Numerical Simulation of Subsidence Due to Pumping Whit Hysteresis Effect Included Cruickshank 1984)

8/3/2019 Numerical Simulation of Subsidence Due to Pumping Whit Hysteresis Effect Included Cruickshank 1984)

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I ? ~ ~ c - - , t ~ V ' - ~ ~ ~ ~ ~ i z . _ , _ ; ~ ~ C - ~ ? ; NUMERICAL SIMULATION OF SUBSIDENCE DUE TO PUMPING WITH HYSTERESIS EFFECTINCLUDEDCarlos Cruickshank-VillanuevaIns t i tu te of Engineering, National University of MexicoMxico, D.F.AbstractHighly compressible impermeable, mater ials have in general dif ferent com-pressibi l i ty indexes in the effect ive stress ranges above and below thepreconsolidation s t ress ; th is i s traduced in different specif ic storage coeff ic ients .

I . Herrera and coworkers (see references) developped a numerical methodto simulate the flow in a system of aquifers separated by aquitards whichincludes a boundary solut ion for the flow in the confining layers and whichpreved to be accurate and more eff ic ient than a numerical integrat ion overa discretization grid of the aquitard. Yet, i t cannot be direc t ly appliedto cycl ic head variat ions because of the afore mentioned hysteresis ef fec t .

An approximate solution is presented here which extends the boundarymethod to such s i tua t ions . The solution was tested against f in i te d i f fe r -ences resul t s , which in general take much more computational effor t andstorage al locat ion, obtaining very good approximation in a l l s i tuat ionstes ted . I t was also tested against f ie ld measurements of subsidence withequally good coincidence.IntroductionI t is a well known property of clayey deposits to have a larger ver t i ca lcompressibili ty index when the effect ive s t ress ( to ta l minus pore waterpressure) acting on the solid st ructure is above the preconsolidations t ress (maximum preconsolidation s t ress occurred in the history of the mate r ia l ) than when i t i s below i t . This larger compressibili ty correspondsto the so called "virgin" s ta te of the deposit in which i t s interna! s t ruc-ture is weak to externa! pressures and deforms rapidly; when the ef fect ivepressure is re l ieved, the structure does not recover i t s or ig inal s ta te butyet behaves elast ical ly to changes in s t ress with a much lower s t ra in-s t ress ra t io; when the preconsolidation s t ress in surpassed, the materia lregains i t s higher compressibili ty. The compressibili ty of the materia l inthe virgin s ta te is in general one to two orders of magnitude larger thanin the preconsolidated s ta te . This is shown schematically in Fig 1 withtypical results from a drained, vert ical consolidation t es t .

This behaviour t rans la te s into geohydrological terms as a larger specific storage coeff icient when the pore water pressure drops below the hystor ica l mnimum (which, keepin g to ta l pressure constant , corresponds to amaximum effect ive ver t i ca l stress) than when i t is above i t . In aquiferswhich are in contact with compressible, low permeabili ty deposits there isan important amount of water relea sed to the aquifer from these depositswhen the pressure drops due to pumping or to a natural decline in rechargeto the aquifer. This water comes from the destro ye d interna! s t ructure ofthe fine grain materials and does not return ba ck to i t as easi ly as i twent out . This behaviour has to be taken into account in order to explainand predict water flow and land subsidence in aquifer systems consist ing of

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o.._o.....

"Oo>Q)

o>.3Log P ( consolidotion stress l

Fig 1 Typical results from drained consolidation testsperme abl e and impermeabl e l aye r s and s ub jec t to a l t e rnare pe riods of headdra1vdown and r ecovery . I t s i mpo r t ance and cons equences ar e rema rkably wellde s crib ed by Lofgr en (1978) fo r t he a quif e r below Pixle y , Calif . where ver vwell des i gned f ield meas urement s hav e been carried on.

