MODELING COMPLEX, NONLINEAR ... - Semantic Scholar

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Annu. Rev. Earth Planet. Sci. 2002. 30:35–64 DOI: 10.1146/annurev.earth.30.082801.150140

MODELING COMPLEX, NONLINEAR GEOLOGICAL PROCESSES1 Greg A. Valentine, Dongxiao Zhang, and Bruce A. Robinson Hydrology, Geochemistry, and Geology Group EES-6, Mail Stop D462, Los Alamos National Laboratory, Los Alamos, New Mexico 87545; e-mail: [email protected]

Key Words multiphase flow, stochastic porous media, magma dynamics ■ Abstract We review selected geological processes to which numerical modeling has been applied, with the aim of describing some of the general approaches and applications of the modeling. All of these examples involve multiphase fluid flow, in some cases coupled with heat transport and phase changes. First, we describe modeling approaches to a human-made geological system—a potential underground radioactive waste repository. Next, we describe recent advances in modeling two-phase flow through random heterogeneous porous media. We review recent modeling studies of fluid processes in magmatic systems, especially focusing on melting and crystallization induced by magma chambers. Finally, several research directions are suggested, including improving our understanding of the linkage between small-scale and fieldscale processes, coupling across regimes (e.g., surface water and ground water), and further developments in the modeling of stochastic geological processes.

INTRODUCTION Our ability to model geologic processes theoretically is becoming increasingly important as humankind places more demands on the environment, requires more efficient tapping of geologic resources, and grows more vulnerable to natural disasters. A solid ability to model complex earth processes can be a benefit in many ways, ranging from improving our basic understanding of geologic systems, to enhancing our ability to integrate data and improve our efficiency in gathering further information, to providing a predictive capability with which planning and mitigation strategies can be enhanced. This paper reviews a selection of complex geologic processes where modeling capabilities are advancing rapidly. Physical earth processes are governed by the principles of solid and fluid mechanics and thermodynamics. These principles are represented mathematically as 1 The U.S. Government has the right to retain a nonexclusive, royalty-free license in and to any copyright covering this paper.

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sets of governing equations, and the behavior of these equations is generally nonlinear; that is, the equations involve variables that are multiplied by themselves or by some function(s) of themselves. Nonlinear equations are inherently difficult to solve analytically and in general are best approached by numerical (approximate) solutions. Beyond the difficulty of the equations themselves for a given process, the earth sciences face four additional challenges that “enhance” complex nonlinear behavior. First, real geologic problems commonly involve a coupled set of nonlinear processes. For example, the eruptive behavior of a volcano depends on both the flow of magma and on the deformation or fracture of the solid edifice, each of which is nonlinear by itself and which are likely to be coupled by a nonlinear process. Second, geologic systems and materials are nearly always heterogeneous and contain some degree of uncertainty. A familiar example is the flow of fluid through porous rocks, where permeability varies even within a given stratigraphic unit and where total characterization of that variation is impossible without destroying the medium. Third, large-scale geologic processes that follow one type of governing equation often depend on small-scale processes that follow quite different equations. Borrowing again from hydrology, a large-scale saturated flow through an aquifer follows a linear equation known as Darcy’s Law; but at the scale of individual pores, the fluid flow is described by the nonlinear Navier-Stokes equations. Finally, unlike engineering sciences where the geometry of objects or flow systems can be carefully controlled, in the earth sciences we must work with the often-complicated geometries that nature has given us. In this paper we discuss examples of recent advances in modeling that have addressed these challenges. Rather than trying to address all types of nonlinear geologic processes, we focus on a few classes of geologic fluid flows: coupled multiphase flow and heat transport through porous media, two-phase flow in heterogeneous porous media, and magmatic systems. We describe recent modeling works on these topics and some of the research issues that have demanded numerical modeling of these nonlinear systems. Our intent is not to give a “recipe” for all nonlinear geologic problems (which would be impossible in any case), but to provide the reader with a flavor of modern approaches to these difficult problems and of some of the commonalities between problems, at least within the fluid flow domain. A major underlying commonality is that these systems all involve multiphase flow, which is responsible for many of the nonlinear aspects of their behavior. We conclude with a brief discussion of important future directions of research. During the past 15 years, there has been a major focus in the physics and applied mathematics communities on some of the special properties of nonlinear systems, including chaotic and fractal behavior. Chaotic behavior occurs when two slightly different initial or boundary conditions produce widely different results. Chaotic nonlinear systems often exhibit some sort of scale invariance, or self-similarity. This is commonly characterized by a fractal dimension; examples of fractals in nature are described both in Turcotte (1992) and in the classic, definitive book on fractals by Mandelbrot (1982). We do not focus on these aspects of nonlinear

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geologic processes in this paper; rather we direct the reader toward these two books, particularly Turcotte (1992).

MULTIPHASE, REACTIVE FLOW AND TRANSPORT IN POROUS AND FRACTURED MEDIA The influence of heat on the transport of water and dissolved chemicals in porous media is a topic rich in nonlinear effects. One of the specific applications for which these effects have been studied in greatest detail is radioactive waste disposal at the potential high-level radioactive waste repository at Yucca Mountain, Nevada. We note that although this is not a “pure” geologic system in the sense that there is human input to the mass and energy balance, it is analogous to other natural systems such as shallow igneous intrusions (e.g., WoldeGabriel et al. 1999, Lichtner et al. 1999). Consideration of the impact of radioactive waste heat is necessary for many aspects of potential repository design. For example, thermohydrologic (TH) considerations near emplacement drifts (the mined shafts that would contain the waste canisters) impact the relative humidity and temperature and the likelihood of dripping water on the waste packages, all of which in turn control the corrosion rate (Buscheck et al. 1996). The solubility and speciation of radionuclides and the dissolution rate of the waste form itself are also strongly influenced by TH considerations (Nitsche et al. 1994, CRWMS M&O 1995). On a larger scale, waste heat might alter the mineralogy of the host rock and may change the character of the zeolitic units below the potential repository that are likely to be the primary natural barriers to radionuclide migration (Bish 1995). The heat effects may also influence the migration of radionuclides from the potential repository to the water table should waste packages fail and radionuclides become dissolved in the percolating fluid (e.g., Viswanathan et al. 1998). Finally, the impact of waste heat on the near-surface temperature within the soil zone is of interest, as there may be an issue of potential impacts on the above-ground ecosystem from increases in soil temperatures (CRWMS M&O 1999). For this reason, modeling studies at the drift scale (Buscheck et al. 1996) and site scale (Buscheck & Nitao 1993, CRWMS M&O 1999, Haukwa et al. 1999) have been conducted to estimate the magnitude of these effects. If high-level radioactive waste is buried in the subsurface, the heat expelled from the waste due to radioactive decay is a time-varying source of heat that potentially impacts the fluid properties, hydrologic properties, and nature of the heat transport mechanism in the vicinity of the waste canisters. To properly account for these effects, a two-phase solution to the mass and energy balance equations must be solved. For higher heat loads, and close to the waste packages, two-phase conditions may exist at or near the boiling temperature, resulting in complex, nonlinear heat and mass interactions controlled by fluid convection, thermal conduction, and transport of latent heat in the vapor phase. In the calculation presented as an example below, we demonstrate the nature of the nonlinear processes that can arise

