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By Lee W. Abramson, William H. Hansmire

The H-3 Project is planning, design, and construction of a 10-mile (16 km)-long final segment of new highway that traverses the Koolau mountain range in Oahu. Interstate H-3 traverses North Halawa Valley on one side of the Koolau Mountains and Haiku Valley on the other side. The project includes a tunnel through the mountain range. The one-mile (1.6 km)-long twin-bore Trans-Koolau Tunnel will have two lanes in each direction. The total roadway width including shoulders in each bore will be 38 ft (11.6 m). The geologic and geohydrologic conditions along the alignment present a variety of tunneling problems. Geologic conditions consist of volcanic materials deposited generally in fairly thin basaltic layers. These volcanic materials generally weather down to "MH" soils which are compressible but relatively strong. Perched ground water that is recharged by heavy rainfall is present in the mountains. The paper presents results of initial geologic surveys and exploratory boring, and plans for exploratory tunneling and testing programs. A site specific classification method developed to characterize the weathered profile is presented.

Jan 1, 1989

By Timothy N. Dunne, Joseph M. Gonzalez

This paper presents the case history of a comprehensive geotechnical investigation for the Crosstown Transport Facility, a 10,880 m long, 3.2 m diameter pressure tunnel proposed for construction in San Francisco, California. The proposed tunnel would be part of the City and County of San Francisco's Clean Water Program, and its purpose would be to convey stormwater and sewage from the east side of the City to the Pacific Ocean. Tunneling conditions were defined by surface geologic mapping, 94 exploratory borings, surface and borehole geophysics, aquifer testing, Goodman Jack testing, pressuremeter testing, and laboratory testing. Through the use of advanced coring techniques and a strictly controlled drilling fluid system, core recovery averaged 90 percent in the sheared and fractured rock of the Franciscan Formation. As a result, over 1,400 linear meters of rock core was recovered and is available for examination.

Jan 1, 1983

By William F. Kane, Eric C. Drumm

Rock slope failures in the Great Smoky Mountains of Tennessee are a continuing problem. A study of the geotechnical factors affecting such slopes was conducted in this geologically complex region. Seven rock slopes, five of which had undergone failure, were identified and mapped. Other parameters were mapped in addition to the standard geologic and geometric parameters. These were joint frequencies across and down the slope, and surface hardness from Schmidt hammer tests. Regression analysis on the various data revealed that Schmidt hammer hardness correlates very well with joint cohesion back-calculated from failure geometries. This approach suggests an alternative means for estimating joint strength.

Jan 1, 1986

By David Pentz, Paul Hodges

GENERAL INTRODUCTION The object of this paper is to discuss those aspects of open pit mine design which are affected by geotechnical factors. The temptation exists for some practitioners of geotechnical engineering to believe that this relatively new engineering discipline is the ‘be all and end all of mining.’ In the opinion of the authors, this is fallacious but, as our ability to put numbers to the behaviour of rocks and soils improves, it is clear that geotechnical engineering will become an integral and important part of designing and operationg mines success- fully. Open pit mines in the context of this paper also includes open cast mines, where the 'ore" is extracted and the waste material is immediately replaced or ' backfilled in the mined portion of the pit. Open pit mines operating today move from very small tonnages a day to in excess of 400,000 tons per day. Several mines are operating and are being designed to capital expenditure for such projects in some cases exceeds five hundred million dollars and in a few cases will exceed one billion dollars. The size of this investment means that financing will have to be provided from banks or joint partnerships. This financial environment means that planning will be carefully scrutinized by experienced representatives of financial institutions or other mining companies. All too often a project has been put in jeopardy because the personal desire of one or two men to make a project proceed has caused incomplete planning or perhaps in some instances deliberately leading observers 'down the garden path.’ In our opinion the day is rapidly arriving when the quality, logic, practicability and completeness of the planning will make the difference between securing financing or not and, in the cases of existing open pit mines, of the difference between survival and closure.

