I’ll start by expressing my most sincere thanks to Martin Blunt, Majid Hassanizadeh, and Brian Berkowitz, for initiating and implementing the idea of a special issue of TIPM to honor my initiative some 37 years ago, of starting a new Springer scientific/professional journal devoted to phenomena of transport of extensive quantities in porous medium domains. Indeed, this is a great honor for me. THANKS! Let me tell you about myself. I was born (1929) in Israel (then British mandate Palestine). My parents immigrated to Israel as Zionists driven by the urge to participate in the establishment of a homeland for the Jews in their ancestral land so that Jews will have at least one place in the world to where they can flee whenever persecuted. I married my wife, Siona, in 1951. We have three children and six grandchildren. Finishing high school (1946), I joined the PALMACH (the Israeli underground fighting units) and fought in the War of Independence (1947–8). In 1949, I started my studies of Civil Engineering at the Technion-Israel Institute of Technology, Haifa. I elected to graduate in the option of water resources, realizing that water would be a central issue and a limiting factor in Israel’s development. I liked the challenge. I received my B.Sc. in 1953 and started to work as an engineer in the Planning Division of TAHAL-Water Planning for Israel Ltd., the government (now private) company in charge of water resources planning and development in Israel. Awarded a scholarship by the Dutch government, I spent 1955/6 at the Government Institute for Water Supply (Scheveningen, NL), conducting research on artificial recharge of groundwater and on seawater intrusion. I used laboratory models, especially the Hele-Shaw (parallel-plate) model, as tools for solving ground water flow and storage problems. During the 1960s (no computers, but practical problems had to be solved!), I used these (horizontal and vertical) models extensively to investigate seawater intrusion into layered coastal aquifers under the sharp interface approximation (my Technion M.Sc., 1957), and artificial recharge of aquifers. Returning from the Netherlands, I moved to the groundwater department of TAHAL. In Israel, at that time, groundwater, primarily from the coastal (sandstone) aquifer and the (limestone) mountain aquifer, constituted the major source of water. However, the total available annual sustainable water yield is rather small, to the extent that it seriously constrains development. Seawater intrusion was high on my agenda. I worked under the

The chapter deals with cases of transport of mass and energy in which the solid matrix, as the entire porous medium domain, undergoes deformation. Models of consolidation as a consequence of construction, and models of land subsidence due to heavy pumping, are developed and presented. Also, the propagation of waves in a porous medium domain, in which soil deformability plays an essential role, is briefly discussed. Models of consolidation as a consequence of construction, and models of land subsidence due to heavy groundwater pumping, are developed and presented. Also, the propagation of waves in a porous medium domain, in which soil deformability plays an essential role, is briefly discussed. The chapter focusses on cases in which fluids are extracted from or injected into geological formations, possibly under non-isothermal conditions. The presented models describe phenomena of deformation of porous medium domains in aquifers in response to imposed stresses. Examples are: (1) Land subsidence as a consequence of pumping water from aquifers, (2) ground surface upheaval, as a consequence of injection, (3) development of fractures as a consequence of injecting fluids into a tight geological formation (4) induced seismicity as a result of fluid injection into confined formations, and (5) soil liquefaction. Change in the flow regime, associated with fluid pressure changes, causes changes within a considered porous medium domain causing formation deformation. The chapter starts by introducing the concepts of stress and strain in a single phase continuum-first at the microscopic level and then at the macroscopic one. Then, following the phenomenological approach, the complete model for stress-strain analysis is developed for the saturated (or multiphase) elastic porous medium, leading to the strain and deformation within the considered porous medium domain.

The chapter introduces some fundamentals of thermodynamics of phases and chemical species that are required for the understanding and constructing the flow and transport models considered in this book. Phases and chemical species are defined. Among the concepts defined and discussed are the pressure, density, chemical and other potentials, Gibbs function, internal energy, enthalpy and capillary pressure. Equations of state and their role in modeling are discussed, as well as the concepts of phase Behavior. The concepts of a tensors, stress, strain ar introduced. They will be used in modeling flow in deformable porous media, as well as for the discussion on poromechanics and deformation in Chap. 9. Onsager?s Theory of Coupled Processes is briefly reviewed. No effort is made to present a complete review of the considered subjects as these can easily be found in texts on Thermodynamics. In this chapter, as in the entire book, we use the International System of units.

