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Well Hydraulics, and Indeed of Groundwater Hydrology in General, Can Be Considered Essentially Linear and Solved Via Superposition

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Well Hydraulics, and Indeed of Groundwater Hydrology in General, Can Be Considered Essentially Linear and Solved Via Superposition ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs Unconfined aquifers figures from Kruseman and de Ridder (1991) 1 Variations from “normal” drawdown hydrographs Unconfined aquifers Early time: when pumping starts, drawdown has not reached the water table; water comes from elastic storage only (Ss) 2 Hydrology Program, Prof. J. Wilson 1 ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs Unconfined aquifers Intermediate time: now, a moving water table is being drained (Sy); there is a strong 3-D component to flow The vertical flow component means that more energy is required to get water out of the aquifer—flowpaths are longer 3 Variations from “normal” drawdown hydrographs Unconfined aquifers Late time: now, most water is coming from far away from the well; large annular area means a small drawdown will yield a lot of water Flow is now close to 1-D Jacob-type flow (or Theis flow)— but storage is still from S y 4 Hydrology Program, Prof. J. Wilson 2 ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs Unconfined aquifers Slopes are determined by T; intercepts are determined by S t t o early o late S = 10-4 S = 10-1 base figure from Kruseman and de Ridder (1991) 5 Variations from “normal” drawdown hydrographs Leakage Flow through aquitards is common An aquifer receiving leakage is a “semi-confined aquifer” or [“leaky aquifer”] 6 Hydrology Program, Prof. J. Wilson 3 ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs Leakage figures from Kruseman and de Ridder (1991) 7 Variations from “normal” drawdown hydrographs Partial penetration (Dawson and Istok, 1991) 8 Hydrology Program, Prof. J. Wilson 4 ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs Partial penetration (Kruseman and de Ridder, 1991) 9 Variations from “normal” drawdown hydrographs Large diameter wells The specific yield (even in a confined aquifer, a pumping well must be open to the atmosphere) of a well bore is essentially 1; this high storage term results in ‘well bore storage’ effects—the water produced in early time is coming mainly from stored water in the well bore, not inflow to the well bore from the aquifer If T is low, S is very low, or Q is very high, even a “small” well bore can be “large” At low T, well bores above ~2 cm can show storage effects; at high T, a well might have to be over 0.5 m in diameter before storage effects are observed 10 Hydrology Program, Prof. J. Wilson 5 ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs Large diameter wells (Kruseman and de Ridder, 1991) 11 Variations from “normal” drawdown hydrographs Boundary effects (Kruseman and de Ridder, 1991) 12 Hydrology Program, Prof. J. Wilson 6 ERTH403/HYD503, NM Tech Fall 2006 Variations from “normal” drawdown hydrographs In-well pumping tests It’s best to avoid using data collected in the pumping well •can be difficult to get measuring instruments in •data can be bad •could damage pump or measuring equipment •permeability adjacent to the well bore can be very different than in the bulk of the aquifer •bentonite clay “skin” of low permeability •gravel pack •well screen 13 Variations from “normal” drawdown hydrographs Skin effects skin or poor screen gravel pack or very well developed well base figure from Kruseman and de Ridder (1991) Can get T (slope is fine), but estimates of S will be off—you should never try to estimate S from a pumping well 14 Hydrology Program, Prof. J. Wilson 7 ERTH403/HYD503, NM Tech Fall 2006 Superposition • Today – Superposition – Image Well Theory – Well Test Analysis Charles Harvey, MIT 15 Motivation: How will the aquifer How can we use the answers respond … to these questions to interpret well tests for model diagnostics and parameter – to pumping at a different rate estimation? from the same well? For example: – to injection at a constant rate through the same well? – Variable pumping rates • especially “Recovery Tests” – if the pumping rate varies in time? – Effects of aquifer boundaries • Especially lateral stream and barrier boundaries – If there are two or more wells pumping or injecting at different locations? 