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in this coupled system. In doing so we focus only on TH effects. The subject of rock-fluid interactions and their possible influence on heat and mass transport is a topic that is beginning to be studied because of its potential to strongly affect the overall system behavior, but it is not examined in this short review. For details on the development of reactive chemical transport model formulations for porous media, the reader is referred to the review volume of Lichtner et al. (1996). Geothermal reservoir simulation provided the application in which some of the first complex multiphase heat and mass transport models were developed. The fundamental equations of multiphase heat and mass transport in a porous medium have been outlined in several publications (e.g., Mercer & Faust 1975, Brownell et al. 1975, Zyvoloski et al. 1979, Zyvoloski 1983) and thus are briefly summarized here, following the development presented in Zyvoloski et al. (1997). Conservation of mass of water is represented by the following equation: ∂ Am + ∇ · f¯m + qm = 0, ∂t

(1)

where qm is a fluid source or sink, and the mass per unit volume, Am, is given by Am = φ(Sv ρv (1 − ηv ) + Sl ρl (1 − ηl )),

(2)

where the mass flux, f¯m , is given by f¯m = (1 − ηv )ρv v¯ v + (1 − ηl )ρl v¯ l .

(3)

In these equations, φ is the porosity, S is saturation, ρ is density, η is the concentration of a noncondensible gas such as air, expressed as a fraction of total mass, v¯ is velocity, and the subscripts v and l refer to the vapor and liquid phases, respectively. A mass balance equation for air is also required. This expression parallels Equations 1–3 but includes an additional term to capture diffusive transport of air in the vapor phase (Zyvoloski et al. 1997, Equation 20). Conservation of energy in the combined fluid/porous-medium system is given by: ∂ Ae + ∇ · f¯e + qe = 0, ∂t

(4)

where qe is energy associated with sources or sinks, and the energy per unit volume, Ae, is given by Ae = (1 − φ)ρr u r + φ(Sv ρv u v + Sl ρl u l ),

(5)

with u r = c pr T , and the energy flux f¯e is given by f¯e = ρv h v v¯ v + ρl h l v¯ l + K ∇T.

(6)

In Equations 4–6, ur, uv, and ul are specific internal energies, cpr is the rock specific heat, hv and hl are specific enthalpies, K is the effective thermal conductivity, T is temperature, and the subscript r refers to the rock.

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The final step in the development of a multiphase heat and mass transfer model is to relate the fluid velocities to state variables of the system, such as pressure. This is almost always done by assuming Darcy’s law: v¯ v =

¢ kv ¡ ∇ Pv − ρv g¯ µv

(7)

v¯ l =

¢ kl ¡ ∇ Pl − ρl g¯ . µl

(8)

In Equations 7 and 8, µ is the viscosity, g¯ is the gravity vector, Pv and Pl are the phase pressures (related to the capillary pressure, Pcap, by Pv = Pl + Pcap), and kv and kl are the permeabilities of the two phases. As presented in detail in the next section (Unsaturated Flow in Randomly Heterogeneous Porous Media), the permeabilities, capillary pressure, and saturations are related through empirical nonlinear constitutive relationships, such as the van Genuchten–Mualem model (van Genuchten 1980). A final note on these constitutive relationships pertains to the simulation of flow in fractured rock, which has been the subject of several investigations on the numerical approaches and resulting system behavior. A pair of papers by Pruess and coworkers (Pruess et al. 1990a,b) effectively lays out the issues associated with TH effects in fractured, unsaturated rocks. Pruess et al. (1990a) performed numerical simulations of a simplified system consisting of a single fracture embedded in a rock matrix containing a heat-emitting waste package. They showed that under certain conditions fractures serve as conduits for vapor-phase transport of steam away from the heat source. When the steam travels far enough away from the source it condenses, thereby affecting the thermal profile significantly when compared to pure heat conduction. A detailed discussion of the nature of this effect is shown in the repository scale calculations presented below. For now, we note that the details of the TH behavior are a strong function of the characteristic curves applied for the fracture and surrounding rock matrix. To make the simulation of large-scale TH processes tractable, the role of fractures must in some way be averaged, rather than simulated at the scale of individual fractures. To this end, Pruess et al. (1990b) introduced the concept of an “effective continuum” model (ECM). This concept allowed the characteristic curve of a representative elementary volume (REV) containing both fractures and matrix to be captured by a single relationship. After applying individual characteristic curve properties (set using a van Genuchten–Mualem or other model) for each continuum, averaging procedures are used to produce a single characteristic curve for the medium based on the assumption that the temperature and capillary pressure are equal in the fractures and matrix. Extreme nonlinearities arise in this type of formulation as the averaged properties (say, liquid permeability versus saturation) capture the fluid flow behavior of media with orders of magnitude differences in permeability. Subsequent to the Pruess et al. (1990b) work, others (e.g., Tseng & Zyvoloski 2000) have applied dual permeability models to relax the assumptions

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of the ECM. Nevertheless, the ECM approach is a useful starting point of complex, multidimensional TH calculations and is used in the calculations presented below. Site-scale models of the TH behavior at Yucca Mountain in the presence of repository waste heat are typically performed on coarse numerical meshes capable of capturing the general, large-scale features of moisture and heat-flow processes within the unsaturated zone, but models of this kind cannot be used to determine fine-scale temperature and moisture-content gradients. To illustrate the nonlinear effects of coupled TH processes, as well as the sensitivity of model predictions to the resolution of the numerical mesh, we present a model from a two-dimensional, East-West cross section at Antler Ridge, used in previous modeling studies to simulate radionuclide transport and TH behavior (Viswanathan et al. 1998). A depiction of the layered stratigraphy and finite-element grid generated for this cross section is shown in Figure 1. Descriptions of the hydrologic properties, the assumed water infiltration rate at the surface, the time-varying heat load at the potential repository horizon, and other boundary conditions are presented in Viswanathan et al. (1998) and CRWMS M&O (1999). To capture the modeled hydrostratigraphy in detail sufficient to simulate the fluid and heat transport through the medium (Figure 1b), sophisticated grid generation techniques are employed (Gable et al. 1996). In addition to capturing the hydrostratigraphy, extremely fine grid resolution near the potential repository is needed to capture the temperature and fluid-saturation profiles near the source of the waste heat. At the repository portion of the model (Figure 1c), heat is supplied to individual nodes of the grid along a series of nodes spaced an appropriate distance apart so that, in these calculations, there are roughly four unheated nodes between each node representing an emplacement drift. The transport equations for heat, mass, and momentum transport developed above are solved for this system using the computer code FEHM (Zyvoloski et al. 1997), a finite element code in two or three dimensions. The nonlinear effects arising from the interplay of boiling water and liquid and vapor transport are interesting and significantly affect potential repository performance. Figure 2a shows the predicted fluid-saturation state of the system at 1000 years after waste emplacement for a 1-mm/year infiltration rate. The figure, including the close-up view of the model results in Figure 2b, illustrates the redistribution of fluid due to heat-up near the emplacement drifts. Over a period of roughly 10,000 years, dryout and rewetting occur near the emplacement drifts. A two-phase, countercurrent flow zone develops above the potential repository, with buoyant and pressure gradient– driven vapor movement in the upward direction counterbalanced by downward fluid percolation, the primary source of which is condensed steam. The resulting heat-transfer mechanism, called a heat pipe, is much more efficient than thermal conduction alone and results in an extended two-phase region of enhanced heat transfer at about 1 atm and 100◦ C. When the heat from the repository region is insufficient to maintain the two-phase zone, the zone collapses and fluid reenters the repository region. The extent of the two-phase region is controlled by the nature of the fracture system because transport of latent heat (as steam) increases with increased fracture permeability. In these simulations, the fracture

