UNCONFINED FLOW THROUGH JOINTED ROCK1

The two-dimensional, steady-state, unconfined flow of a homogeneous fluid through jointed rock is studied for both laminar and turbulent conditions by use of a method which is based on previously developed theoretical and experimental flow relationships. However, only the independent unknowns are selected in order to reduce the complexity of the problem and render it more readily tractable. The intact rock is assumed to be impermeable, and two intersecting systems of plane, parallel joints are used in the mathematical model, taking into account the surface roughness of the joints. The mathematical solution of the resulting nonlinear (due to turbulent flow in some joints) system of equations is obtained by use of a rapidly converging iterative procedure, wherein each iteration takes special advantage of the banded nature of the associated matrix. For the particular case where a free surface exists, the general flow equations are not satisfied, because some of the joints in the vicinity of the free surface do not flow full; therefore, new equations must be established to handle this condition. Once the development of the mathematical model is accomplished, several cases involving different geometric characteristics (width, orientation, and roughness of joints) are solved for a rectangular domain, and graphs are given to illustrate the influence of the various parameters on the manifested flow behavior.     

CASING AND LEAK DEPTHS,AND SOLUTE TRAVEL TIMES TO WELLS1

ABSTRACT: A mathematical solution based on porous media flow is developed for solute travel time to a well as affected by a leak around the upper part of the casing. Consider a well of radius 0.2 meters (m) penetrating, fully, a semiconfined aquifer of thickness 6 m with impermeable casing length of 4.5 m, and screened casing length 1.5 m. Around the upper 1.5 m of the impermeable casing length, there is a highly permeable region (a leak). The radius of influence of the well is 10 m. The porous flow medium has a hydraulic conductivity of 10 m/day and a porosity of 0.25. Between the water table and the water level in the well, there is a steady state pumped down head difference of 0.3 m. Solute travel time from a point at the bottom of the leak to the well is 2.33 days. If the leak is sealed (grouted), the travel time is 6.24 days. Examples of six different geometries are given. Laboratory studies verify the theory. The computations should be useful in the design and protection of water wells from solutes, such as from agriculture, industry, strip mines, or sanitary landfills.     

A LUMPED APPROACH TO THE INVERSE PROBLEM IN GROUND WATER HYDROLOGY1

ABSTRACT: A solution procedure to solve the inverse problem in ground water, based on lumped approach, has been proposed. The method has the following advantages: 1) exact determination of the boundary conditions and the physical laws of flow through porous media is not required; 2) all errors of approximation in describing the boundary conditions, physical laws, and the aquifer properties are lumped into the surrogate parameters; and 3) the same mathematical model can be employed both in the identification process and in the subsequent management studies. The optimal values of the surrogate parameters are found by using a multidimensional unconstrained optimization code devised by Powell. The solution procedure and the convergence characteristics of the proposed algorithm have been illustrated by two hypothetical problems.     

OPTIMIZATION OF WATER ALLOCATION DECISIONS AFFECTING ESTUARINE ECOLOGY1

ABSTRACT .The problem analyzed in this paper is how to allocate optimally the available surface water in a river system among those who compete for its use, while acknowledging explicitly that for coastal states the ecology of bays and estuaries must be numbered among the competitors. The objective is to maximize the benefit resulting from water use while satisfying a set of constraints on flow. Benefit is assumed to be a function of the amount of water used and the time period in which the water is used. A mathematical model of this problem is shown to fit the format of the minimum cost circulation network flow problem. The Out-of-Kilter algorithm of D. R. Fulkerson is proposed as a solution technique. Sensitivity analysis on the input data is described as a means of determining the minimum economic benefit required to justify the allocation of a given volume of water needed to sustain the ecology of an estuary.     

Three-dimensional nickel ion transport through porous media using magnetic resonance imaging

The transport of Ni2+ ions in a column, filled with porous media, was observed in three dimensions and time by magnetic resonance imaging (MRI) in a clinical scanner. For porous media we used glass beads or quartz sand in a saturated continuous flow mode. The magnetic moment of Ni2+ decreased the T1 relaxation time of 1H in aqueous solution. This concentration-dependent effect was used by a fast low angle shot (FLASH) MRI sequence for imaging the concentration of the dissolved ions. Since Ni2+ behaves as a conservative tracer under the chosen conditions, the tracer motion was representative for the water flow in the porous medium. Currently, we can achieve an isotropic spatial resolution of 1.5 mm and a temporal resolution of 170 s. The transport observation gives direct access to hydraulic flow properties of the porous media. The fluid flow velocity field was calculated by a fronttracking method and the statistical properties of the velocities were investigated. We also compared the experimental data with the three-dimensional particle tracking model PARTRACE, which uses the experimental flow field as input.     

