Modeling of Fluid Transport in Geothermal Research Jörg Renner a and Holger Steeb b a Experimentelle Geophysik - Institut für Geologie, Mineralogie und Geophysik, Ruhr-Universität Bochum, Bochum, Germany b Kontinuumsmechanik - Mechanik, Ruhr-Universität Bochum, Bochum, Germany Abstract Extraction of heat stored in the rocks in Earth s upper crust requires a fluid to which the heat can be transferred and that can migrate through the conduits in the rock. Flow of these fluids requires gradients in fluid pressure. Often, a linear relation between flow rate and pressure gradient offers a sufficiently precise description. Fluid pressure in turn couples the flow with the deformation of the solid host. We derive and analyze the governing equations underlying these hydro-mechanical, coupled problems. Our presentation addresses simple conduits and porous media. The latter are tackled from a continuum perspective, specifically mixture theory. Analytical and numerical solutions of the differential equations are analyzed for scenarios from well testing to propagation of elastic waves. Well testing constitutes the traditional tool for hydraulic characterization of geothermal reservoirs. Evaluation of attenuation of elastic waves may allow for hydraulic characterization and monitoring relying on surveys. 1 Introduction Fluid flow entirely inside the earth, for example, convection in the outer liquid core on a spatial scale of 10 6 m (Gubbins 2001), and between its interior and its surface, for example, the rise of magma at mid-ocean ridges or water exchange between the world s oceans and oceanic lithosphere or the continental-scale groundwater flow on a spatial scale of 10 5 m (e.g., Stein and Stein 1994; Garven 1995; Stein et al. 1995; Müntener 2011), is crucial for matter and heat transport and thus for the very dynamics that so much control plate tectonics and ultimately the environment provided for life (e.g., Ingebritsen and Manning 2003).Onasmallerscale(10 1:::4 m), fluid flow is also of eminent importance for provision of liquid resources (e.g., Doré and Sinding-Larsen 1996; May2010; Gleeson et al. 2012) and temporary or permanent underground storage of fluids (e.g., Daniel 1993). Thus, it is not surprising that a large number of theoretical and numerical studies regarding geoscientific fluid-flow problems have been presented in the past, addressing fundamental questions as well as engineering applications. Employed methods range from analytic techniques to numerical analyses on supercomputers. For geothermal research, fluid flow is relevant in two prominent ways. First and foremost, the power that can potentially be provided at the surface in electrical or thermal form simply reads P / 4V f 4c; (1) 4t E-mail: renner@geophysik.rub.de Page 1 of 55
i.e., it scales with the product of the (volume) flow rate of fluid, 4V f =4t [m 3 s 1 ]; the temperature drop experienced by the produced fluid during its usage for energy provision, 4 [K]; and the specific heat of the fluid, c [J m 3 K 1 ]. The temperature difference is essentially controlled by the reservoir temperature and the technical installations at the surface transforming the heat stored in the fluid into electricity and/or transferring the heat of the produced fluid to a secondary fluid circulating in a heating system. Reservoir temperature depends on the depth reached by the production well and the specific geological situation. In the absence of exceptional hydrothermal activity, a depth of about 3 5 km is required to reach a reservoir temperature of close to 150 ı C, the prerequisite for operating a turbine with steam to produce electricity. Specific heat is a material parameter characterizing the working fluid, likely water or steam. Typically the reservoir will contain liquid water, and steam will only be generated when the phase boundary is crossed during depressurization on the fluid s way to the surface. Thus, the flow rate constitutes the central variable parameter defining a successful and economically feasible operation. Central questions such as How does the flow rate evolve over the lifetime of a power plant? or How can the flow rate be affected by pumping protocols before (stimulation) and during plant operation? can only be answered based on a substantial understanding of the fluid flow in the reservoir. Second, the controlled initiation of fluid flow in a potential geothermal reservoir is a mandatory prerequisite for its hydraulic characterization. A key question in this context is How can the longterm flow rate be reliably predicted from short-term hydraulic experiments employing minimal perturbations of the reservoir pressure? Besides the conventional approach of conducting pumping tests in wells reaching the