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Figure 1 Model domain for the Yucca Mountain Unsaturated Zone Thermohydrologic (TH) Calculations. a) Layered stratigraphy; b) finite element mesh; c) expanded view of mesh at the repository.

permeabilities in the nonwelded Paintbrush Tuff (PTn) are low enough that the upward advance of the heat-pipe region is arrested at the base of the PTn and conduction is the primary mode of heat transfer to the surface. The heat-pipe effect is also seen in the temperature field of Figure 2c, which exhibits a preferential upward transport of heat and temperature rise. Waste-package degradation and dissolution of the solid waste are dependent on the time history of temperature, fluid saturation, and relative humidity in the rock near the emplacement drifts. As an example of the model results, Figure 3 shows the predicted temperature versus time in the rock near two emplacement drifts (one closer to the middle of the repository and one near the edge). After an initial warm-up period lasting less than one year, a boiling period lasting on the order of ten years is needed to dry the rock near the drift, after which time the rock

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Figure 3 Predicted temperature versus time at different locations in the model.

heats to values greater than the boiling temperature for a substantial period. The period of complete dryout and elevated temperatures depends most critically on the position of the drift within the repository. In this simulation, temperatures near the emplacement drift approach 190◦ C for inner drifts and only 140◦ to 170◦ C for edge drifts. Furthermore, the time required for two-phase conditions at 100◦ C to be reestablished is about 100 years for edge drifts and 500 to 1000 years for inner drifts. A critical aspect of nonlinear calculations is the accuracy of the results with respect to the resolution of the numerical grid. This is especially true for complex systems for which analytical solutions are only available for simplified model geometries and boundary conditions. The necessity of adequate grid resolution near the potential repository for this model is illustrated in the curves labeled “smeared source” in Figure 3. For these curves, the heat load associated with the waste heat is uniformly distributed within a 5-m-thick rectangular region extending the length of the repository. Thus, although the finely resolved grid is used, the heat distribution is spread (or smeared) in space as it would be in a coarse-grid simulation. With the smeared heat source, temperatures near the potential repository now are predicted to rarely or never exceed the boiling temperature, as the rock

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fails to fully desaturate. It is necessary to use a finely resolved grid in the vertical and horizontal directions to capture the details of fluid saturation and temperature near the emplacement drifts. If the modeling objective requires accuracy near the emplacement drifts, as in predicting canister lifetime or radionuclide source terms, then fine grid resolution is needed, especially in the horizontal direction so that the detailed fluid and thermal patterns at and between drifts can be resolved. Nonlinear heat and mass transfer processes accentuate this effect. Therefore, “standard practice” in simulating complex, nonlinear behavior must include grid resolution studies.

UNSATURATED FLOW IN RANDOMLY HETEROGENEOUS POROUS MEDIA The above section focuses on an example of multiphase flow in which the host porous media within individual geologic units are assumed to be homogeneous with known values of porosity and permeability. This allows a deterministic approach to be followed, based on Darcy’s Law for porous media flow. In reality, though, natural porous media are heterogeneous; this adds a complication to the already difficult problems associated with the nonlinear physics involved. In this section, we summarize recent work aimed at accounting for heterogeneity in two-phase flow through porous media as an example of how these two effects can be modeled. In doing so, we introduce simplifications to the physics of the transport processes to make the examination of the effect of heterogeneity more tractable. Although geologic media exhibit a high degree of spatial variability, medium properties, including fundamental parameters such as permeability and porosity, are usually observed only at a few locations due to the high cost associated with subsurface measurements. This combination of significant spatial heterogeneity with a relatively small number of observations leads to uncertainty about the values of medium properties and, thus, to uncertainty in predicting flow and solute transport in such media. The theory of stochastic processes provides a natural method for evaluating the prediction uncertainty. In the stochastic formalism, uncertainty is represented by probability or by related quantities like statistical moments. Medium properties, boundary conditions, and/or initial conditions are treated as random fields (RFs) whose values are determined by probability distributions. In turn, dependent variables like pressure and flux are RFs, and the equations governing subsurface flow and transport become stochastic differential equations (SDEs) whose solutions are probability distributions of pressure and flux. Generally, we cannot solve an SDE exactly but can only estimate the first few moments of the corresponding probability distribution, specifically its mean, variance, and covariances. However, these moments usually suffice to approximate confidence intervals. In the last two decades, many stochastic theories have been developed to obtain statistical moments for subsurface flow and transport quantities (e.g., Dagan 1989, Gelhar 1993, Dagan & Neuman 1997).

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In this section, we concern ourselves with the situation in which water and air coexist (vadose zone or unsaturated flow). In recent years, consideration of the vadose zone has received increasing attention among scientists, engineers, and regulators. In the context of groundwater pollution, the vadose zone can act as either a buffer or a conveyor belt between the land surface, where most contaminants originate, and groundwater, which is a resource protected under a number of environmental regulations. Unsaturated flow processes play an important role in determining the pathways of a contaminant plume before it reaches the aquifer, particularly in semiarid and arid regions where the vadose zone may be several tens to several thousands of meters thick. The inherent spatial variabilities of rock or soil properties significantly impact fluid flow and solute transport in the vadose zone. In such an unsaturated system, the problem is further complicated by the fact that the flow equations are nonlinear because unsaturated hydraulic conductivity depends on pressure head. Unsaturated flow is a special case of two-phase flow. Another case of two-phase flow involving water displacing oil in random media was recently studied by Zhang & Tchelepi (1999). The multiphase flow and transport equations developed in the previous section can be simplified for many groundwater flow problems through the application of the following assumptions: (a) isothermal conditions, (b) individual phases consisting only of water in the liquid phase and air in the vapor phase, and (c) an infinitely mobile air phase. The latter assumption reduces the two-phase flow problem to one in which only the water movement is computed. This formulation, known as Richards’ equation, is represented by the following expressions (with a change in nomenclature from the previous example to maintain consistency with the groundwater literature): ∂h(x,t) + ∇ · q(x, t) = g(x, t), ∂t ∂ qi (x, t) = −K[h, ·] [h(x, t) + x1 ], ∂ xi

C[h, ·]

(9) (10)

subject to initial and boundary conditions h(x, 0) = H0 (x)

x ∈ Ä,

(11)

h(x, t) = H (x, t)

x ∈ 0D ,

(12)

q(x, t) · n(x) = Q(x, t)

x ∈ 0N .