Book presenting the physics and construction of models for the flow of fluids and the transport of mass/heat in porous and fractured geologic media.

This chapter is devoted to the transport of chemical species dissolved in the fluid phases that occupy the void space, with or without chemical reactions. Solute fluxes are due to diffusion, dispersion, and advection. Various sources and sinks, as well as chemical reactions and interphase transfers are discussed and integrated in the species (first microscopic and then macroscopic) mass balance equation. Like any other model of transport, the core of the solute transport model is the mass balance of the considered chemical species. The discussion leads to a well-posed model of the solute transport problem. (mass balance equations for the chemical species, initial and boundary conditions, constitutive relations, etc. To facilitate the discussion, a review of selected topics of chemistry that describe source/sink and interphase exchange phenomena that occur within the considered fluid(s) is presented in this chapter. One such topic is chemical reactions that occur within the fluids that occupy the void space. Adsorption, solid dissolution and precipitation are also modelled. The presentation considers also sinks and sources in the form of extraction and injection of solute carrying fluids through wells. Obviously, the presentation of the chemical aspects should be considered merely as a brief introduction to some of the essentials and to the employed terminology. Following the methodology presented in this book, once phenomena are understood at the microscopic level, where they really occur, they are incorporated in the macroscopic ones The discussion in this chapter includes the effects of temperature, but the subject of flow and transport under non-isothermal conditions is discussed in Chap. 8 . The objective is to derive models that describe solute transport not only at laboratory scale domains, but also at large natural domains, primarily in heterogeneous geological formations. The material presented in this chapter should be of use for those dealing with phenomena of transport in geological formations. However, the material will also be useful to chemical engineers who design chemical reactors in the Chemical Engineering industry. Appendix A discusses chemical reactors, and the various phenomena of transport that occur in them.

The chapter deals with modeling non-isothermal mass momentum, and energy transport. Here, an additional variable is added-the temperature. To obtain the temperature distribution in the fluid and solid phases that occupy the porous medium domain, we have to write and solve the energy balance equation, obviously with appropriate initial and boundary conditions, as well as with temperature-dependent constitutive relationships. This equation, with appropriate initial and boundary conditions are considered in this chapter. The assumption that underlies this chapter is that thermal equilibrium exists between all phases present at point in the porous medium domain. The presentation in Chaps. 5, 6, and 7, focused on flow and transport under isothermal conditions. Under non-isothermal conditions, all constitutive relations involve also temperature an additional variable, and the complete model should also include the energy balance equation. The solute transport models considered in Chap. 7, may now involve exogenic and endogenic reactions. Within a porous medium domain, thermal energy may be transported by four mechanisms: (1) Advection by a fluid (or fluids) moving in the void space, (2) conduction in all solid and fluid phases (overlooking advection in a deformable solid), (3) mass diffusion in the fluid phases, and (4) thermal dispersion in the fluid phase(s). All these modes of energy transport are discussed in this chapter. The effects of natural thermal gradients, produced by solar radiation at ground surface, on water and water vapor movement, as well as on the chemical and biological behavior in the subsurface, may serve as examples of interest to soil scientists. In dealing with contaminated groundwater, a number of remediation techniques are associated with heating the soil, and injection of steam. Other interesting cases that require knowledge of heat and mass transport in porous media are the storage of energy in aquifers, or in the unsaturated zone, the production of geothermal energy, the disposal of $$\text {CO}_{2}$$ in deep geological formations, the geological storage of high-level nuclear waste, and the thermally enhanced production of petroleum. Finally, in the chemical industry, most processes that take place in reactors occur under non-isothermal conditions. Because of its importance, we have added an appendix (App. A) that presents and discusses modeling of phenomena of transport that take place in chemical reactors. Coupling between the transport of mass and heat is also due to the fact that both the fluid’s density and its viscosity are temperature dependent. Most partitioning and equilibrium coefficients, discussed in Chaps. 6 and 7, are strongly temperature dependent.

