PSF Fitting

Steady Radial Flow is assumed to occur in isotropic – homogeneous aquifer conditions where flow to well is to be equal (radial) in all directions. Consequently the flow to pumping well is steady which implies that the drawdown is a function of location.

- Confined Aquifers

Available equations for estimating aquifer hydraulic properties in a confined aquifer, under steady radial flow are based on the following assumptions (Kasenow, 2010);

The aquifer is confined

The aquifer has infinite aerial extent

The aquifer is homogeneous, isotropic and of uniform thickness The piezometric surface is horizontal prior to start of pumping The aquifer is pumped at a constant discharge rate

The pumping well fully penetrates the aquifer and thus receives water by horizontal flow All flow is radial towards the well and Darcy’s law is valid

Groundwater has a constant viscosity and density

It is important to use more than one piezometer during pumping test in order to avoid drawdown errors due to well losses at the abstraction well. Meanwhile, according to the assumptions earlier stated, the flow in figure 2.14 is expressed by applying Darcy’s law to derive the flow equation that relates drawdown with pumping, thus;

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𝐐𝐐=𝐀𝐀𝐀𝐀 (2.18)

Where Q = Discharge; A = Area of a cylinder(𝟐𝟐𝟐𝟐𝟐𝟐𝐊𝐊); q = Velocity of flow (−𝐊𝐊^{𝛛𝛛𝐡𝐡}_{𝛛𝛛𝟐𝟐})

Meanwhile from Darcy’s Law 𝐀𝐀=−𝐊𝐊^{𝛛𝛛𝐡𝐡}_{𝛛𝛛𝟐𝟐} (2.19)

By eliminating A and q from equation (2.18)

Gives; 𝑸𝑸= −𝟐𝟐𝟐𝟐𝒓𝒓𝟐𝟐𝟐𝟐^{𝛛𝛛𝐡𝐡}_{𝛛𝛛𝟐𝟐} (2.20)

Figure 2.14: Cross-section of a pumped confined aquifer From Figure 2.14; let h = 𝐡𝐡_𝐰𝐰 at r = 𝟐𝟐_𝐰𝐰 ; h = 𝐡𝐡_𝟏𝟏at r = 𝟐𝟐_𝟏𝟏, yields

Rearranging and Integration gives;

𝐐𝐐

𝟐𝟐𝟐𝟐𝐊𝐊𝐊𝐊∫_𝟐𝟐^𝟐𝟐_𝐰𝐰^{𝟏𝟏𝟏𝟏}_𝟐𝟐𝐝𝐝𝟐𝟐= ∫ 𝐝𝐝𝐡𝐡_𝐡𝐡^𝐡𝐡_𝐰𝐰^𝟏𝟏 (2.21)

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Thus, ^𝐐𝐐

𝟐𝟐𝟐𝟐𝐊𝐊𝐊𝐊𝐥𝐥𝐧𝐧(_𝟐𝟐^{𝟐𝟐𝟏𝟏}

𝐰𝐰) =𝐡𝐡_𝟏𝟏− 𝐡𝐡_𝐰𝐰 (2.22)

Therefore, 𝐐𝐐=𝟐𝟐𝟐𝟐𝐊𝐊𝐊𝐊(𝐡𝐡− 𝐡𝐡_𝐰𝐰)

𝐥𝐥𝐧𝐧(𝟐𝟐 𝟐𝟐�_𝐰𝐰 ) (2.23)

The equation (2.23) is known as the equilibrium or Thiem Equation and can be used to estimate transmissivity.

However, transmissivity can be estimated from Drawdown measurement from the field from the equation below;

𝐐𝐐

𝟐𝟐𝟐𝟐𝐊𝐊𝐊𝐊𝐥𝐥𝐧𝐧(^𝟐𝟐_𝟐𝟐^𝟐𝟐

𝟏𝟏) =𝐬𝐬𝟏𝟏− 𝐬𝐬𝟐𝟐 (2.24)

𝐓𝐓=𝐊𝐊𝐊𝐊 = _{𝟐𝟐𝟐𝟐(𝐬𝐬}^𝐐𝐐

𝟏𝟏−𝐬𝐬_𝟐𝟐) 𝐥𝐥𝐧𝐧(^𝟐𝟐_𝟐𝟐^𝟐𝟐

𝟏𝟏) (2.25)

