Theis equation

Details are provided in the supporting information. A solver for equation 4 was successfully implemented in Fortran and tested to be robust for numerous combinations of parameters and conditions [Müller, 2015]. Both functional forms of (equations 2 and 3) rely on the assumption of multilognormal transmissivity. Thus, the semianalytical ...

A stochastic process x(t), tϵI is a family of random variables x(t) defined in a measure space (Ω,ℱ) or in a probability space (Ω,ℱ P); here x(t) is either real valued or n-vector valued and I is an interval, usually [0,∞). Notice that x(t) is a function x(t,ω)), ωϵΩ. The function t → x(t,ϵ) is called a sample path ϵ. if a.e. sample path is continuous (right continuous), we say that …Quartz is a guide to the new global economy for people in business who are excited by change. We cover business, economics, markets, finance, technology, science, design, and fashion. Want to escape the news cycle? Try our Weekly Obsession.Theis Equations for Analyzing Pumping-Test Data. Worked Example 14-2 Use of Transmissivity and Storativity to Predict Impacts of Pumping on an Aquifer. Analysis of Pumping-Test Data: Confined Aquifers: Theis Method. Worked Example 15-1 Theis Method. Cooper and Jacob (Straight-Line) Method. Box 15-1 Cooper and Jacob Approximation to …The Theis equation is one of the most commonly used and fundamental solutions to the groundwater flow equation; it can be used to predict the transient evolution of head due to the effects of pumping one or a number of pumping wells. The Thiem equation is a solution to the steady state groundwater flow equation (Laplace's Equation) for flow to ...After presenting the Sichardt formula [36] and exposing its inconsistency with the Thiem equation [70], the solutions developed by de Glee [71] and Theis [72] are discussed: the former is used for steady flow in a leaky aquifer and the latter for transient flow in a confined aquifer.The method involves an equation based on the principle of Jacob's modified Theis equation by measuring residual drawdown data versus recovery time in the production well.

Hydrogeology 101: Theis Method Geosearch International 9.05K subscribers Subscribe 587 34K views 3 years ago Hydrogeology 101 This video is about the Theis (1935) non-steady-state method of...radial flow into well. Write down the Theis equation for the unsteady radial flow into a well in a confined aquifer. UNIT - 111 Enumerate the different methods which are used for drilling tube wells. Discuss any one of these methods in detail, (OR) 6. 8. 9. Write short notes on : (a) (b) (c) and (d) Gravel pack slotted pipe tube wells Well screens.The Theis groundwater flow equation was proposed under some conditions that do not correspond to the real world observations [107,108,111]. The Theis groundwater equation assumes that, the medium by mean of which the flow is taken place is homogeneous, there is no variation of the medium in space and time.Rearrangement of the Theis equation provides a mathematical relation for evaluating drawdown during nonequilibrium conditions and delayed yield from storage at any time after pumping begins. This relation is obtained from type curves of u plotted against u[W(u)] where u[W(u)] is equal to 1.632 × 10 −2 [(S y) r 2 s/Qt]. The only term that needs to be …The video presents the Theis (1935) non-steady-state method of pumping test analysis in confined aquifers. It shows how the Theis curve fitting method was used in the past to calculate Transmissivity and Storativity, and how a dynamic Excel spreadsheet can instead be used to determine these values based on real water level measurements.The most common equation for speed is: speed = distance / time. It can also be expressed as the time derivative of the distance traveled. Mathematically, it can be written as v = s/t, or v = (ds/dt), where speed is denoted by v, distance is...The course will cover the following topics: Darcy equation, flow nets, mass conservation, the aquifer flow equation, heterogeneity and anisotropy, storage properties, regional circulation, unsaturated flow, recharge, stream-aquifer interaction, well hydraulics, flow through fractured rock, numerical models, groundwater quality, contaminant transport …