In this chapter, we construct mass transport models for the case of multiple (i.e., two or three immiscible) fluid phases that occupy the void space simultaneously. The objective is to lead to complete, well-posed mathematical models, taking into account the coupling that takes place between the phases. The fluids occupy disjoint (microscopic) subdomains that together fill up the entire void space. Because of surface tension phenomena, one of the fluids, called the wetting fluid, tends to adhere to the solid, while the other, called the nonwetting fluid, stays farther from the solid surface. The capillary pressure and its relationship to saturation is discussed. The case of drainage and imbibition of a wetting fluid in an unsaturated vertical column in the unsaturated zone is presented. Another case is that of oil water and gas in a petroleum reservoir. Thus, the material in this chapter should be of interest to those who deal with the unsaturated zone in the subsurface, e.g., in connection with irrigation and drainage in agricultural engineering. It should also be if interest to reservoir engineers, to those who plan the disposal of supercritical CO2 in deep brine-containing geological formations, and to those who consider the injection of air or natural gas into depleted oil and gas reservoirs for storage purposes.

This chapter is devoted to modeling mass transport of a single fluid phase, liquid or gas, that completely occupies the void space. The core of such model is the mass balance equation of the phase. This (macroscopic) equation is obtained from the general macroscopic balance equation when applied to mass, or phenomenologically. The specific storativity is introduced to account for fluid and solid matrix compressibility. Boundary and initial conditions are presented, leading to a well-posed flow (= mass transport) model. A separate model is developed for a constant density fluid and essentially horizontal flow, commonly used to describe flow in aquifers. The entire discussion in this chapter is under isothermal conditions (non-isothethermal conditions are presented I Chap. 8). Two additional subjects are discussed in this chapter. One is flow in a deformable porous medium, primarily as associated with storage of water in aquifers. The general subject of flow and other phenomena of transport in deformable porous media are discussed in detail in Chap. 9. The other subject is an introduction to flow in fractured porous medium domains. Throughout the chapter, we make use of the concepts of stress and shear which are second rank tensors, assuming that the reader is familiar with these concepts. Some introductory remarks about stress and strain are presented at the beginning of the chapter. The last section in this chapter is an introduction to flow in fractured rocks and in fractured porous rock domains.

This chapter starts by discussing the concepts of a point, velocity and flux. The general balance is developed for any extensive quantity, for a point at the microscopic level, i.e., at a point within a fluid phase present in the void space, and at a point in the solid matrix. Then, the phenomenological approach is used to present this balance equation at the macroscopic level. The two approaches are briefly described. Then, the balance equations for the transport at the macroscopic level of mass, momentum and energy are derived. The Advective and diffusive flux appearing in both the microscopic and macroscopic balance equations are introduced, as well as e dispersive flux appearing only the latter. Special attention is devoted to the coefficients appearing the macroscopic balance equations. The macroscopic balance equation is also obtained by volume and mass averaging. Interphase transfers and sources appearing in the balance equations are also discussed, with their appropriate expressions and coefficients. Special attention is devoted to the meaning and role of coefficients. Dimensionless numbers and are discussed and used for determining non-dominant effects.

Chapter 2 discusses, in a descriptive manner, the processes occurring during the geological storage of CO2. In this chapter, the mathematical models Mathematical model describing these processes are described. The chapter starts from the basic properties of the injected CO2 and of the native brine, proceeding to the relevant models Model for multiphase flow Multiphase flow of CO2 and brineBrine , the related chemical and reactive transport Reactive transport processes, the non-isothermal Non-isothermal effects of CO2 injectionCO2 injection and the mechanical deformationMechanical deformation . The concept of degrees of freedomDegrees of freedom , facilitating the selection of a smaller number of equations to be solved, in order to obtain a complete solution for this multifaceted problem, is also discussed. The numerical and analytical approaches for solving these mathematical models Mathematical model are presented in the following Chap. 4 .