To compute aquif er behavi our i t s now a universal practice to use two di-mensional numerical models discret izing the plan extension of the permeablelayers; th is is done so because in general the thickness of these layers issmall compared to i t s horizontal dimensions. The lower permeabili ty of theconfining layers (aquitards) makes i t possible to assume that the inducedflow in them is perpendicular to the flow in the aquifers; in this way, anobvious extension of the horizontal discre t iza t ion i s that of a ver t ica l ,one dimensional grid to compute the flow in the aqui tards, building a quasithree dimensional numerical model. With such a model, the non l inear i tyintroduced by the above de s cribed var iabi l i ty of the specif ic storage coeff i c ien t can be easi ly taken into account by means of i tera t ive solutionmethods. Nevertheless, for re la t ive ly thick aqui tards, the number of gridpoints may need to be very la rge in arder to accurately compute the flow.A method has been developped recently (Herrera et a l . , 1973, 1977, 1980,1982) which avoids the ver t i ca l discretization of the confining layers bycomputing the flow direct ly a t their boundary with the aquifers and addingi f to the mass balance of th e l a t te r , thus reducin g the dimensionality ofthe problem and the computational and storage ef for t . This is done bysolving the one dimensional flow equation in the aquitard (see Fig 2):

with boundary conditions:H(o,t)

and i n i t i a l condition:

s 1


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z.... : Aquifer 2

z=b

. T 1 ,S 1 -: . : . _ : .. . A q u fe r 1 . 'l/ / / /7 / / / T//7/7/7/7// / / / / 7//7/7// /77///7/T///T// /77//

Fig 2 Layered system consideredduct iv i ty - s ' - aquitard specif ic storage; b - aquitard thickness; t -t ime; H - p i ~ z o m e t r i c head of the lower aquifer; H2 - piezometric head ofthe upper aquifer.In thei r solut ion K1 , are assumed to be c o n ~ t a n t from Z=o to Z=b.Eqs (1) are solved and the flow at the boundary w1th each aquifer is addedto i t s mass balance; for aquifer 1 th is is expressed by:

tT (3 2H1 /3X 2+3 2H1 / 3Y2)-(K 1 /b) J 3H/3t / f [SCt-T)j dTo Tt+(K 1 /b) f lHd l t/ h [SCt-T)jdT+Q (2)

o Tand a similar equation for th e upper aquifer. In Eq (2) Q is an externa!uni t source and th e uni t re s ponse functions are:

f ( S t ) 1+2

h( Et ) 1+2

00

n=l

n=l

-n 2 1i 2 8te

where f is called the memory function and vares rapidly from inf ini ty a tthe origin, approaching the value f=l around St = 0.4; h is called the in -fluence function and vares from h=o at the origin, approaching h=l forSt=0.4; i t is for th is dimensionless time that the flow becomes establ ishedin the aquitard for a uni t head difference between the two aquifers and nowater is then realeased from aquitard 's storage; on the contrary, for veryshort dimensionless times most of the water is realeased from storage. This

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zb

0.02

0.01

o o

Numbers fo r curves indicoteo time sequence

0 .5 1.0 HHaFig 3 Heod profiles ofter o sudden recovery

behaviour of the uni t response functions helps also in fixing the conditionof accuracy in a discre t iza t ion of the aquitard i f a numerical scheme i sused to compute the ver t i ca l flow; any such scheme assumes l inear , or nearly so, pressure dis t r ibut ion between grid points , so that for each aquitardsubdivision i t s local dimensionless time interval should be equal or largerthan 0.4: 6 t / ( S ~ LlZ 2 )2. 0.4 or I'IZ2.JK 16t / (0 .4 S ~ ) J112 This condition i svery near to that of numerical s tab i l i ty of an expl ic i t f in i te differencescheme: I'IZ