In this paper, we investigated the non-linear Schrödinger equation (NLS) to extract optical soliton solutions by implementing the extended Sinh–Gordon equation expansion method (ShGEEM ...Jul 4, 2019 · Theis’ (and Cooper and Jacob’s approximation) and Thiem’s equations describe, respectively, the transient and steady state solutions of the groundwater flow equation for confined aquifers. Hantush and Jacob, instead, derived the transient analytical solution for leaky aquifers, while De Glee formalized the steady state solution. In the ... equation (Theis 1935). For an aquifer of infinite extent and with a constant transmissivity, ... is shown in Equation 4, and with a barrier present (v 5u) is shown by Equation 5. These equations are obtained from Equations 1 and 3 by adding an additional expression for drawdown which uses an inflow term with the same, yet negative, rate of extraction Q where[8] developed convenient equations in the forms of a series based on the Theis recovery equation using the residual drawdown data. Agarwal [9] developed a ...Theis-Cooper-Jacob Assumptions Real aquifers rarely conform to the assumptions made for Theis-Cooper-Jacob non-equilibrium analysis • Isotropic, homogeneous, uniform thickness • Fully penetrating well • Laminar flow • Flat potentiometric surface • Infinite areal extent • No recharge The failure of some or all of these …

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Alternative formulations based on the well-known de Glee and Theis equations are presented, and the contested formula that estimates the radius of influence by balancing pumping and infiltration ...The method involves an equation based on the principle of Jacob's modified Theis equation by measuring residual drawdown data versus recovery time in the production well.The Theis equation (Eq. (1) ) is used to determine the radius of pressure reduction (or cone of drawdown) caused by fluid production from a well within a fully confined aquifer. Any well which falls within this radius can be considered at risk of some degree of pressure depletion, taking into account the geological, structural and …Nonleaky Confined Aquifer. Compute well function or drawdown under transient conditions using the Theis Equation.. Compute transmissivity of a nonleaky confined aquifer under steady-state conditions using the Thiem Equation.. Unconfined Aquifer. Compute drawdown around a pumping well in an unconfined aquifer using the Theis Equation and Jacob's correction for partial dewatering.

14 Jan 2018 ... Theis type curve . A log-log plot of the well function, W(u), versus ... Cf: Theis equation; Thiem equation . Tags: geotimes. Website: GeoWord ...Figure 55 – Dupuit’s simplification mathematically approximates unconfined flow as horizontal by using the gradient –dh/dx (blue solid arrows) instead of the gradient along the flow path –dh/dL (orange dashed lines). The calculated heads and flow rates are sufficiently accurate if the slope of the water table is small. If the bottom of the unconfined aquifer is …Theis nonequilibrium equation allows determining the stor age coefficient( S) , and the transmissivity ( T) of the aquifer. The detailed descriptions about Theis eq uation and its limitations can be found in Todd and Mays 2. 1.2 Methods of solving Theis equation Though Theis equation is widely recognized, solving it is a challenging task. MethodsThe following modified Theis equation illustrates one form which the mathematical model may take. r = (2.25 KHt S 10 x) 1 2. where: ... The above equation is based on the following assumptions: (i) The injection zone is homogenous and isotropic; (ii) The injection zone has infinite area extent; (iii) The injection well penetrates the entire thickness of the injection …Theis Method The steady state or Thiem's equation is widely applicable in determining permeability. However, field conditions may be such that considerable time is required to reach steady state flow and this is sometimes considered more in less serious disadvantage of the Thiem's method.Empirical formulas to estimate the radius of influence, such as the Sichardt formula, occasionally appear in studies assessing the environmental impact of groundwater extractions. As they are inconsistent with fundamental hydrogeological principles, the term “radius of influence myth” is used by analogy with the water budget myth. Alternative …Theis Recovery Test The rise in water level after pumping test when the pump is shut down is known as residual drawdown (s’) s’ = residual drawdown S’= recovery storativity values T’ = elapsed times endingof pumping S’ and S are constant and equal , T = constant, the equation can be reduced toHydrogeology 101: Theis Method Geosearch International 9.05K subscribers Subscribe 587 34K views 3 years ago Hydrogeology 101 This video is about the Theis …Also, in the Theis equation, the approximation method of Barry et al. was employed (Eq. (4) rewritten by Vatankhah) (Barry et al., 2000; Vatankhah, 2014). This approximation can easily be used in ...May 27, 2016 · 7.3.3 Cooper–Jacob Modification of the Theis Equation. The Cooper and Jacob modification of the Theis non-equilibrium equation, also known as the ‘straight-line’ method, is perhaps the most widely used method to analyze pumping test data because of its simplicity. A semilogarithmic plot of drawdown-versus-time is prepared, with drawdown ... The method involves an equation based on the principle of Jacob's modified Theis equation by measuring residual drawdown data versus recovery time in the production well.

