In class, we illustrated the use of flow nets to solve simple groundwater flow problems. We described flow nets as graphical versions of the solution to a boundary value problem. What is the name of the equation that we are solving graphically and where does it come from, i.e. what does the book say about its derivation? (5 points)
Flow nets are used to depict patterns of steady two-dimensional groundwater flow and are made up of two families of orthogonal lines: equipotentials and streamlines. Within the flow net, water moves from high to low hydraulic head and cannot cross a streamline. These areas called stream tubes. Flow nets are used to calculate the rates of groundwater flows. We can calculate the discharge through the squares if we separate one of the squares and use Darcy’s law of q=-K dh/dl, which we can eventually calculate the discharge in aquifers. Let say if the side of the square is ds and dm, and the thickness or the square is b, we can apply Darcy’s law to calculate the total discharge of which is Q_s=qA=K(dm b)dh/ds, where Q_s is the total discharge measured in L3/T, dh is the head difference across the square. Since the stream tubes are dominantly squares, we can say that ds=dm, which then becomes Q_s=Kbdh. This means that we can calculate the amount of water going through each stream tube if we know the hydraulic conductivity to find the flow net to figure out the interval is used for the hydraulic head and multiply that value by the transmissivity (T=Kb).
Why is the specific yield for an unconfined aquifer so much higher than the storage coefficient for a confined aquifer? Where does the water actually come from when a well pumping a confined aquifer draws down the potentiometric surface? What are the potential environmental consequences? (10 points)
The specific yield for an unconfined aquifer is so much higher than the storage coefficient for a confined aquifer because declines in head in unconfined aquifers are accompanied by drainage of a portion of the aquifer; in confined aquifers, the material is not drained and water is only released from storage due to expansion of the water and compression of the aquifer. Water and most aquifers materials are not very compressible. If water is being withdrawn from a well within a confined aquifer, the hydraulic head and the fluid pressure is being reduced at that point. As a result, the fluid will expand slightly. This is because water is slightly compressible. This is one mechanism by which water is released from storage in a confined aquifer. The additional volume of water produced may flow to the well and be withdrawn by the pump. Equation dp= -dσ_e indicates that the decrease in fluid pressure must be accompanied by an increase in effective stress (the upward stress exerted by aquifer solids). A part of the weight of the overlying materials us being transferred from the fluid to solid. This then resulted in the compression of the aquifer material, as decreasing the fluid pressure resulted in an expansion of the fluid. We can think to compress the aquifer material as squeezing a sponge, this produces water that may be pumped from the well. This mechanism by which water is removed from storage in a confined aquifer. Since the specific yield for an unconfined aquifer is so much higher than the storage coefficient for the confined aquifer, the drawdown in the unconfined aquifer will be less than the storage coefficient of the confined aquifer. This means that the volume of the cone of depression in a confined aquifer is equal to the water pumped divided by the storage coefficient. Therefore, by removing water from confined aquifers produces a substantial drawdown of the potentiometric surface. The cone of depression can be pretty huge in ca confined aquifer since it is heavily pumped over a very long period of time. Another consequence is that this affects the balance between water pressure and effective stress in aquifers and thus affecting the water levels in the wells. Atmospheric pressure changes as low pressure and high-pressure systems move across the landscape. In a confined aquifer, an increase in atmospheric pressure resulted in a decrease in the water level and a decreasing in atmospheric pressure resulted in an increase in water level. Unlike in an unconfined aquifer, changes in atmospheric pressure are transmitted equally to the water table and the water in the well, thus do not make any changes in the water levels.
Look at the flow net below and answer the following questions. The heavy solid line is the water table; the land surface is not actually shown.
Label the equipotential lines with the correct value of h.
Draw arrows on the streamlines indicating the correct direction of groundwater flow. (Formatting issues in Word may not allow you to do this in the document but you can print this out and draw them by hand and just take a picture or scan the result.)
Provide a number for the elevation of the water level in piezometers A and B.