Jan 1, 1977

By Louis A. Panek

INTRODUCTION Block caving is distinguished from other types of mining in its requirements that the mineral body to be extracted must cave after a relatively large tabular open¬ing is created underneath it, and that the caving must be predictable and controllable for both operating and safety reasons. A mineral body that caves readily may present special problems with respect to ground support in the mining access openings, where stability, rather than failure, is the desired objective. The complete ex¬traction of a caving block, which may measure 50 to 100 m on a side and 100 to 300 m in height, tends to create a large void, resulting in significant changes in the ground stress system and a migration of rock from all directions toward the void; these events begin very early in the life of the mine and lead to additional spe¬cial problems with respect to control of ground move¬ments in the mine working areas, in the rock adjacent to the caving stope, and on the subsiding surface above the mine. Thus, the mechanics of an undercut-cave operation involve a broad range of geotechnical subject matter, including the determination of the lithostatic earth stresses, the structural properties of a large jointed rock mass, the flow properties of the fragments after caving, the stability of tunnel-like openings, and the stability and displacements of the rock structure (1 to 2 km in size) that surrounds the ore body. The objectives are achieved by in-place measurement of rock properties, laboratory tests, theoretical structural analyses of the related prob¬lems, measurements of structural behavior in and above the mine, and comparison of theoretical predictions with actual performance in order to interpret and refine the predictive capability. Undercut-cave mining (Julin, et al., 1973) differs from most other methods in that the caving is an essen¬tial and inherent part of the excavation process and in¬volves the movements of millions of cubic meters of rock over distances of hundreds of meters, as compared to the gradual 2-m settling of the overburden typical of full¬caving longwall coal mining. Undercut-cave mining may involve a rock mass on the scale of a 1-km diam open¬pit mine, but mining is several times deeper than that in the open-pit mine; furthermore, a large portion of the rock mass is in a failed, rather than a stable, state. Ground-control problems are inevitable in a situa¬tion where ground movements are occurring on such a large scale, and where the rock mass is expected con¬veniently to cave at one location but to be stable at an¬other. To devise effective responses to these ground¬control problems, one needs to know first of all what is happening, not merely at the point where the symptoms are observed, but more broadly with respect to the be¬havior of the entire mine rock-mass structure over the total area of extraction, including the effects on the sur¬face as well as those underground. Monitoring structural behavior on this scale would require a sizable group of people, if they attempted to measure all the significant mining-induced changes of ground displacement and pressure, even on a 100-m grid of points. The practical approach is to conduct the mea¬surements activity by a judicious sampling, making in¬termittent, concentrated efforts in selected areas of lim¬ited extent to study problems of immediate importance; simultaneously a minimum level of surveillance is main¬tained over the total structure to delineate long-term be¬havior trends in the major structure, which provide the baseline reference data that are so essential for interpre¬tive purposes. Mine-structure behavior depends not only on the mine extraction configuration and sequence, but on the rock structural properties and how they change from one location to another in the mine. Increasing attention is being given to determinations of rock strength and de¬formation by performing mechanical properties tests in place and in the laboratory, and by drawing inferences from the mapping of geologic structure in mine openings as they are driven and from the logging of cores ob¬tained from exploration drilling that may be performed primarily for delineating ore reserves. Monitoring of mine-structure behavior and inter¬preting it in terms of the growing excavations and the structural properties of the rocks have significant im¬pacts on designing the major undercut-cave operations of undercutting and drawing the ore, supporting the ac¬cess openings, and controlling surface subsidence. The purpose of this chapter is to call attention to the com¬mon ground-control problems encountered in undercut¬cave mining, to relate them to the mining process, and to explain the geomechanical approaches to analyzing these problems with emphasis on methods that are most readily available to the mine staff. UNDERCUTTING, CAVING, AND DRAWING A distinguishing characteristic of an undercut-caving method of extraction is the objective that the rock mass fails of its own weight in a predictable and controllable manner, after it is undercut by excavating a tabular¬shaped void, horizontal or inclined. Induced or forced caving refers to procedures that may be resorted to when the ore will not self-cave over the desired span. Block caving, panel caving, and mass caving are terms used to denote the particular sequence of undercutting and ex¬tracting the total area of the ore body on a single min¬ing level. The discussion herein is not specialized to any of these variations; the word block as used does not imply anything as to sequence. CREATING THE UNDERCUT A common undercutting method is to drive a series of drifts and blast the intervening pillars. Theoretical structural analysis and practical experience show that ground-support problems in the undercutting drifts and in underlying access openings as well (the same princi¬ple applies even if undercutting is done by belling out drawpoints from the slusher drifts) are minimized by retreat-mining across the undercut slot according to a sequence such that the growing undercut area retains a

