## Application of DP_LAQ to Confined Fractured Karstic Aquifer Pumping Test Data

### By Darrel Dunn, Ph.D., PG, Hydrogeologist. (Professional Synopsis 🔳)

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### Introduction

The purpose of this web page is to describe the results of an application of DP_LAQ pumping test analysis software (Moench, 136) to actual data. The pumping test selected for this study is described by Crouch (135). I selected this test because the quality of the data presented in the paper is good, and the pumping test provides an extreme case for exploring the applicability of the fractured aquifer capability of DP_LAQ. The case is extreme because the aquifer is karstic with properties that deviate considerably from the assumptions of the analytical solution used. An interesting aspect of the test is that an observation well (ZS-1) located 247 feet northeast of the pumped well had less drawdown than a more distant observation well (ZS-13) located 409 feet southwest of the pumped well. ZS-13 was reported to have penetrated “caves,” but ZS-1 did not.

### Description of the Aquifer Tested

The aquifer is a buried karstic formation (San Andres Limestone, Permian) underlain by and interconnected with fractured sandstone (Glorietta Sandstone, Permian). The two hydraulically connected units are called the San Andres-Glorietta Aquifer. Most of the wells involved in the test encountered “caves,” which I suspect are driller's descriptions of open fissures or conduits formed by dissolution of limestone by groundwater under karst conditions. Karst developed when the limestone was exposed at the ground surface in late Permian or early Triassic time. These karst features are now buried beneath a thick aquitard composed of mudstone and siltstone in the lower part of the Chinle Formation (late Triassic). At the pumped well the San Andres-Glorietta Aquifer is 406 feet thick. It is underlain by more than 900 feet of interbedded sandstone, siltstone, limestone, and evaporites of the Abo and Yeso Formations (early and middle Permian) which act as a subjacent aquitard. The top of the aquifer is at a depth of 602 feet at the pumped well, but rises westward to crop out on an anticline about three miles from the pumped well.

### Pumped Well Description

The pumped well has 18-inch casing to the top of the San Andres Limestone. Below the casing, 16-inch open hole extends through most of the 135 foot thickness of the San Andres Limestone, and 6-inch open hole extends 45 feet into the Glorietta Sandstone. The non-pumping water level was 120 feet above the top of the San Andres Limestone.

### Observation Well Descriptions

Data are reported for four observation wells and an enclosed spring. The spring is on the aforementioned anticline 4.83 miles northwest of the pumped well. The observation well distances are 247 (ZS-1), 409 (ZS-13), 6,470 (ZS-101), and 15,950 (ZS-100) feet. They all encountered “caves” except ZS-1, and they all extend into the top of the Glorietta Sandstone. The present analysis only uses the two closest observation wells (ZS-1 and ZS-13).

### Description of the Pumping Test and the Original Analysis

The pumping test and the original analysis are described by Crouch (135). The well was pumped at approximately 2540 gallons per minute for 13,380 minutes (9.3 days). Time-drawdown data were not reported for the pumped well, but the average drawdown was reported to be about 4.3 feet. Most of the drawdown occurred within a few minutes after pumping started.

The original analysis used the Jacob semilogarithmic plot method to analyze the pumping test data. This method yielded a transmissivity of 640,000 ft^{2}/day and a storativity of 4.8E-4 from the ZS-13 data, and transmissivity of 750,000 ft^{2}/day and storativity of 7E-2 from the ZS-1 data. The data from the more distant wells were not analyzed.

### DP_LAQ Fractured Aquifer Pumping Test Analysis

The present analysis of the data used software titled DP_LAQ, which is described by Moench (136). DP_LAQ calculates type curves for pumping tests in fractured aquifer dual-porosity groundwater systems. It allows the user to choose between three cases for the simulation of dual-porosity. In the input file, the cases are labeled 4, 5, and 6:

4. Horizontal fractures separated by slabs of matrix rock,

5. Vertical and horizontal fractures with matrix rock in the interstitial cubes (simulated as spheres), and

6. Simplified matrix to fracture flow.

I used case 5 because outcrops of the aquifer show both horizontal and vertical fractures. Case 5 is depicted in Figure 1. The matrix blocks are mathematically treated as spheres. The spheres have a hydraulic conductivity of K_{m} and specific storage S_{sm}. The head in the blocks can vary with dimensionless distance (ρ) from the center of the block, and with time in accordance with flow to the fissure. The block can be separated from the fissure by a thin skin which produces a resistance to flow from the block to the fissure.