(13)

In these equations, q is the specific discharge (flux) vector with components qi; h(x, t) + x1 is the total head; h is the pressure head; i = 1, . . . , d (where d is the number of space dimensions); g(x, t) is the fluid source/sink term; H0(x) is the initial pressure head distribution in the domain Ä; H(x, t) is the prescribed head on Dirichlet boundary segments 0 D at time t; Q(x, t) is the prescribed flux across Neumann boundary segments 0 N at time t; n(x) = (n1, . . . , nd)T is a unit vector outward normal to the boundary; C[h, ·] ≡ dθe /dh is the specific moisture capacity; and K[h, · ] is the unsaturated hydraulic conductivity (assumed to be

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isotropic locally). Both C and K are functions of pressure head and soil properties at x. For convenience, they are written as C(x, t) and K(x, t) in the sequel. Without loss of generality, H0(x), Q(x, t), and g(x, t) are assumed to be deterministic (i.e., known with certainty). The elevation x1 is directed vertically upward. In these coordinates, recharge has a negative sign. Equations 9–13 are incomplete without specifying constitutive relationships of K versus h and C versus h. No universal models are available for the constitutive relationships. Instead, several empirical models are commonly used, including the Gardner-Russo model (Gardner 1958, Russo 1988), the Brooks-Corey model (Brooks & Corey 1964), and the van Genuchten–Mualem model (van Genuchten 1980). Among these three models, the Gardner-Russo constitutive relation is the simplest and is often used for deriving analytical solutions and for stochastic modeling. The simple Gardner-Russo model reads as K (x, t) = K s (x) exp[α(x)h(x, t)],

(14)

θe (x, t) = (θs − θr ){exp[0.5α(x)h(x, t)] · [1 − 0.5α(x)h(x, t)]},

(15)

C(x, t) = −

θs − θr 2 α (x)h(x, t) exp[0.5α(x)h(x, t)], 4

(16)

where α is the soil parameter related to pore size distribution, θ r is the residual (irreducible) water content, θ s is the saturated water content, and θ e is the effective water content. However, it is well known that the more complex van Genuchten model and the Brooks-Corey model usually fit measured data better. The spatial variability of saturated hydraulic conductivity, KS, is well documented in the literature (e.g., Freeze 1975, Hoeksema & Kitanidis 1985, Sudicky 1986, Neuman & Depner 1988). However, there are relatively few studies that investigated the statistical properties of rock or soil unsaturated parameters such as α. Recently, Tartakovsky et al. (1999) gave an excellent review on the variability of α. The values of α strongly depend on soil texture and vegetation: ranging from 0.05 cm−1 for clay to 0.71 cm−1 for gravely loam fine sand (White & Sully 1987), 0.15–1.34 cm−1 for grassland, 0.36–0.37 cm−1 for woodland, and 0.28–0.89 cm−1 for arable land (Ragab & Cooper 1993a,b). The variance of log transformed α, ln α, can be either larger or smaller than that of f = ln KS. Unlu et al. (1990a) reported ln α variances in the range of 0.045–0.112, corresponding to 21.5%–34.4% for the coefficient of variation of α, Cvα = σ α/hαi. They reported the variances for f in the range of 0.391–0.96, which are equivalent to 69.2%–127% for CvKs, the coefficient of variation of KS. Russo et al. (1997) found the ln α variance to be 0.425 (i.e., Cvα = 72.8%), compared to 1.242 for the variance of f (i.e., CvKs = 156.9%). White & Sully (1992) and Russo & Bouton (1992) found the variances of ln α and f to be of similar order, whereas Ragab & Cooper (1993a,b) observed the variance of to exceed that of f. When the saturated hydraulic conductivity and the unsaturated rock (soil) parameters are treated as random fields, the Richards’ equation becomes a nonlinear SDE. The method of moment differential equations (MDEs) is frequently used for

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solving SDEs of unsaturated flow. The essence of this approach is to first formulate equations governing the statistical moments of flow quantities of interest from the original SDEs and then solve these moment equations, either analytically or numerically. Recently, Zhang (2002) discussed various moment equation methods. Commonly used moment equation methods include perturbative expansions, spectral analyses, and adjoint state equations. The results from the moment equation methods are usually the expected values, variances, and covariances of the flow quantities such as pressure head, effective water content, flux, and velocity. The resultant moments provide a prediction of the flow field and a measure of prediction uncertainty caused by incomplete knowledge of medium heterogeneity. Many earlier stochastic studies focused on steady-state, gravity-dominated unsaturated flow in unbounded domains (e.g., Yeh et al. 1985a,b; Russo 1993, 1995a,b; Yang et al. 1996). Under these conditions, the unsaturated flow field is stationary; hence analytical or semi-analytical solutions are possible. Some researchers investigated the effects of boundary conditions on steady-state flow and hence the effects of flow nonstationarity in one-dimensional bounded or semibounded domains (Andersson & Shapiro 1983, Indelman et al. 1993). Recently, Zhang & Winter (1998) developed a general nonstationary stochastic approach for steady-state unsaturated flow in bounded domains. Based on some one- and twodimensional examples, the latter authors found that the simpler, gravity-dominated flow models may provide good approximations for flow in vadose zones of large thickness and/or coarse-textured soils, although they may not be valid for vadose zones of fine-textured soils with a shallow water table. In most of the previous studies, soils are assumed to be described by the Gardner-Russo constitutive model (Gardner 1958, Russo 1988) due to its simplicity. However, it is well known that the more complex van Genuchten model (van Genuchten 1980) and the Brooks-Corey model (Brooks & Corey 1964) are usually in better agreement with observed data. Zhang et al. (1998) recently studied the impact of different constitutive models on steady-state unsaturated flow in both bounded and unbounded domains. Mantoglou & Gelhar (1987) extended the spectral perturbation approach of Yeh et al. (1985a,b) to transient unsaturated flow. However, due to the nature of the spectral approach, it was necessary to make some restrictive assumptions such as spatial and temporal stationarity of local flow characteristics (e.g., slowly varying mean gradient), stratified soil formation, and infinite flow domain. Unlu et al. (1990b) studied one-dimensional transient redistribution of unsaturated flow through Monte Carlo simulations and found a good agreement with the spectral perturbation approach except near the boundaries. Mantoglou (1992) developed a more general stochastic framework for transient unsaturated flow using the distributed parameter estimation theory of McLaughlin & Wood (1988a,b). The theoretical framework consists of a set of partial differential equations for mean head and head covariances. These equations are nonlinearly coupled and have a high dimensionality of 2d + 1 (where d is the number of space dimensions). Due to the nonlinear coupling and dimensionality problems, Mantoglou (1992) was not able to implement the theoretical equations except for some simplified cases.