Jan 1, 1982

By H. Maleki

This paper recognizes the 100th year of research accomplishments by the mining industry resulting from close cooperation among mining companies, geotechnical consultants and researchers from the former United States Bureau of Mines in stability assessments and development of improved and productive mine designs. The paper provides an overview of dramatic developments in geotechnical instrumentation, and its applications in the U.S. and around the world for enhancing the understanding of ground response to mining-induced changes in stress, resulting in improvements in ground control technologies.

Jan 1, 2010

By Michael Schultz, Gregory Sanders

In 2017, the North Texas Municipal Water District (NTMWD) initiated the design of the Sister Grove Regional Water Resource Recovery Facility (SGRWRRF) to provide additional treatment capacity for future population growth within the Lavon Lake watershed. To provide a discharge location for SGRWRRF approximately 4.5 miles of 2.4- meter (96-inch) internal diameter pipeline will be constructed, including 2.3-kilometer (7,800) feet and 487-meter (1,600 foot) long tunnel segments. This paper will discuss the anticipated geology of the two tunnel segments, selection of the tunnel alignment, the anticipated initial tunnel support requirements and other aspects of the project design

Jun 13, 2021

By Julian Prada

Tunneling requirements are presenting designers with greater challenges. The design and construction of the 2000-ft long unidirectional twin highway tunnels for the approach to one of two bridges of the Louisville-Southern Indiana Ohio River Bridges (LSIORB) Project is no exception. Excavation under low cover, beneath an historic site, with each tunnel of 70-feet wide and 50-feet high dimensions will set precedents for US highway tunnel excavation size. To investigate ground conditions to supplement final design of the highway tunnels, an exploratory tunnel excavated through karstic limestone and shale will be located within the confines of the proposed final alignment. With the LSIORB Project, Jefferson County, Kentucky and Clark County, Indiana will have improved cross-river mobility between them. The Indiana Department of Transportation (INDOT) and the Kentucky Transportation Cabinet (KYTC) are developing the LSIORB Project jointly, in coordination with the Federal Highway Administration (FHWA). Section 4, one of six sections, as shown in Figure 1 is the East End Kentucky Approach and Connections to I-265. Key elements of Section 4 are the twin three-lane highway tunnels under the historic Drumanard Estate. The property is located on the north side of US Highway 42 (US 42) at the intersection with the Gene Snyder Freeway (also KY841) as shown in Figure 2. One aspect of the project consists of the excavation of an Exploratory Tunnel along the southbound alignment tunnel. At the time of preparation of this paper, KYTC has not awarded the Exploratory Tunnel Contract. This paper will focus on the planned geotechnical investigation program that will occur during excavation of the exploratory tunnel and how the design of the twin highway tunnels will incorporate the findings to achieve an innovative and efficient configuration.

Jan 1, 2008

By Robert P. Bell, John T. Waggoner

The Metropolitan Water District of Southern California is designing new outlet facilities for Lake Mathews, including a junction shaft, connecting tunnels, and an ear-shore outlet tower. Since Lake Mathews must remain operational during construction, groundwater inflow potential is a concern. This paper describes the challenges and methods used to investigate rock mass and groundwater conditions at an operational reservoir, where offshore and onshore geotechnical exploration was performed. Several investigative and analytical methods were used to characterize rock mass and groundwater conditions, including rock coring, downhole geophysical and video surveys, hydraulic conductivity testing, automated groundwater level monitoring, and laboratory testing.