An exact equation, allowing for the vertical velocity-component of the flow and delayed yield from storage, is given in Appendix 3. ... In isotropic aquifers these conditions may not greatly restrict the use of the Theis equation, but in anisotropic aquifers, having much greater horizontal than vertical permeability, a well-function is required which takes …

May 27, 2016 · 7.3.3 Cooper–Jacob Modification of the Theis Equation. The Cooper and Jacob modification of the Theis non-equilibrium equation, also known as the ‘straight-line’ method, is perhaps the most widely used method to analyze pumping test data because of its simplicity. A semilogarithmic plot of drawdown-versus-time is prepared, with drawdown ... Figure 55 – Dupuit’s simplification mathematically approximates unconfined flow as horizontal by using the gradient –dh/dx (blue solid arrows) instead of the gradient along the flow path –dh/dL (orange dashed lines). The calculated heads and flow rates are sufficiently accurate if the slope of the water table is small. If the bottom of the unconfined aquifer is …drawdown equations. A simple model of the hydraulic head drawdown as a mathematical boundary-value problem will be developed in Section 3. In conclusion, Section 3.1 presents an analytical solution in the form of the Theis Equation in the specific example of the drawdown in a confined aquifer of a non-steady flow.Aquifer tests are typically interpreted by using an analytical model of aquifer flow (the most fundamental being the Theis solution) to match the data observed in the real world, then assuming that the parameters from the idealized model apply to the real-world aquifer.The new fractional Theis equation solution type curves (drawdown and radius of influence) are evaluated for relationships between aquifer properties and a ...... Theis equation in Matlab. Tuba City, AZ is within the Navajo Nation with the highest population and has been experiencing water resource issues with the ...6.1 verification with the theis nonequilibrium equations 6-1 6.2 verification with 2-dimensional analytical solution, paths. 6-3 6.3 verification with hantush solution for nonsteady flow to flowing wells in leaky aquifers 6-14 7.0 sample illustrative problems . 7-1 7.1 sutter basin, california. 7-1 7.2 hypothetical waste repository in a representative salt …The Theis groundwater flow equation was proposed under some conditions that do not correspond to the real world observations [107,108,111]. The Theis groundwater equation assumes that, the medium by mean of which the flow is taken place is homogeneous, there is no variation of the medium in space and time.The Theis equations are based on the assumption that the water is released from storage in the aquifer in immediate response to the decline in the water table. In unconfined aquifers, gravity drainage may not be immediate, particularly in fine grained formations and with stratifications of slow yielding materials.

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The Theis equation has become the most widely used equation in transient groundwater hydraulics and the solution in terms of drawdown is where = drawdown at distance(r) at time (t) after the start of pumping [L] Q = discharge rate [L T] = well function of Theis Specifically where Euler`s constant = .577215664901532860606512 Conventional Units ...The drawdowns are worked out based on the Theis equation for 1, 10, 50, 100, 500 and 1000 m distances from pumping well. The calculated drawdown data versus distance from pumping well are plotted on semi-logarithmic graph paper with distance on the logarithmic scale. The best fit line plotted through the points is the configuration of the …To solve equation we need initial conditions (ICs) and boundary conditions (BCs) IC: h(r,0) = h 0 ⎛ BCs: h( ∞,t)=h 0 and lim r Æ0 ⎜ r ∂h⎞ ⎟ = Q for t > 0 ⎝ ∂r ⎠ 2πT which is just the application of Darcy’s law at the well Theis Equation – transient radial flow 1935 – C.V. Theis solves this equation (with C.I. Lubin ...Theis Equation Calculation for Aquifer Testing and Well Drawdown Last Revision: May 18, 2010 An aquifer test (or a pumping test) is conducted to evaluate an aquifer by "stimulating" the aquifer through constant pumping, and observing the aquifer's "response" (drawdown) in observation wells.HA/648 Porosity is high materials . in one (3) of the following 6. About 80% of the rainfall contributes from 2. 3. 