16 Hydrology Program, Prof. J. Wilson 8 ERTH403/HYD503, NM Tech Fall 2006 Linearity • The answers to these and other questions relies on the concept of linearity – We “superpose” the effects of each event or well separately, and then – sum their effects. • Many problems of well hydraulics, and indeed of groundwater hydrology in general, can be considered essentially linear and solved via superposition. • It’s a very useful concept – But should be tested to see if the system is actually behaving approximately linear. – Testing linearity is not a subject of this class. 17 Reminder: Linear Models Recall the concept of an operator L( ) Given '( ) T ' '( ) eg, L( ) S &r # = ( $ ! – constants a & b 't r 'r % 'r " – states (dependent variables) s1 & s2 then an operator L( ) is linear iff L(a s1 ) = a L(s1 ) L(s1 + s2 ) = L(s1 ) + L(s2 ) L(a s1 + b s2 ) = a L(s1 ) + b L(s2 ) 18 Hydrology Program, Prof. J. Wilson 9 ERTH403/HYD503, NM Tech Fall 2006 Example application of Linear Model Rules Let 3 – s1 & s2 be the drawdowns due to unit (e.g. 1 m /d) pumping at each of two wells, – Let constants a & b, respectively, represent the multiplier to find the actual pumping rate at each well. Then the total drawdown … r1 due to pumping rate a at well 1 is s = a s1 r2 Pumping well 1 due to unit pumping at the two wells is s = s1 + s2 Pumping well 2 Point of interest due to arbitrary pumping at the two wells is s = a s1 + b s2 19 Model Define si as the drawdown caused Due to linearity: by a single well, well index i, – say due to a “unit” pumping rate 3 Drawdown due to arbitrary (eg, Q=Qu=1m /d) – is sometimes called the influence pumping at two wells is or response function. s = a s1 + b s2 Let’s illustrate how we use Q1 Q2 linearity to answer our = W (u1 ) + W (u2 ) questionsfor well hydraulics 4!T 4!T aQ bQ – where the response of an aquifer = u W (u ) + u W (u ) to pumping by a well at a constant 4!T 1 4!T 2 rate is described by the Theis model, or its Cooper-Jacob logarithmic approximation. r1 Most of what follows can be generalized to other aquifer/well where : 2 2 models; you can find this in your r1 S r2 S u1 = ; u2 = r text and various references. 4Tt 4Tt 2 Pumping well 1 r1, r2 = distance to Pumping well 2 wells 1 & 2, respectively Point of interest 20 Hydrology Program, Prof. J. Wilson 10 ERTH403/HYD503, NM Tech Fall 2006 Answers to questions: How will the aquifer respond … – to pumping at a different aQ rate from the same well? s = u W (u) 4!T We’ll take a look at a simple version of this – to injection at a constant $ Q problem, step changes in rate through the same s = u W (u) pumping, not involving a 4 T convolution integral. well? ! r 2 S #W ( ) – if the pumping rate varies 1 t 4T (t $" ) s = Q(" ) d" in time? 4!T %0 #t – If there are two or more aQu bQu wells pumping or injecting s = W (u1 ) + W (u2 ) at different locations? 4!T 4!T 21 Varying pumping rate Can be handled by superposing pumping and injection in our well so as to recreate the desired pumping history. Q In practice assume pumping history is Q2 piecewise constant, i.e. step changes in pumping rate. Q1 0 Suppose, e.g.: 0 t1 t Q = 0, t<0, 1 1 0 Q = Q , 0 <t <t , pumping period 1 2 1 Q = Q , t <t , pumping period 2 s where Q1 and Q2 are, respectively, constant pumping rates during the two pumping periods. Then superpose drawdowns. This is most 0 t1 clearly seen in the simple case of a t recovery test (next slide). 22 Hydrology Program, Prof. J. Wilson 11 ERTH403/HYD503, NM Tech Fall 2006 Well Test Analysis: What happens when we shut off a How can we use the well that has been pumping? answers to these Does drawdown recover instantly? questions to interpret well tests for model Q diagnostics and parameter estimation? 0 t For example: – Variable pumping rates 0 especially “Recovery Tests” ? – Effects of aquifer boundaries s especially lateral stream and barrier boundaries t 23 For a realistic recoveryR model,ec weo useve superposition.ry Test Pretend the well keeps pumping at the same constant rate +Q, but, at the time it shuts off add an imaginary well injecting water into the aquifer at constant rate –Q at the same location; use superposition to add the results. Because Q + -Q = 0 for t’>0, we are modeling the situation where then Q=0. 24 (Freeze and Cherry, 1979) Hydrology Program, Prof. J. Wilson 12 ERTH403/HYD503, NM Tech Fall 2006 Recovery Test s t’ -s Harvey, MIT 25 Recovery Test Recovery tests can be important, as there is no problem maintaining a constant pumping rate. Using semilog model, the forward model is: Q & 2.25 T t # Time 0 < t s = ln$ 2 ! 4 ' T % r S " Time 0 < t’ Superposing solutions: Q & 2.25 T t 2.25 T t' # s s s ln, ) ln, ) total = pumping + "injection" = $ * 2 ' - * 2 '! 4 . T % + r S ( + r S (" Q & t # stotal = ln$ ! 4 ' T % t' " t: time since pumping started t’: time since pumping stopped 26 Hydrology Program, Prof.
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