Jan 1, 1999

By G. Avolio, M. Gilbert

INTRODUCTION The city of Alexandria, Egypt is located on the Mediterranean Sea at the northwestern edge of the Nile River delta. The city is approximately 26 km long and 3 km wide bounded by the sea to the north and Lake Maryut to the south. The Alexandria Project is split into 2 phases: phase one included designs to update two existing primary wastewater treatment plants, five pump stations and over 75 km of sewer pipe, 18 km o f which is tunneled, (1500mm to 2750mm ID pipe). Currently the pipeline and pump stations are under construction. Phase two of the project includes expanding both the capacity of the treatment plants and the limits of the collection system. Included in this system expansion is the 3.5 km long, 2250mm ID Central Zone Conveyance, which is the subject of this paper. The project is under the direction of The Alexandria General Organization for Sanitary Drainage (AGOSD). It is an USAID funded joint venture of two American firms, CH2M - Hill and Metcalf & Eddy who share equally the major portion of the work and two Egyptian firms, Engineering Consultants Group (ECG) and Wairth Associates who are also partners. The venture, which is both design and construction management, worked under the name Wastewater Consultants Group, WWCG. Michael Gilbert was Chief o f the Geotechnical Department and Project Engineer for the Central Zone Conveyance. Gennaro Avolio was Geotechnical Exploration Manager for the Alexandria Wastewater Project.

Jan 1, 1987

By Jerome C. Neyer

Within most of Canada and the northern third of the United States, the soils overlying bedrock are of glacial origin. Tunnelling in glacial soils presents numerous challenges to the contractor, including abrupt changes in soil type, boulders and glacial erratics, water bearing fine sands and silts, toxic or explosive gases, and glacial till. This paper examines the available techniques for geotechnical exploration in glacial soils; the relationship between subsurface conditions predicted by various techniques and conditions actually encountered during tunnel construction; and potential improvements in the planning and implementation of geotechnical explorations. Case histories of geotechnical exploration and tunnel construction in glacial soils are also reviewed.

Jan 1, 1985

By Greg Sanders, Michael Gilbert, James G. Shaughnessy

"A new raw water intake from the South Holston River to the Kingsport water treatment plant is under construction. The exploration for the 335 meter long horseshoe shaped tunnel design took advantage of the site topography along the alignment to better define the complex geology consisting of Knox Limestone and Sevier Shale formations separated by a fault zone. To assess the geological conditions and to determine the extent of the fault zone, a 183 meter long horizontal boring and geophysical investigation was performed along a portion of the tunnel’s horizontal alignment. Explorations defined the fault zone width at approximately 5 meters and packer testing along the alignment provided a means of estimating groundwater inflows. As the tunnel excavation advances, comparisons of exploratory data versus encountered conditions will be made. This paper provides details on the geotechnical investigation, lessons learned, and the influence of the geology on the tunnel design. PROJECT OVERVIEW The Kingsport WTP is supplied with raw water via the South Holston River. The City of Kingsport (City) owns and operates the Kingsport Water Treatment Plant (WTP). The existing raw water intake system, consisting of pumps, power supply, raw water piping and appurtenant facilities has been determined to be vulnerable to failure. The original Raw Water Pump Station (RWPS) was constructed in 1928 as part of the original Kingsport WTP project. The 1928 RWPS operated until 1970 when it was replaced by the existing RWPS. This existing system still utilizes the 1928 piping and piping tunnel. Improvements to the RWPS hydraulics and capacity are required to meet the City’s future water treatment needs. The system extracts water from the South Fork of the Holsten River and coveys it via three pressure pipelines to the WTP. The conveyance piping exits the RWPS, ascends approximately 23 m up a near vertical rock outcropping then enters a 120 m long tunnel. Upon exiting the tunnel the piping continues underground to the WTP. The drill and blast tunnel is horseshoe shaped and nominally 3.3 m high by 3.3 m wide. The easternmost 62.3 m of the tunnel is without any type of support. The remainder is lined with concrete. According to the City of Kingsport, the concrete lining was installed in the 1970’s as part of upgrades to the facility. An evaluation of the current facilities concluded that the conveyance piping was in poor condition and required replacement."