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More recently, Liedl (1994) proposed a slightly different perturbation model for transient unsaturated flow. Similar to Mantoglou (1992), the results of the model are a set of partial differential equations governing the statistical moments of saturation. Liedl implemented the model in one dimension. Protopapas & Bras (1990) and Li & Yeh (1998) studied transient unsaturated flow in heterogeneous porous media using a vector state-space approach. This approach was originally proposed by Dettinger & Wilson (1981) for saturated flow in heterogeneous porous media. The essence of the approach is to first discretize the governing equations by a numerical scheme such as finite differences or finite elements and then expand random variables contained in the system by Taylor series to derive approximate statistical moments of flow quantities. Protopapas & Bras (1990) studied flow and solute transport in one-dimensional soil columns. Li & Yeh (1998) investigated the behavior of head variances for transient unsaturated flow in two dimensions. More recently, Zhang (1999) extended the nonstationary stochastic approach for steady-state unsaturated flow (Zhang & Winter 1988) to transient unsaturated flow. He first derived partial differential equations governing the statistical moments of the flow quantities by perturbation expansions and then implemented these equations under general conditions by the method of finite differences. The developed model is applicable to the entire domain of a bounded, multidimensional vadose zone in the presence of sink/source. This moment equation approach is different from the state-space approach of Protopapas & Bras (1990) and Li & Yeh (1998) in that the former first derives the moment equations and then solves them numerically, whereas the latter expresses the statistical moments based on the spatial and temporal discretizations of a particular numerical scheme. Therefore, unlike the state-space approach, the moment equations derived by Zhang (1999) are independent of the specific numerical scheme to be used and can be solved on numerical grids to be determined a posteriori based on the characteristics of the moment functions. This moment equation approach is similar to the approach of Mantoglou (1992) in that both approaches result in a set of partial differential equations governing the statistical moments of flow quantities, which are to be solved numerically. However, unlike those of Mantoglou (1992), the equations derived by Zhang (1999) for the second moments are linear and decoupled with the nonlinear first moment (mean) equation and can thus be solved sequentially and efficiently. The other widely used method is Monte Carlo simulation. This approach is conceptually straightforward and is based on the idea of approximating stochastic processes by a large number of equally likely realizations. The Monte Carlo approach can handle complex geometry and boundary conditions and requires fewer assumptions than does the moment equation approach. Most importantly, the Monte Carlo approach can, in principle, deal with extremely large variability in the independent variables so long as the number of realizations is large. However, this leads to the main disadvantage of the Monte Carlo approach: It must solve many realizations of a given aquifer or vadose zone. This approach requires considerable computation and a careful examination of the results. In general, two

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types of error are associated with Monte Carlo simulation results: numerical and statistical. The former depends on the numerical method and the particular solver used as well as the spatial and temporal discretizations. The larger is the spatial variability, the finer is likely the required spatial discretization. Statistical errors arise from the method used to generate realizations and the number of realizations. On the other hand, moment equations are derived under the assumption of small perturbations or with some kind of closure approximation, either of which may introduce error. Nevertheless, the moment equation approach has some important advantages. For example, the coefficients of the moment equations are relatively smooth because they are averaged quantities. Thus, the moment equations can be solved on relatively coarse grids. In addition, the moment equations are available in analytical form, even though they are usually solved numerically in applications. This holds the potential for increased physical understanding of the mechanisms of uncertainty through qualitative analysis. Last but not least, the results from the moment equation approach are less ambiguous than those of the Monte Carlo approach. The ambiguity in the Monte Carlo simulation results may stem from its multiple sources of potential error, which are difficult to either ascertain or control (Neuman 1997). Because of this ambiguity, different sets of Monte Carlo simulations may give significantly different results even for the same physical problem. Based on this observation, Dagan (1998) recently suggested that “. . . (Monte Carlo) simulations may serve as a reliable tool for validating approximate theoretical results only after making sure that various authors arrive at similar results albeit by different methods.”

MAGMA CHAMBERS AND VOLCANIC ERUPTIONS One of the most challenging sets of complex, nonlinear geologic problems are those associated with the generation, ascent, evolution, and eruption of magmas. There are many reasons for this complexity, including the complex rheology of magmas, the range of flow conditions (creeping to supersonic) in volcanic systems, and the fact that most magmatic processes involve multiphase flow. Numerical modeling of igneous processes has rapidly evolved over the past 15 years but is still in its infancy compared to the level of complexity that the natural systems exhibit. Bergantz (1995) provides a good description of the importance of such modeling in magma dynamics. Although advances have continued over the six years since Bergantz’s paper, most of his points remain important today. Many magma chamber and eruption processes have also been addressed by laboratory analog experiments; much of this work was recently reviewed by Jaupart & Tait (1995). Analog experiments have an important role in understanding such geological problems as magma dynamics in that they help us to understand basic “subprocesses” within the chambers and provide test cases against which theoretical/numerical models can be validated. As Bergantz (1995) describes, though, there are many complications in magmatic systems (e.g., nonlinear, variable rheology; geometry