Nuclear magnetic resonance imaging for studies of flow and transport in porous media

Nuclear magnetic resonance imaging (NMRI) methods for visualization of fluid flow and transport in porous media are reviewed in this paper. They are illustrated with experiments showing applications of velocity imaging, NMRI measurements of multiphase flow, and NMRI measurements of density flow. The latter two are compared with numerical simulations. The examples show the capacity of NMRI to give structural information both of the medium and the fluid distributions as well as their temporal development. The resulting data can be used in a black box-white box comparison and as benchmarks for numerical models.     

A VOLUME BALANCE SOLUTION FOR WATER FLOW OVER FLAT SOIL SURFACES1

ABSTRACT: This paper presents the development of a mathematical model to compute the advance of water flowing over flat soil surfaces. The solution is of interest to the design and management of irrigation systems, and the model can also be applied to overland flow problems. The hydraulics of water flow during the advance phase is simulated by the Lewis-Milne integral equation. The general solution to this equation is obtained by using the Laplace transform theory. A particular solution was developed, based on series expansions, that uses the modified Kostiakov equation to predict infiltration. The solution is given by a double infinite series that has terms of alternate sign. Results from this model show satisfactory agreement when compared with field data collected by the author.     

ADVANCES IN THE BOUNDARY INTEGRAL EQUATION METHOD IN SUBSURFACE FLOW1

ABSTRACT: This paper explores some of the advances of the boundary element method, as applied to ground-water problems, during the last five years. Although the method is still somewhat limited compared to solution by finite elements, the range of solutions has increased considerably. Diffusion and advection-diffusion solutions are done efficiently. These include the incorporation of inhomogeneity, anisotropy, and nonlinear diffusion. The difficult problem of stream-aquifer interaction is an important application as it is much easier to follow a free surface with its multiple configurations. The application must be able to cycle between ground-water connection and disconnection with the stream and include seepage surfaces. Flow in fractured media is a natural application where the flow in fractures can usually be treated without a computational exception in spite of extremely high aspect ratios. The case of seawater intrusion forms a type of free surface problem and thus is a case for which the method has special advantages. For these and other applications the boundary element method provides an inexpensive technique for calculation where the data preparation and setup time is minimal. In most of these cases, programs can and have been written on microcomputers.     

CRITICAL EVALUATION OF CERTAIN METHODS OF UNSTEADY GROUNDWATER HYDRAULICS1

The basic theories and fundamental assumptions usually employed in the solution of unsteady groundwater flow problems are reviewed critically. The best known method of analysis for such problems is based on the Dupuit-Forchheimer approximation and leads to a nonlinear parabolic differential equation which is generally solved by linearization or numerical methods. The accuracy of the solution to this equation can be improved by use of a different approach which does not employ the Dupuit Forchheimer assumption, but rather is based on a semi-numerical solution of the Laplace equation for quasi-steady conditions. The actual unsteady process is replaced by a sequence of steady-state conditions, and it is assumed that the actual unsteady flow characteristics during a short time interval can be approximated by those associated with “average” steady state flow. The Laplace equation is solved by a semi-discretization method according to which the horizontal coordinate is divided into subintervals, while the vertical coordinate is maintained continuous. The proposed method is applied to a typical tile drainage problem, and, based on a comparison of calculated results with experimental data, the method is evaluated and practical conclusions regarding its applicability are advanced.     

DYNAMIC FLUID LOSS DURING VISCOUS FLOW THROUGH A POROUS VERTICAL SLOT1

ABSTRACT. An experimental study of two-dimensional viscous flow through a vertical slot with one highly resistant porous wall was made. The fluid loss area of the porous wall was divided into five sections. The fluid loss rate for the various subareas was measured as a function of the bulk flow rate through the slot and the viscosity of the fluid. Static flow tests through the porous media were also conducted for each fluid viscosity. The results indicate that the experimental data can be correlated in terms of the difference between the static flow rate and the dynamic fluid loss rate as a function of the bulk Reynold's number and the bulk flow rate. Stream function profiles were determined for each experimental run to visualize flow through the length of the slot. An empirical correlation was developed between the superficial entrance width, δ, and the ratio of bulk Reynold's number to the Reynold's number based on flow through the porous wall.     