Jan 1, 2016

By D. L. Johnson

A number of small but rich copper sulfide (chalcocite) ore deposits are located on Michigan's Keweenaw Peninsula. These deposits are considered too small to warrant the capital investment required for conventional mining. However, they are being considered for in-situ solution mining as a way to revitalize the area's once prosperous copper extraction industry. Geology, mineralogy and geohydrology of the deposits have been investigated to determine their suitability for leaching; Three conclusions were drawn from the work. First, the sulfide mineralization is found primarily in fractures (65%) and, therefore, is accessible to injected leach solutions. Second, the ores are shallow, occurring from surface exposure to about 200 meters depth; injection and recovery wells would not have to be excessively deep. Third, the rock was found to have low permeability, in the neighborhood of thirty millidarcys or less. However, preliminary hydraulic fracturing experiments indicate that this will be an effective method for permeability enhancement. In one experiment a permeability increase of 100 times was achieved using a very low injection pressure at shallow depth. Additionally, fracture aperture spacings measured from drill core were correlated with fair success to field permeabilities.

Jan 1, 1989

By R. D. Hendry, R. A. Pearce, Allyn J. Steward, Dennis C. Martin

This paper outlines the geotechnical exploration and design approach utilized for the Km 80 and Km 86 Tunnels on the British Columbia Railway Tumbler Ridge Branchline. Information from surface mapping, trenching and diamond drilling was used to prepare a rational support design using accepted rock mass classification systems. Four possible support types were developed and used as a basis for preparing a cost estimate and contract documents for the tunnels. A unit price contract was selected to provide flexibility and to enable rational selection of support based on actual ground conditions encountered during excavation. Although ground conditions along the tunnels are variable, the general geology and quantities of support installed were consistent with those estimated from the geotechnical investigation and design.

Jan 1, 1985

By B. M. Ghadiali, J. B. Putnam, T. P. Smirnoff, M. A. Elioff, P. F. Ryan

Geotechnical investigations for the Mid-City Extension of the Los Angeles Metro revealed high concentrations of toxic hydrogen sulfide (H2S) gas as well as methane. Testing techniques employed to determine the extent of gasses, and to identify potential treatment measures, included gas sampling with cone penetrometers and soil-gas extraction with air injection testing. A gas reservoir model was developed to predict inflows into excavations. Other studies included a hazard evaluation and a general data search for other H2S occurrences and geologic conditions that could facilitate the accumulation of H2S. The paper describes the investigations and proposed design and construction alternatives.

Jan 1, 1995

By Seth Pollak, Jon Hurt, Mark Havekost, David Vardiman, James Schick

"INTRODUCTION The Long Baseline Neutrino Facility (LBNF) Far Site Conventional Facilities are proposed to be sited at the Sanford Underground Research Facility (SURF) located within the former Homestake Gold Mine in Lead, South Dakota. The project involves excavation and support of three large caverns which will house four 10 kT fiducial volume liquid argon (LAr) neutrino detectors and cryogenic equipment, as well as numerous ancillary connecting drifts and chambers. This paper describes the geological setting of the project, the geotechnical investigation undertaken nearly a mile underground and the challenges associated with collection of reliable data in a high stress and geologically complex environment. The layout for the new facility will be presented, along with a description of some of the constraints that will impact construction, including shaft access and working within an operating research facility. THE LBNF PROJECT The global neutrino physics community is coming together to develop a leading-edge, dual-site experiment for neutrino science and proton decay studies, the Deep Underground Neutrino Experiment (DUNE). The facility required for this experiment, the Long-Baseline Neutrino Facility (LBNF), will be an internationally designed, coordinated and funded program, comprising the world's highest-intensity neutrino beam at the Fermi National Accelerator Laboratory (Fermilab) in Batavia, IL and the infrastructure necessary to support massive deep underground cryogenic detectors 800 miles (1,300 km) downstream at the LBNF Far Site at the Sanford Underground Research Facility (SURF), in Lead, South Dakota. The Deep Underground Neutrino Experiment is expected to achieve transformative discoveries, making definitive determinations of neutrino properties, the dynamics of the supernovae that produced the heavy elements necessary for life, and the possibility of proton decay. Neutrinos created by the LBNF beamline will travel on the 1,300 km (800 mi) path shown in Figure 1 leading straight through the earth's crust to intercept DUNE's massive, cutting-edge neutrino detector at the Sanford Lab. No tunnel is needed as neutrinos pass easily through soil and rock rarely interacting with the matter. The DUNE far detector will consist of four modules, each of which will be housed in a cryostat containing 17,000 metric tons of liquid argon target material, providing a 10kT fiducial volume. The LBNF Far Site Conventional Facilities project will excavate and fit out a set of four caverns 4850 ft underground in which to place them."