Meanwhile,

K = Hydraulic conductivity

b = Aquifer thickness

𝟐𝟐_𝟏𝟏, 𝟐𝟐_𝟐𝟐 = Distances from the two respective observation wells to the pumping well

𝐡𝐡𝟏𝟏, 𝐡𝐡𝟐𝟐 = Heads of the respective observation wells 𝐬𝐬𝟏𝟏, 𝐬𝐬𝟐𝟐 = Drawdown at the respective observation wells

- Unconfined Aquifers

The basic assumptions for estimating aquifer hydraulic properties in a steady state flow to well in unconfined aquifers are the same with that of the confined aquifer except that the aquifer must be unconfined.

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Flow in figure 2.15 is also expressed by applying Darcy’s law to derive the flow equation that relates drawdown with pumping, thus from equation 2.18;

𝐐𝐐= 𝐀𝐀𝐀𝐀

Thus, from Darcy’s Law and continuity equation;

𝐐𝐐=−𝟐𝟐𝟐𝟐𝟐𝟐𝐊𝐊𝐡𝐡^{𝛛𝛛𝐡𝐡}_{𝛛𝛛𝟐𝟐} (2.26)

Figure 2.15: Cross-section of a pumped unconfined aquifer (steady-state flow) From the figure 2.15 let h = 𝐡𝐡_𝐰𝐰 at r = 𝟐𝟐_𝐰𝐰 ; h = 𝐡𝐡_𝟏𝟏at r = 𝟐𝟐_𝟏𝟏, yields

Rearranging and Integration,

𝐐𝐐

𝟐𝟐𝟐𝟐𝐊𝐊∫_𝟐𝟐^𝟐𝟐_𝐰𝐰^{𝟏𝟏𝟏𝟏}_𝟐𝟐𝐝𝐝𝟐𝟐= ∫ 𝐡𝐡𝐝𝐝𝐡𝐡_𝐡𝐡^𝐡𝐡_𝐰𝐰^𝟏𝟏 (2.27)

Thus, ^𝐐𝐐

𝟐𝟐𝟐𝟐𝐊𝐊𝐥𝐥𝐧𝐧(_𝟐𝟐^{𝟐𝟐𝟏𝟏}

𝐰𝐰) = ^{𝐡𝐡𝟏𝟏𝟐𝟐−𝐡𝐡}_𝟐𝟐 ^{𝐰𝐰𝟐𝟐} (2.28)

52 Therefore, 𝐐𝐐= 𝟐𝟐𝐊𝐊 _{𝐥𝐥𝐧𝐧(}^{𝐡𝐡𝟏𝟏𝟐𝟐}𝟐𝟐^−𝐡𝐡_𝟏𝟏 ^{𝐰𝐰𝟐𝟐}

𝟐𝟐_𝐰𝐰

� ) (2.29)

𝐐𝐐=𝟐𝟐𝐊𝐊 _{𝐥𝐥𝐧𝐧(}^{𝐡𝐡𝟐𝟐𝟐𝟐}𝟐𝟐^−𝐡𝐡𝟐𝟐�𝟐𝟐^{𝟏𝟏𝟐𝟐}_𝟏𝟏 ) (2.30)

The equation (2.30) which is identical to the Theim equation is called Dupuit Formula. This formula is used to estimate Transmissivity in unconfined aquifer.

In estimating Transmissivity (T) from the equation (2.29) establish Q as the subject of the formula;

Thus,

𝐐𝐐=^{𝟐𝟐𝐊𝐊(𝐡𝐡𝟐𝟐}_{𝐥𝐥𝐧𝐧(}^+𝐡𝐡𝟐𝟐_𝟐𝟐^{𝟏𝟏)(𝐡𝐡𝟐𝟐−𝐡𝐡𝟏𝟏)} 𝟐𝟐𝟏𝟏

� ) (2.31)

In unconfined aquifer, 𝐓𝐓=𝐊𝐊^{𝐡𝐡𝟐𝟐+𝐡𝐡}_𝟐𝟐 ^𝟏𝟏 (2.32)

Hence, 𝐊𝐊=_{𝟐𝟐(𝐡𝐡}^𝐐𝐐

𝟐𝟐𝟐𝟐−𝐡𝐡_𝟏𝟏^𝟐𝟐)𝐥𝐥𝐧𝐧(𝟐𝟐_𝟐𝟐 𝟐𝟐𝟏𝟏

� ) (2.33)

Meanwhile, Dupuit and Forchheimer assumed that;

The slope of water in pumped well in an unconfirmed aquifer is equal to the hydraulic gradient of flow.

