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The spatial responses of two types of time domain refiectometry (TDR) probes used for measuring the water content of soils and other porous materials are examined. We investigate the effect of spatial variation in the water content of the sample on the measured value of the water content for both the coaxial cylinders probe and the two parallel wires probe. The response of the instrument to water content distributions that are perturbed slightly from a uniform distribution is calculated, using the conventional electrostatic treatment of electromagnetic wave propagation in the transverse electromagnetic (TEM) mode. Under this small perturbation assumption, we show that the spatial weighting function is approximately proportional to the distribution of electromagnetic energy between the electrodes for uniform water content, calculated from a solution of Laplace's equation for each probe type. For coaxial probes, most of the energy (and hence most of the measurement sensitivity) is concentrated around the inner cylinder in a "skin effect" if the ratio of the radii of the inner and outer cylinders is too small. For parallel wire probes, most of the measurement sensitivity is close to the wires if the wire diameter is too small compared to the spacing between them. This can cause significant errors if there is an air gap close to the wires or soil around the wires has been compacted by the process of inserting them into the soil.

This paper introduces the general theory of water exclusion from, or entry into, subterranean holes from steady uniform downward unsaturated seepage. Buried holes serve as obstacles to the flow and so increase water pressure at parts of the hole surface. When downward seepage is fast enough and/or the hole is large enough, water pressure increases to the point where a seepage surface forms and water enters the hole. Contrary to the conventional picture drawn from capillary statics, hydrodynamics shows that the larger the hole the more vulnerable it is to water entry. Cavity shape is important also. Applications include optimal design of configurations of tunnels and underground repositories (e.g., for nuclear wastes) against entry of seepage water. The theory also embraces the disturbance of seepage flows by buried impermeable obstacles such as stones and structures. The quasi‐linear exclusion problem for circular cylindrical cavities is solved. Both exact solutions and simple asymptotic results are found, and graphs and tables presented. The moisture field about the cavity exhibits upstream and downstream stagnation points and retarded regions: “roof‐drip lobes” in which the moisture content and downward flow velocity are augmented by water essentially deflected from the cavity roof, and the “dry shadow” region of reduced moisture content and flow velocity, essentially sheltered by the cavity. A practical consequence is that we can establish, for any given combination of downward seepage velocity, cavity radius, saturated hydraulic conductivity K1, and sorptive number α, whether or not seepage water will enter a cylindrical cavity.

Abstract. We define the sample area in the plane perpendicular to the long axis of conventional and alternative time domain reflectometry (TDR) probes based on the finite element numerical analysis of Knight et al. [1997] and the definition of spatial sensitivity of Knight [1992]. The sample area of conventional two-and three-rod probes is controlled by the rod separation. Two-rod probes have a much larger sample area than three-rod designs. Low dielectric permittivity coatings on TDR rods greatly decrease the sample area. The sample area of coated rod probes decreases as the relative dielectric permittivity of the surrounding medium increases. Two alternative profiling probes were analyzed. The separation of the metal rods of Hook et al. [1992] probes controls the size of the sample area. Reducing the height or width of the rods improves the distribution of sensitivity within the sample area. The relative dielectric permittivity of the probe body does not affect the sample size. The sample size of the Redman and DeRyck [1994] In addition to the ability to accurately measure a soil property, the volume of porous medium sampled is an important characteristic of any sampling method. We use the spatial In addition, we examine the influence of changes in both the probe design and the relative dielectric permittivity of the surrounding medium on the sample areas of these probes. Definition of the Spatial Weighting FunctionIt was shown by Knight [1992] that the spatial weighting function Wo(X, y) for a TDR probe surrounded by a medium with a near-uniform distribution of relative dielectric permittivity, K(x, y), is given by Wo(x,y) = .The weighting function has the property that ff•wo(x,y) dA=l.The electrostatic potential distribution, •o(X, y), corresponds to a uniform value of K o of the relative dielectric permittivity in the region, 1•, surrounding the probe. Knight [1992] showed that when K is not uniform, the weighting function, w(x, y), depends on the distribution of the relative dielectric permittivity, K(x, y), and is given by w(x,y) = , Knight et al. [1997] showed that the weighting factor, w(x, y), can be determined for any given distribution of dielectric permittivities in the transverse plane. In brief, the method followed is to determine numerically the potential distribution, •o(X, y), for the metallic rods forming a probe with none of the nonmetallic probe components present if those rods were placed in a homogeneous porous medium. Then the potential distribution, •(x, y), is determined numerically for the same geometry with all of the probe components present. The gradients of these potential distributions are used in (3) to define the spatial weighting factors throughout the transverse plane for each probe design. Heterogeneous K DistributionsA number of probe designs incorporate nonmetallic probe components of known K that lie within the sampling volume of the probe. To compare the performance of probes and to optimize their designs, it is important to know how these materials affect the spati...