A = 60 m, B = 80 m
Assuming that K = 10-4 ms-1, find the approximate value of q for the stream tube along the lower boundary of the flow field between x = 50 m and x = 10 m. (15 points – 4 for a, 2 each for b and c, 7 for d)
Q= Kbdh = (10-4 ms-1)(100 m)(2.5 m)= 0.025 m3s-1
What is the moisture characteristic and why is it important to understanding the behavior of water in the unsaturated zone? Which of the trends illustrated in the figure below is the moisture characteristic? Define the variables , K() and () represented by each of these curves and plotted on the x- and y-axes, and explain the physical reasons why the curves follow these trends. (The correct physical explanation provides a link between the two trends.) (20 points)
The moisture characteristic is important to understanding the behavior of water in an unsaturated zone because a suction must be applied to withdraw water from the unsaturated zone above the water table. The greater the applied suction, the more water we can withdraw and the lower soil moisture content when the soil reaches equilibrium with applied suction. The moisture characteristic is the relationship between this external suction applied to a rock and the amount of water per bulk volume (moisture content) that the rock retains against that de-watering suction. The variable is defined as the water content, the variable is defined as hydraulic conductivity measured in cm/day, and variable ( is defined as tension head measured in cm. The curve shows that, as the water content decreases, the hydraulic conductivity increases. As the water content increases, tension head increases. The hydraulic conductivity decreases rapidly as the medium becomes unsaturated. Negative pressure heads grow due to the capillary force acting on curved air-water interfaces in unsaturated soils. The gradient in hydraulic head drives flow in the unsaturated zone, just like in a saturated zone. The hydraulic head is calculated by h= φ+z , by which φ is the negative capillary pressure head. This explains the trend for the pressure head curve above. Hydraulic conductivity’s trend can be explained that mean velocity is proportional to the second power of the diameter of a cylindrical tube, therefore, the intrinsic permeability of a porous medium is proportional to the square of the pore size. From Poiseuille’s law, discharge is equal to the mean velocity times the cross-sectional area of the flow. The discharge will be proportional to the fourth power of the tube diameter. The large pores of soil become air-filled first as suction is applied to soil, move down to the smaller pores which conduct water in way lower rates that could be handled by the emptied pores. This is also the reason why the hydraulic conductivity of an unsaturated soil with much more clay might be higher than the unsaturated sandy soil.sa
In class we discussed two different approaches to describing or simulating the trend in infiltration capacity vs. time (and its relation to runoff) – there are many others (e.g. the Horton and Philip equations, to name just two others) but these are the ones we used. The equations used in each approach are listed below as items a and b.
a. Weff = (W-Vi)2/(W+0.8 Vmax) where VI = 0.2 Vmax and Vmax = 1000/CN – 10
b.
Identify each approach by name; define the variables in each; and explain how these approaches are different and what they have in common. How does each method incorporate the influence of soil characteristics on the infiltration process? How do we explain the shape of the infiltration capacity curve over time during a storm using these approaches? What assumptions are made in each approach about what is happening during the infiltration process? (20 points)
This approach is called The Soil Conservation Service or SCS method for estimating precipitation excess and runoff response during a storm. W_effis the effective rainfall, Wis the cumulative precipitation depth, V_i is the initial abstraction, V_max, is the potential abstraction measured in inches and CN is the curve number (dimensionless). The SCS procedure was empirically introduced from studies of small agriculture watersheds. The SCS method consists of selecting a storm and computing the direct runoff by the use of curves founded on field studies of the amount of measured runoff from numerous soil cover combinations. A selection of the runoff CN is depending on antecedent condition and the types of cover. Soils are classified A, B, C, or D according to these criteria:
Soils with wetted and consist of a deep well to excessively drained sands or gravels are soils with high infiltration rates. They have a high rate of water transmission which means have potential low runoff rate.
Soils with wetted and consist of moderately deep to deep, moderately well to well-drained soils with moderately fine to moderately coarse textures are soils with moderate infiltration rates. These soils have a moderate rate of water transmission.
Soils with wetted and consists of soils with a layer that impedes the downward movement of water or soils with moderately fine to fine texture are soils with slow infiltration rates. These soils have a slow rate of water transmission.
Soils with wetted and consist of mostly clay with a high swelling potential, soils with a permanent high water table and a claypan or clay layer near the surface, and shallow soils over nearly impervious material are soils with a very slow infiltration rate. These soils have a very slow rate of water transmission.
The SCS method assumes that the calculations are based on the watershed that is homogeneous in CN. If a watershed has several areas with different curve numbers, the weighted curve number can be used. If the difference in CN is more than 5, it is better to subdivide the watershed into subareas, analyze them individually, and weight their runoff values rather than the curve numbers. The CN method should only be used when CN exceeds 50 and the concentration-time is greater than 0.1 hour and less than 10 hours.
This approach is called The Green-Ampt equation. Observation of infiltration into dry soils, especially sandy soils, indicates that the water tends to progress downward as a “slug.” This means that a sharp wetting front separates the unsaturated soil below from the saturated soil below from the saturated soil above and it is this front that progresses downward as infiltration proceeds. This first equation i= -K_s (Ψ_f+L_f)/L_f , where, i is the infiltration rate also equal to the specific discharge, Ψ_fis the capillary-pressure head at the wetting front, L_f is the depth to the wetting front, and K_s is the saturated conductivity. The Green-Ampt equation presented an analysis of the infiltration problem assuming that this wetting front was infinitely sharp, that is, horizontal. In this case, the flux everywhere in the saturated upper portion must equal the infiltration rate and so the hydraulic gradient is uniform. The Green-Ampt method assumes that the soil column is homogeneous and the water table is deep in a way that the wetting front does not reach the water table during s rainfall event. In the presence of a shallow water table or of bedrock at a depth Ls, infiltration water may saturate the soil column. A ring infiltramometer is a simple device to measure infiltration rates. A metal ring is inserted into the ground and water is added to maintain a level pool of water in the ring. The amount of water required to maintain the level is the rate at which water enters the soil surface, that is, the infiltration rate. Data recorded are typically time (in a minute) and cumulative depth of water infiltrated (in millimeter).
Explain what baseflow separation is (also sometimes referred to as “hydrograph separation”); why we do it; how it is typically done; and what it actually tells us. [You might want to look up what chapter 2 of Beven has to say about this issue. And you don’t need to provide the details of all of the different separation techniques, just provide an explanation of how they generally work or use one of the typical methods as an example and draw a figure to illustrate your answer.] In class we talked about the use of tracers in the Hornberger et al. textbook and the application of the equation below:
Q_n= (C_t-C_o)/(C_n-C_o )× Q_t
Explain the meaning of this equation, how it is used, and the nature of the “paradox” that was discovered empirically once the equation was applied to real data. How might the paradox be explained? [Chapter 1 of Beven, posted in the Runoff and Streamflow folder, provides at least one possible explanation; there are others, but any potential explanation you can find can be cited.]` (20 points)
Hydrologists often wonder problems what makes up the quick flow and baseflow reservoirs? What are the characteristics of catchments that determine the relative amounts of quick flow and baseflow? Does the water in different reservoir consist of different chemical elements from their contact with different geological materials? These questions are still unanswered until now. The main difficulty is to determine the components of streamflow hydrograph since we do not measure the hydrographs of streamflow and baseflow. We measure the total flow. How can we separate the measured streamflow hydrograph? Hydrograph separation technique is the answer to this question. Baseflow separation techniques have been used by hydrologists for many years, but all the methods are empirical. The empirical methods are used to enable the calculation of quick flow from catchments. Once the hydrographs have been separated into quick flow and baseflow. A very simple computation, known as the unit hydrograph method can be used to route the portion of the precipitation that goes through the quick flow “reservoir” to the stream, a procedure that has application in engineering hydrology where the size of pipes to carry stormwater is decided on the basis of such routing calculation. One of the methods for separating hydrographs is based on differences in the chemical composition of waters in the reservoirs thought to store and release water. With this method, we assume that the streamflow during a storm event is the mixture of water that has been precipitated into the catchment during the particular storm and water that was stored in the catchment prior to the onset of the storm. Provide that the concentration of some chemical component of these two waters naturally differs and does not vary during the storm or vary spatially, measurement if the concentration of the tracer in the two components and in the mixture through time can be used to back-calculate the contribution from the old and new reservoir. This then come up with the equation above where Qt is the total measured streamflow, Qn is the flow contribution to the hydrograph associated with the new water are Qo is the flow contribution to the hydrograph associated with the old water. We assume that there are no other flow components at any time so Qt = Qo + Qn. If we assume that the concentration of both chloride in precipitation and in old water in the catchment are relatively constant, we can represent concentration as Cn and Cn for concentration in new water and old water respectively. We will have QtCt = QoCo + QnCn. Given the measurement of Qt, Ct (at several times), Cn (assume constant), and Co (assume constant), we can solve Qn (shown on the equation above). This calculation yields a hydrograph separation in terms of new and old water.
In class, we discussed the SCS approach to predicting the runoff hydrograph as well as the TOPMODEL approach. The latter is described in the Hornberger text and the former is described in other textbook chapters that are posted in the “Runoff and streamflow” folder and in the set of notes that were handed out in class. Both of these approaches are predicated on certain assumptions about how runoff is generated. (The conceptual models of runoff generation are discussed in chapter 10 of the Hornberger text, in chapters 1 and 2 from Beven, in Tarboton’s “Rainfall-runoff process,” in chapter 9 from Dunne and Leopold, and in chapter 9 from Dingman 2nd edition or chapter 10 from Dingman 3rd edition.)
Describe the major modes of runoff generation that were discussed in class, in the textbook, and/or in-class handouts and posted materials. As part of your answer, include explanations of the partial area contribution and variable source area concepts and how they are related to antecedent watershed conditions. Also, explain the circumstances under which different modes of runoff generation might occur. (20 points)
There are four modes that water precipitated onto catchment ultimately can be discharged into a stream channel. The four modes are direct precipitation onto stream channel, overland flow, shallow subsurface stormflow, and groundwater flow. The contribution direct precipitation to an active stream channel is relatively small because the surface area of a lost lasting channel system in most catchments is a small percentage of the catchment area. Measurable expansion of the channel system mostly occurs during rainy seasons for many catchments. A significant fraction of storm runoff can result from precipitation falling on this expanded channel network in upland catchments. The second process is the overland flow. The overland flow is water that flows across the ground surface and discharges into the stream channel. In order for this process to occur, the water must accumulate at the surface rather than infiltrate into the soil. Overland flow occur caused by the catchment surface may be nearly impermeable due to the presence of exposed bedrock, the infiltration rate through the pervious surface may be exceeded by the rainfall rate onto the catchment surface, causing ponding of water at the surface, and the catchment soil upon which the rainfall is precipitated may be saturated to the soil surface, which then causes pending due to the precipitated water cannot infiltrate into an already saturated soil. Overland flow can contribute into quick flow due to being one of the fastest paths that precipitated water can follow to the stream channel. There are two types of mechanisms that overland flow produced in catchments: Infiltration excess overland flow, also known as Hortonian overland flow, and saturation excess overland flow. Infiltration excess overland flow is considered dominant in systems where the soil profile or soil surface has been disturbed, such as agriculture or settlement, in arid and semiarid regions where vegetation density is low and in urban areas where the surface is made essentially impermeable by paving or construction. Another mechanism, saturation excess overland flow is most common in humid areas with dense vegetation and topographic condition that causes the water table to be located relatively close to the surface. Another process is the shallow subsurface stormflow. This process may occur when permeable surficial soils become saturated. Water may then flow to the stream through these soils. Some of the water in the subsurface stormflow moves very slowly through soil and creating a baseflow of streams, usually occur during wetter winter or spring time. Subsurface stormflow also occur along specific flow pathways called macro-pores and the flow through these pathways are relatively fast. Therefore, subsurface stormflow can contribute sufficiently to quick flow. Some of the infiltrated water does not become subsurface stormflow, in fact, it goes further down and reaches the water table. This water then becomes groundwater. Another generation is the groundwater flow. Groundwater flow is the slowest of all flow paths through a catchment. Baseflow in low flow periods almost entirely comes from groundwater discharge. When the subsurface water flowing downslope to the stream entering the saturated water near the stream, some water is forced to reemerge to the ground surface due to the soil’s capacity and rocks to transmit water flowing through downslope is insufficient. This reemerge water is called return flow. Groundwater can discharge to return flow, stream or river during precipitation period, this means that groundwater can contribute to increasing streamflow during storm periods.
What does the SCS approach assume about runoff generation? What does the TOPMODEL approach assume? What is the significance of the topographic index (ln(a/tan β)) and what does it have to do with runoff? (20 points)
The SCS method uses the curve number procedure to predict runoff. The curve number quantifies the hydrologic response of the soil to precipitation events. In another word, this approach is measuring to what extent a soil is likely to generate Hortonian overland flow or infiltration excess. In reality, Beven mentions that the method includes any runoff generation process that may be active without making an assumption of what is happening physically in the soil. Routing of water through a catchment can be done using computer simulation modes. The TOPMODEL approach is a catchment model that is based on the idea that topography exerts a dominant control on flow routing through upland catchments. TOPMODEL uses the equation for conservation of mass for reservoirs in a catchment. Rainfall contributes the input to the interception reservoir. The output from the interception reservoir are evaporation and using the evaporation formula to calculate it. This then produces the input to the soil reservoir. The conservation of mass equation gives a process for calculating the water balance for the soil reservoir. A routing computation can be completed linking the water balance equations. The significance of the topographic index is that it is important to know the characteristics of a hillslope that influence the likelihood of areas of saturation. The ratio of contributing area per unit contour length to the local slope of the topography TI=ln(α/tanβ ), provides a useful measure of the likelihood of saturation in a section of a hillslope. A map of topographic indices for a catchment reveals areas where runoff processes such as saturation-excess overland flow are likely to occur. High values of topographic index represent wide contributing areas and flat slope. This can be found usually at the base of the slope, near the stream. These areas are typically the groundwater discharge areas. Low values of topographic index represent areas with little upslope contributing area and steep slope. These areas can be found at the top of hills.