Flow lines are horizontal and parallel to the impermeable layer.

However, in thick unconfined aquifers, drawdown (s) is negligible compared to 𝐡𝐡_𝟏𝟏, while 𝐡𝐡_𝟐𝟐+ 𝐡𝐡𝟏𝟏 is assumed to equal 2h. Therefore; (𝐡𝐡_𝟐𝟐^𝟐𝟐− 𝐡𝐡_𝟏𝟏^𝟐𝟐) = (𝐡𝐡𝟐𝟐+𝐡𝐡𝟏𝟏)( 𝐡𝐡𝟐𝟐− 𝐡𝐡𝟏𝟏) and 𝐡𝐡𝟐𝟐− 𝐡𝐡𝟏𝟏=𝐬𝐬𝟏𝟏− 𝐬𝐬𝟐𝟐. From equation (2.31); T = K𝐡𝐡𝟏𝟏 .

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In the figure 2.15; 𝐡𝐡𝟐𝟐=𝐡𝐡𝟏𝟏− 𝐬𝐬𝟐𝟐 and 𝐡𝐡𝟏𝟏=𝐡𝐡𝟏𝟏− 𝐬𝐬𝟏𝟏 , consequently, substituting these values in the equation (2.33) and multiplying both sides of the equation by 2𝐡𝐡𝟏𝟏 gives;

𝐊𝐊= ^{𝐐𝐐 𝐥𝐥𝐧𝐧(𝟐𝟐𝟐𝟐�𝟐𝟐𝟏𝟏 )}

𝟐𝟐𝟐𝟐𝐡𝐡_𝟏𝟏 ��𝐡𝐡𝟏𝟏𝟐𝟐−𝟐𝟐𝐬𝐬𝟐𝟐𝐡𝐡𝟏𝟏+𝐬𝐬𝟐𝟐 𝟐𝟐 � – �𝐡𝐡𝟏𝟏𝟐𝟐−𝟐𝟐𝐬𝐬𝟏𝟏𝐡𝐡𝟏𝟏+𝐬𝐬𝟏𝟏 𝟐𝟐 �

𝟐𝟐𝐡𝐡𝟏𝟏 �

(2.34)

Therefore Transmissivity is,

𝐓𝐓=𝐊𝐊𝐡𝐡𝟏𝟏= ^𝐐𝐐

𝟐𝟐𝟐𝟐{�𝐬𝐬𝟏𝟏−_{𝟐𝟐𝐡𝐡𝟏𝟏}^{𝐬𝐬𝟏𝟏𝟐𝟐}�−�(𝐬𝐬_𝟐𝟐−^{𝐬𝐬𝟐𝟐𝟐𝟐}_{𝟐𝟐𝐡𝐡𝟏𝟏})�}𝐥𝐥𝐧𝐧(𝟐𝟐_𝟐𝟐 𝟐𝟐𝟏𝟏

� ) (2.35)

- Leaky Aquifer

There are two distinctive methods that are widely used in the analysis of steady state drawdown data in leaky aquifers in order to determine the aquifer characteristics. The two methods are the De Glee’s method and Hantush- Jacob’s method.

- De Glee’s Method

De Glee (1930, 1951) derived the equations below based on the following assumptions; all the assumptions for steady radial flow to well conditions and the flow to well must be in steady state.

