Tien-Chang Lee

Numerical evaluation of exponential integral: Theis well function approximation

This paper reviews various approximation methods for evaluating one form of the exponential integral known as the Theis well function W(u) and presents an algorithm that can be easily implemented to ones desired accuracy for 0 < u < ∞. The algorithm is based on a combination of a fast-converging series representation for small u and an easy-implementing Gauss-Laguerre quadrature formula when u becomes large. The partition point for u varies with the desired program accuracy and efficiency. The error analyses suggested that a partition point in the interval of 1 ≤ u ≤ 4 can yield sufficiently accurate results for practical applications of groundwater problems with minimum computational effort. A comparison of the results revealed that the popular Stehfest inverse Laplace transform technique generally yielded a relative error several orders of magnitude greater than that obtained by the proposed algorithm. For large u, the approximation accuracy increased rapidly using the quadrature formula while the results of the Laplace inversion technique showed numerical oscillation between some positive and negative values.

Validation of aquifer parameter determination by extrapolation fitting and treating thickness as an unknown

Abstract A genetic algorithm is used here to guess-estimate a close-to-true set of trial values as input to a three-staged quasi-linear inverse modeling scheme for the determination of aquifer parameters. To validate the parameter determination, in addition to the conventional measures of misfit root mean squares (rms) and distribution, the aquifer thickness is treated as an unknown parameter and the model parameters are further evaluated by comparing the expected drawdown with the observed drawdown at wells which are not used for parameter determination (extrapolation fitting). The method is tested with synthetic and observed drawdown data from five partially screened monitoring wells in a water-table aquifer. Test results for synthetic data doped with random errors indicate that modeling based on two or more well data can yield satisfactory parameter values and extrapolation misfits in an ideal aquifer. For field data, the results indicate that a model misfit on par with the standard error of the data is achievable for each individual well or a combination of two wells but the extrapolation misfit distributions are generally biased and their rms are far greater—possibly due to aquifer heterogeneity. Consistent parameter values can be obtained from the geometric means for multiple runs of the genetic-inverse modeling of one-, two-, three-, and four-well data. Our test aquifer can be represented by a set of parameters with 10 to 15% consistency, including transmissivity, storativity, vertical-to-horizontal conductivity ratio, and storativity-to-specific yield ratio, as affirmed by model aquifer thicknesses that deviate less than 10% from the actual thickness.

Gravitational attraction of solids of revolution: Part 1: Vertical circular cylinder with radial variation of density

Integral expressions are derived for the vertical component of gravitational attraction arising from vertical circular cylinders and horizontal circular disks with radial variation of density. The expressions contain a single elliptic integral of either the first or second kind and are evaluated easily by numerical integration. Various closed-form solutions are obtained for special cases. For constant density, the expressions reduce to well-known solutions as well as approximations that are based on the solid angle and line element, thus providing a unified theme for the derivations of those solutions and approximations. Numerical implementation is discussed in terms of practical application.

Distortion in resistivity logging at shallow depth

Owing to the proximity of an insulating ground surface, normal resistivity logging at shallow depths (less than 30 m) can yield an apparent resistivity that exceeds 200% of the formation resistivity for a homogeneous medium. The distortion is more acute for long‐normal than for short‐normal logging. Three examples from a landfill site in southern California are presented to show such distortion. The patterns of distortion are similar for logging devices consisting of either two point‐source electrodes or one point‐source and one finite length, line‐source electrode. The former electrode array is a generally accepted approximation of the latter. However, the simulated apparent resistivity for the line‐source array is greater than that for the point‐source array at any given depth. A resistivity contrast between the formation and the borehole fluid can shift the magnitude of the background apparent resistivity but does not significantly alter the pattern of distortion. The magnitude of the distortion can be...

A simple method for the absolute measurement of thermal conductivity of drill cuttings

We present a method for absolute measurement of thermal conductivity of drill cuttings. The simplicity of the apparatus makes it suitable for nondestructive use of cuttings and for sample sizes too small to be measured with a needle probe. Because the measurement is absolute, no calibration standards are required. Samples are placed in a Plexiglas cup with a lid containing an electric heat source. The base of the cup is placed in good thermal contact with an aluminum‐block heat sink. Upward and radial heat losses are minimized with styrofoam insulation surrounding the cup. The accuracy of the method was estimated by cross‐measurement of selected samples with a well‐calibrated needle probe. Results indicate that errors in measurement are less than 5 percent for sample conductivities greater than 0.8 W/m ⋅ K if the heat loss through the styrofoam insulation is accounted for. Reproducibility is typically within 3 percent. An axisymmetric finite‐element model which simulates the temperature distribution of th...

Groundwater level and hydrochemistry in the San Jacinto Basin, Riverside County, California

Abstract The San Jacinto basin, structurally a sediment-filled graben (4 km by 40 km) formed by right-stepping en echelon faulting in the right-slipping San Jacinto fault zone, provides most of the water demand in the San Jacinto valley. Pumping of groundwater in the past 80 yeas had caused water level to drop at some locations by 90 m. Analysis of seasonal and long-term fluctuations of water levels suggests at least four hydrologic subregimes in the basin. Stiff and Piper diagrams indicate four major types of subsurface water. Major ion concentrations appear stable over the recorded period from 1951 to 1986 despite repeated changes of water levels in the past 48 years. Hydrologic discontinuities are recognized from contour maps of four-year mean winter water elevations and four-year mean concentrations of total dissolved solids (TDS). Some discontinuities coincide with faults while others probably signify sedimentary facies boundaries. A trough of low TDS (reflecting ancient channel deposits of high hydraulic conductivity) lies southwest of the present channel of the San Jacinto River. This implies the San Jacinto River has migrated northeast probably in response to tectonic tilting of the basin. That TDS trough also suggests that most of the subsurface water leaves the basin about 7 km south of the present river exit. Because of its mixing with high-TDS water from the northwest, the northwest moving subsurface water appears to have been displaced southward before it exists the basin. Contours of sulfate, which is almost absent in the water from the northwest, provide a better indication of exit location.

A seismic refraction and reflection study across the central San Jacinto Basin, Southern California

The San Jacinto Basin is a northwest‐trending, pull‐apart basin in the San Jacinto fault zone of the San Andreas fault system in southern California. About 24 km long and 2 to 4 km wide, the basin sits on a graben bounded by two strands of the San Jacinto fault zone: the Claremont Fault on the northeast and the Casa Loma Fault on the southwest. We present a case study of shallow structure (less than 1 km) in the central basin. A 2.75-km refraction line running from the northeast to southwest across the regional structural trend reveals a groundwater barrier (Offset I). Another line, bent southward and continued for 1.65-km, shows a crystalline basement offset (Offset III) near an inferred trace of the Casa Loma Fault. Although a basement refractor was not observed along the 2.75-km line, a mismatch between the estimate of its minimum depth and the basement depth determined for the 1.65-km line suggests that an offset in the basement (greater than 260 m) exists around the junction of the two refraction lin...

Gravitational attraction of solids of revolution: Part 2: General expressions

A general integral expression is formulated for the gravitational attraction due to any solid of revolution based on the attraction of vertical, semi-infinite, circular, cylindrical shells. The expression accommodates solids with radial variation of density. For solids with constant density, further generalization to arbitrary orientation is made through coordinate transformation that considers both vertical and radial components of attraction. Special solutions for solids with simple geometry are listed. Practical applications are demonstrated by forward gravity modelling the effects of volcanic islands (cylindrical core bounded by an outer exponentially trending surface) and inclined finite-length cylinders. Corresponding expressions for the magnetic case are discussed in terms of using Poissons relation.

Determination of thermal properties and formation temperature from borehole thermal recovery data

Thermal recovery in boreholes cooled by circulation of drilling mud has been modeled for estimating formation temperature and thermal conductivity. Coupled with a finite‐element simulation of heat conduction, inverse modeling for the desired parameters starts with a genetic algorithm that feeds initial estimates of model parameters to an iterative quasi‐linear inversion scheme. In addition to using the rms misfit between the computed and observed borehole temperatures, the results are assessed by comparing or constraining the model formation temperature with a value obtained conventionally from an asymptotic temperature–time relation for a steady line source. The model conductivity is further evaluated for equality with a conductivity value, which is estimated through simulation of heat exchange between the formation and circulating mud. Test results on synthetic data and two sets of highly noisy borehole data from Lake Baikal in Russia indicate that the two equality criteria in temperature and conductivi...