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of crystal mushes and solid-liquid interfaces; system geometry; coupling with surrounding processes such as hydrothermal circulation) that cannot be duplicated or scaled adequately in laboratory settings. This necessitates numerical modeling. Obviously, because both analog experiments and numerical modeling have limitations, rapid progress will require integration of the two approaches. Magma chambers are “holding tanks” for melt in the lithosphere. The chambers may range in size and shape from thin (tens of meters) sill-like bodies to large “vats” that contain thousands of cubic kilometers of magma, and in some cases they may actually be collections of closely spaced, partially interconnected pockets of magma. Much of the chemical and mineralogical diversity of igneous rocks originates by magma chamber processes that are inherently nonlinear; some of these are also key in triggering and influencing the dynamics of eruptions. Here we summarize numerical modeling work aimed at understanding these processes. Figure 4 illustrates the overall mass and energy balance for a magma chamber (note that the shape is purely schematic). Mourtada-Bonnefoi et al. (1999) developed a simple mathematical model for this mass and energy balance that accounts for most of the factors illustrated in Figure 4: magma replenishment, magma loss through eruption, cooling to surrounding rocks, and fractionation of crystals and its effects on composition of the remaining melt. Their model forms a set of four coupled ordinary differential equations for conservation of mass, chemical species, energy, and solidification rate. Although many complex processes that occur within magma chambers are not represented in the model, it does capture some basic features of magma chamber evolution and illustrates some of the potential nonlinear behavior of magmatic systems. Mourtada-Bonnefoi et al. show that a magma chamber will achieve steady-state behavior within a few times the residence time (equal to the mass of magma in the system divided by the mass flux into the chamber) if the mass and energy flux into and out of the chamber are held constant. If the fluxes vary with time, there are conditions under which a magma chamber may undergo a sudden change in composition even as the fluxes change gradually. Mourtada-Bonnefoi et al. (1999) suggest that such a catastrophic change may explain some gaps in the composition of erupted products from a chamber or perhaps may cause a major eruption that would otherwise have no apparent cause. At a smaller scale, within magma chambers, a number of important coupled fluid flow-phase change phenomena (e.g., see reviews by Sparks et al. 1984, Martin et al. 1987, Trial & Spera 1990, Valentine 1992, Marsh 1996) have been studied by numerical modeling. The fluid in a magma chamber is a silicate melt with properties that are typically between 2400–2800 kg/m3 in density and 109–102 Pa s in viscosity (not accounting for suspended crystals), which are the ranges for rhyolitic to basaltic magmas, respectively (e.g., Trial & Spera 1990, Spera 2000). Magma chambers may contain magmas of different compositions either because of injection of a new composition into a chamber or from fractionation as components leave the melt phase to form crystals. The compositional variations within a chamber provide buoyancy forces that may drive convective flow. In addition, temperature variations and heat flux out through surrounding rocks can also cause

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Figure 4 Diagram from Valentine (1992) schematically showing the overall heat and mass balance of a magmatic system (not drawn to scale).

convection. The combined effects of compositional and thermal buoyancy are referred to as double-diffusive convection (DDC) because diffusion of both heat and chemical species are important. Thermal diffusivities of magmas are commonly around 10−6 m2/s, whereas binary chemical diffusivities are mostly less than 10−10 m2/s. This means that as convective flow progresses, thermal variations

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will diffuse over a relatively short time frame while compositional variations remain sharp, although they may be stretched and distorted by fluid motion. Another result of the large ratio of thermal to compositional diffusivity is that while the thermal Rayleigh number, RaT = gα1Th3/(νκ), is much smaller than the compositional Rayleigh number, Rac = gβ1Ch3/(νD), they both roughly contribute equally to the convection (g is gravitational acceleration; α is the isobaric thermal expansion coefficient; 1T is the temperature difference from top to bottom of the chamber; h is the height of the chamber; ν is the kinematic viscosity of the magma; κ is the thermal diffusivity; β is the isobaric compositional expansion coefficient; 1C is the compositional difference from top to bottom in the chamber; and D is the compositional diffusivity). The high ratio of diffusivities and large values of Rayleigh numbers (>107 for thermal Rayleigh number defined across the whole thickness of a chamber) result in thin, vigorous plumes and highly distorted composition interfaces. Both of these factors make numerical modeling of DDC very difficult because of the extremely fine meshes necessary to resolve flow structures and because some fluid dynamics algorithms result in artificial diffusion that acts against the low chemical diffusivity of magmatic components (an important technique to overcome this problem is front tracking, which has seen some application in mantle convection modeling but little, to our knowledge, in magma chamber modeling) (Travis 1990). Nevertheless, some important insights have been obtained in the past several years. Oldenburg et al. (1989, 1990) solved the mass, momentum, energy, and chemical species conservation equations for two-dimensional, two-species convection. As with most magma chamber models, the major simplifications include that of creeping flow (acceleration and advection terms neglected in the momentum equations), infinite Prandtl number (ratio of kinematic viscosity to thermal diffusivity), and a Boussinesq equation of state in which density is a linear function of composition and temperature and varies over a sufficiently small range that the flows can be considered incompressible. The boundary and initial conditions for simulations by Oldenburg et al. consisted of a box-shaped magma chamber with insulating sides, specified heat flux into the bottom, and constant temperature at the top. Various aspect ratios were used for the chamber, and the “magma” consisted of a thin, compositionally buoyant layer on top of a thicker, compositionally denser layer (i.e., representing a layered magma chamber with more silicic magma on top) as the initial condition. Figure 5 shows snapshots of a simulated magma chamber during early times in its evolution. By a dimensionless time of 0.1, a convection cell with counterclockwise rotation has been established, drawing the initially horizontal upper layer of compositionally low-density magma downward along the left side of the domain (see contours in right-hand column of Figure 5). At a time of 0.15, however, the flow has reversed. This reflects the competing, nonlinear effects of thermal buoyancy driving fluid upward from the base of the domain and local compositional buoyancy working to maintain the initial configuration. The more rapid diffusion of heat compared to composition is shown in the contours; compositional differences

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Figure 5 Early phases of double-diffusive convection simulations for an aspect ratio of unity, from Oldenburg et al. (1989). Time is given in dimensionless units (a value of 1 equals the characteristic thermal diffusion time for the height of the chamber). The columns show, from left to right, dimensionless velocity vectors (scale shown by vertical arrow), temperature, and composition contours (both temperature and composition range in value between 0–1).

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tend to maintain a sharp, interface-like configuration long into the simulation (the ratio of thermal to chemical diffusivities in this simulation is 200). By a time of 89.8 (Figure 5d), a counterclockwise motion is reestablished. One purpose of these simulations was to test the stability of simple, vertically layered magma chambers that have been inferred from volcanic eruption sequences. The work of Oldenburg et al. (1989) indicates that, within the parameter regimes they studied, such layering may not be stable or long lived. Rather, any initial compositional layering that exists will likely get caught up in the whole-scale convection of the magma chambers. Initial compositional contrasts can be very long lived, but instead of maintaining simple layers, they form lenses or ocularpods (Oldenburg et al. 1990) within a chamber. Swirls or stringers of different compositions may flow together but only gradually lose their compositional integrity due to the low diffusivity. Oldenburg et al. (1989) developed statistical methods for characterizing these features in their simulations, and it is likely that some of the statistics could be used on carefully studied volcanic products in order to infer more quantitatively the nature of the parent magma chamber. Information from the simulations can also be used to track the overall heat transfer out the top of the numerical magma chamber. The complicated convection with flow reversals produces large fluctuations in heat flow. This can aid in interpreting heat flow measurements at actual volcanic systems. The reader is referred to Hansen & Yuen (1990) for a review of a broad range of nonlinear, double-diffusive convection phenomena associated with a variety of geophysical applications. The work described in the above paragraphs studies convection in multicomponent (i.e., at least two chemical species) systems. In addition, magmas are almost always multiphase systems that include liquid, solid (crystals), and at shallow depths, gases. The role of phase change in the form of crystallization is critical to understanding magma chambers. Ultimately, crystallization must be considered hand-in-hand with convection because of the key role of the latter in heat and mass transport; in turn, the presence of crystals affects the bulk “convectibility” of a magma chamber (e.g., Brandeis & Marsh 1989) or can drive convection and affect flow patterns (e.g., Bergantz & Ni 1999, Koyaguchi et al. 1990). Here we briefly describe numerical modeling aimed at coupled convection and melting/crystallization and at multiphase effects on convection. These processes overlap greatly with many aspects of metallurgical sciences, although some of the parameter regimes are different, and natural magmas are much more complicated than most engineered materials. A large fraction of research aimed at magmatic crystallization and convection is being published in the engineering literature, demonstrating the interdisciplinary nature of geological dynamics. One set of interesting problems involving coupled flow and phase change is that having to do with the formation of crustal magma chambers. Some magma chambers are caused (partly or wholly) by the melting, or anatexis, of crustal rocks that are composed of low-melting-temperature mineral assemblages. The source of heat for this melting is from rapid intrusion of hot basaltic magma that originates at mantle depths. Fountain et al. (1989) explored this process by solving