Figure 1. Diagram of case 5 dual-porosity fractured system implemented in DP_LAQ (from Moench, 136)

DP_LAQ also includes the effects of the fissure system hydraulic conductivity and specific storage, effective radius of the pumped well, water storage in the pumped well casing, and pumped well skin. DP_LAQ generates type curves for the pumped well based on these parameters and for observation wells based on these parameters plus the distance from the pumped well.

I developed type curves for the two closest monitoring wells ZS-13 and ZS-1 based on successive trials of the unknown variables. I treated the drawdown in ZS-13 as fissure system drawdown because it was described as penetrating “caves.” I treated the drawdown in ZS-1 as drawdown in a matrix block because it showed less drawdown that ZS-13 even though ZS-1 was closer to the pumped well. The resulting type curve match is shown in Figure 2. In this figure, “TYPE CURVE DIMENSIONLESS TIME” is the expression t_{D}/r_{D}^{2} defined by Moench (136). “TYPE CURVE DIMENSIONLESS DRAWDOWN” is h_{D} for ZS-13 and h_{D}' for ZS-1, and “ADJUSTED TIME” is t/r^{2} (t is time since pumping started, and r is distance to an observation well. The variable h_{D} is dimensionless drawdown in the fissure system, and h_{D}' is dimensionless drawdown in the matrix (Moench, 136). Putting the observed data on the secondary axes and putting dimensionless time multiplied by Ss/K or Ss_{m}/K_{m} and dimensionless drawdown multiplied by Q/4πKb on the primary axis allows type curve matching by successive trial values of K and Ss (and the other unknown parameters, K_{m}, Ss_{m}, S_{F}, that I treated as calibration variables). The parameter S_{F} is dimensionless fracture skin (Moench, 136). This technique allows the matching to be performed on a spreadsheet rather than manually or electronically superimposing graphs and picking a match point.

Figure 2. Type curve match for observation wells ZS-13 and ZS-1 with T=902,880 ft2/day and S=2.85E-1.

The values of the other parameters varied in successive trials to produce this type curve match were:

K', matrix hydraulic conductivity, 0.2246 ft/day,

Ss', matrix specific storage, 0.1, 1/ft,

S_{F}, fracture skin, 1000, dimensionless.

Values of parameters that were held at the same value for all trials were:

r_{w}, effective radius of pumped well, 0.3125 ft,

r_{c}, internal radius of pumped well casing, 0.73 ft,

S_{w}, well skin, 0, dimensionless,

b', radius of matrix blocks, 31.67 ft.

ρ, dimensionless distance from the center of the block, 0.5.

Figure 2 shows that I was not able to develop a precise match. I think this is due to the difference between the assumptions of the solution and the actual aquifer system (prototype). The solution assumes a regular geometry for the fissures, as illustrated in Figure 1, but the prototype is much more complex. The fissures are probably irregularly spaced and do not have a uniform hydraulic conductivity. The separation of the fissures is probably much greater than implied by the 31.67 radius of matrix blocks (b'), which was constrained by the capability of the Fortran program (overflow occurred at greater radii). I simulated a restricted connection between well ZS-1 and the fissure system by using a large value for fracture skin S_{F}. Thus the fact that drawdown in ZS-1 was less than drawdown in ZS-13 even though ZS-1 was closer to the pumped well than ZS-13 was simulated. Some of the complexity of the dual-porosity karstic aquifer was captured even though the prototype differs considerably from the model. This could not have been done with a single-porosity analytical solution.

However, the arithmetic drawdown plot presented in Figure 3, shows that the calculated drawdown begins to depart from the observed drawdown early in the pumping test.

Figure 3. Extended arithmetic plot of calculated and observed drawdown for ZS-13 and ZS-1.

The reason for this departure of the calculated and observed drawdowns can be deduced from the semilogarithmic plot shown in Figure 4. The drawdown plot for both observation wells curves upward at about 700 minutes. The observed drawdown plots roughly as a straight line before 700 minutes, and as a straight line with twice the slope after 700 minutes. This behavior suggests a sharp reduction in hydraulic conductivity ("impermeable boundary") at some distance from the pumped well and observation wells. One would have to add an image well to get the calculated drawdown to follow the observed drawdown after about 2000 minutes. Figure 4 shows that the calculated curve for ZS-13 is flat from about 0.5 to 300 minutes. This flat portion of the curve suggests the effect of flow from the matrix to fissures inhibiting drawdown. It is interesting to note that if one calculates the transmissivity from the early part of the drawdown curve of ZS-13 using the semilog method (see Freeze and Cherry, 66), the value is 783,000 ft^{2}/day. The storativity calculated using the semilog method is 7.3E-5.

Figure 4. Semilog plot of calculated and observed drawdown ZS-13 and ZS-1.

Figure 5 shows the result of eliminating the part of the arithmetic plot after 1000 minutes that is affected by the "impermeable boundary." The calculated curve for ZS-13 closely matches the observed drawdown after 300 minutes. Consequently, the transmissivity calculated from the type curve match is probably a good representation of the transmissivity in the area around the pumped well and ZS-13. The drawdown in the matrix rock at ZS-1 is overestimated by 0.2 feet at 1000 minutes.

Figure 5. Early arithmetic plot of calculated and observed drawdown for ZS-13 and ZS-1.

### Derivative Analysis

Figure 6 shows the results of a derivative analysis using DERIV (134). DERIV calculates the approximate derivative of drawdown with respect to the natural logarithm of time at successive points on a drawdown curve. This derivative is sensitive to small variations in the rate of change of drawdown, which are often difficult to see on arithmetic or logarithmic curves like the ones shown above. The derivative also has the property that it plots as a horizontal line when drawdown is increasing in proportion to the logarithm of time. This horizontal portion of a derivative curve is consistent with the Theis curve (often called infinite radial flow, IRF, in derivative analysis).

Figure 6. Derivative analysis of calculated and observed drawdown for ZS-13 and ZS-1.

The calculated drawdown derivative curve in Figure 6 corresponds to the calculated curves in Figures 2, 3, 4, and 5, and is just another representation of the same data. This derivative curve follows the typical plot of a curve affected by wellbore storage in the pumped well and delayed yield from storage in the intrafracture matrix blocks (137). The hump in the curve prior to 0.002 minutes represents the effect of wellbore storage. This hump is followed by a nearly horizontal segment to about 0.5 minutes representing initial IRF in the fissure system system not much affected by delayed yield from the matrix rock. The depression in the curve from 0.5 minutes to about 300 minutes represents the effect of delayed yield. After 300 minutes the curve is horizontal, representing IRF.

The observed drawdown derivative curve in Figure 6 does not begin until the time is equal to 1 minute, so the effect of wellbore storage is missed. The part of the curve from 1 minute to 8 minutes may represent initial IRF. The depression in the curve from 8 minutes to about 30 minutes may represent the effect of delayed yield. This depression looks small, but this appearance is partly an artifact of the log-log graph. The actual decline in the derivative is about 0.06 feet compared to about 0.03 feet for the calculated drawdown derivative. The slightly declining part of the curve from 30 to 100 minutes may be an approach to IRF, followed by IRF to about 700 minutes. After 700 minutes the curve is affected by a reduction in hydraulic conductivity at some distance from the pumped well ("impermeable boundary"). A small decline in the derivative at the end of the curve may signal the presence of an increase in hydraulic conductivity or a recharge boundary at a large distance from the pumped well.

### Conclusions

The following conclusions are based on my experience applying DP_LAQ to this pumping test of a complex confined karstic fractured aquifer:

Inhomogeneity of transmissivity in the fissure system affected the observed drawdown after 1000 minutes and the type curves depart from the observed drawdown after this time.

A useful transmissivity was obtained by matching the early data that were not affected by aquifer heterogeneity.

The results are consistent with drawdown in an observation well (ZS-13) in the fissured system penetrated by the pumped well that was greater than the drawdown in an observation well (ZS-1) in the matrix rock located closer to the pumped well.

Fracture skin S

_{F}was more effective for producing a difference in drawdown between the fissure system and the matrix than matrix hydraulic conductivity (K'), or position of the observation well in the matrix (ρ).Most of the drawdown in the fissure system occurred within the first minute of the test.

I had to use a high value (0.001) for specific storage (Ss) of the fissured system to get a type curve match.

Pumping test data analysis for complex systems should include logarithmic, semilogarithmic, and arithmetic plots with appropriate time scales.

Derivative analysis is useful for identifying and diagnosing subtle features of the drawdown curves.

It is unsafe to project pumping test drawdown into the future for a karstic fractured aquifer system (numerical modeling might be used).

DP_LAQ may provide useful results in complex dual domain systems when the configuration of the fissures and matrix blocks deviates from the simple assumptions in the mathematical development of the drawdown equations.

**Addendum**

Constant discharge aquifer testing methods for water table aquifers, and leaky aquifers are also described on this website. Such tests are often preceded by a step test.

Posted 5/13/2015. Revised 11/14/2018