What is the rational method and how is it applied? What do you have to know about your watershed in order to find the correct values for the parameters needed for calculations? [The method is described at length in Viessman and Lewis ch. 11 in the Urban Hydrology folder, which also refers back to some information in Viessman and Lewis ch. 9 on hydrographs, also in the Runoff and Streamflow folder; Dunne and Leopold’s chapter on Calculation of flood hazard also has an extensive description.] (10 points)
The rational method is used to estimate peak runoff rates and it is commonly used for designing drainage facilities for small urban and rural watersheds. The rational method is typically used for relatively frequent storms and the peak flow rate should be increased for very extreme storms. In determining peak flow rates, most applications of the rational formula utilize the following steps:
Estimate the time of concentration if the drainage area
Estimate the runoff coefficient (there is a table that you can use to find. For example, downtown areas for business watershed have a runoff coefficient from 0.7 to 0.95 while neighborhood areas for business have runoff coefficients from 0.5 to 0.7. Parks and cemeteries have a lower coefficient of 0.1 to 0.25. It all depends on what settlements characteristics, soil types or road styles.)
Select a return period T_r and find the intensity of rain that will be equaled or exceeded, on the average, once every T_ryears. To produce equilibrium flows, this design storm must have a locally derived IDF curve using a rainfall duration equal to the time of concentration.
Determine the desired peak flow Q_p. Peak flow rate is found from Q_p=CIA, Where Q_p is the peak runoff rate in cfs, C is the runoff coefficient (dimensionless), I is the average rainfall intensity in inches/hour for a storm with a duration equal to a critical period of time t_c, t_c is the time of concentration, A is the size of the drainage area in acres.
We did not have time in class to get into the details of the unit hydrograph or how it is applied. But it is still worth knowing what it is and how it is used. Without a blow-by-blow description of all the details, provide a short description of what the unit hydrograph is, what assumptions are made in using it, and why it is so widely applied. Chapter 10 of Dunne and Leopold, chapter 9 of Viessman and Lewis, and chapter 2 of Beven all have relevant information. (10 points)
Unit hydrography was introduced by Sherman in 1932. He defined it as “if a given one-day rainfall produces a 1-in. the depth of runoff over the given drainage area, the hydrograph showing the rates at which the runoff occurred can be considered a unit graph for the watershed.” This means that a unit hydrograph is a hydrograph of direct runoff (does not include base flow) for any storm that produces exactly 1.0 inch of net rain (the total runoff after abstractions). To develop a unit hydrograph, it is important to acquire as many rainfall records as possible within the study area to ensure that the amount and distribution of rainfall over the watershed are accurately known. Preliminary selection of storms to use in deriving a unit hydrograph for a watershed should be restricted to the following:
Storms have to be a simple storm structure and occur individually
Storms have to be uniformly distributed throughout the period of rainfall excess.
Storms are spatially distributed over the entire watershed.
To further unit-hydrograph analysis, suitable storms should meet more restrictive criteria such as:
Rainfall duration should be about 10-30 percent of the drainage are lag time.
Direct runoff for the selected storm should range from 0.5 to 1.75 in.
A suitable number of storms with the same duration should be analyzed to obtain an average of the ordinates (approximately five events). Modifications may be made to adjust different unit hydrographs to a single duration by means of S-hydrographs or IUH procedures.
Direct runoff ordinates for each hydrograph should be reduced so that each event represents 1 in. of direct runoff.
The final unit hydrograph of a specific duration for the watershed is obtained by averaging ordinates of selected events and adjusting the result to obtain 1 inch of direct runoff.