Response Functions

What are Response Functions?

When an aquifer is pumped, ground water levels in the vicinity of the well decline, creating a cone of depression around the well.  This cone of depression is the response of the system (the aquifer) to the pumping that has been introduced (see illustration above).  Aquifer drawdown varies with distance from the well; consequently, the system response is location-dependent.  The longer the well is pumped, the greater the drawdown becomes, making the system response time-dependent as well.  If we select a specific location within the cone of depression, and by field measurements or theoretical means we quantify how drawdown varies with time in response to pumping, we have developed a simple response function for that particular location and well.  In most cases, the drawdown will be proportional to the discharge of the well, so our response function, if expressed as a ratio of drawdown to well discharge, can be used to predict the system response to any pumping rate.  Cones of depression from multiple pumping wells normally can be superimposed, assuming that the system response is independent of other events occurring simultaneously.  These are the basic concepts of a response function.

Response functions are a means by which we can express cause and effect relationships for an aquifer or a river-aquifer system.  The functions may be thought of as a series of values or ratios that express the response at a specific location to aquifer recharge or discharge at another location.  Since the response changes with time, a different value is required not only for each pair of locations, but also for each time period of interest.  The following two examples illustrate the concept of response functions for a) drawdown in an aquifer, and b) for depletion of a stream.

EXAMPLE 1: AQUIFER DRAWDOWN RESPONSE

Let's assume that we have two wells, A and B which are 1000 feet apart (see illustration above).  We measure water levels in both wells and start pumping well A at a rate of 1.0 cubic foot per second.  After 1 hour, we measure the water level in well B and note that the water level is 4 feet deeper than before we started pumping well A.  The reponse ratio expressing the response of the aquifer at point B to pumping at point A would be the ratio of the drawdown at point B (4.0 feet) to the pumping rate (1cfs) at point A.  In this example, the ratio 4 ft./1.0 cfs expresses the effects from pumping at one location on water levels in the aquifer at another location.  It is assumed that the ratio at location B holds regardless of the pumping rate at location A (see section on assumptions).  Consequently, if the pumping rate were doubled to 2 cfs, the drawdown would double to 8 feet.  It must also be noted that the drawdown increases as the pumping time increases.  For example, say that we continue pumping at 1 cfs and measure water levels in well B that were 6, 8 and 10 feet deeper at 2, 4 and 10 hours after we began pumping at well A, respectively.  There must, therefore, be a different response ratio for each time period of interest.  After one day of pumping, the ratio may have increased to 15 ft./1.0 cfs.  The term response function is used to represent the series of ratios describing responses at different times.

In addition to describing the response of water levels in an aquifer to ground water pumping, response functions may be used to describe the effect of ground water pumping (or recharge) on surface water resources.  When surface water resources are in hydraulic connection with an aquifer, ground water pumping or recharge can affect lake levels or flows in springs and streams (see Surface Water/Ground Water Interactions).  An example of a response function for a stream-aquifer system follows.

EXAMPLE 2: RIVER DEPLETION RESPONSE

Let's assume that we have a stream that traverses a plain and is in hydraulic connection with an underlying aquifer (see figure at left).  That stream (a gaining stream) is being fed by the aquifer at a rate of 3 cubic feet per second (cfs)  in our reach of interest between two gaging stations.  A new well installation in the aquifer begins continuous pumping at a rate of 2 cfs.  After 100 days of pumping, the stream is now only gaining 2 cfs rather than the 3 cfs it used to gain in this reach.  The stream flow in this reach has therefore been depleted by one cfs after 100 days of pumping.  The response ratio representing this pumping location to this river reach at 100 days is calculated as the ratio of the depletion (1 cfs) to the pumping rate (2 cfs) or 0.5.  This ratio will change over time.  After 200 days, the ratio may increase to 0.6.  The series of ratios representing responses of this river reach to pumping (or recharge) at this location, at different times, comprise the response function for this well and river reach.  A different response function will exist for different river reaches and for different pumping locations.

It is assumed that the ratios determined between the pumping well and the river reach are valid for all pumping rates.  Pumping from the well for 100 days at a rate of 4 cfs rather than 2 cfs would deplete the stream by 0.5 x 4 cfs or 2 cfs.  This means the stream would be gaining 1 cfs rather than the 3 cfs gain that existed before pumping started.  If the rate of depletion were to exceed the rate of river gain (3 cfs in this example) the reach would become a losing rather than gaining reach of the stream. In this case, the ratio would hold only as long as 1) the river stays in hydraulic conection with the aquifer (the reach does not become perched) and 2) there is enough water flowing in the river to meet the loss (the river does not go dry).

Although response functions can be developed to represent the drawdown response of an aquifer as in the first example, this description is focusing on ground water and surface water interactions and, therefore, on stream-aquifer response functions as illustrated in the second example.

The stream depletion response to continuous aquifer pumping has been illustrated for the Eastern Snake River Plain aquifer by Johnson, and others (1993) and Hubbell, and others (1997).  These descriptions portray the response of four reaches of the Snake River to ground water pumping at selected locations in the Snake River Plain aquifer.  The response varies dramatically with location.  Graphs of response in two river reaches to continuous pumping at five locations can be seen in the above figure.  Although this example shows response functions as graphs of proportioned depletion (depletion rate/pumping rate) vs. time, the response functions could also have been expressed as a matrix of response ratios (for each time, pumping location, and river reach), or as equations representing depletion as a function of time and location.

What Factors Affect Response Functions?

System response is controlled by the physical characteristics of the system.  The stream depletion effects resulting from pumping a given well will be controlled by 1) the proximity of the stream and well, 2) the degree to which the stream and aquifer are interconnected (see  Surface Water/Ground Water Interaction ), 3) the distribution of aquifer properties such as transmissivity and storativity, 4) layering of the aquifer and the depth and open interval of the well, 5) the distance to other hydraulically connected surface water bodies within the same aquifer, and 6) the distance to low-permeability aquifer boundaries.  Obviously, the estimation of response functions can become very complicated.  Since we never fully understand the characteristics and physical properties of the aquifer, generated response functions represent only an approximation of actual system response.  The reliability of estimated response functions depends on the degree of complication of the real system and the degree to which we understand and represent the complexities in the method we choose to calculate response functions.  There are several ways, with varying levels of accuracy, to calculate response functions.  These are described in the section on "How are Response Functions Determined?".  Regardless, our understanding of the real system and our estimates of response functions are far from perfect.

What if Wells are not Continuously Pumping?

The above examples illustrated how response functions can be developed and used when wells are operating continuously.  In reality, however, this is seldom the case.  We need to be able to evaluate stream depletion from wells that are used intermittently.  Response functions can be determined for wells that are pumped for a brief period and then turned off.  The procedure is described in the section "How are Response Functions Determined?".

Wells that are pumped for a brief period may produce very minor stream depletion effects that persist long after pumping has ceased.  The accompanying graph shows that pumping at a rate of 1 cfs has greatly attenuated effects on a hydraulically connected stream reach.  Although the illustration is hypothetical, it is important to recognize that ground water pumping impacts may persist for years or even decades into the future, depending upon the scale and properties of the system.

Response Functions and Aquifer Recharge

Most of the above discussion and examples related to depletion of streams resulting from aquifer pumping.  In conditions where response functions can be used, pumping and aquifer recharge create equal but opposite effects on stream and spring discharge.  Where ground water pumping results in a depletion of surface water resources, aquifer recharge will result in an increase in flows.  The rate of increase will be proportional to the rate of recharge.

In the second example, if the well were recharging at a rate of 2 cfs, rather than pumping, then the stream gains would have increased, rather than decreased, by 1 cfs (the response ratio was 0.5) after 100 days.

Combinations of pumping and recharge can be evaluated by adding responses from individual activities on the stream reach of interest.

How Can Response Functions be Used?

Response functions describe the response of specific river reaches to ground water pumping or recharge at specific locations for specific times.  In physical situations where response functions are applicable (see section on assumptions), response functions may have several uses.

One of the most basic uses of response functions may be to obtain and convey a better understanding of the way in which surface water systems interact with ground water systems.  Graphs showing how response (stream depletion) changes with time are one way of gaining an understanding of the attenuation of pumping effects on surface water resources.  Superimposing these graphs on a map provides additional information on how impacts change depending on the location of pumping (see figure above).  Contouring response ratios (response functions at a single point in time) for an entire aquifer can portray how different pumping locations impact a given river reach.  In the Snake River Plain aquifer, contours of steady state response functions (response after many years of continuous pumping) can provide guidance to water managers as to the degree that pumping in any area impacts different reaches of the Snake River:  Upper Snake reaches, Blackfoot to Neeley reach, Neeley to Minidoka reach, and Kimberly to King Hill reach.

In many states, ground water and surface water supplies are being conjunctively managed (see Water Rights and Conjunctive Management page).  This integrated management philosophy may result in ground water users being held partially responsible for surface water shortages.  If this is the case, then the quantities that individuals or groups of ground water users are accountable for must be determined.  This determination is an essential step in development of mitigation plans.  Mitigation in the form of managed recharge will be designed to offset injury resulting from pumping.  Just as pumping impacts may be readily determined from response functions, the counter-effect of recharge may be assessed.  Response functions may be incorporated into spreadsheet water accounting programs to determine debits, credits, and balances associated with ground water pumping and recharge.

In a simple analytical expression, or a table of numbers, response functions can describe the response of a surface water body to ground water pumping or recharge at a selected location. Subdividing an aquifer into a series of zones allows representation of cause and effect relationships by a simple series of independent equations or tables.  These simplified relationships can then be incorporated into regional surface water or multi-disciplinary models that represent the aquifer as one component of a larger system (see figure below).  This allows the inclusion of ground water systems into more holistic ecosystem models.

The additive nature of response functions also makes them applicable to optimization techniques.  Optimization methods may be used to identify schemes to accomplish goals such as minimizing pumping cost or drawdown at specific locations.  Development of response functions is a necessary part of optimization in ground water systems.
How are Response Functions Determined?

Response functions may be determined by several methods.  The methods may be divided into three basic categories: 1) direct measurement, 2) analytical methods, and 3) numerical methods.  These methods vary in the level of assumtions employed and accuracy of the results. The method selected should be commensurate with the purpose of the evaluation and the information available for each situation. Each method is described in the following paragraphs.

Direct measurement of stream or spring depletion by pumping is possible in relatively few cases.  Measurement of depletion resulting from pumping of a specific well is possible only when the well is relatively close to the stream or spring and well pumping is a significant proportion of the stream or spring flow.  It is also sometimes difficult to isolate the effects of a single well from other influences that may be causing changes in discharge.  When it is possible to measure the relationships between pumping and stream or spring discharge, the proportion of depletion can be determined from measurements of stream or spring flow and well discharge.  The depletion changes with time, so observations are required at different times.  The response function can be represented by the proportion of well pumping appearing as depletion of the stream or spring.  The response function will be valid within some limits of pumping rate, stream flow, and aquifer water level.  These limits are site-specific and related to the assumptions identified in the section "Assumptions Supporting Response Functions".

Analytical methods have been developed and used to estimate the effects of ground water pumping on stream depletion.  Methods such as those of Jenkins (1968) and Glover (1968) are burdened with intensive assumptions such as straight and fully penetrating streams, fully penetrating wells, and homogeneous and infinite aquifers.  These methods, however, can be employed to develop response functions when data are inadequate to use more sophisticated techniques.  In these cases, response functions involve little more than plugging the site-specific data into explicit equations.

Response functions can be developed from numerical models in situations where more data are available and numerical models have been developed. Some model codes generate response functions directly, in other situations it may be necessary to run multiple simulations, each simulating response to ground water pumping at a specific location (model grid with well).  MODFLOW is the most widely used code for ground water modeling.  It is accepted as the industry standard for numerical modeling of ground water systems.  A detailed description of the use of MODFLOW for developing response functions can be found in the technical article Use of MODFLOW for Development of Response Functions.  Simulated changes, when divided by the magnitude of the pumping rate, yield response functions.  The simulations must be based on linear equations.  This means that changes in aquifer thickness from pumping must be small relative to total thickness, and head-dependent boundary conditions must be linear functions of aquifer head.  This is perhaps the most restrictive condition relative to the use of response functions.  If springs dry up, streams transition between hydraulically connected and perched, or other non-linear conditions develop, and if these features significantly affect the operation of the system, then response functions should not be used.  Their use should be limited to the range of conditions where these sort of changes do not significantly impact the system.  This topic is also addressed in the section "Assumptions Supporting Response Functions".

Regardless of the technique used to develop the response functions, it must be recognized that the response functions are only as good as our conceptual understanding of the system and the data used in the development of the models!

Response functions describe the changes in the system (ground water levels or stream and spring discharge) resulting from ground water pumping or recharge at a specified site for a specified time duration.  The response functions become useful tools because of the ability to the add effects of multiple pumping or recharge events.  The events may be occurring at the same location sequentially, at multiple locations simultaneously, or both.  This makes it most effective to generate response functions for the shortest duration of pumping that is of interest.  The response to longer periods of pumping can then be developed by adding the effects from one pumping period to that of a second period immediately following the first.  These sorts of operations are ideally suited to spreadsheets and data bases.

Assumptions Supporting Response Functions

Response functions cannot be universally applied to all situations.  The validity of response functions is linked to the concept that responses, such as river depletion from pumping, are additive and proportional to the pumping or recharge rate.  This assumption is valid in many ground water environments.  Each situation should be evaluated by a qualified professional hydrologist, however, to determine if any of the following excluding conditions are likely to have a significant effect on the ground water flow system in the area of interest.

EXCLUDING CONDITIONS

1) Saturated aquifer thickness changes, resulting in a significant change in aquifer properties, normally transmissivity.

2) Springs either begin or cease flowing as a result of natural or man-imposed changes in aquifer water level.

3) Streams transition between perched and hydraulically connected with the aquifer.

4) Evapotranspiration or other water consumption varies non-linearly with aquifer head.

5) Enough water is flowing in the river to meet demands in losing reaches.

These conditions need to be evaluated, not just for the development of response functions, but for their application to each specific circumstance.  If ground water elevations are changing dramatically in a given situation, then the above conditions are more likely to occur, and response functions should not be used.

Response functions generated by field measurements, analytical techniques, or numerical models will be burdened with all of the assumptions of the method used in their development.  For example, if the Jenkins (1968) analytical technique is used to generate the response functions, then one of the assumptions underlying the response functions is that the stream is straight and fully penetrating.  If a numerical model is used to generate the response functions, then the accuracy of the response functions is directly dependent on the accuracy of the numerical model's calibration and the underlying conceptual model.  In many cases, this will be the greatest source of error in application of response functions.