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a set of equations that conserve the mass of a solid phase and of a melt phase (melt fraction determined by phase relationships), accounting for advection of melt by porous flow and of the solid matrix by compaction. Momentum balance within the melt phase is described by Darcy’s Law. Their heat transport model accounts for advection and conduction, with latent heat accounted for in effective heat capacities. In their model, a thick layer of mafic magma is instantaneously emplaced at the bottom of the domain and then allowed to cool by conduction. The constraint that the mafic layer cools by conduction means that the rate of heat transfer into the overlying rocks is minimized compared to the situation in nature, where it is likely that convection within the mafic layer would speed up heat transfer (e.g., Huppert & Sparks 1988). Overlying rocks begin to melt, producing an increasingly open porous network through which melt can begin to migrate. Upward, buoyant migration of melt, interacting with heat transfer processes, results in the formation of a layer of completely molten, silicic magma that migrates slowly upward (Figure 6). The absence of thermally driven convection (either in the mafic magma layer or in the resultant silicic melt layer) in the Fountain et al. (1989) calculations limits their application to real systems. Bergantz (1992) conducted a series of onedimensional simulations aimed at heat transfer into and melting of an initially solid rock by a hot, convecting magma below. Bergantz’s calculations show that heat transfer into the initially solid rock from a convecting magma is up to 35% greater than the conduction only case. Barboza & Bergantz (1998) extended this work to study how different transitional melt fractions, which define the transition from crystal-supported porous media flow to fluid flow with suspended crystals, affect the melting of country rocks above basaltic sills. Building on the work of Oldenburg & Spera (1992), who defined a switching function that accounts for rheological changes from pure melt, to melt with suspended crystals, to a crystal mush through which the small melt fraction flows according to Darcy’s law, Barboza & Bergantz solved equations for conservation of mass, momentum, heat, and chemical species in two dimensions (see also Barboza & Bergantz 1996, 1997). Figure 7 illustrates the effects of varying the width of the rheological transition zone on propagation of a melting front and convection within the growing fluid region. The interesting result is that a broad transition zone (i.e., porous media flow at relatively larger melt volume fractions) significantly slows the melting front and the initiation of convection. This is due to the complex interplay between nonlinear melting as a function of temperature, compositional effects, and conductive and advective heat transport. While the Fountain et al. (1989) and Barboza & Bergantz (1998) papers explore coupled convection and phase change from the perspective of melting and the formation of magma chambers, Spera et al. (1995) modeled processes associated with the solidification of an existing magma chamber. Spera et al. (1995) also applied the switching function of Oldenburg & Spera (1992) to account for flow regime and rheological changes in going from pure melt to crystal mush. As with descriptions of many nonlinear geologic systems, the work by Spera et al. illustrates

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Figure 6 Volume fraction of melt as a function of distance above the contact between crustal rocks and mafic magma, for four different times after initial intrusion. In this simulation, the intrusion thickness is 5 km. A layer of large melt fraction forms some hundreds of meters above the contact, as refractory minerals are compacted and squeeze out pore melt. From Fountain et al. (1989).

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Figure 7 Snapshots of two-dimensional melting calculations, from Barboza and Bergantz (1998). The computational domain consists of an initially cold rock at 600◦ C, with a melt fraction versus temperature curve of metapelite. The right half of the bottom boundary is set at a constant temperature of 1000◦ C, representing a basaltic sill (the right side of the domain is reflective, due to symmetry, while the other sides are thermally insulating). The snapshots show contours of melt volume fraction and velocity vectors in the viscous flow (mainly fluid) region at a time of 79.3 years. Each snapshot shows the results for a different value of ζ , which is a measure of the sharpness of the transition from a crystal-supported network with porous flow to a pure fluid flow (small values of ζ represent cases where porous media flow dominates to larger melt volume fractions).

the often nonintuitive complexities in behavior when mass, chemistry, momentum, and energy conservation are all coupled. For example, their calculations of a crystallizing, convecting magma chamber with two mineral components (leucite and diopside) show the development of “spontaneous” heterogeneities in mineral abundance and time-dependent variations in heat flux out of the chamber that result from sudden crystallization.

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Magma chamber dynamics set the stage for volcanic eruptions. Much attention has been focused on modeling explosive eruptions because of their intrinsic danger and the difficulty in making direct measurements of associated phenomena. Two recent books (Freundt & Rosi 1998, Gilbert & Sparks 1998), particularly the chapters by Papale (1998) and Valentine (1998), provide thorough reviews of explosive eruption modeling, so we do not further discuss this topic here. A characteristic common across nearly all magmatic processes, from melting in the deep earth through evolving magma chambers to eruption, is that they involve multiple phases; in other words, the processes involve coupled motion of some combination of liquid (normally a silicate melt), solids (mineral crystals), and gases. Coupling between the phases drives many nonlinear processes. For example, a parcel of magma near the roof of a magma chamber may contain an increasing fraction of crystals as it cools. These crystals may begin to settle downward due to their relatively higher density, and as they do, they will drag surrounding melt downward with them. As the resulting downwelling plume moves into progressively warmer surroundings, the crystals may begin to melt, reducing their drag on the melt and causing feedback processes that affect the flow. Similarly, a pyroclastic flow consists of solid particles mixed with gases that are dragged downward with the particles. As particles migrate downward as the flow moves across the landscape, a denser bottom zone is formed that may favor high mobility. Simultaneously, though, upper parts of the flow have fewer particles and therefore less downward drag on the hot gases, which now may be able to rise into the atmosphere. This in turn causes a “headwind” as air is entrained into the rising batch of gas; the headwind may substantially retard the lateral motion of the pyroclastic flow. Such multiphase feedback mechanisms are a trademark of many nonlinear geologic processes. A very effective framework for modeling the complicated nonlinear processes associated with such multiphase flows is the multifield approach. This approach treats each phase (liquid, gas, and particle ensembles) as a fluid field. The fields occupy the same control volume in proportion to their volume fractions, which sum to unity, and each have their own set of mass, momentum, and energy conservation equations. Special terms in these equations account for the coupling between phases. Examples of such coupling include movement of chemical components from melt into solid crystals, drag (momentum coupling) between rising gas bubbles and melt, and heat transfer from hot particles to surrounding gas in an eruption plume. Note that accurate representation of these coupling processes requires detailed knowledge of the processes at the scale of individual bubbles and particles, while the multifield approach itself is aimed at modeling ensemble behavior. Many of the modeling studies reviewed above in this section use multifield approaches to varying degrees. Valentine (1994) proposed a multifield framework for all magmatic processes, with the goal of using the framework to guide experimental and modeling studies and to build predictive models of whole volcanic systems (e.g., as proposed by Dobran 1993, 1994).

