Государственный комитет Республики Узбекистан

Advantages and Disadvantages of Pumping Tests
The determination of aquifer constants through pumping tests has become a standard step in the evaluation of groundwater resource potential. In practice, there is much art to successful pump testing and the interested reader is directed to Kruseman and de Ridder (1970) and Stallman (1971) for detailed advice on the design of pumping-test geometries, and to Walton’s (1970) many case histories.
The advantages of the method are probably self-evident. A pumping test provides in situ parameter values, and these values are, in effect, averaged over a large and representative aquifer volume. One obtains information on both conductivity (through the relation K = T/b) and storage properties from a single test. In aquifer-aquitard systems it is possible to obtain information on the very important leakage properties of the system if observations are made in the aquitards as well as the aquifers.
There are two disadvantages, one scientific and one practical. The scientific limitation relates to the nonuniqueness of pumping-test interpretation. A perusal of Figure 8.23(b), (c), and (d) indicates the similarity in time-drawdown response that can arise from leaky, unconfined, and bounded systems. Unless there is very clear geologic evidence to direct groundwater hydrologists in their interpretation, there will be difficulties in providing a unique prediction of the effects of any proposed pumping scheme. The fact that a theoretical curve can be matched by pumping test data in no way proves that the aquifer fits the assumptions on which the curve is based.
The practical disadvantage of the method lies in its expense. The installation of test wells and observational piezometers to obtain aquifer coefficients is probably only justified in cases where exploitation of the aquifer by wells at the test site is contemplated. In most such cases, the test well can be utilized as a production well in the subsequent pumping program. In geotechnical applications, in contamination studies, in regional flow-net analysis, or in any flow-net approach that requires hydraulic conductivity data but is not involved with well development, the use of the pumping-test approach is usually inappropriate. It is our opinion that the method is widely overused. Piezometer tests are simpler and cheaper, and they can provide adequate data in many cases where pumping tests are not justified.
8.7 Estimation of Saturated Hydraulic Conductivity
It has long been recognized that hydraulic conductivity is related to the grain-size distribution of granular porous media. In the early stages of aquifer exploration or in regional studies where direct permeability data are sparse, this interrelationship can prove useful for the estimation of conductivity values. In this section, we will examine estimation techniques based on grain-size analyses and porosity determinations. These types of data are often widely available in geological reports, agricultural soil surveys, or reports of soil mechanics testing at engineering sites.
The determination of a relation between conductivity and soil texture requires the choice of a representative grain-size diameter. A simple and apparently durable empirical relation, due to Hazen in the latter part of the last century, relies on the effective grain size, d₁₀, and predicts a power-law relation with K:
(8.47)
The d₁₀ value can be taken directly from a grain-size gradation curve as determined by sieve analysis. It is the grain-size diameter at which 10% by weight of the soil particles are finer and 90% are coarser. For K in cm/s and d₁₀ in mm, the coefficient A in Eq. (8.47) is equal to 1.0. Hazen’s approximation was originally determined for uniformly graded sands, but it can provide rough but useful estimates for most soils in the fine sand to gravel range.
Textural determination of hydraulic conductivity becomes more powerful when some measure of the spread of the gradation curve is taken into account. When this is done, the median grain size, d₅₀, is usually taken as the representative diameter. Masch and Denny (1966) recommend plotting the gradation curve [Figure 8.25(a)] using Krumbein’s φ units, where φ = –log₂d, d being the grain-size diameter (in mm). As a measure of spread, they use the inclusive standard deviation, σ₁, where
(8.48)
For the example shown in Figure 8.25(a), d₅₀ = 2.0 and σ₁ = 0.8. The curves shown in Figure 8.25(b) were developed experimentally in the laboratory on prepared samples of unconsolidated sand. From them, one can determine K, knowing d₅₀, and σ₁.
Figure 8.25 Determination of saturated hydraulic conductivity from grain-size gradation curves for unconsolidated sands (after Masch and Denny, 1966).
For a fluid of density, ρ, and viscosity, μ, we have seen in Section 2.3 [Eq. (2.26)] that the hydraulic conductivity of a porous medium consisting of uniform spherical grains of diameter, d, is given by
(8.49)
For a nonuniform soil, we might expect the d in Eq. (8.49) to become d_m, where d_m is some representative grain size, and we would expect the coefficient C to be dependent on the shape and packing of the soil grains. The fact that the porosity, n, represents an integrated measure of the packing arrangement has led many investigators to carry out experimental studies of the relationship between C and n. The best known of the resulting predictive equations for hydraulic conductivity is the Kozeny-Carmen equation (Bear, 1972), which takes the form
(8.50)
In most formulas of this type, the porosity term is identical to the central element of Eq. (8.50), but the grain-size term can take many forms. For example, the Fair-Hatch equation, as reported by Todd (1959), take the form
(8.51)
where m is a packing factor, found experimentally to be about 5; θ is a sand shape factor, varying from 6.0 for spherical grains to 7.7 for angular grains; P is the percentage of sand held between adjacent sieves; and d_m is the geometric mean of the rated sizes of adjacent sieves.
Both Eqs. (8.50) and (8.51) are dimensionally correct. They are suitable for application with any consistent set of units.
8.8 Prediction of Aquifer Yield by Numerical Simulation
The analytical methods that were presented in Section 8.3 for the prediction of drawdown in multiple-well systems are not sophisticated enough to handle the heterogeneous aquifers of irregular shape that are often encountered in the field. The analysis and prediction of aquifer performance in such situations is normally carried out by numerical simulation on a digital computer.
There are two basic approaches: those that involve a finite-difference formulation, and those that involve a finite-element formulation. We will look at finite-difference methods in moderate detail, but our treatment of finite-element methods will be very brief.
Finite-Difference Methods
As with the steady-state finite-difference methods that were described in Section 5.3, transient simulation requires a discretization of the continuum that makes up the region of flow. Consider a two-dimensional, horizontal, confined aquifer of constant thickness, b; and let it be discretized into a finite number of blocks, each with its own hydrogeologic properties, and each having a node at the center at which the hydraulic head is defined for the entire block. As shown in Figure 8.26(a), some of these blocks may be the site of pumping wells that are removing water from the aquifer.
Let us now examine the flow regime in one of the interior nodal blocks and its four surrounding neighbors. The equation of continuity for transient, saturated flow states that the net rate of flow into any nodal block must be equal to the time rate of change of storage within the nodal block. With reference to Figure 8.26(b), and following the developments of Section 2.11, we have
(8.52)
where S_S5 is the specific storage of nodal block 5. From Darcy’s law,
(8.53)
where K₁₅ is a representative hydraulic conductivity between nodes 1 and 5. Similar expressions can be written for Q₂₅, Q₃₅, and Q₄₅.
Figure 8.26 Discretization of two-dimensional, horizontal, confined aquifer.
Let us first consider the case of a homogeneous, isotropic medium for which K₁₅ = K₂₅ = K₃₅ = K₄₅ = K and S_S1 = S_S1 = S_S2 = S_S3 = S_S4 = S_S. If we arbitrarily select a square nodal grid with Δx = Δy, and note that T = Kb and S = S_Sb, substitution of expression such as that of Eq. (8.53) into Eq. (8.52) leads to
(8.54)
The time derivative on the right-hand side can be approximated by
(8.55)
where Δt is the time step that is used to discretize the numerical model in a time-wise sense. If we now convert to the ijk notation indicated on Figure 8.26(c), where the subscript (i, j) refers to the nodal position and the superscript k = 0, 1, 2, . . . indicates the time step, we have
(8.56)
In a more general form,
(8.57)
where
(8.58)