7. the (1) (2) (3) Northwest monsoonThe Theis equation (Eq. (1) ) is used to determine the radius of pressure reduction (or cone of drawdown) caused by fluid production from a well within a fully confined aquifer. Any well which falls within this radius can be considered at risk of some degree of pressure depletion, taking into account the geological, structural and …installations, Theis was able to derive an equation for optimum well spacing. In many situations, however, economics may be of secondary importance, or other considerations may be involved. One or more of the following factors may be of importance in determining the proper spacing of pumped wells: (a) physical limita­Jan 7, 2022 · equation [70], the solutions developed by de Glee [71] and Theis [72] are discussed: the former is used for steady flow in a leaky aquifer and the latter for transient flow in a confined aquifer. Based on these solutions, alternative formulas for the radius of influence are presented. Then the Theis equation can be applied to an unconfined aquifer with storage coefficient for confined aquifer replaced by specific yield, S y, s Q T wu u rS Tt ==y 44 2 π (), This approach is reasonable only when the drawdown is significantly smaller than the saturated thickness, that is, Approach #2: (Concept of delayed yield) T ≈Kb s ≤bJan 7, 2022 · equation [70], the solutions developed by de Glee [71] and Theis [72] are discussed: the former is used for steady flow in a leaky aquifer and the latter for transient flow in a confined aquifer. Based on these solutions, alternative formulas for the radius of influence are presented. Theis, C. V.(1935). “The relation between lowering of piezometric surface and the rate and duration of discharge of a well using ground water storage.” Trans., Am. Geophys.Jul 4, 2019 · Theis’ (and Cooper and Jacob’s approximation) and Thiem’s equations describe, respectively, the transient and steady state solutions of the groundwater flow equation for confined aquifers. Hantush and Jacob, instead, derived the transient analytical solution for leaky aquifers, while De Glee formalized the steady state solution. In the ... ….

30 Nov 2022 ... Summary of formulas for calculation of influence radius. Number. Name. Formula. Aquifer type. Theoretical. 3. Theis. = √.An exact equation, allowing for the vertical velocity-component of the flow and delayed yield from storage, is given in Appendix 3. ... In isotropic aquifers these conditions may not greatly restrict the use of the Theis equation, but in anisotropic aquifers, having much greater horizontal than vertical permeability, a well-function is required which takes …Theis equations, distance-drawdown, and time-drawdown are three methods commonly employed to determine S and T. To simplify things here we will focus on the distance-drawdown method. Distance-Drawdown Developed by Cooper and Jacob, the distance drawdown method is a modification of the Theis equation. The equationTheis Equation Calculation for Aquifer Testing and Well Drawdown Last Revision: May 18, 2010 An aquifer test (or a pumping test) is conducted to evaluate an aquifer by "stimulating" the aquifer through constant pumping, and observing the aquifer's "response" (drawdown) in observation wells.The first term (aquifer loss) on the right hand side of Eqn. (13.2) can be calculated by using Thiem equation (if the well drawdown is steady) or Theis equation (if the well drawdown is unsteady). However, the second term (well loss) can be determined by step-drawdown test (described later in this lesson) or can be estimated from distance-drawdown data.The Theis Equation. The Theis equation (Theis, 1935) describes radial confined groundwater flow in a uniformly thick horizontal, homogeneous, isotropic aquifer of infinite areal extent. S = (Q / 4πT) ∫ (r 2 s/4Tt) ∞ [(e-u /u) du] (1) The radius of the pumped well is assumed negligible (line source or sink approximation).Anthony Lane reviews Martin Scorsese’s “Killers of the Flower Moon,” an epic drama about the Osage murders, starring Leonardo DiCaprio, Robert De Niro, and Lily …May 18, 2010 · Theis Equation Calculation for Aquifer Testing and Well Drawdown Last Revision: May 18, 2010 An aquifer test (or a pumping test) is conducted to evaluate an aquifer by "stimulating" the aquifer through constant pumping, and observing the aquifer's "response" (drawdown) in observation wells. The new approximation (equation 6) differs from that of Agarwal (equation 1) as the delay in the aquifer response after pumping shutdown is taken into consideration. In fact, instead of a constant value s ( t p ) , an extrapolated function s ap ( t ) has been considered to evaluate the pumping test drawdown that would have occurred at t > t p if …the Theis equation. This spreadsheet is not designed to esti­ mate aquifer parameters. It can be used only in forward mode and may be helpful in planning aquifer tests. All the spreadsheets are designed for