A COMPARISON OF SOME DYNAMIC,LINEAR AND POLICY ITERATION METHODS FOR RESERVOIR OPERATION1

Within the past few years, a number of papers have been published in which stochastic mathematical programming models, incorporating first order Markov chains, have been used to derive alternative sequential operating policies for a multiple purpose reservoir. This paper attempts to review and compare three such mathematical modeling and solution techniques, namely dynamic programming, policy iteration, and linear programming. It is assumed that the flows into the reservoir are serially correlated stochastic quantities. The design parameters are assumed fixed, i.e., the reservoir capacity and the storage and release targets, if any, are predetermined. The models are discrete since the continuous variables of time, volume, and flow are approximated by discrete units. The problem is to derive an optimal operating policy. Such a policy defines the reservoir release as a function of the current storage volume and inflow. The form of the solution and some of the advantages, limitations and computational efficiencies of each of the models and their algorithms are compared using a simplified numerical example.     

LABORATORY AND NUMERICAL INVESTIGATION OF LONGITUDINAL DISPERSION IN OPEN CHANNELS1

ABSTRACT: The non-Fickian nature of the longitudinal dispersion in natural channels during low flow has been investigated using both laboratory experiments and the numerical solution of the proposed mathematical model which is based on a set of mass balance equations describing the dispersion and mass exchange mechanisms. Laboratory experiments, which involved collection of channel geometry, hydraulic, and dye dispersion test data, were conducted to obtain sets of experimental data on a model of four pool and riffle sequences in a 161-ft long tilting flume in the Hydrosystems Laboratory at the University of Illinois at Urbana-Champaign. The experimental results indicate that flow over the model pool-riffle sequences is highly nonuniform. Concentration-time curves are significantly skewed with long tails. The mixing and dispersion in the laboratory channel was simulated using a numerical solution of the mathematical model in which the finite difference method developed by Stone and Brian (1963) was used as a solution technique. The comparison between measured and predicted concentration-time curves shows that there is a good level of agreement in the general shape, peak concentration, and time to peak. The proposed model shows significant improvement over the conventional Fickian model in predicting dispersion processes in natural channels under low flow conditions.     

Combustion in porous media and its applications – A comprehensive survey

The rapid advances in technology and improved living standard of the society necessitate abundant use of fossil fuels which poses two major challenges to any nation. One is fast depletion of fossil fuel resources; the other is environmental pollution. The porous medium combustion (PMC) has proved to be one of the technically and economically feasible options to tackle the aforesaid problems to a remarkable extent. PMC has interesting advantages compared with free flame combustion due to the higher burning rates, the increased power dynamic range, the extension of the lean flammability limits, and the low emissions of pollutants. This article provides a comprehensive picture of the global scenario of research and developments in PMC and its applications that enable a researcher to decide the direction of further investigation. The works published so far in this area are reviewed, classified according to their objectives and presented in an organized manner with general conclusions. A separate section is devoted for the numerical modeling of PMC.     

THEORY,DEVELOPMENT, AND UTILIZATION POTENTIAL OF THE BIOMILIEU CONCEPT1

The paper presents a quantitative engineering approach to analysis of total environment allowing for simultaneous consideration of a theoretically infinite number of quality indicators and physiological requirements. It discusses theory and fundamentals of a two-dimensional space and time function solution concerning a small estuarine-type environment. A three dimensional solution is indicated. Input data may range from reconnaissance-type to the outputs of mathematical transport models. Applications are discussed with respect to environmental quality problems, availability of suitable data, and some areas of research where results could find immediate application.     