Jan 1, 2016

By Earle J. Klohn

INTRODUCTION The paper presents a broad overview of the geotechnical site studies required for siting uranium tailings dams. These studies, which are interdisciplinary in nature, touch upon such related fields as: hydrogeology, hydrology, and seismicity. Tailings dams are commonly used to develop tailings ponds for the storage of the waste products generated by mine milling operations. In the past, most tailings ponds have been developed either above ground on dry land, or in existing lakes. In the United States a majority of the uranium mines are located in arid or semi-arid regions and past practice has been to store the wastes above ground on dry land (1,2). Currently, the United States Nuclear Regulatory Commission (N.R.C.) recommends below ground disposal of uranium waste products, although they indicate that they will consider above ground alternatives (3). In Canada many of the existing uranium mines are located in a wet environment and past practice has been to use small lakes and depressions as tailings ponds (1,4). However, current practice in Canada leans towards above ground, dry-land storage (5,6,7), and the Atomic Energy Control Board (AECB) guidelines (5) indicate this to be the preferable method. In B.C., where a moratorium on uranium mining has been declared by the Provincial Government, there are no existing uranium tailings ponds. However, both the formerly proposed Birch Island and Blizzard uranium developments would have used dry land storage. Most of the existing non-uranium mill wastes in B.C. are stored behind tailings dams on dry land. The waste products delivered to the tailings pond comprise, the ground-up rock that remains after most of the uranium values have been removed and the excess chemicals used in the milling process. These waste products, which are called "tailings", are normally mixed with water to produce a slurry and then pumped to the tailings pond. Since the uranium recovered constitutes only a small percentage of the ore (usually less than 1%) (8), large quantities of waste rock and associated transportation water are discharged into the tailings impoundment area. Consequently, relatively large tailings pond areas are required for the uranium tailings. In recent years, major advances have been made in the design, construction, and operation of non-uranium tailings storage facilities (9,10,11,12,13,14, 15). These improvements have been based on combining the latest advances

Jan 1, 1980

By David Chapman, Glen Frank

Due to the transport mechanics of the depositional environment, glacial soils display a very wide range of grain sizes and behavior. Several nonstandard drilling, sampling, and analytical techniques are therefore required to characterize the subsurface. The paper summarizes the geotechnical investigation for the design of a large project in glacial geology, which will include both large and small diameter pressure balance tunneling. The paper focuses on the challenges presented by the glacial environment and the techniques required to identify and characterize it for design and contract document preparation.

Jan 1, 2001

By John Keefner, Charles Kocsis, Kyle A. Scalise

Geotechnical modeling of mines is crucial for understanding the many factors that can pose potential hazards or limit production opportunities. The process of back analyses for modeling is often used, but in many instances, the back analyses can be cumbersome and time inefficient.While thesemodels and back analyses may be accurate, these types of back analyses may not be ideal to be used at mines and by engineers because the process is so time-consuming. This paper introduces event-based back analyses that utilize prior events to create more accurate models while eliminating large modeling time because the models are back analyzed from previous events. The completed analyses supply a more detailed insight into the potential hazards at Round Mountain, an open pit operation, as well as provide a better understanding of how improper risk analysis results in missed opportunities. The more we understand about mining activities and processes through modeling, the safer and more productive mines will be.