single-well tests except for one that is based on the distance-drawdown method for analysis of multiwell test data once the draw­ Theis equation, Jan 7, 2022 · Because of the low computational cost, axisymmetric single- and multi-layer models are well-suited to perform these kinds of calculations. Using the de Glee Equation (6) [71,86] and the Theis Equation (9) , respectively, it is even possible to determine the maximum radius of influence R max, which is independent of the aquifer transmissivity KD: , Then the Theis equation can be applied to an unconfined aquifer with storage coefficient for confined aquifer replaced by specific yield, S y, s Q T wu u rS Tt ==y 44 2 π (), This approach is reasonable only when the drawdown is significantly smaller than the saturated thickness, that is, Approach #2: (Concept of delayed yield) T ≈Kb s ≤b, 23 Des 2009 ... The Theis (1935) equation, which is a solution of the partial differential equation (pde) describing the drawdown caused by radial flow to a ..., ante equation with Darcy's flow equation. The assump tions implicit in the ... from the Theis solution would be substantial. We wish to know the pressure ..., ... equation look like? a) How can you calculate drawdown according to Theis? b) What does W(u) mean in the theis equation? a) drawdown as a function of ..., Solve equation (6) for K: K = Q 2πb (h-hw) ln r rw (7) If we drill two observation wells at some distance from the pumping well, we can get measurements (h1,r1) and (h2,r2). r 1 r 2 h1 h2 well 1 well 2 Since h and ln r have a straight-line relation, we can substitute h1 and h2 for h and hw, and r1 and r2 for r and rw. That is: K = Q 2πb (h2 ..., Question 8.What is the Theis Equation for Unsteady flow into a well fully penetrating a confined aquifer? Elaborate on the Cooper-Jacob approximation for solving this equation and to estimate the transmissivity (T) , storativity (S)., Abstract EN. The presented study is tried to analyze the main difference between the application of Theim (1906) and Thies (1935) equations for steady and non- steady conditions to determine the property of layer such as (Transmissivity , Specific yield and storativity). Thies and Theim methods can be applied for pumping test, using a single ..., The course will cover the following topics: Darcy equation, flow nets, mass conservation, the aquifer flow equation, heterogeneity and anisotropy, storage properties, regional circulation, unsaturated flow, recharge, stream-aquifer interaction, well hydraulics, flow through fractured rock, numerical models, groundwater quality, contaminant transport …, An exact equation, allowing for the vertical velocity-component of the flow and delayed yield from storage, is given in Appendix 3. ... In isotropic aquifers these conditions may not greatly restrict the use of the Theis equation, but in anisotropic aquifers, having much greater horizontal than vertical permeability, a well-function is required which takes …, Aug 9, 2021 · The Theis equation is an important mathematical model used for analyzing drawdown data obtained from pumping tests to estimate aquifer parameters. Since the Theis model is a nonlinear equation, a ... , Jun 18, 2012 · The Theis Equation. The Theis equation (Theis, 1935) describes radial confined groundwater flow in a uniformly thick horizontal, homogeneous, isotropic aquifer of infinite areal extent. S = (Q / 4πT) ∫ (r 2 s/4Tt) ∞ [(e-u /u) du] (1) The radius of the pumped well is assumed negligible (line source or sink approximation). , The Cooper-Jacob method is a simplification of the Theis method that is valid for greater time values and smaller distances from the pumping well (i.e. smaller values of u). This method ... to this truncation, not all early time measured data is considered to be valid for this analysis method. The resulting equation is: where: is the drawdown at the …, Charles Vernon Theis was the first to develop mathematical expressions that allow hydrogeologists to determine the characteristics of an aquifer and predict how water levels in the aquifer change during pumping. Early Work and the Theis Equation. Theis was born on March 27, 1900 in Newport, Kentucky. A very intelligent child, Theis graduated ... , The following modified Theis equation illustrates one form which the mathematical model may take. where: r = Radius of endangering influence from injection well (length) k = Hydraulic conductivity of the injection zone (length/time) H …, Question 8.What is the Theis Equation for Unsteady flow into a well fully penetrating a confined aquifer? Elaborate on the Cooper-Jacob approximation for solving this equation and to estimate the transmissivity (T) , storativity (S)., Theis Equation; Radial Dependence; These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. Download chapter PDF ... Theis, C. V., “The Relation Between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well …, Theis nonequilibrium equation allows determining the stor age coefficient( S) , and the transmissivity ( T) of the aquifer. The detailed descriptions about Theis eq uation and its limitations can be found in Todd and Mays 2. 