GROUND WATER MANAGEMENT OPTIMIZATION USING GENETIC ALGORITHMS AND SIMULATED ANNEALING: FORMULATION AND COMPARISON1

ABSTRACT: Genetic algorithms (GA) and simulated annealing (SA), two global search techniques, are coupled with MODFLOW, a commonly used groundwater flow simulation code, for optimal management of ground water resources under general conditions. The coupled simulation-optimization models allow for multiple management periods in which optimal pumping rates vary with time to reflect the changing flow conditions. The objective functions of the management models are of a very general nature, incorporating multiple cost terms such as the drilling cost, the installation cost, and the pumping cost. The models are first applied to two-dimensional maximum yield and minimum cost water supply problems with a single management period, and then to a multiple management period problem. The strengths and limitations of the GA and SA based models are evaluated by comparing the results with those obtained using linear programming, nonlinear programming, and differential dynamic programming. For the three example problems examined in this study, the GA and SA based models yield nearly identical or better solutions than the various programming methods. While SA tends to outperform GA in terms of the number of forward simulations needed, it uses more empirical control parameters which have significant impact on solution efficiency but are difficult to determine.     

NUMERICAL AND ANALYTICAL SOLUTIONS OF DISPERSION IN A FINITE,ADSORBING POROUS MEDIUM1

ABSTRACT Numerical and analytical solutions are developed for the distribution of a contaminant within an adsorbing porous medium in a unidirectional flow field subject to a step function for input concentration. The medium is considered to be homogeneous, isotropic, and areally finite. As a by-product, solutions are also obtained for the case of a non-absorbing porous medium. An example that demonstrates the applicability of the solutions is presented.     

INTERFACE BETWEEN TWO DISSIMILAR FLUIDS BENEATH A SHEETPILE COFFERDAM1

ABSTRACT: A closed form solution is presented for determining the shape and location of the interface between two dissimilar fluids (having different densities) when steady flow takes place through a homogeneous and isotropic porous medium, into a sheetpile cofferdam; the interface is assumed to be sharp and the lower fluid stationary. The solution is obtained using the inverse hodograph. Numerical results are presented in nondimensional form for various parametric conditions in the physical plane; the interface pattern, as also the seepage discharge and exit gradient distribution are shown. The critical conditions of the interface are studied.     

A one-dimensional model of vertical gas plume migration through a heterogeneous porous medium

This work is motivated by the growing interest in injecting carbon dioxide into deep geological formations as a means of avoiding its atmospheric emissions and consequent global warming. Ideally, the injected greenhouse gas stays in the injection zone for a geologic time, eventually dissolves in the formation brine and remains trapped by mineralization. However, one of the potential problems associated with the geologic method of sequestration is that naturally present or inadvertently created conduits in the cap rock may result in a gas leakage from primary storage. Even in supercritical state, the carbon dioxide viscosity and density are lower than those of the formation brine. Buoyancy tends to drive the leaked CO₂plume upward. Theoretical and experimental studies of buoyancy-driven supercritical CO₂ flow, including estimation of time scales associated with plume evolution and migration, are critical for developing technology, monitoring policy, and regulations for safe carbon dioxide geologic sequestration.In this study, we obtain simple estimates of vertical plume propagation velocity taking into account the density and viscosity contrast between CO₂ and brine. We describe buoyancy-driven countercurrent flow of two immiscible phases by a Buckley–Leverett type model. The model predicts that a plume of supercritical carbon dioxide in a homogeneous water-saturated porous medium does not migrate upward like a bubble in bulk water. Rather, it spreads upward until it reaches a seal or until it becomes immobile. A simple formula requiring no complex numerical calculations describes the velocity of plume propagation. This solution is a simplification of a more comprehensive theory of countercurrent plume migration [Silin, D., Patzek, T.W., Benson, S.M., 2007. A Model of Buoyancy-driven Two-phase Countercurrent Fluid Flow. Laboratory Report LBNL-62607. Lawrence Berkeley National Laboratory, Berkeley, CA]. In a layered reservoir, the simplified solution predicts a slower plume front propagation relative to a homogeneous formation with the same harmonic mean permeability. In contrast, the model yields much higher plume propagation estimates in a high-permeability conduit like a vertical fracture.     

A DYNAMIC MODEL FOR WATER AND RELATED LAND RESOURCE PLANNING1

ABSTRACT. Planning an optimal system of activities for generating economic goods and services within an existing natural resource capacity is a difficult problem to solve. A mathematical programming model with the capacity to check multiple resource demand and supply compatibility over many time periods was developed for the solution to this type of problem. The characteristics of natural resource supply and the demand of activities were utilized to reduce the number of time periods and to minimize the loss of the dynamic reality of the problem. Reduction in the number of time periods extended the capability of the model to the solution of complex resource planning problems without oversimplification.