L > 3D

Thus; 𝐒𝐒_𝐦𝐦=_{𝟐𝟐𝟐𝟐𝐊𝐊𝟐𝟐}^𝐐𝐐 𝐊𝐊_𝟏𝟏(_𝐒𝐒^𝟐𝟐) (2.36)

L = √𝐊𝐊𝟐𝟐𝐜𝐜 (2.37)

Where;

𝐒𝐒_𝐦𝐦= Steady state drawdown in a piezometer from distance ‘r’ from the well (L) L = Leakage factor (L); Q = Discharge (L³/T)

c = ^𝟐𝟐′

𝐊𝐊^′ : Hydraulic resistance of the aquitard (T)

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𝐊𝐊^′ = Hydraulic conductivity of the aquitard for the vertical flow (L/T) 𝟐𝟐^′ = Saturated thickness of the aquitard (L)

𝐊𝐊_𝟏𝟏(𝐱𝐱) = Hankel function (obtained from a table)

However, after some of the variables are plotted on a log-log paper, KD can be calculated by substituting the known value of Q and the values of 𝐒𝐒_𝐦𝐦 and 𝐊𝐊_𝟏𝟏(r/L) into equation (2.35). From substituting the calculated value of KD and the values of r and r/L into equation (2.36), c can be calculated, thus;

𝐜𝐜=_{𝐊𝐊𝟐𝟐}^{𝐒𝐒𝟐𝟐} = _{(𝟐𝟐/𝐒𝐒)}^𝟏𝟏 _𝟐𝟐 =_{𝐊𝐊𝟐𝟐}^{𝟐𝟐𝟐𝟐} (2.38)

- Hantush-Jacob’s method

Hantush and Jacob (1955) modified the equation (2.36) as;

𝐒𝐒_𝐦𝐦≈^{𝟐𝟐.𝟑𝟑𝟏𝟏𝐐𝐐}_{𝟐𝟐𝟐𝟐𝐊𝐊𝟐𝟐}�𝐥𝐥𝐥𝐥𝐥𝐥𝟏𝟏.𝟏𝟏𝟐𝟐^𝐒𝐒_𝟐𝟐� (2.39)

The Hantush and Jacob’s method can be used practically if only the following assumptions and conditions are fulfilled;

i. All the assumptions for steady radial flow to well conditions ii. The flow to the well is in steady state

iii. L > 3D iv. ^𝟐𝟐_𝐒𝐒≤0.05

When 𝐒𝐒𝐦𝐦 is plotted against ‘r’ on a semi-log paper, with r on the logarithmic scale, the resultant graph will be a straight- line within the range where r/L is small. However, in the range where r/L is large, the resultant graph will be curved as the zero-drawdown axis is asymptotically approached. Thus, the drawdown difference ∆𝐒𝐒_𝐦𝐦 per log cycle of ‘r’ which is the slope of the

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straight portion of the curve (i.e., range where r/L is small) is expressed by (Hantush, 1956 and 1964),

∆𝐒𝐒_𝐦𝐦= ^{𝟐𝟐.𝟑𝟑𝟏𝟏𝐐𝐐}_{𝟐𝟐𝟐𝟐𝐊𝐊𝟐𝟐} (2.40)

Meanwhile, 𝐒𝐒𝐦𝐦= 0 and r = 𝟐𝟐𝟏𝟏 at the point of interception at the r- axis where drawdown is zero.

Thus, equation (2.39) becomes;

𝟏𝟏=^{𝟐𝟐.𝟑𝟑𝟏𝟏𝐐𝐐}_{𝟐𝟐𝟐𝟐𝐊𝐊𝟐𝟐}�𝐥𝐥𝐥𝐥𝐥𝐥𝟏𝟏.𝟏𝟏𝟐𝟐^𝐒𝐒_𝟐𝟐� (2.41)

Hence; 𝟏𝟏.𝟏𝟏𝟐𝟐_𝟐𝟐^𝐒𝐒

𝟏𝟏 =^{𝟏𝟏.𝟏𝟏𝟐𝟐}_𝟐𝟐

𝟏𝟏 √𝐊𝐊𝟐𝟐𝐜𝐜=𝟏𝟏 (2.42)

And therefore; 𝐜𝐜= ^{(𝟐𝟐𝟏𝟏/𝟏𝟏.𝟏𝟏𝟐𝟐)}_{𝐊𝐊𝟐𝟐} ^𝟐𝟐 (2.43)

2.11.2 Unsteady Radial Flow/ Non-equilibrium Well Pumping Equations

In document Beyond the Blur: Construction and Characterization of the First Autonomous AO System and an AO Survev of Magnetar Proper Motions (Page 136-144)