CHAPTER 1
INTRODUCTON
Natural aquifers are never homogeneous, but their hydrologic properties are often
assumed to be uniform in order to use analytical solutions for evaluating well tests, and
because the data needed to characterize heterogeneities are often unavailable (Eykholt, et
al. 1999). Lateral heterogeneities associated with geologic structures are probably
common in aquifers (Park, et al. 2000) and may affect the results and interpretation of
well tests (Dawson and Istok, 1991). Moreover, lateral heterogeneities can affect
horizontal ground water flow; therefore, a tool to characterize heterogeneities will have
important applications in a variety of hydrogeological scenarios.
Analyses for interpreting well tests conducted in the vicinity of lateral
heterogeneities have been known in the petroleum industry for around 40 years
(Maximov, 1962), but with a few exceptions (Nind, 1965; Fenske, 1984; Butler, 1991)
they have been largely overlooked by the hydrogeology community. As a result, the
basic effects of lateral heterogeneities on the type curves used to analyze well tests are
poorly known, and the opportunities for characterizing lateral heterogeneities using well
tests have been under utilized by hydrogeologists.
The purpose of this research is to describe the basic effects that idealized
heterogeneities have during well tests, and evaluate the feasibility of using well tests to
recognize and characterize heterogeneities. This chapter contains an overview of
geologic features that may create simple heterogeneities in aquifer properties, along with
a review of analyses of well tests that include the effects of heterogeneities. The
objectives and approach of this investigation are also described in more detail in the
following pages.
Geologic Examples of Idealized Heterogeneities
Geologic features that can produce lateral heterogeneities in aquifers include
faults, steeply dipping beds, facies changes, and igneous intrusions. Certainly these
features may span a diverse range of forms, but in many cases they can be characterized
by vertical, planar contacts separating aquifer properties. Some features may resemble a
single vertical contact, where one aquifer material is juxtaposed against another. Other
features more closely resemble a vertical strip of one material embedded in another
aquifer material that is aerially extensive. As a result, many geologic features that are
complex in detail can probably be approximated using one of two idealized forms:
A Two Domain heterogeneity consists of two aquifer materials of different
properties separated by a vertical, planar contact.
A Vertical Strip heterogeneity consists of a vertical band of one aquifer material
embedded in another material with different properties where two vertical, planar
contacts bound the vertical strip.
Faults in rock or unconsolidated material are common features that cause lateral
heterogeneities in aquifers. Faults can be either barriers or conduits, and they can
influence groundwater flow from the pore to the regional scale (Rojstaczer, 1987). Two
different lithologies can be juxtaposed by displacement along a fault (Fig. 1.1a). Vertical
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a
b
Figure 1.1
Faulting producing heterogeneities. a) Juxtaposition of two aquifer
materials such as sand and clay creating a lateral change in hydraulic
properties. b) Damage zone created during faulting producing a strip with
different hydraulic properties than the aquifer.
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displacements along faults can juxtapose flat lying beds, and strike-slip displacements
can juxtapose dipping beds of differing hydrologic properties.
The faulting process itself may change the hydrologic properties of a formation.
Vertical displacement through an assemblage of flat-lying beds may trap fine-grained
sediments in the fault plane and smear them across coarse-grained units (Antonellini and
Aydin, 1994; Yielding, 1997). By trapping clay, the permeability of the fault may be less
than the adjacent formations. This effect can also occur in sedimentary rocks where
interbedded shales can be smeared along the fault by drag folds (Yielding, 1997).
Deformation associated with faulting may either reduce or increase permeability
over a finite band. Faults that cut sand can cause cataclasis, where the sand grains are
crushed to produce a material of fine, angular grains that can be tightly packed
(Antonellini and Aydin, 1994), and thus lowering the permeability with respect to the
surrounding material (Yielding, 1997). As a result of grain-size reduction, the porethroat radii are reduced and cause the permeability and porosity of the fault rock to be
less than that of the matrix (Engelder, 1974). Cataclasis is the dominant process of
porosity and permeability reduction in siliciclastic rock (Engelder, 1974)
Faults that cut crystalline rock or well-cemented sands are commonly
characterized by a damage zone that is bounded by a planar contact on either side of the
fault zone (Fig. 1.1b). Open, permeable fractures may develop where there is minor
displacement because of less opportunity for gouge to form between the moving surfaces.
This will give the fault zone a high porosity and permeability and thus serve as a conduit
for groundwater movement.
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Steeply dipping beds can also create lateral heterogeneities in areas where
significant folds or ramp faulting has occurred. Two thick, steeply dipping formations
may create a lateral change in aquifer properties at their contact (Fig. 1.2a). This creates
two aquifers laterally adjacent to each other. Permeable sands between thick shale units,
or shale layer between sandstone will create long, strip-like features (Fig. 1.2b).
Facies changes can produce large, lateral variations in the hydraulic properties of
sedimentary units. For example, carbonate reef facies that grade laterally into marine
clay may result in noticeable changes in hydraulic conductivity (Fig. 1.3a). Material
deposited as a karst limestone reef facies can be highly permeable, 10-4 to 102 cm/s
(Freeze and Cherry, 1979), due to primary porosity or fractures and dissolution features.
In contrast, the permeability of marine clay is many orders of magnitude less, 10-11 to 10-7
cm/s (Freeze and Cherry).
In a terrestrial setting, permeable channel sands embedded in fine-grained
floodplain deposits can also form lateral heterogeneities (Fig. 1.3b). Deposits of channel
sand typically resemble the river channel in which they were deposited. They are a long
permeable strip bounded on either side be lower permeability material. In some
locations, the sandy strip resembles the meanders of a river, whereas in other locations
the channels form a broad band of sand flanked by fine-grained sediments (Sharp, 1988).
Areas underlain by igneous rocks may be characterized by lateral heterogeneities
that result either from patterns of fractures, or from the geometry of the igneous bodies
themselves. The contact between a batholith or stock and adjacent country rock could
resemble a vertical, planar contact in aquifer properties (Fig. 1.4a). The contact between
a salt dome and enveloping sedimentary rock would have a similar geometry.
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a
b
Figure 1.2
Dipping Beds. a) Thick steeply dipping beds creating lateral
heterogeneity. b) Series of steeply dipping beds creating a strip.
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Marine
Clay
Reef
a
Floodplain
CHANNEL
Deposit
sand
b
Figure 1.3
Facies changes creating lateral heterogeneities. a) Reef facies grading
into marine clay creating a lateral change in hydraulic properties. b)
Channel sands embedded in floodplain deposits creating a strip of
differing hydraulic properties.
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The hydraulic conductivity of shallow igneous dikes may be significantly different than
their country rock. Fresh igneous rock forming a dike may create a relatively low
permeable layer where it intrudes fractured basalt (Hunt et al., 1988). However, dikes
may fractured and weather more rapidly than country rock. In these cases, dikes may be
form bands of relatively high permeability (Fig. 1.4b).
Confined aquifers may become unconfined under certain conditions, where a
confining layer pinches out. The transition from a confined to an unconfined aquifer
creates lateral change in storativity. Another feature that can change an aquifer from
confined to unconfined is where the confining layer has been eroded, exposing the
confined aquifer (Fig. 1.5).
Previous Work
This research builds upon a body of previous work in the hydrogeology and
petroleum engineering literature. There have been several transient and steady state
solutions that deal with determining the drawdown from a well test in the presence of an
idealized lateral heterogeneity. The early analyses focus on the two-domain problem, and
only recently have the effects of an arbitrarily located, vertical strip been evaluated. .
An early analysis of a well test conducted near a two-domain heterogeneity was
published by Maximov (1962). Maximov’s paper is in Russian, but was translated for
this research by Alla Khramtsova at Clemson University.
Maximov treats the aquifers as thin, two-dimensional sheets. He uses Laplace
and Fourier transforms to reduce the two-dimensional, transient problem to a onedimensional ordinary differential equation that can be solved exactly. He describes an
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Batholith Country
rock
a
Dike
b
Figure 1.4
Igneous rock intrusion producing lateral heterogeneities. a) Batholith
intrusion into country rock creating lateral change in hydraulic properties.
b) Dike intrusion into country rock producing a strip with different
hydraulic properties than the country rock.
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Confined Aquifer
Figure 1.5
Unconfined Aquifer
Confined aquifer changing laterally to unconfined due to pinching out or
erosion of the confining layer.
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analytical inversion of the Fourier and Laplace transforms to derive a closed-form,
approximate expression for the hydraulic heads in the two domains. I conducted a
numerical analysis to verify Maximov’s analytical solution. The results where contoured
to evaluate the drawdown across the contact; however, the analytical solution was unable
to maintain continuity at the boundary between the two regions.
Streltsova (1988) applied the solution described by Maximov to determine the
drawdown at a well near a two-domain heterogeneity. She also evaluates the drawdown
at a well for multiple discontinuities such as two boundaries (no-flow or constant head)
intersecting at right angles, 45-degree angles, a closed square, a circle, and two parallel
boundaries of infinite length with the well between the contacts.
A similar approach was used by Bixel et al. (1963) whose solution utilizes the
diffusion equation to approximate the pressure behavior of compressible fluids. The
solution was developed using a Laplace transform and the exponential Fourier transform
in y. Bixel et al. present explicit expressions for the transform of the solution, but they
are in the form of integrals that are cumbersome to evaluate. Therefore, the diffusivity on
either side of the contact were set equal to each other to simplify the integrals and invert
the solution analytically. This solution is limited to the drawdown on the same side of
the contact as the well, and in some cases, only on a line perpendicular to the contact and
in the pumping well region.
This differs from Maximov’s solution in that Bixel et al. only evaluates the
drawdown field in the pumping well region whereas Maximov characterizes the
drawdown in both regions. Bixel et al. noted that early portions of the drawdown curve
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were independent of the contact. They also points out that long pumping periods
areneeded in order to characterize the drawdown field in the pumping well region in the
vicinity of a two-domain heterogeneity.
Another solution, predicts the drawdown in two aquifers separated by a planar
contact for both the steady state and transient cases (Nind, 1965). This solution is
performed using the method of images. The analytical solution accurately predicts the
drawdown in the aquifer domains, provided the diffusivity (transmissivity divided by
storativity) for each domain is equal. Nind noted that if the radial distance between the
observation point and real well were nearly the same as the radial distance between the
observation point and the image well, there is only a semi-log straight-line at late times.
He also noted that observations made in the region across the contact were similar to
those made from an observation point near the contact and that the presence of the
heterogeneity is not reflected in the results of the well test.
An analytical solution for a transient case of an arbitrary diffusivity ratio was
developed by Fenske (1984) to determine the drawdown at any location where two
aquifers are separated by a vertical contact. This solution applies the method of images
to the transient solution for hydraulic head in two confined, homogeneous aquifers
separated by a planar contact. The solution for hydraulic head in an aquifer with a lateral
change in hydraulic properties requires that three conditions be satisfied at the contact.
The drawdown at the contact between the two aquifers must be equal on both sides. The
discharge from one aquifer across the discontinuity must be equal to the recharge into the
second aquifer, and the tangent law of refraction of equipotential lines must also be
obeyed.
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A general analysis that considers the effects of a vertical strip of arbitrary location
was described by Butler and Liu (1991). The Dirac delta function is used to represent the
source term for the pumping well (Butler and Liu, 1991). This solution determines the
hydraulic head in the strip as well as the surrounding matrix for differences in diffusivity
ratios. The solution is semi-analytical with the inversion of the Laplace and Fourier
transform done numerically. The semi-analytical solution allows for the pumping well to
be located on either side of the strip or within the strip itself.
Objectives
There are three main objectives of this research. One objective is to characterize
how well tests are affected by geologic features that can be idealized as either a twodomain or a vertical strip heterogeneity. Another objective is to determine how idealized
lateral heterogeneities affect the interpretation of apparent properties of the formation.
The third objective is to develop a method for estimating the hydraulic properties of
idealized heterogeneities and test it using a field example.
Approach
The objectives will be met by describing drawdown fields in the vicinity of
idealized heterogeneities, and by describing type cures for these scenarios. Those results
will be interpreted using graphical methods, such as the semi-log straight-line method
(Cooper and Jacob, 1947), along with methods developed during this research. The
graphical methods will be used to interpret a well test conducted in the Piedmont
province of South Carolina.
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The following thesis is organized into seven chapters. Chapter 2 develops the
analytical and numerical solutions used to produce the type curves and numerical well
tests. Chapter 3 presents and discusses the patterns of drawdown fields produced by
different types of heterogeneities. Time-drawdown type curves are presented in Chapter
4 to evaluate the affect of lateral heterogeneities. Chapter 5 presents a method for
determining the hydraulic properties of idealized heterogeneities using drawdown curves.
A field example is presented in Chapter 6 to test the feasibility of characterizing idealized
lateral heterogeneities using time-drawdown curves. The results are summarized in
Chapter 7.
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