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FUTURE DIRECTIONS As mentioned in the Introduction, we have focused on three examples of complex, nonlinear geologic processes. The underlying commonality of these examples lies in the fact that they all involve multiphase fluid flow, as do a large fraction of geological problems. For such problems there are, in our opinion, four broad areas of research that will have a major impact: upscaling and constitutive models, coupling across regimes, stochastic processes, and model validation.

Upscaling and Constitutive Models The work that we have reviewed above all focuses on the large-scale, or macroscopic, behavior of systems. However, all of this macroscopic behavior depends on processes at the microscale that may be described by quite different governing equations. One example has already been provided in the Introduction: that of saturated flow through porous media, where the large-scale flow described by linear equations ultimately depends on flow around individual grains that is described by nonlinear equations. Although saturated porous media flow is not typically thought of as a multiphase flow, it can be considered multiphase flow where one phase (the solid grains or matrix) is stationary. Recent studies by Zhang et al. (2000) are quantifying the link between the micro- and macroscale for this type of flow. Other examples of different physics at different scales abound in geology. For instance, the mobility of pyroclastic flows over tens of kilometers, a scale at which multifield Navier-Stokes equations are appropriate, probably is strongly linked to the momentum exchange between colliding particles of millimeter size. The upscaling from these small-scale processes to the fieldscale processes of interest will be essential to our full understanding of geologic problems and will require both experimental and theoretical studies. A rigorous understanding of upscaling will in turn be the foundation for appropriate constitutive models that average the microscale processes for field-scale models.

Coupling Across Regimes Studies of geologic processes have typically involved isolation of a phenomenon of interest for intensive research. For example, many studies have been conducted separately on magma chambers, volcanic conduit flow, and eruptions. The complexity of each one of these phenomena or regimes warrants such separate study, and this needs to continue. However, our understanding of each of these regimes has evolved to the point where it is appropriate to also study the coupling between them. For example, in the case of a volcanic system, one can envision eruption phenomena that are determined by the interplay between convection of magmas of different compositions and evacuation through a conduit. Within the realm of hydrology, studies of the coupling between ground and surface waters are

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increasing. Research on coupled geologic systems are sure to reveal further complex, nonlinear behaviors. In general, geologic systems involve multiple length and time scales. It is a scientific challenge to couple processes of various length and time scales.

Stochastic Processes We have already seen, in one of the sections above, that techniques are being developed to model multiphase flow through heterogeneous porous media. Nearly all geologic processes involve heterogeneous materials and uncertainty in properties and processes. A very fruitful area for future research is continued development of theories that combine first principles with our understanding of the stochastic nature of geologic processes.

Model Validation The design and execution of field experiments or detailed observations of natural systems are necessary to provide confidence in the validity of numerical models for capturing the physical and chemical processes being simulated. Experimental studies often require large expenditures of money and detailed planning to achieve even a partial validation, but when successful, they provide critical confirmation of model results. Examples in the Yucca Mountain project are the experimental program to examine TH processes (e.g., Lin et al. 1997, Tsang & Birkholzer 1999) and the field experiments to establish the validity of unsaturated flow and transport models (Robinson & Bussod 2000). In some cases, rather than conducting an experiment to study processes and validate a model, observations of a natural system are used to constrain the model, and in such cases, the “validation” is achieved through a demonstration that the model captures the essential features of the measurements. In such cases, more robust models will be developed when projects take more multidisciplinary approaches in data collection. The application of a variety of different measurements (hydrologic, geochemical, geophysical, etc.) must then be combined with innovative methods for synthesizing the information and incorporating it into numerical models. Advanced visualization techniques and, for larger systems, the use of Geographic Information Systems (GIS) tools will assist in these efforts. Studies of nonlinear geologic processes are important in their own right in improving our understanding of how Earth works. In addition, there are many applications of this understanding that are of key importance to society. Examples include the subsurface sequestration of carbon dioxide, environmental remediation and mitigation, and the reduction of geologic hazards. The few areas of research that we have highlighted in this review are but a drop in the bucket, but we hope that they will aid in focusing interest in bringing numerical modeling into the mainstream of geology.

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ACKNOWLEDGMENTS Included in this review is work derived from several U.S. Department of Energy programs, most notably Laboratory Directed Research and Development and the Yucca Mountain Project. We thank Paula Geisik for her expert help in preparing the manuscript. Visit the Annual Reviews home page at www.annualreviews.org

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ber: a fluid dynamic investigation. Nature 339:613–16 Brooks RH, Corey AT. 1964. Hydraulic properties of porous media. Hydrol. Pap. 3. Colo. State Univ., Fort Collins Brownell DH, Garg SK, Pritchett JW. 1975. Computer simulation of geothermal reservoirs. Proc. Calif. Reg. Meet. Soc. Pet. Eng. AIME, 45th, Ventura, Paper SPE 5381 Buscheck TA, Nitao JJ. 1993. Repository-heatdriven hydrothermal flow at Yucca Mountain: part I. Modeling and analysis. Nucl. Technol. 104:418–48 Buscheck TA, Nitao JJ, Ramspott LD. 1996. Localized dryout: an approach for managing the thermal-hydrological effects of decay heat at Yucca Mountain. Mater. Res. Soc. Symp. Proc. 412:715–22 CRWMS M&O (Civilian Radioactive Waste Management System Management and Operating Contractor). 1995. Total System Performance Assessment-1995: An Evaluation of the Potential Yucca Mountain Repository. B00000000-01717-2200-00136, Rev. 01. Las Vegas, Nevada CRWMS M&O (Civilian Radioactive Waste Management System Management and Operating Contractor). 1999. Impact of Radioactive Waste Heat on Soil Temperatures. BA0000000-01717-5700-00030, Rev. 0. Las Vegas, Nevada Dagan G. 1989. Flow and Transport in Porous Formations. New York: Springer-Verlag Dagan G. 1998. Comment on “Renormalization group analysis of macrodispersion in a directed random flow” by U. Jaekel and

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Figure 2 Predicted TH model results at 1000 years after waste emplacement. (a) Fluid saturation, (b) expanded view of fluid saturation, (c) temperature.

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