The dual‐probe heat‐pulse (DPHP) method is useful for measuring soil thermal properties. Measurements are made with a sensor that has two parallel cylindrical probes: one for introducing a pulse of heat into the soil (heater probe) and one for measuring change in temperature (temperature probe). We present a semianalytical solution that accounts for the finite radius and finite heat capacity of the heater and temperature probes. A closed‐form expression for the Laplace transform of the solution is obtained by considering the probes to be cylindrical perfect conductors. The Laplace‐domain solution is inverted numerically. For the case where both probes have the same radius and heat capacity, we show that their finite properties have equal influence on the heat‐pulse signal received by the temperature probe. The finite radius of the probes causes the heat‐pulse signal to arrive earlier in time. This time shift increases in magnitude as the probe radius increases. The effect of the finite heat capacity of the probes depends on the ratio of the heat capacity of the probes (C0) and the heat capacity of the soil (C). Compared with the case where C0/C = 1, the magnitude of the heat‐pulse signal decreases (i.e., smaller change in temperature) and the maximum temperature rise occurs later when C0/C > 1. When C0/C < 1, the magnitude of the signal increases and the maximum temperature rise occurs earlier. The semianalytical solution is appropriate for use in DPHP applications where the ratio of probe radius (a0) and probe spacing (L) satisfies the condition that a0/L ≤ 0.11.

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Calculations for the transient electromagnetic (TEM) method are commonly performed by using a discrete Fourier transform method to invert the appropriate transform of the solution. We derive the Laplace transform of the solution for TEM soundings over an N‐layer earth and show how to use the Gaver‐Stehfest algorithm to invert it numerically. This is considerably more stable and computationally efficient than inversion using the discrete Fourier transform.

Abstract. Fluid-filled gaps or dielectric coatings around parallel-wire transmission lines affect the ability of time domain reflectometry (TDR) to measure the water content of soils and other porous materials. We use a steady state, two-dimensional, finite element numerical solution of Laplace's equation to analyze these effects. We prove that the numerically determined electrostatic potential distribution and boundary fluxes can be used to calculate the equivalent relative dielectric permittivity measured by TDR by comparing the results of the numerical model with those obtained using existing analytical solutions for special cases. We then analyze the effects of fluid-filled concentric gaps that completely or partially surround TDR rods. The results show that an analytical solution due to Annan [1977b] for nonconcentric gaps can be used as a good approximation to predict the effect of concentric gaps or coatings that completely surround the rods. Coatings or gaps filled with low relative dielectric permittivity materials have a greater impact on the measured relative dielectric permittivity than those filled with high dielectric media. An increase in the thickness of the gap or coating for given rod diameters and separations increases the impact of the coating. To a lesser degree, the impact of a coating of a given thickness decreases with an increase in the ratio of the rod diameter to the rod separation. A gap or coating of a given thickness and relative dielectric permittivity will have a greater impact on the response of a three-rod probe than on that of a two-rod probe with the same rod diameter and separation of the outermost rods. Partial air gaps surrounding less than 30 ø of the rod circumference are not likely to affect the probe response significantly.