USGS Professional Paper 1703—Ground-Water Recharge

in the Arid and Semiarid Southwestern United States—

Appendix—1

Thermal Methods for Investigating Ground-Water Recharge

By Kyle W. Blasch, Jim Constantz, and David A. Stonestrom

Introduction

Recharge of aquifers within arid and semiarid environments is defined as the downward flux of water across the 

regional water table. The introduction of recharging water 

at the land surface can occur at discreet locations, such as in 

stream channels, or be distributed over the landscape, such 

as across broad interarroyo areas within an alluvial groundwater basin. The occurrence of recharge at discreet locations 

is referred to as focused recharge, whereas the occurrence of 

recharge over broad regions is referred to as diffuse recharge. 

The primary interest of this work is focused recharge, but 

regardless of the type of recharge, estimation of downward 

fluxes is essential to its quantification.

Direct measurement of water flux in the subsurface is 

difficult, prompting investigators to pursue indirect methods. 

Geophysical approaches that exploit the coupled relation 

between heat and water transport provide an attractive class 

of methods that have become increasingly reliable and 

widely used in investigations of recharge—for example, 

in chapters D–H, J, and K of this volume. This appendix 

reviews the application of heat to the problem of recharge 

estimation. Geophysical methods other than heat are presented in appendix 2.

Tutorials on measuring and modeling heat as a tracer of 

hydrologic processes have appeared elsewhere (Stonestrom 

and Blasch, 2003; Niswonger and Prudic, 2003). The objective of this appendix is to provide a fairly complete account of 

the theoretical underpinnings together with a comprehensive 

review of thermal methods in practice.

Like chemical tracers, heat can come from natural 

sources or be intentionally introduced to infer transport properties and aquifer recharge. The admission and redistribution 

of heat from natural processes such as insolation, infiltration, 

and geothermal activity can be used to quantify subsurface 

flow regimes (fig. 1). Heat is well suited as a ground-water 

tracer because it provides a naturally present dynamic signal and is relatively harmless over a useful range of induced 

perturbations; however, artificially exchanging heat with 

ground-water systems can change the hydraulic properties 

and fluid fluxes of interest. Specific properties that depend on 

temperature include viscosity, density, and surface tension, all 

of which affect hydraulic conductivity and fluid flow.

Thermal methods have proven valuable for recharge 

investigations for several reasons. First, theoretical descriptions of coupled water-and-heat transport are available for 

hydrologic processes most often encountered in practice. 

These include land-surface mechanisms such as radiant heating from the sun, radiant cooling into space, and evapotranspiration, in addition to the advective and conductive mechanisms 

that usually dominate at depth. Second, temperature is theoretically well defined and readily measured. Third, thermal 

methods for depths ranging from the ground surface to depths 

of hundreds of meters are based on the same physical principles. Finally, numerical codes for simulating heat and water 

transport are widely available.

Investigators began using subsurface temperatures to 

delineate recharge areas and infer directions of ground-water 

flow about the turn of the 20th century. During the 1960s, 

analytical and numerical solutions for simplified heat- and 

fluid-flow problems became available. These solutions, though 

one-dimensional and otherwise restricted, provided a strong 

impetus for applying thermal methods to problems of liquid 

and vapor movement in systems ranging from soils to geothermal reservoirs.

The combination of fast processors, massive data storage, 

and efficient matrix techniques now provide numerical solutions to complex, three-dimensional transport problems. These 

solutions allow researchers to take advantage of the considerable information content routinely achievable in high-accuracy 

temperature work.

Theoretical Framework

Darcy (1856) demonstrated that the movement of water 

through a porous medium is the product of the medium’s 

hydraulic conductivity and the gradient of total head, where 

total head is composed of a gravitational and a pressure component. Electrical, chemical, and thermal gradients represent 

potential-energy fields that also are capable of inducing fluid 

flow. In most cases, induced fluid, chemical, and energy fluxes 

can be approximately conceptualized as linearly independent 

summations of the products of gradients acting on a given 

domain times a set of corresponding coupling (“phenomenological”) coefficients, or constants of proportionality (Freeze 

and Cherry, 1979; Warrick, 2003). Following Warrick (2003), 

the hydraulic flux Q (L t–1) is given as the summation of pressure, gravitational, and thermal gradients, X (L L–1), multiplied 

by the coupling coefficients, L (L t–1):
354 Geophysical Methods for Investigating Ground-Water Recharge

Heat flow (conduction) 

Heat flow (evapotranspiration) 

Heat flow (radiation) 

Heat flow (conduction 

plus advection) 

Focused recharge 

Diffuse recharge 

Unconfined aquifer 

EXPLANATION 

Bedrock 

Fault 

Aquitard 

Partly confined 

aquifer

(Minimal) Diffuse Basin Recharge 

Recharge Channel Focused 

Block Mountain 

Front 

Mountain 

Figure 1. Conceptual model of coupled water and heat transport through focused (for example, channel and mountain block) recharge 

and diffuse (for example, basin) recharge processes. Thermal energy is sustained by radiant heating from the sun and geothermal 

heating from the Earth’s interior.

, (1)

where the subscript j denotes pressure, gravitational, and thermal gradients, and the subscript i denotes water in the present 

case or, with appropriate changes to units, heat.

Darcy’s law is thus a special case of equation 1. The 

flow of liquid water through isothermal but variably saturated 

media is induced by a hydraulic gradient composed of pressure and gravitational terms:

. (2)

Substitution of gradients and coupling coefficients produces:

, (3)

where KH is the hydraulic conductivity (L t–1), which depends 

strongly on soil-water content θ (L3

 L–3), ψ is pressure head 

(L), and z is elevation head (L). The symbol ∇ is the del operator denoting (here) the gradient of a scalar in three dimensions.

Equation 3 can be generalized to include temperature 

gradients and vapor flow: 

, (4)

where Q is now the total water flux (vapor plus liquid) due 

to hydraulic, thermal, and temperature gradients (L t–1), Dθ

is the moisture diffusivity (L2 t

–1), and DT

 is the sum of vapor 

and liquid thermal moisture diffusivity (L2t

–1T–1). The first 

term on the right-hand side of equation 4 is a diffusionformalized hydraulic flux that is actually driven by a gradient in pressure head, which—with some restrictions—can 

be related to the gradient in moisture content by means of 

the soil-moisture characteristic ψ(θ). The second term is a 

liquid-plus-vapor diffusive flux driven by the temperature 

gradient. The third term is the gravitational hydraulic flux, 

written here for liquid water only. This latter simplification 

is possible because the specific density of liquid water is 

some 40-thousand times higher than water vapor at standard 

temperature and pressure.
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 355

Temperature Gradients

Conditions frequently encountered in nature produce 

large gradients in hydraulic head that cause water at an initial 

temperature to flow into a new location with a contrasting 

temperature. For example, alpine streams can rapidly transport 

near-freezing meltwater to warmer portions of the channel at 

lower altitudes. Irrigation water applied to agricultural fields 

often has a sharply contrasting temperature from that of ambient soils and subsoil sediments.

Fluid movement is usually dominated by pressure and 

elevation gradients rather than by temperature gradients, 

at least for high-to-medium water contents. Nevertheless, 

Hoekstra and Miller (1967) documented thermally induced 

migration of liquid water at the pore scale in partially frozen 

soils. And at larger scales, Bredehoeft and Papadopulos (1965) 

described redistribution of water in the saturated zone due to 

density gradients arising from spatial variation in temperature.

Arid and semiarid sediments are often far from saturation. Because the hydraulic conductivity of dry materials 

is many orders of magnitude below that of water-saturated 

materials, naturally occurring hydraulic gradients—though 

often large—are insufficient to drive appreciable amounts 

of liquid movement. Over an appreciable range of low-tointermediate water contents, temperature gradients tend to 

dominate the movement of water, predominantly as vapor 

(Philip and de Vries, 1957; Rose, 1963; Walvoord and others, 

2004). Scanlon and Milly (1994) simulated thermally driven 

vapor fluxes in near-surface sediments of a semiarid basin in 

the Chihuahuan Desert. Acting over many years, thermally 

induced vapor movement accounted for the deepening of 

tritiated water (3

H–O–1H) relative to radioactive chloride ions 

(36Cl–) generated by above-ground nuclear-weapons testing 

in the 1950s and 1960s (Scanlon, 1994; Scanlon and Milly, 

1994). As a practical matter, early designs for heat-pulse 

probes produced problematic amounts of moisture movement 

due to induced temperature gradients (Ren and others, 1999; 

Ren and others, 2000).

Distribution of Heat in the Subsurface

The background distribution of temperature in the 

subsurface reflects the flows of thermal energy from two 

primary sources—the cyclic radiant heating and cooling from 

above, and nearly constant geothermal heating from below 

(fig. 1). Spatial flows of heat are strongly influenced by 

climate and geology. Shallow temperature regimes vary with 

latitude, altitude, topography, soil and sediment properties, 

shading, and vegetation, in addition to sources and sinks of 

heat. Soil and sediment properties include composition—particularly soil-organic-matter content—and structure, layering, and water content.

Subsurface sources of heat include microbial metabolism, radioactive decay, magmatic intrusions, and tectonic 

activity. Geothermal energy provides heating from deep 

within the Earth’s crust, mantle, and core. The outward 

propagation of heat from the planet’s interior results in a 

characteristic geothermal gradient. The shape of the geothermal profile reflects (1) the local thickness and thermal 

properties of the crustal rocks through which the heat is 

escaping towards space, (2) radiogenic additions of heat from 

local crustal rocks, and (3) regional magmatic emplacement 

(Lachenbruch and Sass, 1977).

At the land surface, temporal fluctuations in temperature due to changes in deep geothermal heating are negligible 

compared to those due to (1) diurnal fluctuations in incoming 

(predominantly shorter wave) and outgoing (predominantly 

longer wave) radiation and (2) sensible heat exchanges with 

overlying air. Bare-soil surface temperatures typically lag 

the radiant diurnal cycle by approximately one hour. Surface 

heating of wet soils is decreased appreciably by the large 

latent heat of evaporating water. For soils with vegetative 

cover, soil surface temperatures are reduced as a consequence of shade, transpiration, and the thermal mass of the 

canopy. Under a vegetated canopy, soil-surface temperatures 

change mostly by contact with air, and thus lag the daily 

cycle of air temperature rather than the radiant diurnal cycle.

The cyclical nature of vertical heat exchange is manifested in characteristic downward-pinching temperature 

envelopes (Heath, 1964; Lapham, 1987). As oscillating temperature signals propagate downward from the surface, heat 

is exchanged with a progressively greater mass of sediments. 

This results in signal attenuation manifested in a decrease in 

the width of temperature envelopes with depth (fig. 2). Annual 

temperature envelopes capture the range and persistence of 

annual cooling and heating and extend wider and deeper than 

diurnal temperature envelopes. Temperature envelopes at 

depth narrow to a point where cyclic variations are no longer 

detectable. As shown in subsequent sections, the depths of 

temperature envelopes depend sensitively on hydraulic fluxes 

within the sediments (fig. 2).

Heat Propagation Through Conduction

If the movement of fluids is temporarily not considered, 

then the flow of thermal energy from a radiatively heated or 

geothermal source, for example, can be described by Fourier’s 

first two laws of heat conduction. Fourier’s first law states that 

heat flow in a homogeneous material is directly proportional 

to the temperature gradient:

, (5)

where qT is the thermal (heat) flux (E L–2 t–1) and KT is the 

thermal conductivity (E L–1 t–1 T–1). Dimensions E, L, T, and t 

represent energy, length, temperature, and time. The thermal 

conductivity is approximately independent of temperature but 

varies with water content, as indicated. Equation 5 is analogous to Darcy’s law (eqn. 3).
356 Geophysical Methods for Investigating Ground-Water Recharge

Annual (or daily) streambed temperature range 

Stream 

Downward flux 

Upward flux 

Z = 10 m (or 0.5 m) 

DEPTH (

Z ) 

TEMPERATURE 

Minimum 

annual (daily) 

temperature 

Maximum 

annual 

(daily) 

temperature 

Figure 2. Ranges of sediment temperatures versus depth, Z, 

for gaining stream conditions (green lines) compared with losing 

stream conditions (red lines), over daily or annual cycles. The 

depth at which the temperature becomes constant depends upon 

the upward or downward flow of water through the sediments. 

For annual profiles, this depth may be 10 meters or more for 

downward flow versus less than a few meters for upward flow. 

For diurnal cycles, the depths at which temperatures become 

constant are shallower by the square root of (365/1) for a neutral 

flux. (Modified from Constantz and Stonestrom, 2003.)

Fourier’s second law of heat conduction states that the 

rate of change in thermal energy stored in a volume element of 

thermally conducting medium is equal to the divergence of the 

heat flux:

, (6)

where s is the mass density (M L–3) and cs is the specific heat 

capacity (E M–1 T–1) of the medium (in this case soil or sediment).

Combining equations 5 and 6, and defining thermal diffusivity as

, (7)

produces an equation for the conductive flow of heat in four 

dimensions:

. (8)

Conductive heat transport occurs by molecule-to-molecule transfer of thermal energy in solids and semi-solids, in 

which the molecules have no long-range motion. As with all 

forms of heat, the movement of energy is from higher temperature to lower temperature.

Equation 8 can be solved analytically or numerically 

depending on the complexity of the boundary conditions and 

system geometry. In one dimension, the governing equation 

can be written (Carslaw and Jaeger, 1959; van Wijk and de 

Vries, 1963a):

, (9)

where z is the depth below the land surface (L). Equation 9 

assumes a homogeneous, isotropic medium in which the thermal conductivity is constant. The diurnal temperature signal at 

the upper boundary, T(zo, t), is frequently approximated by a 

sinusoidal forcing function:

, (10)

where Ta is the amplitude of the temperature wave at the surface (T), Tm is the mean temperature (T), P is the period of the 

wave (t), and t

o

 is the time that the amplitude is zero (t).

The solution to equation 9 with an upper-boundary condition represented by equation 10 and a lower-boundary condition T = Tm at infinite depth is:

, (11)

where D is the damping depth (L):

. (12)

The damping depth conveys several useful attributes of 

the thermal wave, including its speed of propagation:

. (13)

In addition, the amplitude of the temperature wave falls to e–1

(≈ 0.37) at a depth D. The phase of the temperature wave shifts 

180 degrees with respect to the surface at a depth z = Dπ.

Equation 13 is suitable for describing annual as well as 

diurnal cycles. The thermal wave described by the annual cycle 

can penetrate about 19 times deeper than the diurnal wave, the 

square root of 365 being approximately 19. Peerklamp (1944) 

provides solutions for amplitude and phase of thermal waves as 

a function of depth for the case of layered systems.

The addition of nonflowing water in the pore space 

causes an increase in the volumetric heat capacity as well as 

the thermal conductivity. To the degree that heat is transported 

only by conduction, the solution just given remains applicable. The change in thermal conductivity is more pronounced 

than the change in the thermal heat capacity, resulting in an 

increased thermal diffusivity. As shown above, the damping 
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 357

depth of the thermal wave is directly proportional to the square 

root of the thermal diffusivity. Thus, contrary to what might be 

expected, increasing saturations result in a faster (and deeper) 

propagation of a thermal signal into the profile (fig. 3).

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

0 0.5 1 1.5 2 2.5 3 3.5 

TIME, IN DAYS 

S U I S L E C SE E R G E D NI ,E R U T A R E P M E T 

20-cm, dry 

10-cm, dry 

20-cm, saturated 

10-cm, saturated 

Figure 3. Conductive heat transport in a sandy soil at 10-cm and 

20-cm depths for dry (dashed lines) and saturated (solid lines) 

water contents, computed with typical thermal properties of Rillito 

Creek sediments (chapter H, this volume).

Heat Propagation Through Conduction plus 

Advection 

The previous section considered only conductive heat 

transport. Advective heat transport associated with pore-water 

movement changes the distribution of temperatures described 

by equation 11. Advection is the transport of heat by the 

organized movement of fluid from one location to another. 

The term “convection” as used especially in the older literature is synonymous with the term “advection” as used here. In 

the context of ground-water recharge, the advection of heat is 

primarily through the flow of liquid water and thus depends 

primarily on hydraulic-flux density, water content, and initial 

and boundary conditions.

The nonisothermal transport of water through a variably 

saturated porous medium can be described by the following 

generalization of the Richards equation:

, (14)

where DT is the thermal moisture diffusivity (L2 t

–1T–1), Dθ

is the hydraulic moisture diffusivity (L2 t

–1), and t is time. 

Richards’ original equation combined the equation of continuity with Darcy’s law for isothermal conditions to describe 

the change of water content due to capillary and gravitational 

gradients alone (Richards, 1931).

The corresponding heat-transport equation balances the 

change in stored thermal energy with the inflow and outflow 

of heat through conduction, advection, dispersion, and sources 

and sinks within the porous medium (Kipp, 1987; Nasser and 

Horton, 1992; Healy and Ronan, 1996). The following equation represents contributions from these sources in homogeneous and isotropic sediments:

, (15)

where  is the sediment porosity (L3 L–3), Cw and Cs are the 

volumetric heat capacities of water and bulk sediment, respectively, KH is the bulk thermal conductivity, DH is the hydrodynamic dispersion tensor (L2

 t–1), Q is the hydraulic flux (L t–1), 

q* is the rate of water added per volume of porous medium 

from an external or internal source (t–1), and T*is the temperature of that water source (T).

The left-hand side of equation 15 represents the change 

in thermal energy stored in the liquid and solid phases; heat 

stored in the gas phase is assumed to be negligible. The 

change in stored thermal energy is balanced by the terms on 

the right, beginning with the energy transported by thermal 

conduction followed by the energy transported by thermomechanical dispersion. The remaining two terms represent 

the energy transported by advection and sources or sinks of 

heat within the domain. Thermomechanical dispersion caused 

by mixing of water within the porous medium is treated by 

a hydrodynamic dispersion tensor term DH analogous to that 

from solute-transport theory, defined as follows (Healy and 

Ronan, 1996):

, (16)

where T is the transverse dispersivity of the porous medium 

(L), |v| is the magnitude of the hydraulic flux-density vector 

(L t–1), vi is the ith component of the hydraulic flux-density 

vector (L t–1), ij is the Kronecker delta operator (equal to 1 if 

i=j; otherwise equal to 0), L is the longitudinal dispersion (L), 

and i and j are indices (i = 1, 2, 3; j = 1, 2, 3) representing the 

orthogonal Cartesian directions. Heat transport is coupled to 

fluid flow not only through the thermal and hydraulic gradients, but also through the water-content- and temperaturedependent phenomenological coefficients.

The equations for coupled water and heat transport, 

14–16, provide the mathematical foundation for the application of thermal methods to recharge investigations. Thermal 

methods can be grouped into shallow-, intermediate-, and 

deep-application ranges, although the boundaries between the 

depth ranges are not distinct. Similar mechanisms and principles apply to all ranges.
358 Geophysical Methods for Investigating Ground-Water Recharge

A key factor differentiating shallow, intermediate, and 

deep thermal profiles with respect to recharge analysis is the 

periodicity or stability of the temperature profile. At shallow 

depths, aperiodic fluctuations in weather and radiation along 

with periodic fluctuations on diurnal and annual periods 

generate large transients in temperature. At intermediate 

depths temperature profiles are more stable, changing cyclically with the seasons, such that changes due to vertical 

water fluxes combine with the annual temperature cycle to 

produce temperature profiles intermediate between those of 

shallow, diurnally varying profiles and the nearly constant 

temperature profiles at depth (fig. 4). Temperature profiles 

at depth are mostly controlled by the interplay of water 

movement with deep-seated, upwardly directed geothermal 

gradients. Deep profiles tend to be stable, changing only on 

decadal or longer time scales in most cases. Shifts in temperature regimes at large depths indicate substantial changes 

in recharge conditions.

Zero flux Upward flux Downward flux 

Surface 

TEMPERATURE 

H T P E D 

A. B. Downward background 

thermal gradient 

Upward background 

thermal gradient 

Figure 4. The influence of vertical water flow on the shape of 

background thermal gradients. Upward, zero, or downward fluxes 

of water superimposed on A, an upward (geothermal) background 

gradient, or B, a downward background thermal gradient. Example 

constructed from data obtained at Rillito Creek, Arizona (chapter 

H, this volume).

Emergence of Thermal Methods

Thermal methods of soil-moisture and ground-water 

investigations came into widespread use in the early to mid 

1960s. Whereas earlier investigations, such as those of Smith 

(1910) and Rorabaugh (1956), used heat qualitatively to 

locate sources of ground-water recharge and ground-water 

flow paths, it was not until Suzuki (1960), Stallman (1965), 

and Bredehoeft and Papadopulos (1965) published analytical 

solutions to the coupled heat and water transport equations 

that thermal methods for estimating rates of subsurface water 

movement came into general use. Suzuki (1960) used temperature measurements to estimate infiltration and deep percolation in rice paddies by assuming saturated, vertical, steady 

state flow in a homogeneous medium with a sinusoidal daily 

surface temperature. Suzuki presented an approximate analytical solution to a simplified form of equation 15, assuming the 

temperature at depth was equal to the mean surface temperature. For one-dimensional, vertical transport, the governing 

equation is (Suzuki, 1960):

u

u  u

u 
 u

u

¥

§

¦  ́

¶

μ T

t

K

c

T

z

c

c

v

T

z

T

s s

w w

s s

z 





2

2 . (17)

Stallman (1960; as reported in Stallman, 1963) independently developed a three-dimensional generalization of equation 17 from first principles and remarked that an analytical 

solution was available for the one-dimensional case, attributed 

to Robert E. Glover. Stallman (1965) published an exact solution for the one-dimensional problem solved approximately by 

Suzuki. Like Suzuki, Stallman represented the upper boundary condition as a sinusoidal temperature fluctuation; however, Stallman investigated the propagation of annual as well 

as diurnal temperature signals. Stallman estimated that his 

exact analytical solution could be used to detect percolation 

rates as low as 0.1 cm/d by using annual temperature signals 

and 0.3 cm/d by using diurnal signals. At lower velocities, 

heat transport through conduction would become dominant. 

Taniguchi and Sharma (1993) employed Stallman’s (1965) 

one-dimensional solution to determine recharge beneath two 

forested sites near Perth, Australia. Ground-water recharge 

rates obtained by this method were consistent with results 

obtained with chemical tracers.

Additional early approaches to solving the one-dimensional coupled transport equation include a type-curve method 

developed by Bredehoeft and Papadopulos (1965), discussed 

in the following section, and a heat-balance method developed 

by Wierenga and others (1970). Wierenga developed the heatbalance method to model temperatures in irrigated soils. For 

purposes of analysis the soil is discretized into horizontal layers, and different mathematical approaches are used to simulate early and late-time behavior. For periods of infiltration and 

appreciable soil-water redistribution, interlayer transfers of 

heat are adjusted to match measured temperatures by algebraically adjusting transfers of water. After the transfer of water 

between layers becomes negligible, subsequent (conductiveonly) heat flow is modeled with an explicit finite-difference 

form of the one-dimensional heat equation. Taniguchi and 

Sharma (1993) evaluated the heat-balance method and found 

that, while useful for estimating water fluxes near the surface, 

its accuracy deteriorated quickly with depth. They concluded 

that the heat-balance method is limited to the upper two meters 

because changes in temperature below this depth could not be 

resolved with sufficient accuracy.
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 359

Early Thermal Methods for Estimating Deep 

Recharge

The fact that ground-water movement in deep sediments 

causes anomalies in the Earth’s geothermal gradients was realized by the 1930s (Van Orstrand, 1934; Bullard, 1939). Deviations from linearity in vertical thermal profiles were proposed as 

a means to identify the extent and thickness of low-permeability 

layers near the production interval of an aquifer (Norris and 

Spieker, 1962). The discovery of subterranean hot springs thousands of feet below land surface enabled Lovering and Morris 

(1965) to estimate amounts of fresh water coming from the 

surface versus geothermal water coming from depth on the basis 

of a simple conductive heat-flow and water-mixing model. Schneider (1972) used deviations in thermal profiles to qualitatively 

infer the presence of aquifer recharge beneath a losing reach of 

a river in the presence of ground-water pumpage.

Bredehoeft and Papadopulos (1965) presented a general 

analytical solution to the differential equation describing 

the steady flow of heat and fluid in the deep subsurface and 

showed how it could be used to estimate vertical ground-water 

velocities in and hydraulic conductivities of confining layers. The solution assumes vertical steady-state ground-water 

flow through a homogeneous layer. The authors presented 

type curves corresponding to various combinations of medium 

properties and flow rates. They estimated that with available 

thermometry one could detect Darcian water fluxes as small as 

0.1 cm/d in a 10-m thick semiconfining layer. The type-curve 

method quickly became popular for determining ground-water 

velocities because it required minimal data collection and was 

easy to apply (Cartwright, 1970; Sorey, 1971; Cartwright, 

1979, Boyle and Saleem, 1979).

Cartwright (1970) constructed contour maps of temperatures at 500 feet by linear interpolation of surface data and data 

from deep exploration boreholes throughout Illinois and compared them with theoretical maps derived from extrapolations 

of geothermal gradients measured at the bottom of the boreholes and the assumption of no fluid flow. The residuals of predicted minus interpolated temperatures showed regions of positive and negative anomalies corresponding to areas of inferred 

recharge and discharge. Cartwright (1970) applied the analysis 

of Bredehoeft and Papadopulos to estimate water movement 

at the basin scale and, in later work, to estimate ground-water 

movement between two aquifers (Cartwright, 1979). Similarly, 

Sorey (1971) calculated vertical flow velocities on the order 

of 0.3 m/yr hundreds of feet below the land surface in semiarid basins in New Mexico and Colorado. Velocities from the 

type-curve method were similar to those from water-budget 

and pump-test analyses. Sorey recommended that the typecurve method be restricted to cases where the geology was 

well characterized and relatively simple. Stevens and others 

(1975) recommended that temperature profiles used in thermal 

analysis be measured in a borehole that has sat idle for several 

years, allows no circulation from one interval to another inside 

or along the outside of the casing, and has a diameter no greater 

than 5 cm. Furthermore, they warned against using metal 

casing, which conducts heat preferentially, and cement-based 

grout, which generates heat while curing.

Numerical methods were also introduced starting in the 

1970s to interpret temperature anomalies in terms of water 

movement. Supko (1970) measured temperatures in the 

unsaturated zone to delineate ground-water flow in the Tucson Basin of Arizona by using a “valley mapping function” 

developed from a numerical heat- and fluid-transport model. 

Areas of recharge and paths of subsequent ground-water 

movement were delineated by tracks of relatively low ambient 

subsurface temperatures.

Expansion of Thermal Methods

During the 1970s and early 1980s, field and numerical 

studies were conducted in Europe and the United States to 

determine the feasibility of storing thermal energy in confined 

and unconfined aquifers (Tsang and Hopkins, 1982). Data 

from field experiments greatly improved the understanding 

of fluid flow resulting from temperature gradients. Sophisticated numerical models of multiphase heat and fluid transport 

emerged. A large number of thermal investigations undertaken 

during this period, including modeling techniques and calibration methods, are summarized in Beck and others (1989).

During the past two decades, thermal methods have 

been increasingly applied to problems of water availability in 

arid and semiarid communities. Many of these communities 

depend on ground water as a primary source for agricultural, 

domestic, and industrial uses. As shown by the examples 

in chapters D–H, J, K (this volume), thermal methods have 

been increasingly applied to identify and enumerate channel 

infiltration and basin recharge. A major factor has been the 

improvement in field instrumentation, including temperature 

sensors and data-storage technology. A recent development is 

the use of fiber-optic-cable techniques—which, while relatively expensive, allow nearly continuous measurements of 

temperature in space and time (Day-Lewis and others, 2006; 

Selker and others, 2006; Gungle, 2007).

Investigations of Channel Infiltration and 

Recharge

Channel recharge is defined as recharge that originates as 

channel infiltration. Infiltration rates measured at the streambed 

provide upper bounds for channel recharge. They are upper 

bounds because some of the infiltrating water will be subsequently lost to evapotranspiration. Water evaporates directly 

from the channel following the cessation of ephemeral flow. In 

addition, water that moves laterally from the channel may also 

not reach the water table, returning instead to the atmosphere 

through evapotranspiration adjacent to the channel.

Thermal methods are advantageous for determining seepage losses because useful measurements can be obtained from 
360 Geophysical Methods for Investigating Ground-Water Recharge

the sediments beneath the channel surface. Buried sensors 

are less prone than surface instruments to loss by scour and 

deposition or damage by flood-borne debris. Measurements 

can be obtained during large flows that render surface measurements impractical and unsafe. Infrequent events are more 

easily captured. Traditional seepage losses are determined as 

the difference in stage-based discharge at two or more points 

along a reach. Stage-based estimates of discharge tend to be 

inaccurate, especially for large, non-steady flow events. Infiltration rates inferred by using thermal methods are essentially 

point measurements, differentiating them from seepage losses 

inferred by discharge measurements, which involve integration 

of infiltration rates along a reach.

Lapham (1987) monitored temperature profiles beneath 

streams to determine the magnitude and direction of water 

exchanges between streams and underlying aquifers. He 

modeled the coupled transport of water and heat by obtaining 

solutions to explicit finite-difference representations of equation 17. This approach allowed the use of measured (nonsinusoidal) temperature boundary conditions at the channel surface 

as well as a ground-water temperature not equal to the mean 

surface value. Lapham’s approach provided simultaneous estimates of effective hydraulic conductivities and vertical flow 

rates for water.

Silliman and Booth (1993) identified characteristic thermographs of gaining and losing reaches of channels by measuring streamflow and sediment temperatures along a stream 

in Indiana. Gaining and losing reaches were readily identified 

by inspection of the amplitude of the diurnal temperature signal in the streambed versus in flowing water. Losing reaches 

were characterized by large thermal amplitudes associated 

with radiant daytime heating and nighttime cooling (fig. 5). 

Gaining reaches were characterized by much smaller thermal 

amplitudes because ground water entering the stream from 

an underlying aquifer is buffered from diurnal forcing. Thus, 

temperature profiles within the streambed of a channel provide 

immediate information on gaining and losing portions of 

streams. Lee (1985) had previously used a similar analysis to 

identify areas of ground-water inflow and discharge in lakes. 

Constantz (1998) applied a similar analysis to identify gaining and losing reaches in alpine environments. Lawler (2002) 

identified reaches in a semiarid stream that were gaining and 

losing during different parts of the year. During winter the 

water table was above the altitude of the streambed, producing 

gaining conditions. During spring and summer ground-water 

pumping for agriculture together with riparian evapotranspiration dropped the water table below the streambed, producing 

losing conditions.

Silliman and others (1995) developed solutions to equation 17 for arbitrarily varying upper-boundary conditions 

by using the mathematical technique of superposition. The 

approach required the assumption of fully saturated conditions. Silliman and others used this approach in estimating 

stream-bed losses from temperatures measured at the base 

of the water column and at a depth of 25 cm in underlying 

stream-bed sediments.

1 m 

e g a G ma e r t S 

r e t e m o z e i P 

PERENNIAL STREAM, 

LOSING REACH 

time 

time 

e r u t a r e p m e T 

w o l f m a e r t S 

1 m 

e g a G ma e r t S 

r e t e m o z e i P 

PERENNIAL STREAM, 

GAINING REACH 

time 

time 

e r u t a r e p m e T 

w o l f m a e r t S 

Figure 5. Idealized stream interactions with ground water, with 

sediment thermographs. A, A perennial stream gaining water 

from the underlying sediments. B, A perennial stream losing 

water to the underlying sediments (modified from Constantz and 

Stonestrom, 2003).

Pure Advection Solution

Infiltration rates in the streambed vary widely. Wankiewicz (1984) analyzed thermal profiles beneath two streams 

in the Northwest Territories of Canada and concluded that 

thermal transport beneath one stream was almost entirely by 

conduction, whereas thermal transport beneath another stream 

was almost entirely by advection. The Peclet number can be 

used to assess the relative magnitude of advection and conduction for heat transport in ground-water recharge applications. 

The Peclet number is defined as follows (van der Kamp and 

Bachu, 1989):
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 361

, (18)

where l is the characteristic length of the medium (mean 

grain diameter) (L),  is the thermal diffusivity of the saturated soil sediments (L2

 t–1),  is the porosity (L3 L–3), and Q

is the hydraulic flux density (L t–1). Values much less than 

2×10–4 indicate that heat transport is dominated by conduction (Silliman and others, 1995). Conversely, values much 

greater than 2×10–4 indicate that heat transport is dominated 

by advection.

Several investigators considered the limiting case of 

purely advective heat transport (Nightingale, 1975; Taniguchi and Sharma, 1990; Constantz and Thomas, 1996). These 

studies determined water fluxes from measurements of the 

vertical propagation velocity of temperature fluctuations 

into saturated or nearly saturated sediments. Constantz and 

Thomas (1996) used the propagation velocity of diurnal temperature signals to estimate seepage rates beneath a strongly 

losing channel. Their analysis starts by expressing equation 

17 as (Stallman, 1965):

, (19)

where the first term is the gain or loss of heat by conduction 

(the divergence of the conductive heat flux given by Fourier’s 

first law), the second term is the gain or loss of heat by water 

movement (advection), and the right-hand side is the change in 

stored heat per unit volume of sediment per unit time. Here Q

is the component of the hydraulic flux density in the z direction (volume of water crossing a unit area of sediment per 

unit time; L t–1). If the conductive flux of heat is negligible in 

comparison to the advective flux of heat, equation 19 can be 

simplified by eliminating the first term and solving for hydraulic flux Q (Constantz and Thomas, 1996):

, (20)

where VT is the propagation velocity of the temperature signal 

(L t–1). Note that the specific heat capacity of sediment scs

in equations 19 and 20 is expressed per unit volume of bulk 

(wet) sediment (total bulk volume of all phases), whereas 

the specific heat capacity of water wcw is per unit volume of 

water (liquid only). Thus the term in parenthesis in equation 

20 has the units of volumetric water content, making equation 20 parallel to the well-known equation relating the Darcy 

flux (hydraulic flux density) to the (macroscopic) propagation 

velocity VC of an ideal chemical tracer, Q = VC (see chapter 

E, this volume, for example). Here  is the volume of water 

per bulk volume of sediment (water content). The propagation velocity of a chemical tracer will thus exceed that of the 

temperature signal by a factor of about two:

, (21)

 with parameter values typical of saturated sandy sediments 

(scs = 2.6 MJ m–3  ̊C–1; ρwcw = 4.2 MJ m–3  ̊C–1; and θ = 0.3; 

see table 1 in Stonestrom and Blasch, 2003).

Constantz and Thomas calculated seepage losses from 

the arrival times of the daily temperature peak measured in the 

upper 3 m of channel sediments. Infiltration rates determined 

with equation 20 were about twice estimates determined from 

streamflow losses measured in the adjacent (upstream) reach 

of the channel, after correcting for evaporation. Analysis of the 

temperature data showed that seepage losses from intermittent 

streamflows were rapidly converted to deep percolation, rather 

than spreading laterally into near-surface storage that could 

later be lost to evapotranspiration.

The pure advection solution is readily applicable to

multiday streamflow events with small variations in flow rate 

and large variations in diurnal temperature (fig. 6). Analysis 

becomes increasingly difficult as flow rates become more

highly variable and as temperature variations at the land surface diminish. Stream segments receiving contributions from

tributaries, surface runoff, and snowmelt can have erratic temperature signatures in the flowing water. As a consequence,

subsurface thermographs are also erratic, making it difficult to

conduct an analysis based on diurnal peaks.

Numerical Techniques for Variably Saturated 

Flow, Multiple Layers, and Nonperiodic Forcing

Numerical solutions to the coupled heat- and fluidtransport equations are required for most cases involving 

variably saturated flow, nonperiodic boundary temperatures, 

or non-trivial geometries. Sophocleous (1979) used a numerical model to simulate variably saturated flow of water and 

temperature through laboratory and field plots. Jaynes (1990) 

employed a similar model to evaluate the influence of layering 

on infiltration in agricultural soils.

TIME, IN DAYS 

S U I S L E C SE E R G E D NI ,E R U T A R E P M E T 

10 

12 

14 

16 

18 

20 

22 

Infiltration flux 1.13 meters per day 

302 303 304 305 306 

D epth - 0.5 m 

D epth - 1.0 m 

D epth -1.25 m 

Figure 6. Example of the use of thermal amplitudes to estimate 

Darcy flux for predominantly advective heat transport with 

negligible conductive heat transport (unpublished data from 

Rillito Creek, Arizona).
362 Geophysical Methods for Investigating Ground-Water Recharge

The ability to simulate the transport of heat has been 

incorporated into popular numerical models for solving problems of variably saturated water movement. Well-documented, 

publicly accessible examples include VS2DH (Healy and 

Ronan, 1996), SUTRA (Voss, 1984) and HYDRUS (Simunek 

and others, 1998). These models solve the Richards (1931) 

equation for unsaturated water movement coupled to the 

advection-diffusion equation for heat transport by either finitedifference or finite-element methods. Because of nonlinearities, 

solutions are obtained by iterative approximation. The models 

differ somewhat in their treatment of conduction, advection, 

and dispersion, and they differ considerably in their representation of thermal and hydraulic parameters. Comparing results 

from independently developed codes thus reduces uncertainties 

related to the numerical solution of governing equations.

Thermal and Hydraulic Properties

Applying thermal methods to problems of water-flux measurement requires the specification of thermal and hydraulic 

properties. The study of Lapham (1987) included a sensitivity 

analysis of stream-aquifer interactions to thermal and hydraulic 

properties. Thermal properties depend on the water content of 

the sediments, with thermal conductivity and volumetric heat 

capacity both increasing as soil-water content increases (fig. 

7). Stonestrom and Blasch (2003) present tables of thermal 

conductivity, thermal diffusivity, and volumetric heat capacity 

for several water contents and discuss techniques for measuring and calculating thermal properties. Hydraulic properties are 

discussed at length in van Genuchten and others (1992).

Because the viscosity and surface tension of water both 

vary with temperature, hydraulic properties vary with temperaWATER CONTENT, IN VOLUME OF WATER 

PER VOLUME BULK SOIL 

clay 

peat 

sand 

0.0 

0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

THERMAL CONDUCTIVITY, IN WATTS 

PER METER PER DEGREE CELSIUS 

clay 

peat 

sand 

VOLUMETRIC HEAT CAPACITY, IN MILLIONS OF 

JOULES PER CUBIC METER PER DEGREE CELSIUS 

1 

0 

2 

3 

4 

Figure 7. Dependence of volumetric heat capacity and thermal 

conduction on water content for selected materials. Dashed 

lines are volumetric heat capacities calculated as described 

in Stonestrom and Blasch (2003). Points are experimentally 

determined thermal conductivities from de Vries (1963). Solid 

curves are empirical fits to the thermal-conductivity data. 

(Modified from Stonestrom and Blasch, 2003.)

ture as well. For example, hydraulic conductivity increases 

with increasing temperature. The dependence of hydraulic 

properties on temperature was investigated in the laboratory by 

Constantz (1982) and Nimmo and Miller (1986). These studies 

found that temperature dependencies of unsaturated hydraulic 

parameters tend to be larger than predicted by viscosity and 

surface tension alone. Jaynes’ (1990) study of infiltration in 

field plots included an evaluation of temperature-dependent 

hydraulic conductivities on infiltration rates. Mitchell and 

others (1990) and Duke (1992) documented large temperature 

effects on infiltration fluxes in agricultural settings. Despite 

the greater-than-expected temperature sensitivity of hydraulic 

properties, the water-content dependence still dominates (van 

Genuchten and others, 1992).

A main reason that thermal techniques are well suited to 

determining water fluxes is the relative lack of dependence of 

thermal conductivity on sediment texture and water content 

relative to the large dependencies of hydraulic conductivity 

on these same parameters (fig. 8; Constantz and Stonestrom, 

2003). Because of this, hydraulic conductivity largely controls 

both the percolation flux and the propagation of thermal signals 

in strongly advective systems (Niswonger and Prudic, 2003).

The large diurnal variations in stream temperature typical 

of arid and semi-arid environments strongly influence streamflow losses and associated ground-water recharge. Constantz 

and others (1994) showed that seepage losses from streams in 

Colorado and New Mexico closely tracked diurnal variations in 

stream temperature, with loss rates fluctuating on the order of 

25 percent with daily temperature cycles that fluctuated in the 

range of 4–18 ̊C and 10–25 ̊C for Colorado and New Mexico 

streams, respectively. These changes were attributed to the 

inverse temperature dependencies of both water viscosity and 

water density. Both streams were located near mountain fronts. 

Measurements of streamflow loss and evapotranspiration at 

the New Mexico site indicated that 95 percent of the seepage 

loss became ground-water recharge. Decreased temperatures 

at night simultaneously reduced infiltration and extended the 

distal range of streamflow. Similar effects were observed in a 

stream in Nevada by Ronan and others (1998) and in a stream 

in Arizona by Blasch and others (2000). In the latter study, postflow drainage and redistribution rates increased as much as 50 

percent with a change in temperature from 14 ̊C to 28 ̊C. The 

temperature effects on seepage velocities were modeled using 

VS2DH by Ronan and others (1998), Bartolino and Niswonger 

(1999), and Bailey (2002). Bailey (2002) showed that seepage 

velocities were more sensitive to hydraulic conductivity than 

stage for conditions encountered in Rillito Creek, Arizona.

Further Considerations for 

Investigation of Specific Recharge 

Processes

Thermal methods offer a variety of approaches for investigating recharge processes, whether it involves infiltration at 
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 363

GRAIN SIZE 

10-3

10-2 

10-1 

100 

101 

102 

103 

THERMAL OR HYDRAULIC CONDUCTIVITY 

Uncertainty in hydraulic conductivity 

versus thermal conductivity 

QT = KTT (Fourier)

QH = KH H (Darcy)

Textural range of 

example 

Thermal conductivity

(KT ) in watts per meter 

per degree celsius

Example range of

thermal conductivity

Hydraulic conductivity

(KH ) in centimeters per day

Corresponding range

in hydraulic conductivity

Figure 8. Uncertainty in hydraulic conductivity versus 

thermal conductivity. The hydraulic conductivity of saturated 

sediments (blue band) is strongly dependent on sediment texture 

(represented here by the stippled 10–90 percent range in grainsize distribution), whereas the thermal conductivity (tan band) is 

almost independent of texture. The vertical width of the hydraulic 

and thermal conductivity bands gives an approximate range of 

each parameter for the example texture. (From Constantz and 

Stonestrom, 2003).

the land surface or percolation at depth. The following section 

reviews pertinent considerations for three main applications of 

thermal methods, the first two of which involve measurements 

at relatively shallow depths—the characterization of ephemeral 

streamflow, and the estimation of streambed seepage fluxes. 

The third section discusses considerations involving the application of thermal techniques to the deep unsaturated zone.

Streamflow Detection

Measuring the duration and extent of streamflow in large 

semiarid and arid basins is important for identifying channels 

that contribute to ground-water recharge. Methods employed 

for monitoring inputs to basin-scale and regional-scale sized 

recharge models need to be inexpensive and reliable. Traditional streamflow-gaging techniques are generally inapplicable 

because of the flashy nature of streamflow, shifting channel- geometries, the wide dynamic range of flow rates, and 

the potential importance of low-flow conditions. Transients in 

flow rates and shifting channel geometries invalidate stagedischarge relations.

Constantz and others (2001) used streambed temperatures to indicate the presence of streamflow in central New 

Mexico and southeastern Nevada. The presence of streamflow 

was identified with large perturbations in temperatures in the 

shallow subsurface due to the transport of heat by advection. 

Benchmark temperature measurements made at similar depths 

adjacent to channel helped identify false positives caused, for 

example, by precipitation without streamflow.

Subsequent studies of ephemerally and intermittently 

flowing channels in the Southwest refined methodologies of 

sensor placement in streambed sediments and developed automated techniques for interpreting thermographs (Lawler, 2002; 

Stewart, 2003; Blasch and others, 2004). Optimal placement of 

sensors depends on the details of flow geometry and thermal 

forcing as well as general meteorologic, geomorphologic, 

and hydrologic conditions. The following section describes 

optimal placement for inferring streamflow timing and extent. 

These considerations are largely the same as for measuring 

sediment temperatures to estimate percolation fluxes.

Optimizing Sensor Placement

When streamflow is not present, the thermal signal 

from radiant heating and cooling of the streambed sediments 

travels into the profile primarily by conduction (fig. 9). As this 

signal—idealized as a sinusoidal wave—propagates into the 

sediments, its amplitude decreases and its time lag increases 

with respect to the surface forcing (van Wijk and de Vries, 

1963a). The magnitude of the ideal diurnal temperature wave 

as a function of depth for a homogenous single layer is (van 

Wijk and de Vries, 1963a):

, (22)

where Tconduction(z) is the amplitude of the sediment temperature 

variation at depth z, To is the amplitude of the diurnal temperature wave at the sediment surface, and D is the damping depth 

(eq. 12). The typical damping-depth value for diurnal fluctuations in sandy soils is about 0.15 m, and in clay soils about 

0.12 m (van Wijk and de Vries, 1963b).

When streamflow is present, the amplitude of the diurnal 

temperature fluctuation at the streambed surface is almost 

always reduced (fig. 9). During daylight hours, a portion of the 

thermal energy from incoming solar radiation is used to raise 

the temperature of the water column and to drive evaporation. 

The result is a lower maximum temperature at the sediment 

surface when streamflow is present compared to dry channel 

conditions. Similarly, during the evening, an overlying water 

column buffers the sediment surface from cool air temperatures, resulting in higher minimum temperatures than when 

streamflow is absent. In addition, infiltration and percolation 

of water through streambed sediments increases the amount of 

heat transported downward due to the advection of heat. As a 

result, thermographs at greater depths show a marked increase 

in the diurnal temperature amplitude in the presence of streamflow (fig. 9).
364 Geophysical Methods for Investigating Ground-Water Recharge 1 m e g a G ma e r t S r e t e m o z e i P 

EPHEMERAL STREAM 

WITHOUT FLOW 

Time 

Time 

e r u t a r e p m e T 

w o l f m a e r t S 0 

A 

1 m 

e g a G ma e r t S 

r e t e m o z e i P 

EPHEMERAL STREAM 

WITH FLOW 

Time 

Time 

e r u t a r e p m e T 

w o l f m a e r t S 0 

B 

Figure 9. Streamflow hydrographs and corresponding 

streambed thermographs for an idealized ephemeral-stream 

channel A, before, and B, after onset of flow (from Constantz and 

Stonestrom, 2003).

As mentioned earlier, Stallman (1965) developed an exact 

analytical solution to the coupled heat- and water-transport 

equation for one-dimensional vertical transport in a uniform 

medium with a steady percolation rate and a sinusoidal temperature variation at the land surface. The diurnal temperaturewave amplitude for the advection-conduction equation 17 is 

(Stallman, 1965):

, (23)

where Tadvection is the amplitude of the sediment temperature 

variation due to the combined influence of advection plus 

conduction, and a (L) is a coefficient based on the hydraulic 

flux density (Darcy velocity) as well as the thermal properties 

of the sediment:

, (24)

where

, (25)

and

. (26)

As discussed previously, the variable cw is the specific 

heat of the fluid and sediment in combination (E M–1 T–1), s

is the density of the fluid and sediment in combination (M 

L–3), KT is the thermal conductivity of the fluid and sediment 

in combination (E L–1 t–1 T–1), Q is the hydraulic flux density 

(L t–1), cw is the specific heat of pore water (E L–3 T–1), and w

is the density of pore water (M L–3). If Q is zero, then equation 24 reduces to the conduction equation and  = D–2. As Q

increases, the transport of heat due to advection increases. The 

predominant heat-transport mechanism within the sediments 

depends upon percolation rate, thermal parameters, and depth. 

For simplicity the following discussion will refer to the transport of heat during the presence of percolation as “advection” 

even though conduction contributes to transport of heat.

Figure 10 contrasts the purely conductive case (conduction without water movement) to the advective case (conduction with water movement). The profiles were generated 

using equations 22–26 with assumed hydraulic and thermal 

parameters for coarse-grained sediments typical of ephemeral streams. The same thermal and hydraulic properties 

were used in each case. Material properties were assumed to 

be homogeneous.

The attenuation of diurnal temperature waves are plotted 

as a function of depth for three different cases of streamflow in 

figure 10A. The first (no-damping) case assumes no reduction in the amplitude of diurnal temperature at the streambed 

surface compared to the air-sediment interface outside of the 

channel. The 50-percent-damping case assumes that the amplitude of the diurnal temperature signal at the streambed surface 

is reduced by 50 percent in the presence of streamflow. The 

third case assumes a 50-percent reduction in fluid flux in addition to the 50-percent water-column damping to investigate 

the relative importance of advective heat flux. Comparison of 

the conductive (no percolation) and advective (percolation) 

diurnal temperature waves as a function of depth shows that 

the conductive thermal amplitude is larger than the advective 

thermal amplitude, but only near the surface (fig. 10). It is also 

shows that percolating water transports heat deeper into the 

profile than conduction alone. Obviously, a requirement for 

using thermographs at any depth to infer streamflow is that 

the hydrologic flux be sufficiently large to cause measurable 

deviations in the amplitude of the diurnal temperature wave 

at that depth when streamflow is present compared to when 

streamflow is absent.
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 365

S R E T E M NI ,H T P E D 

0 

2.5 

0.5 

1.0 

1.5 

2.0 

–0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 

AMPLITUDE DIFFERENCE, IN DEGREES CELSIUS n o i t i s n a r T 

50% dampening, 

 reduced fluid 

 flux by 50% 

50% dampening 

No dampening 

B 

S R E T E M NI ,H T P E D 

0 

0 0.2 1.0 

AMPLITUDE , IN DEGREES CELSIUS 

A 

2.5 

0.5 

1.0 

1.5 

2.0 

0.4 0.6 0.8 

Conduction—No infiltration 

Advection—No dampening, 

 infiltration 

Advection—50% dampening, 

 infiltration 

Advection—50% dampening, 

 reduced infiltration flux by 

 50% 

Figure 10. Amplitudes of thermal waves in a coarse-grained 

streambed. A, Conductive-only (dry channel) and advectiveplus-conductive (flowing channel) thermal-wave amplitudes 

(single solid line and dotted line, respectively). The amplitude at 

the surface is 1°C. The 50-percent dampening case assumes that 

half of the heat signal is lost to flowing water. The 50-percent 

dampening–50-percent fluid-flux reduction considers in addition 

a fluid flux that is half of the previous two cases. B, Difference 

between advective and conductive thermal wave amplitudes as 

a function of depth (from Blasch and others, 2004).

To characterize depth-dependent changes in amplitude 

of the temperature wave resulting from percolation, the conductive temperature wave amplitude (Tconducton) at each depth 

is subtracted from the advective temperature wave amplitude 

(Tadvection) (fig. 10B). When there is no reduction (damping) 

in the amplitude of the temperature wave at the sediment 

surface due flowing water in the channel, the difference in 

conductive and advective temperature-wave amplitudes at the 

surface is zero. At depth, however, there is a marked change 

between conductive and advective temperature-wave amplitudes. When the overlying water column damps the amplitude of the temperature wave at the sediment surface, there 

is a change in the amplitude of the temperature wave both 

at the surface and at depth. Thus, temperature monitoring 

within the profile is in general more reliable and informative 

than monitoring at the surface, because changes in the sign 

of the temperature-wave amplitude are less sensitive to possible thermal buffering by the overlying water column. This 

is especially true for ephemeral streams in which stage can 

quickly vary over a large range.

Equations 22–26 can be used with knowledge of the 

sediment profile and percolation range to estimate optimal 

measurement depths for inferring streamflow. The optimal 

measurement depth for inferring streamflow is that depth at 

which streamflow-induced percolation causes the greatest 

change in temperature-wave amplitude relative to no-flow 

conditions. For the case illustrated, the optimal depth is 

about 0.45 m (fig. 10B). The least-optimal depth is that 

for which the presence of streamflow does not appreciably 

change the temperature amplitude (Tconduction ≈ Tadvection). For 

a given percolation rate, transition depths zt can be identified as the shallowest depth at which the conduction and 

advection temperature-wave amplitudes are equal with and 

without streamflow. The transition depth represents the depth 

at which the temperature-wave amplitude with flow becomes 

larger than the amplitude without flow.

Depths closer to the surface than zt may be subjected 

to scour or influenced by temperature anomalies induced by 

precipitation or cloud shadowing in the absence of flow. In 

extreme scour conditions, sensors can be removed entirely 

from the channel. Even in less extreme cases, scour and 

deposition complicate analysis of streambed thermographs 

(Constantz and others, 2001; Stewart, 2003; Gungle, 2006). 

Scour and fill during flow events dynamically affects the 

depth of the sensors below the sediment surface. This variable depth introduces uncertainty into the analysis even if 

sensor depths are surveyed after each flow event. 

In addition to considerations just discussed, optimal 

depth selection for temperature sensors depends on channel cross-sectional shape, wetted perimeter, and location of 

adjacent vegetation and bank features. Stewart (2003) used a 

numerical two-dimensional model to determine the influence of cross-sectional shape on the propagation of heat both 

vertically and horizontally into the sediments. Rectangularshaped channels produced more transport of heat through a 

combination of advection and conduction than triangularshaped channels for the same constant-head upper boundary 

conditions. Blasch and others (2004) suggested placement of 

sensors in channel constrictions after observing timing errors 

caused by channel migration in otherwise optimal locations. 

Additionally, shading by vegetation, bridges, and steep banks 

anomalously reduced the diurnal amplitude, obscuring analysis (Lawler, 2002; Stewart, 2003).
366 Geophysical Methods for Investigating Ground-Water Recharge

Constantz and others (2001) monitored temperature 

at the near surface (15 cm depth) to successfully infer the 

presence of streamflow. The method was appropriate for the 

streams studied because two essential conditions for the success of this method were met. First, the diurnal temperaturewave amplitude at the surface was sufficiently large that 

temperature fluctuations caused by the passage of clouds, for 

example, did not obscure the diurnal signal. Second, during the presence of streamflow, the overlying water column 

produced a large reduction of the amplitude of the diurnal 

temperature wave at the streambed surface compared to the 

magnitude of the diurnal temperature wave in the dry streambed. The magnitudes of flow-induced perturbations can be 

quite large, especially for summer-monsoonal flows (fig. 11). 

On the other hand, changes in the near-surface diurnal temperature-wave amplitude during streamflow alone are often 

insufficient to infer the presence of streamflow. If antecedent 

pore-water temperatures in the bed sediments match temperatures in the flowing water too closely, it can become difficult 

to detect streamflow (Stewart, 2003).

Constantz and others (2001) and Stewart (2003) considered the effects of precipitation, cloud cover, and sudden 

changes in air temperature on heat transport through the bed 

sediments. In general, precipitation-induced fluid fluxes and 

abrupt air temperature changes do not penetrate as deeply 

below the sediment surface or as quickly as streamflow events, 

so will have a greater influence on sensors near the surface. 

Stewart (2003) compared sensors placed at 0-, 15-, 30-, 50-, 

and 100-cm depths and showed that burial to 15 cm was sufficient to differentiate rainfall infiltration from streamflow 

infiltration in the cases considered. S U I S L E C SE E R G E D NI ,E R U T A R E P M E T 

A 

B 

10 

20 

30 

40 

0 SEPTEMBER OCTOBER 

2000 

10 

20 

30 

40 

0 

NOVEMBER DECEMBER 

Figure 11. Thermographs for A, a depth above the transition depth, and B, a depth below the transition depth. The gray areas represent 

periods of streamflow (from Blasch and others, 2004).

Analysis of Streamed Thermographs to Detect 

Presence of Streamflow

Analysis of streambed thermographs to detect streamflow 

can be as simple as visual identification of streamflow-induced 

temperature anomalies. Although visual inspection has proven 

useful, the technique is subjective and is time consuming for 

large data sets. Also, visual inspection is less effective for 

identifying streamflow events shorter than 24 hours in duration, which are typical of ephemeral flows in southern Arizona 

(Blasch, 2003; chapters H and J, this volume). To overcome 

these limitations, automated techniques were developed that 

incorporate comparative and statistical analyses.

Benchmark measurements made outside of the flowing 

portion of the channel were evaluated by Stewart (2003) for 

automating detection of streamflow. Benchmark comparison 

requires that environmental conditions for the in-channel and 

benchmark sensors be similar. Differences caused by shading, precipitation rates, soil types, and exposure to wind can 

obscure interpretation (Lawler, 2002; Stewart, 2003). Benchmark methods proved less useful for short duration events 

than for long-term events. Data from in-channel probes during dry periods were also evaluated for their ability to provide 

a virtual or surrogate benchmark. Recorded diurnal oscillations and amplitudes during known dry periods were used to 

construct monthly and seasonal dry-benchmark signatures, 

which were then compared to the remaining record. However, 

variations caused by short-term meteorological events proved 

even more difficult to distinguish from streamflow events 

with in-channel benchmarks than with out-of-channel benchmarks (Stewart, 2003).
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 367

Analysis of a streambed thermograph to infer the timing of flow is based on identification of the aforementioned 

temporal changes in the thermograph (fig. 9). As an example, 

two thermographs for a coarse-grained alluvial stream at 

depths above and below the transition depth are presented in 

figure 11. When streamflow is present (indicated in this case 

from a nearby streamflow-gaging station), the diurnal temperature-wave amplitudes in the sensor above the transition 

depth decrease, whereas below the transition depth the diurnal 

amplitudes increase. Streamflow timing can be inferred from 

the thermographs by identifying these changes in thermal 

amplitude above and below the transition depth. The analysis 

can be conducted visually (Constantz and others, 2001) or by 

using statistical analysis (Stewart and Constantz, 1999; Stewart, 2003; Blasch, 2003).

Gungle (2006) reported a small but distinctive additional 

temperature drop in the sediments of ephemeral channels 

immediately after at the cessation of streamflow, which dissipated as the sediments regained the diurnal temperature pattern in effect before streamflow began (fig. 12). He attributed 

this brief temperature minimum to evaporative cooling of 

freshly exposed wet sediments. Gungle used the local temperature minimum as a marker for the cessation of streamflow. 

Similar observations had been previously reported by Geiger 

(1965) and Wierenga and others (1970).

Analysis of Sensor Networks to Detect Extent of 

Streamflow

The deployment of multiple sensors along a single channel or within a network of channels allows relatively inexpensive detection of the geographic extent and temporal duration 

of ephemeral flow events. Several examples have already been 

discussed (Lawler, 2002; Stewart, 2003; Gungle, 2006; chapters D–H, J, and K, this volume). Exemplifying this approach, 

Coes and Pool installed temperature sensors at many locations in stream channels draining the Huachuca and Mule 

10 

8/7/02 8/8/02 8/9/02 8/10/02 8/11/02 8/12/02 

15 

20 

25 

30 

35 

40 

0 

50 

100 

150 

200 

S U I S L E C SE E R G E D NI ,E R U T A R E P M E T 

250 

, E G R A H C S I D N I C I B U C T E E F R E P D N O C E S 

Temperature 

Discharge 

Rapid temperature drop 

Local temperature 

minimum 

Streamflow 

event 

Figure 12. Thermograph and streamflow record from Greenbush 

Draw, southeastern Arizona, indicating the decline in temperature 

at the cessation of streamflow (data and interpretation from Bruce 

Gungle, U.S. Geological Survey, written commun., 2005).

Mountains in southeastern Arizona (chapter J, this volume). 

Information on the temporal and spatial extent of ephemeral 

streamflow was used to estimate the total mountain-front 

recharge for the basin.

Estimation of Percolation Fluxes

While streamflow timing alone provides qualitative 

information on stream-channel infiltration losses, quantifying infiltration rates and subsequent percolation fluxes and 

ground-water recharge amounts requires additional analysis. 

Of interest are not only the easily measured infiltration losses 

from the wetted channel but also the net (or recharging) 

percolation flux (infiltration minus subsequent evapotranspiration losses). Thermal methods have been widely used for 

estimating infiltration and percolation fluxes in small to large 

channels across a range of flow regimes—ephemeral, intermittent, and perennial (Constantz and Thomas, 1996; Constantz 

and Thomas, 1997; Ronan and others, 1998; Bartolino and 

Niswonger, 1999; Constantz and others, 2002; Blasch, 2003; 

Conlon and others, 2003).

Ronan and others (1998) developed a method to determine the vertical percolation flux beneath a streambed by 

solving the coupled equations for heat and water flow within 

the context of a publically accessible numerical model. Adaptation of the model required assigning appropriate thermal 

and hydraulic parameters to the bed sediments and establishing boundary conditions. As a simplification, the modeled 

domains were limited to fully saturated conditions, which 

eliminated the need to specify unsaturated parameters and 

the dependence of thermal and hydraulic parameters on soilwater content.

A common additional simplification is the restriction to 

that portion of the domain characterized by one-dimensional 

vertical fluxes. This restriction is often assumed to be met at 

all times beneath the center of the channel, and elsewhere after 

fully saturated conditions have been established and horizontal 

pressure gradients have decayed. Nevertheless, infiltration at 

the onset of streamflow is multidimensional, as the wetting 

front initially travels subhorizontally into the stream bank as 

well as downward. Water-content measurements in sediments 

of Rillito Creek at the beginning of ephemeral flow showed 

an initially quasi-radial progression of the wetting front, with 

lateral flow velocities nearly as large as vertical flow velocities 

(chapter H, this volume).

Instrumentation for Percolation-Flux Estimates

Estimation of vertical fluxes requires deployment of 

temperature sensors at multiple depths to obtain spatial and 

temporal coverage (fig. 13). Stream temperatures or sedimentsurface temperatures are required to provide data on the upper 

boundary condition for subsequent analysis. Constantz and 

Thomas (1996, 1997) installed probes at five depths within 

the streambed sediments in the Middle Rio Grande Basin in 
368 Geophysical Methods for Investigating Ground-Water Recharge

Possible line of symmetry 

Ground−water 

level 

Stream 

stage 

Temperature− 

measurement 

point 

Well with 

screen 

Figure 13. Conceptual modeling framework when flow through 

streambed is downward and outward (from Niswonger and 

Prudic, 2003.)

addition to water-temperature probes at the surface. Sensors 

were installed directly into streambed sediments at depths of 

30, 60, 105, 150, and 300 cm by auger. Temperatures were 

recorded every 30 minutes. Because the optimal installation 

depths of sensors depend on factors that are unknown a-priori, 

the number of depths in vertical sampling arrays is typically 

about five to seven. A larger number of observation depths 

reduces inaccuracies caused by heterogeneity and also provides insurance against the loss of data due to sensor failure 

or removal by scour.

Bartolino and others installed lines of nested piezometers 

crossing the perennial Rio Grande and manually monitored 

ground-water temperatures over a 24-month period (Bartolino 

and Niswonger, 1999; Bartolino, 2003). An advantage to using 

a cable-logging temperature sensor was the ability to obtain 

high spatial resolution. Measurements were obtained at 15-cm 

intervals for piezometers about 9–16 meters in depth. An 

assumption of this method is that the water temperature inside 

the piezometers equaled the temperature of saturated sediments outside the piezometers.

Bailey (2002) and Coes and Pool (chapter J, this volume) 

applied a similar approach to the unsaturated zone beneath an 

ephemeral stream prone to scour and deposition. Piezometers 

were installed about 10 m into the bed sediments, and an array 

of self-contained temperature sensors was mounted on a cable 

and inserted into the piezometers. The temperature sensors 

included self-contained data loggers. This system enabled continuous data collection within the sediments and ready access 

to probes without having to repeatedly disturb the sediments. 

Temperatures inside the borehole were assumed to be equal to 

temperatures outside the borehole.

To investigate installation considerations related to 

the assumption of vertical fluxes, Ronan and others (1998) 

deployed two-dimensional arrays of temperature sensors 

beneath an ephemeral stream and used thermal modeling techniques to estimate two-dimensional infiltration and 

percolation fluxes. The authors observed vertical and lateral 

flow variations within the sediments due to layering as well 

as to the geometry of channel meandering relative to hillslope 

direction. As previously discussed, Stewart (2003) simulated 

two-dimensional flow in a rectangular channel bottom and a 

triangular channel bottom. Flow was predominantly vertical 

beneath most of the width of the rectangular channel, with lateral components of flow increasing near the edges of channel 

flow. The same pattern was observed beneath the triangular 

channel. Comparison of the two channel geometries indicated 

a greater proportion of vertical flow beneath the rectangular 

channel than beneath the triangular channel for a given head or 

wetted channel cross-sectional area.

Hydraulic and Energy Transport Properties

Hydraulic parameters required for numerical modeling of 

saturated conditions include hydraulic conductivity (a tensor for anisotropic materials), specific storage, and porosity, 

whereas modeling unsaturated conditions requires additional 

parametric functions describing soil-moisture retentivity and 

unsaturated hydraulic conductivity. Required thermal-transport 

parameters include heat capacity and thermal conductivity of the sediment, both as functions of water content, as 

well as thermal dispersivity. Hydraulic and energy-transport 

parameters can be determined from core samples (Bailey, 

2002; chapter J, this volume), literature values (Bartolino and 

Niswonger, 1999), expert judgment (Ronan and others, 1998), 

or model calibration (Bartolino and Niswonger, 1999). Often 

a combination of methods is used to obtain the full set of 

required parameters.

Sensitivity analysis has shown that hydraulic conductivity is the most important parameter controlling infiltration flux 

(Smith and others, 1989; Wang and others, 1989; Ronan and 

others, 1998; Bartolino and Niswonger, 1999). Inverse calibration using automated procedures, such as PEST (Doherty 

and others, 1994) and UCODE (Poeter and Hill, 1998) have 

become routine in channel-recharge investigations (Bartolino 

and Niswonger, 1999; Bailey 2002; Stewart, 2003; Constantz 

and others, 2003). Niswonger and Prudic (2003) provide 

guidance on the application of inverse calibration for thermal 

studies of surface-water–ground-water interactions.

Boundary and Initial Conditions

Infiltration and associated percolation fluxes largely 

depend on the product of the hydraulic gradient and the 

hydraulic conductivity of the sediments. Temperature data 

alone can be insufficient to determine both hydraulic conductivity and water fluxes through inverse modeling procedures. 
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 369

Solutions become better constrained with the consideration of 

additional types of data. Pressure observations are the most 

commonly used auxiliary data for constraining solutions of the 

coupled transport equations. Piezometers (often outfitted with 

recording transducers) can be used to measure pressure gradients in saturated sediments. Similarly, tensiometers can be 

used in unsaturated sediments. Measurement of matric pressure or water content is often required to obtain the required 

parameters for partially saturated conditions. At a minimum, 

hydraulic—as well as thermal—initial and boundary conditions must be specified to run the transport model.

The measured depth of water in the stream channel, if 

available, is usually used as the upper hydraulic boundary condition (Ronan and others, 1998; Bailey, 2002). As discussed 

above, Bartolino and Niswonger (1999) employed rows of 

piezometers to study surface-water–ground-water exchanges 

along the Rio Grande. Hoffmann and others (chapter H, this 

volume) used subsurface pressure sensors to measure hydraulic head during ephemeral flow of Rillito Creek.

The bottom boundary condition can be represented in 

several ways. The bottom boundary is often represented as a 

zero-pressure head boundary corresponding to the measured or 

assumed location of the water table (Ronan and others, 1998; 

Bailey, 2002). Another option is the use of measured pressure 

heads at the bottom of the model domain (Bartolino and Niswonger, 1999). Alternative approaches are to place the lower 

boundary at a great enough distance from the active domain 

that assumed stationary conditions do not affect the active 

domain, or to assume constant vertical gradients of temperature and pressure at depth. The latter two approaches simulate 

percolation as occurring into a semi-infinite half-space.

Initial conditions include pressure-head and temperature profiles in the sediments. For unsaturated media, initial 

water-content conditions have been determined directly from 

core samples (Ronan and others, 1998). Initial water-content 

conditions can also be determined from time-domain-reflectometry measurements (Blasch, 2003; appendix 2, this volume).

Sensitivity of Fluxes to Hydraulic Conductivity

Since modeled fluxes reflect hydraulic gradient and 

hydraulic conductivity, solutions are sensitive to fluctuations in 

stream stage and hydraulic conductivity. While stream stage can 

be measured independently, changes in hydraulic conductivity 

over the course of an ephemeral flow event can be difficult to 

monitor. Bailey (2002) estimated infiltration rates along Rillito 

Creek by using temperature monitoring and inverse simulation 

of heat and water transport through stream sediments. Modeling 

of two sequential streamflow events suggested that the hydraulic 

conductivity of the streambed surface layer changed by four 

orders of magnitude due to sediment redistribution from one 

event to the next. During a single event, the hydraulic conductivity changed by two orders of magnitude.

The data of Ronan and others (1998) showed generally decreasing infiltration rates throughout the duration of a 

multiple-day ephemeral flow event with superimposed diurnal 

oscillations, representing the net effect of several influences. 

The dominant influence was the rapid decrease in hydraulic 

gradient at the sediment-water column interface as infiltration 

proceeds. An opposing influence was the initial increase of 

hydraulic conductivity as water content increases from preflow to saturated conditions. Hydraulic conductivity fluctuated 

diurnally with fluctuating streamflow temperature.

Blasch and others (2006) simulated transient infiltration 

fluxes at the onset of the streamflow events by using measured field data from alluvial channels (fig. 14). The multiple 

transient and steady-state fluxes observed after the onset of 

streamflow were attributed to the layering of streambed sediments. The time between transient and steady-state periods 

was inversely proportional to the antecedent moisture content. 

The results of Coes and Pool (chapter J, this volume) indicated 

that even their longest recorded streamflow event, 5.5 hrs, was 

not long enough for infiltration to reach steady state.

Accuracy of Infiltration and Percolation Fluxes

Infiltration and percolation fluxes calculated from thermal monitoring and numerical models have been compared to 

fluxes calculated by using data from streamflow-gaging stations, stage recorders, chemical tracers, and soil-water content 

measurements. Constantz and Thomas (1996) estimated a 

factor-of-two difference between vertical fluxes calculated by 

using a thermal method and fluxes calculated from streamflow 

data. Ronan and others (1998) reported simulated infiltration 

rates that were within an order of magnitude of seepage rates 

determined from flume data. The difference was partially 

attributed to the difference in the point-scale measurements 

of the thermal method and reach-scale measurements of the 

seepage runs. To evaluate the accuracy of simple heat-based 

flux estimates, Blasch and others (2006) compared thermallydetermined infiltration fluxes near the onset of streamflow 

with continuously monitored soil-water content data. Because 

initial transient rates are higher than eventual quasi-steady 

state rates, errors in cumulative infiltration caused by considering only the quasi-steady behavior ranged from 9 to 25 percent 

for the cases tested. Cumulative error for more typical, shortduration (8-hour) events was about 90 percent.

Niswonger and Prudic (2003) provide guidance for collecting field data and modeling measured temperatures to infer 

hydraulic fluxes. These procedures are designed to broaden 

the applicability of thermal methods while reducing measurement uncertainty.

Thermal Methods for Quantifying Deeper 

Recharge

Focused recharge occurs at discreet locations, such as 

stream channels, whereas diffuse recharge is spread over the 

landscape. Percolating water generally becomes more diffuse with depth (Nimmo and others, 2002); however, focused 

recharge in arid settings can reach great depths. This is par-
370 Geophysical Methods for Investigating Ground-Water Recharge

ticularly true for large volume, spatially discreet inputs, such 

as those that occur within large ephemeral streams, and areas 

that receive large volumes of artificial recharge from irrigation 

(Stonestrom and others, 2004) and percolation ponds (Izbicki 

and others, chapter G, this volume). Nimmo and others (2005) 

provide an overview of thermal methods for the investigation 

of deeper recharge.

A comparison of thermal methods to track shallow, 

moderate, and deep percolating water on progressively slower 

paths toward recharge was presented by Constantz and others 

(2003). Temperature-estimated fluxes were as large as 20 m/d 

for shallow depths immediately beneath a stream channel 

and as small as 0.01 m/d for depths greater than 100 m below 

the surface beneath an alluvial fan. A trend to smaller fluxes 

with increasing depth was partly explained by the spreading 

of downward percolating moisture with depth, although deep 

percolation rates may reflect arrested paleorecharge (Walvoord and others, 2004). In any case, flux divergence beneath 

focused sources of water due to sediment layering generates 

additional attenuation of thermal signals beyond the amount 

expected in homogeneous sediments. The greater the depth of 

the recharge investigation, the greater the need for precision 

in temperature measurements. A precision of 0.1o

C is usually more than adequate for estimating fluxes near the land 

surface, whereas a precision of at least 0.01oC is required to 

detect thermal perturbations at depths greater than a few tens 

of meters.

Studies that used thermal data to examine recharge in 

moderate-to-deep unsaturated zones include those of Reiter 

(1999, 2001), Izbicki and Michel (2002), Constantz and 

others (2003), and Dowman and others (2003). Examining 

unsaturated zones to depths of 100 m and saturated zones to 

depths of 350 m, Reiter used high-precision temperature logs 

to evaluate vertical moisture fluxes in the Rio Grande Rift in 

central and southern New Mexico. Thermal gradients were 

measured with precisions of 0.005o

C/m. Vertical water fluxes 

as low at 1×10–6 m/yr were obtained by fitting quadric and 

cubic polynomials to measured temperature data and matching 

analytical solutions.

Izbicki and Michel (2002) used temperature profiles to 

depths of 30 m for estimating recharge rates and cumulative 

recharge volumes beneath ephemeral and intermittent streams 

across the western Mojave Desert. Air temperatures were 

monitored in cased and grouted boreholes in normally dry 

A 

1.E+ 04 

1.E+ 03 

1.E+ 02 

1.E+ 01 

1.E+ 00 

1.E- 01 

1.E- 02 

1.E+ 04 

1.E+ 03 

1.E+ 02 

1.E+ 01 

1.E+ 00 

1.E- 01 

1.E- 02 

0 

1 

2 

3 

4 

5 

6 

7 

0 

1 

2 

3 

4 

5 

6 

7 

B 

Y A D RE P SR E T E M NI ,N O I T A R T L I F N I Y A D RE P SR E T E M NI ,N O I T A R T L I F N I 

S R E T E M NI ,N O I T A R T L I F N I EV I T A L U M U C S R E T E M NI ,N O I T A R T L I F N I EV I T A L U M U C 

1st steady state 

2nd steady state Possible 3rd stready state 

1st steady state 2nd steady state 

11/4/00 0:00 11/4/00 3:36 11/4/00 7:12 11/4/00 10:48 11/4/00 14:24 11/4/00 18:00 11/4/00 21:36 

4/5/01 21:50 4/6/01 2:38 4/6/01 7:26 4/6/01 12:14 4/6/01 17:02 4/6/01 21:50 

Simulated infiltration fluxes 

Simulated cumulative infiltration 

Calculated infiltration fluxes (using TDR) 

Figure 14. Simulated infiltration fluxes and cumulative infiltration for A, November 4, 2000 and B, April 6, 2001. Infiltration 

fluxes at the onset of streamflow were calculated by using time-domain reflectometry (TDR) water-content measurements 

(from Blasch and others, 2006).
Appendix 1—Thermal Methods for Investigating Ground-Water Recharge 371

washes to estimate the sediment-temperature profiles. Companion boreholes outside the channel were used for no-flow 

controls. Average annual temperatures beneath the washes 

were as much as 1.5oC lower than the average temperatures for 

the control sites. Assuming one-dimensional heat transport by 

conduction and advection, calculated recharging fluxes from 

intermittent runoff ranged from 0.22 to 0.84 m/yr depending 

on distance downstream from the mountain front.

In the study alluded to above, Constantz and others 

(2003) used thermal geophysical logs from boreholes as deep 

as 300 m to estimate recharge in southern Nevada beneath an 

alluvial-fan surface. Recharging fluxes were estimated from 

perturbations of assumed geothermal profiles on the basis of 

analytical solutions to the coupled heat-and-water flow equations as well as numerical solutions generated by TOUGH2 

(Preuss and others, 1999). Estimates of recharging fluxes 

ranged from undetectable to 0.011 m/yr.

Dowman and others (2003) investigated temperature perturbations beneath Rillito Creek during constant-flux streambed infiltration experiments. A 5-m by 5-m bermed area was 

instrumented with buried thermocouples to a depth of 5 m, and 

a 15 m-borehole adjacent to the buried area was logged for 

temperature using a cable-mounted logging thermistor. Temperatures measured by the buried sensors agreed with borehole 

temperatures at depths greater than about 1 m. An imposed 

percolation flux of 0.34 m/d produced a thermal anomaly that 

reached 11o

C of cooling at shallow depth (above the transition point) and 0–2o

C of heating between 4 and 10 m depth 

(below the transition point). Numerical sensitivity analysis 

using VS2DH showed that the temperature of the percolating 

water and percolation rate were the most important factors in 

determining the temperature-profile perturbation. The analysis 

also showed that having the deeper temperature measurements substantially reduced the uncertainty of recharge-rate 

estimates compared to estimates that were based on shallow 

temperature measurements alone.

Conclusions

Thermal methods have emerged as a versatile class of 

geophysical tools for monitoring focused recharge in arid and 

semiarid settings. Compared to other geophysical techniques 

and chemical tracers, thermal methods are relatively simple 

and inexpensive. Heat flows through hydrologic systems 

naturally. Heat can also be artificially introduced and is benign 

over a useful range of applications. The theory for coupled 

heat-and-water transport is well established and incorporated 

into numerical models of variably saturated flow.

Recent investigations have led to improved methods for 

experimental design and data collection. Instrumentation has 

been steadily improving. Additionally, the treatment of unsaturated thermal- and hydraulic-transport processes through 

numerical models provides flexibility not possible 50 years 

ago, when Suzuki first estimated percolation rates beneath a 

field of rice (Suzuki, 1960).

Like other methods for determining recharge rate, thermal 

techniques are limited by sampling accessibility and scale of 

measurement, which often become limiting factors in recharge 

investigations. Nevertheless, thermal methods are particularly 

well suited for monitoring focused recharge beneath unconsolidated alluvial channels that remain dry much of the time.

The coupled relation between heat and water transport 

has been used to identify the occurrence, extent, and timing of 

streamflow. Temperature measurements have also been used 

to indicate gaining and losing reaches of stream channels as 

well as of areas of inflow to lakes. Temperature observations 

beneath the depth of seasonal fluctuations have been used to 

identify the presence and direction of water movement through 

analyses of anomalies in the background geothermal gradient.

Infiltration and percolation estimates determined by thermal techniques are similar to values determined by channelflow-loss measurements and chemical tracers. In addition, 

thermal techniques provide the ability to continuously monitor infiltration and percolation fluxes throughout streamflow 

events. Thermal studies have shown that infiltration rates 

can vary by four orders of magnitude from the onset to late 

stages of ephemeral streamflow not only because of hydraulicgradient decay, but also because of flow-induced changes in 

the hydraulic conductivity of streambed sediments. Fluxes as 

high as 100 m/d have been detected in coarse alluvial sediments, whereas fluxes as low as 0.01 m/d have been detected 

in clayey sediments.

Thermal methods have been used to estimate groundwater recharge rates hundreds of meters beneath stream channels and basin floors. Lateral spreading of percolating water 

often reduces local vertical-flux densities to values substantially lower than those prevailing near the surface. Estimated 

fluxes at depths exceeding 100 m determined by thermal 

methods range from 0–0.01 m/d. Detecting small fluxes at 

such great depths requires extraordinary precision. Measurement depth is, therefore, a major consideration when applying 

thermal methods to estimate recharge in arid environments.

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