This book offers readers a comprehensive overview, and an in-depth understanding, of suitable methods for quantifying and characterizing saline aquifers for the geological storage of CO2. It begins wi

In many parts of the world, groundwater resources are under increasing threat from growing demands, wasteful use, and contamination. To face the challenge, good planning and management practices are needed. A key to the management of groundwater is the ability to model the movement of fluids and contaminants in the subsurface. The purpose of this book is to construct conceptual and mathematical models that can provide the information required for making decisions associated with the management of groundwater resources, and the remediation of contaminated aquifers. The basic approach of this book is to accurately describe the underlying physics of groundwater flow and solute transport in heterogeneous porous media, starting at the microscopic level, and to rigorously derive their mathematical representation at the macroscopic levels. The well-posed, macroscopic mathematical models are formulated for saturated, single phase flow, as well as for unsaturated and multiphase flow, and for the transport of single and multiple chemical species. Numerical models are presented and computer codes are reviewed, as tools for solving the models. The problem of seawater intrusion into coastal aquifers is examined and modeled. The issues of uncertainty in model input data and output are addressed. The book concludes with a chapter on the management of groundwater resources. Although one of the main objectives of this book is to construct mathematical models, the amount of mathematics required is kept minimal.

Introduction Israel is a narrow long country of a semiarid climate, located at the eastern end of the Mediterranean. Rainfall, occurring only in the winter months (primarily November–February), varies on average between 900 mm/year in the North to less than 100 mm/year in the South. It is characterized by large fluctuations (coefficient of variation 30%), with drought years (i.e., less than 70% of average) about 20% of the time. This regime, combined with the absence of appropriate sites for surface storage reservoirs (due to the country’s geological structure) and high evaporation losses, has, to a great extent, determined the development of groundwater in Israel. This article covers the period starting with the establishment of the State of Israel in 1948, when the population was around 800 thousands, until the present days when it reached about 8.5 million, due to large immigration and natural growth. In view of this increase, as well as due to the modernization of the country on one hand, and water scarcity on the other, it was decided already at the outset that water has to be recognized as a national natural resource, to be managed centrally. Thus, a Water Law , stipulating that water belongs to the state, which is responsible for its development and management, was promulgated quite early. In parallel, the Water Authority , a government agency in charge of monitoring, regulation, and allocation of water, and public companies dealing with planning, development, and water supply were established. This has led to a comprehensive approach to all water resources and their incorporation into a single management system. In parallel, intensive research was initiated at universities and government institutes, to provide the scientific support and training of required professionals. These features make groundwater

Poster presentado en la European Geosciences Union General Assembly, celebrada en Viena del 7 al 12 de abril de 2013.

The capillary pressure–saturation relationship, P c(S w), is an essential element in modeling two-phase flow in porous media (PM). In most practical cases of interest, this relationship, for a given PM, is obtained experimentally, due to the irregular shape of the void space. We present the P c(S w) curve obtained by basic considerations, albeit for a particular class of regular PM. We analyze the characteristics of the various segments of the capillary pressure curve. The main features are the behavior of the P c(S w) curve as the wetting-fluid saturation approaches zero, and as this saturation is increased beyond a certain critical value. We show that under certain conditions (contact angle, distance between spheres, and saturation), the value of the capillary pressure may change sign.

Highly controlled field injection experiments are necessary for demonstration, for scientific understanding and for quantification of the relevant processes of CO2 geological storage.The preparation of such an experiment requires reliable information on both the hydraulic, thermal and chemical properties of the target layer and the formation fluid as well as on the injection discharges and their associated pressure build-up in the reservoir. For this, there is a need to determine the state variables of CO2 in the injection tube near the well head, which can produce the desired mass flow rates given the condition at the reservoir, while respecting pressure buildup constraints. A model connecting the multiphase flow and transport processes in the target layer (based on the well-known TOUGH2/ECO2N model) at the vicinity of the injection well with those occurring in the injection tube (solving the one dimensional equations mass, momentum and energy conservation) has been developed. To this model the injection tube is a boundary condition. Once the reservoir pressure build-up resulting from the injection discharge is known, there is a need to determine the necessary injection conditions at the wellhead. For this purpose we apply the 1-D tube model, which provides the solution of the conditions in the injection pipe, given the injection rate and the pressure at the reservoir. These two linked models, the porous medium model and the pipe model, are applied to the planning of the Heletz injection experiment to be carried out in the frame of the EU-FP7 funded MUSTANG project. Sensitivity analyses are carried out with regard to uncertainty in the target layer permeability and the temperature of the injected CO2, which depends on the thermal heat transfer coefficient in the injection tube.

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.