1.2 Methods of solving Theis equation Though Theis equation is widely recognized, solving it is a challenging task. Methods, Next: Reynolds Number Up: No Title Previous: Theis Equation. Hydraulic Conductivity The Hydraulic Conductivity is the factor of proportionality in Darcy's flow equation. It is a function of both the fluid and the porous medium through which the fluid is flowing. It can be expressed as: where K = hydraulic conductivity [LT] k = intrinsic permeability [L] g = …, ... Theis equation in Matlab. Tuba City, AZ is within the Navajo Nation with the highest population and has been experiencing water resource issues with the ..., 2.3.1 Derivative of Theis Groundwater Flow Equation. To derive the groundwater flow equation, we make use of the principle of continuity equation of the flow, that is the difference between the rate inflow and the rate outflow from annular cylinder which is the equation of change of volume of water within the annular space. , 11.2.1 Theis Equation. Theis (1935) was the pioneer in solving Eqn. (11.1) and obtained an analytical solution for the transient flow of groundwater to pumping wells tapping a confined aquifer (Fig. 11.1). For solving Eqn. (11.1), Theis made the following assumptions in addition to the basic assumptions mentioned in Lesson 10:, 公式简介. 1935年美国人C.V.泰斯（C.V.Theis）在数学家C.I.卢宾帮助下，导出定流量抽水时的单井非稳定流计算分式。. 其假设条件为：①含水层为等厚且均质各向同性而无限延伸 …, To apply the Theis recovery method given by , plot s ′ as a function of log (t / t ′) on semi-logarithmic axes and draw a straight line through the data . Determine T using the following equation: T = 2.303 Q 4 π Δ s ′ (2) where. Δ s ′ is the slope of the fitted line (change in residual drawdown per log cycle equivalent time) , The Theis equation has been widely used to study the transient movement of groundwater as a result of pumping in a confined aquifer. It is well known that the observed drawdown at early times has an obvious departure from the theoretical drawdown based on the Theis equation. The Theis equation was derived under the assumption that total stress ..., The general solution of that equation is a rather complicated differential equation which symbolically, and in analogy to the Theis equation, may be written as: (1) s = Q 4 π T W (U AB, r/B) where s = Drawdown, [L] Q = Pumping rate, [L 3 /T] T = Transmissivity of the aquifer, [L 2 / T] W(U AB,r/B) = Well-function of Boulton, …, Aug 9, 2021 · The Theis equation is an important mathematical model used for analyzing drawdown data obtained from pumping tests to estimate aquifer parameters. Since the Theis model is a nonlinear equation, a ... , Jul 4, 2019 · Theis’ (and Cooper and Jacob’s approximation) and Thiem’s equations describe, respectively, the transient and steady state solutions of the groundwater flow equation for confined aquifers. Hantush and Jacob, instead, derived the transient analytical solution for leaky aquifers, while De Glee formalized the steady state solution. In the ... , The area of review for each injection well or each field, project or area of the State shall be determined according to either paragraph (a) or (b) of this section. The Director may solicit input from the owners or operators of injection wells within the State as to which method is most appropriate for each geographic area or field. , Jun 15, 2013 · Also, in the Theis equation, the approximation method of Barry et al. was employed (Eq. (4) rewritten by Vatankhah) (Barry et al., 2000; Vatankhah, 2014). This approximation can easily be used in ... , Aug 14, 2020 · This video is about the Theis (1935) non-steady-state method of pumping test analysis in confined aquifers. We will look at how the Theis curve fitting metho... , Oct 1, 2013 · Though Theis equation is widely recognized, solving it is a challenging task. Methods have been proposed by Theis 5 , Cooper and Jacob 6 , and Chow 7 to solve Theis equation. , In this form, the equation describes a flow through a saturated anistrophic porous medium. Incorportating the two basic assumptions of essentially horizontal flow obeying Darcy's Law which reduces the equation to a two dimensional form and that the flow is in an aquifer that is homogeneous, saturated and isotropic (i.e. ), Equation 2 reduces to: