chapter 5
graphical Analysis of drawdown curves
Graphical analysis has long had an important place in the evaluation of well tests; e.g. the semi-log straight-line methods used for the Jacob (1946), Hvorslev (1951), and Bouwer and Rice (1976) analyses. Today, inverse methods are widely used to estimate hydrologic parameters from well test data. Inverse methods are fast and simple to use if appropriate codes are available, but they may be confounded if an inappropriate model is selected for the inversion. Graphical methods are more time intensive, but they are readily implemented in the field and are not limited by software availability.
Graphical techniques promote inspection of field data and encourage a preliminary interpretation of well test data that can be used to identify a model that would be appropriate for more rigorous analysis. As a result, graphical analysis still has an important place in the analysis of many well tests, and the analysis of tests conducted near idealized heterogeneities is no exception.
Graphical methods were developed to determine the properties of the regions in 2-Domain and 3-Domain models, and a method was also used to determine the distance to the contact. These analyses make use of drawdown plotted as a semi-log function of time.
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Type of Discontinuity Contrast
Discontinuity contrasts occur where either the transmissivity of the region containing the well is greater than the neighboring region, or where the transmissivity of the region containing the well is less than that of the neighboring region. The type of contrast can be determined by the shape of the time-drawdown curves for both 2-domain and 3-domain heterogeneities, although there are some limitations.
2-Domain Model
A 2-Domain heterogeneity can be identified by two straight lines on semi-log plots of drawdown and time, and the relationship between the slopes of the curves gives the type of contrast. An increase in slope from one straight segment to the next indicates that the transmissivity of the neighboring region is greater than that of the region near the well, whereas a decrease indicates that the opposite is true (Fig 5.1). It is important to point out that the drawdown record from observation wells that are outside of the critical region will lack an early straight line. A variation in apparent storativities among different monitoring wells will be one indication of heterogeneities, but it will be difficult to recognize the type of heterogeneity using simple graphical methods when a limited number of piezometers are available from outside the critical region.
Figure 5.1 Semi-log dimensionless time-drawdown curve. Increase in slope where Tr > 1.0 and decrease in slope where Tr < 1.0.
3-Domain Model
The type of strip contrast, either low or high, can also be determined by the shape of the time-drawdown curve. A 3-Domain model can be identified by two semi-log straight-lines of equal slope separated by a transition period that either shifts the curve up or down, provided aquifer properties in region 1 and 3 are equal and the observation point is within the critical region. When the strip transmissivity is greater than the surrounding matrix, the transitional portion shifts the curve up. When the strip transmissivity is less than the surrounding matrix, the transitional portion shifts the curve down (Fig 5.2). The magnitude of the shift is related to transmissivity ratio, Tr = Tm/Ts. Curves resulting from a conductive strip resemble those from a storativity contrast (Figs. 4.3 and 4.14).
Estimate Properties of the Formations
The semi-log straight-line method is well suited to analyzing drawdown data from a simple confined aquifer. This is because on a semi-log plot, the time-drawdown data for a well in a homogeneous, infinite aquifer will be a straight-line. It has long been known that it is possible to determine the T and S of an infinite aquifer by matching a straight line to the time-drawdown data (Cooper and Jacob, 1946). The x-intercept of the straight-line, along with the related change in drawdown and slope is used to determine the aquifer properties. The transmissivity is obtained from
(39)
Figure 5.2 Dimensionless semi-log time-drawdown curves for 3-Domian model showing a high and low transmissivity strip. The low transmissivity strip shifts upward to a late-time straight-line where as the high transmissivity strip shifts downward.
and storativity from
(40)
where Q is the pumping rate, Ds is the change in drawdown over one log10 cycle, to is the x-intercept of the straight-line along the slope of the drawdown, and r is the
distance between the pumping well and the observation point (Cooper and Jacob, 1946). It seems possible to extend the graphical approach of Cooper and Jacob (1946) to determine the properties of idealized heterogeneous aquifers. Numerical well tests were conducted using the grid design and solution presented in Chapter 2. Data for the 2-Domain and 3-Domain models were plotted on semi-log graphs using dimensionless drawdown and dimensionless time. This semi-log straight-line analysis was performed on each observation point for a variety of transmissivity ratios for both the 2-Domain and 3-Domain model.
2-Domain Model
When the observation point is within a critical region near the well (defined in Chapter 4, Fig. 4.13), the data consist of two straight-lines with different slopes. However, only one straight-line occurs when the observation point is outside of the critical region. The semi-log straight-line analysis was used at observation points inside and outside of the critical region for several transmissivity ratios to estimate the hydraulic properties of an idealized heterogeneous aquifer.
Observation Inside Critical Region
The drawdown curve forms a semi-log straight-line at both early and late times when observation points are within the critical region (Fig. 4.1). The slope of the early-time semi-log straight-line is inversely proportional to T1, whereas the slope of the late-time semi-log straight-line is inversely proportional to the arithmetic average of the transmissivity of the two regions (Nind, 1965; Fenske, 1984; Streltsova, 1988). The transmissivity and storativity of the local region can be calculated from the early-time semi-log straight-line segment using equations (39) and (40) when the time-drawdown data is plotted on semi-log axis, provided the local region is relatively homogeneous (Maximov, 1962; Fenske, 1984).
Properties used in the 2-Domain numerical well test are T1 = 1, S1 = 0.017, T2 = 0.1, and S2 = 0.017. Observation points defined in Chapter 2 (Fig. 2.1) were used as well as two additional observation points at (x = L/2, y = L) and (x = L, x = L). The example within the critical region is for a radial distance, r, of 0.25L = 3.75. The x-intercept for the early-time semi-log straight-line, to1, was 0.1 and the change in drawdown over 1 log cycle, Ds1, was 2.2 (Fig. 5.3). Substituting these into equations (39) and (40) results in T1 = 1.04 and S1 = 0.016. These results are within a relative percent error of 4% for T and 6% for S. The x-intercept for the late-time semi-log straight-line, to2 = 0.65, and the drawdown over 1 log-cycle was Ds2 = 4.1. Substituting these into equations (39) and (40) results in T2 = 0.56 and S2 = 0.104. These results are within a relative percent error of –44% for T and 512% for S.
From these results it seems that the early straight-line closely predicts the properties of the local region, however, the late straight-line is a poor prediction of the
Figure 5.3 Dimensionless time-drawdown for a 2-Domain model with x-intercepts (to) and change in drawdown (Ds) from an observation point within the critical region. Tr = 10.
properties of the neighboring region. The arithmetic average of T is 0.55, which is essentially the same as the value obtained by analyzing the well test data.
Observation Outside Critical Region
The early-time semi-log straight-line segment is absent and only a late-time segment is present when observation points are outside of the critical region (Fig. 4.5). The semi-log straight-line analysis was used on the late time straight segment to evaluate hydraulic properties. Data was used from the numerical well test discussed above for an observation point located at r = L. The x-intercept was to = 2.7, while the Ds remained the same as that for the observation point in the critical region, Ds = 4.1. Substituting these into equations (39) and (40), gives T = 0.56, and S = 0.025 (Fig. 5.4). These results are within a relative percent error of –44% for T and 1488% for S. The T determined using the late straight line and equation (39) for r = L, is essentially the same value determined for r = 0.25L at late time.
The above procedure was carried out at several observation points in both the local and neighboring regions (Fig. 5.5). The x-intercept and Ds determined for each observation point is in Table 5.1 along with the T and S values determined using equations (39) and (40) for both early and late time. It is clear that the apparent transmissivity at all locations using the late-time semi-log straight-line is the arithmetic average of the two regions. The storativity calculated at each location is different from the S that was used in the model that generated the data. The apparent storativity can be higher, lower, or equal to the true storativity value depending on the difference in
Figure 5.4 Dimensionless time-drawdown for a 2-Domain model with x-intercept (to) and change in drawdown (Ds) from an observation point outside the critical region. Tr = 10.
Fig. 5.3. Transmissivity and Storativity determined using the Jacob method for various piezometer locations. Subscript E denotes properties determined using the early-time semi-log straight-line. Subscript L denotes properties determined using the late-time semi-log straight-line.
Figure 5.5 Map view of 2-Domain model with T and S calculated using the semi-log straight-line method for each observation points. Points within the critical region have two semi-log straight-lines, the subscript E denotes values calculated from the early-time semi-log straight-line and L denotes values calculated from the late-time semi-log straight-line.
ObservationPoint / r / toE / DsE / TE / SE / toL / DsL / TL / SL
0.125L / 1.94 / 0.1 / 2.2 / 1.04 / 0.018 / 0.65 / 4.2 / 0.55 / 0.25
0.25L / 3.88 / 0.12 / 2.3 / 1 / 0.018 / 0.91 / 4.2 / 0.55 / 0.136
0.5L / 7.75 / -- / -- / -- / -- / 1.6 / 4.2 / 0.55 / 0.06
L / 15.5 / -- / -- / -- / -- / 2.7 / 4.2 / 0.55 / 0.025
1.1L / 17.05 / -- / -- / -- / -- / 2.7 / 4.2 / 0.55 / 0.021
2L / 31 / -- / -- / -- / -- / 29 / 4.2 / 0.55 / 0.068
0.5L, L / 17.33 / -- / -- / -- / -- / 3.9 / 4.2 / 0.55 / 0.029
L, L / 21.92 / -- / -- / -- / -- / 4.5 / 4.2 / 0.55 / 0.021
Table 5.1 x-intercept, to, and change in head, Ds, with corresponding apparent T and S value of early (E) and late (L) straight-line for each observation point. Actual values are T1 = 1.0, T2 = 0.1, S1 = S2 = 0.017.
diffusivity and location of the observation point (Maximov, 1962; Streltsova, 1988; Fenske, 1984).
Capabilities and Limitations
The semi-log straight-line method can provide useful information regarding the hydraulic properties of an aquifer containing two homogeneous regions if the observation point is located within the critical region. The transmissivity and storativity of the local region can be determined from the early-time straight-line when the observation point is within 0.25 times the distance between the well and discontinuity. The arithmetic average of the T can be determined from the late-time straight-line. Thus, in order to determine the properties of an aquifer, measurements must be taken within the critical region. With the average transmissivity, Ta, from the late-time semi-log straight-line, and T1 from the early-time semi-log straight-line, the transmissivity of the neighboring region can be determined
(41)
In addition to having the observation point within the critical region, the well test must run long enough for a second semi-log straight-line to occur in order to determine the properties of the neighboring region. The average transmissivity will be the only property that the semi-log straight-line method can predict if the only observation points are outside of the critical region. Thus, an observation point should be located within the critical region to obtain the most information about an aquifer with a planar discontinuity. The storativity of the neighboring region will be inaccurate using the semi-log straight-line method and equation 40. The apparent storativity can be predicted analytically by rearranging (18), although details are beyond the scope of this thesis.
3-Domain Model
There are two semi-log straight-lines during simulated pumping tests in a 3-domain aquifer, similar to those of the 2-Domain model, when the observation point is within the critical region. However, unlike the 2-Domain model, the slope of the line at late time is equal to that of the early-time semi-log straight-line. It is noteworthy that properties of the aquifer on either side of the vertical strip are the same for all the analyses conducted in this work. As with the 2-Domain model, there is only one semi-log straight-line when the observation point is outside of the critical region. The semi-log straight-line analysis was used at each observation point for several transmissivity ratios to estimate the hydraulic properties of the heterogeneous aquifer.
Observation in the Pumping Well Region
The semi-log straight-line method was used on both segments for an observation point in the critical region and for an observation point outside the critical region. Within the critical region, data were used from a curve generated when Tm = 1 and Ts = 0.1, Sm = Ss = 0.017. The distance r for this example is 0.125L = 1.94. The x-intercept for the early-time semi-log straight-line, to1, was 0.028 and the change in drawdown over 1 log cycle, Ds1, was 2.3 (Fig. 5.6). Substituting these values into equations (39) and (40) results in Tm = 0.999 and Sm = 0.0167. These results are within a