May 24, 2021

By Christopher M. St. John, Michael P. Hardy

INTRODUCTION Geotechnical models, particularly those based on the finite element method, have been available to aid in en¬gineering design of underground mining excavations for over ten years. Despite this fact there are remarkably few cases of their use in mine design documented in the literature. It is therefore to be anticipated that many potential users of these models are relatively unaware of their capabilities and limitations and also of the form and detail of geotechnical data needed for their success¬ful application. This subsection attempts to address this problem by discussing the different types of numerical models now available and by noting how some of them have been used to study a variety of problems associated with underground mining. The subsection concludes by discussing the applica¬tion of the displacement discontinuity method to the de¬sign of possible mining systems for the copper-nickel deposits of northeastern Minnesota. The object of the analyses, which were nonsite specific, was to determine the significance of geotechnical parameters, such as ini¬tial stress and rock structure, on the stability of under¬ground excavations and hence to provide guidance for future geotechnical investigation. NUMERICAL MODELS In the geomechanical design of underground mining openings, use is made of numerical models that repre¬sent or simulate the large-scale mechanical behavior of rock. There has been less interest in analysis involving fluid flow and heat transfer, but with increasing interest in such areas as in-situ retorting and solution mining it is likely that there will be a growing need for numerical models embracing and coupling all three physical proc¬esses. However, the emphasis in this subsection will be on mechanical behavior. Models which simulate such behavior will be divided into two groups: continuum models and discontinuum models. These will be dis¬cussed in turn in order that some insight into alternative solution strategies and their merits may be gained. Continuum Models Almost all geomechanical numerical models must be classed as continuum models even though particular computer codes incorporate special provisions for rep¬resenting discontinuities such as faults, bedding planes, or joints. They are continuum models because they pro¬vide solutions for cases where material behavior is governed by the differential equations of continuum mechanics. Two basic solution strategies for such equa¬tions may be identified immediately: the differential ap¬proach and the integral approach. In the differential approach a means of approximating the differential equations over the entire region of interest is sought. In the integral approach use is made of fundamental solutions from continuum mechanics, and these are used to construct a solution to the whole problem, making approximations only on the boundaries of the region of interest. The several differential and integral methods are identified in Table 1. Differential Methods: Problems in continuum me¬chanics involve the solution of three types of partial differential equations. Two of these govern the behavior in so-called initial value problems, in which variables change both in time and in space. Examples of such problems include nonsteady heat transfer and fluid flow, and stress wave propagation. The last type of partial differential equation governs the behavior in boundary value problems. In these, variation is in space but not in time. Solution of initial value problems may be achieved in two significantly different methods: implicit and ex¬plicit. The differences between these two methods will be illustrated by considering a very simple initial value problem, that of one-dimensional heat diffusion. The equation governing this process may be written as: [ ] where T is temperature, K is thermal conductivity, p is density, c is specific heat, t is time, and x is the spacial coordinate. In finite difference form this equation might be written as: [ ] where the superscripts refer to the time and the sub¬scripts to the spacial location. Several solution strategies for this equation have been used. Two important ones may be illustrated very simply by discussing the signifi¬cance of the superscript * on the right-hand side of the equation. If i + 1 is substituted for * then the second derivative of the temperature with respect to distance is evaluated at the end of the next time step (using tem¬peratures not already known). Such an approach leads to a set of equations involving unknown temperatures and a solution procedure which is referred to as being implicit. The important characteristic of the implicit procedure is that it leads to a set of equations that must be solved for each time at which the temperature dis¬tribution is required. If instead of substituting (i + 1) for the superscript *, i is substituted, the following equation is obtained: [ ] {In this case the new temperature is defined in terms of an already known temperature distribution. The solution procedure is now known as explicit and has the impor¬tant characteristic that there are no equations that have to be stored or solved. A practical advantage of this

Jan 1, 1982