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0 .10zb0.08

0.06

0.04

0.02

o

10

o 0.05

Numbers for curves indicatetime interval

0.1 HbFig 4 Heod profile going from recovery ( 10) to drowdown (11)

than the preconsolidation head profi le (s tate 1) but has had an evolut ionthrough s ta te 2 (see Fig 4).Hypothesis and solutionThese two observations lead to adventure the hypothesis tha t the speci f ics torage coeff ic ien t could be considered to be mainly determined by thes t ress s ta te of the region of the confining layer nearer to the aqui fer ,for computational purposes. In the case of head recovery the storage coeff ic ient would be the one corresponding to the higher than preconsolidationhead (s tate 2) and should remain there provided the head a t the boundarydidn ' t f a l l below the mnimum of the head his tory; i f the head should dropbelow tha t value af te r a previous recovery, then part of the deplet ionwould take place under s ta te 2 and par t of i t under s ta te l . These assumpt ions avoid the need to take the variat ion of with the ver t i ca l coordinate in the analysis , but s t i l l leaves i t s time dependence which in ' i tsturn i s dependent on the s t ress s ta te of the boundary.

The equations to solve are s t i l l Eqs (1) with a function of time;Eq ( l a ) is non l inear , but may be solved considering i t piecewise l inearbetween f in i te time in tervals and then making the in terva l tend towards zero . The solution is very s imilar to that expressed in Eq (2) for boundaryflows, with the difference that the argument of the uni t response functionsf and h is now i t se l f also an in tegra l : ins tead of 8(t-T) one gets as thetargument f 8 (T)dT

TNumerical in te grat ionThe change in the solution does not introduce any ser ious difference in thenumerical in tegrat ion previously proposed (Herrera and Yates, 1977) and is

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E 5 Heod history ot the oquiferoOo 5 10 15 20 25Q) -5::z: Time in yeors

-lO...Q)....s 60O"o Finite differences2Q) 40oo

Proposed method_;:; ou ,..-


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Heod histo ry ot the oqu ifer5 10 15 20 Time in yeors

Ee"' -10Q)I

.... 40) ::JCJo.E 20Q)O>.... o oS:::. ou . -Ul"' )( Time in yeors- Ul -20........e E; ) F inite differancas Propasad method

Fig 6 Comparison of results . Cycl ic drawdown

Dapth to piezomatric levels ot the oquifer1959 1960 1961 1962 1963 1964 Timeo

Ee

2.0.S:::.a.Q)o 40

E oee 0.5o-oa.E 1.0o X Computad with K'= 2 x O-e m/s( .)

Fig 7 Observed and computed subsidence. Aquifer at Pixley ( Lofgren ,1978)85


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f i le out of s ta te 2 into s ta te 1, contrary to the method hypothesis ; thegood agreement with f in i t e differences resul ts indicates tha t the lowerpar t of the head prof i le controls the flow, as presupposed. The secondcase (Fig 6) refers to a cycl ic head change with growing mnimum heads; theagreement there i s also good with f ini te differences resul ts except in thepeaks; th is might be due to the rounding off tendency of the implici t nu-merical scheme needed in the f in i te difference solut ion.After t es t ing the method in these extreme condi t ions i t is not surpr i s -ing tha t i t f i t s resu l t s for more slowly varying condi t ions as are rea lboundary condi t ions . h ~ a t is probably more reassuring is that i f f i t s alsof ield measurements. The method was used to compute so i l consol idat ion reg-is tered in the aquitard overlying the aquifer below Pixley, Cal i f . as re -ported by Lofgren (1978). He presents a six year corre la t ion among piezo-metric levels in the confined aquifer and compactation (consol idat ion) ofthe confining layer of lOO m thickness. Taking the water well hydrbgraphsas boundary heads, consolidation was computed with the proposed methodusing the reported aqui tard character i s t ics ; only the ver t i ca l perrneabil i tyhad to be assumed (s ince no value was reported) to reproduce the observedconsolidation evolut ion with good agreement as shown in Fig 7. An impor-tant character i s t ics of the aquitard at the Pixley locat ion i s the largera t io of elast ic to virgin deformation modulus which causes tha t head var i -ations within the e las t i c range (over preconsolidat ion head) have l i t t l eeffect on so i l consol idat ion or expansion.ConclusionsThe proposed method to compute flow and land subsidence due to pumping withcyclic head varia t ions preved to give good resul ts when compared to themore computing time and storage consuming discre t iza t ion methods. Althoughbased on a part ia l ly t rue hypothesis the method is supported by the forma-t ion of consolidated layer near to the aquifer control l ing largely the ver -t i ca l flow interchange between aquifer and aqui tard, be i t in recovery headperiods or in drawdowns a f te r recoveries. This was confirmed tes t ing themethod as described for extreme head varia t ions and in reproducing sorne lo -cal f ie ld consolidation measurements.Thus, the method i s recommended to be used in the evaluation of flow andland subsidence due to pumping in aquifers in contact with compressiblelayers and with cycl ic head variat ions, or to evaluate pump suspension pol-i c ies in those aquifers . For this purpose the solution has to be coupledwith a two dimensional model for the permeable layers as in Eq (2 ) . Thesolut ion of the coupled system of equations has to be i t e ra t ive due to thethe i r nonlinear i ty . The i terat ive procedure has also been tested showingto be rapidly convergent.Appendix. Numerical evaluat ion of convolutionsFollowing Herrera and Yates ' (1977) development with the new solut ion toEqs (1) with s a funct ion of time, the aquitard unit flow contr ibution toaquifer 1 i s giben by:

q Kt t

J (3H/ 3t )T h( ! BdT)dTb 1 o TConsider the f i r s t in tegra l only and subst i tute the expression for f(Eqs. 2):

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(3)


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tJ caH;ar)T (1+2 zo n=1

SdT 1 )dT

The f i r s t term can be evaluated easi ly between time s teps taking a mean val-ue for (aH/at)T in each interval . Now, write the second term for the nexttime:

00

SdT dTt+(lt

"'2 n=l"'dT+2 t + ~ t 2 2f (aH/at) -n n f SdTt dTT e Tn=l t

Considering S constan t during the ti:,e in terva l :

2 zn=1 J n, k+l

"'

This provides a recurrence formula s ince:

J n, k SdT dT

is known form the preceding time. I ts f i r s t value (af ter the f i r s t timein terval ) is :

(4)

The method to compute the inf ini te ser ies af ter the mentioned authors i s ,for large and intermediare times ( S ~ t > 0.005), to truncare the ser ies t o anumber of f in i te number of terms and t ranslate the res t of them to the or i -gin to maintain to ta l yield and, for shorter dimensionless times, to usethe same procedure with a change in the kernel argument (Chen and Herrera,1982). An equivalent to this l a s t approach was used here, modified towiden the appl icabi l i ty l imits of the computations to account for the greatvar iabi l i ty of the specif ic s torage coeff ic ient with time. The method i sbased on the approximation of the ser ies by an in tegral which in i t s turnis evaluated numerically by quadrature formulae. This wil l be describedelsewhere.

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ReferencesChen, B. and Herrera, I . , 1982, Nurnerical Treatrnent of Leaky Aquifers inthe Short Time Range, Water Resources Research, v. 18, no. 3, p. 557-562Herrera, I . and Rodarte, L., 1973, Integrodif ferent ia l Equations for Systerns of Leaky Aquifers and Applications. l . The Nature of Approxirnate

Theories, Water Resources Research, v. 9, no. 4, p. 995-1005Herrera, I . , and Yates, R., 1977, Integrodif ferent ia l Equations for Systernsof Leaky Aquifers and Applications. 3. A Nurnerical Method of UnlimitedApplicabil i ty , Water Resources Research, v. 13, no. 4, p. 725-732Herrera, I., Hennart, J.P. and Yates, R., 1980, A Cri t ica l Discussm ofNurnerical Models for Multiaquifer Systems, Advances in Water Resources,v. 3, no. 12, p. 159-163

Lofgren, B.E., 1978, Changes in Aquifer-System Propert ies with Ground WaterDepletion, Proceddings, International Conference on Evaluation and Predict ion of Subsidence, Pansaccola Beach, Florida, p. 26-46

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Numerical Simulation of Subsidence Due to Pumping Whit Hysteresis Effect Included Cruickshank 1984)

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