wherethere is something happening in the aquifer and whichgives a betterbasis to estimatehydrogeologicalparameters (exciting the system)

  • Typically in a pumpingtest the pumpingwell is pumped at a constantrate and the drawdownvaluesarerecorded as a function of time in observationwells at differentdistances

  • The pumpingtestresultscanbeinterpretedeitherusinginversemodelling, orwhensomesimplifiedassumptionsarevalidusinganalyticalsolutions


Theis method

TheisMethod

  • Theis (1935) constructed a method to estimatetransmissivity and storagecoefficient of an aquiferfrompumpingtestresults

  • Assumptions

    • Homogeneous, isotropic, confined aquifer

    • The well is fullypenetrating

    • A constantpumpingrate

    • Infinitearealextent

    • Horizontalflow

    • Impermeable and horizontaltop and bottom boundaries of aquifer


Theis method1

TheisMethod

H0 - H = drawdown, Q = pumping rate, T = transmissivity and W(u) on the well function, where

r = distance from pumping well, S = storage coefficient and t is time from start of pumping

Now

The graphwheredrawdown is plotted as a function of timehas the sameshape as the graphwhere W(u) is plottedagainst 1/u.


Theis method2

TheisMethod

  • Plot the well function W(u) against 1/u:ta in a log-log scale (type curve)

  • Plot the measured drawdown values H0-Hagainst time t in a log-log scale (field curve)

  • Overlay the type curve and field curve – keeping the axes parallel – in such a way that the curves overlap as closely as possible

  • Select a match point (does not need to be on the curves) and read values W(u), 1/u, H0-Hand t

  • Use the equations shown a couple of slides back to calculate values for T and S


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