reservoir, the transmission of seismic waves through the reservoir provides a means of initiating fluid movement, and we therefore consider the relation between fluid flow and propagation of elastic waves, a relevant aspect of fluid flow in geothermal research. However, we restrict to such highly dispersive wave phenomena that are characterized by a relative movement between the fluid and the solid. We feel requested to analyze pressure diffusion and hydromechanical coupling, too, under the overarching theme of fluid flow. The analysis of hydraulic well testing in particular of fractured reservoirs likely mandates accounting for hydromechanical coupling, i.e., coupling of changes in pore fluid pressure and stress state of the solid framework. The mobilization of fluid during the passage of a seismic wave is a coupled phenomenon by itself. Volume flux (flow rate per unit area) and fluid pressure are inseparable field variables in poro-elasticity. Yet, pressure diffusion may reach spatial extents quite different from the ones characteristic for the transport of substantial amounts of fluids. Small fluid volumes may have a significant effect on fluid-pressure levels that in turn may alter the local stress state of the reservoir to the extent that inelastic deformation, e.g., seismic activity, is induced or triggered (see McGarr et al. 2002, for terminology and examples). Seismicity, i.e., inelastic failure of underground volumes documented by the occurrence of earthquakes, associated with pumping operations to some degree wanted and necessary for a successful stimulation but unwanted during continuous plant operation will however not be tackled in detail. As in most geoscientific fields of problems, the analyses of data records associated with subsurface fluid flow require the iterative development of subsurface models. Since data in the form of time series of, for example, flow rate and pressure are typically only available at a limited number of points in space, one has to rely on concepts such as hydraulically equivalent scenarios or materials with a limited number of structural parameters and explicit physical properties. Laboratory experiments on rock samples may plainly serve the purpose of material characterization or may aim at the investigation of analogue models on miniature scale. The transport of heat by fluids in motion, the very prerequisite for geothermal energy provision, exhibits parallels to the transport of solutes. Thus, tracer tests play a crucial role in the identification of preferential flow paths. Page 2 of 55
The contribution is structured in a way that aims at serving the likely disparate readership spanning from mathematicians eager to be informed about the model concepts employed in geothermal research and geoscientists hoping to expand their understanding of the theoretical concepts required to solve fluid flow problems. We provide two rather formal introductions of the fundamental relations (second and fourth section). The reader who is unfamiliar with the formalities of continuum mechanics may happily skip these sections (possibly apart from comments on notation at the start of the first one) and come back to them only after reading the third and fifth section that intend to slowly increase complexity. In the third section, we first analyze steady-state flow in hydraulic conduits with simple geometry and then proceed to complex networks. The subsequent presentation of aspects of transient flow in the fifth section similarly evolves from individual conduits with simple geometries to homogeneous poro-elastic media to heterogeneous media. Each subsection is headed by a tool box comprising the basic equations involved in the sequel. Working through these formal preambles in the light of the subsequent examples may facilitate appreciating the introductions of the fundamental relations in retrospect. We conclude with a collection of notes on current research developments and related fields. Our presentation of basic relations leans heavily on mixture theory. We restrict however to a single fluid phase dispersed in a solid phase since liquid-gas mixtures are not typical for deep geothermal reservoirs. 2 Fundamental Laws: A Primer We present the basic equations of (linear) continuum mechanics. Focus is set on the governing kinematic relations, balance equations, and constitutive equations for single-phase solids or fluids. The aim is to recapitulate basic physical principles and discuss briefly resulting linear field equations, e.g., Lamé-Navier equations for linear elastic solids or the Navier-Stokes equations for Newtonian fluids. 2.1 Kinematics 2.1.1 Position Vectors and Displacements of a Material Point The positions of a material point P in its deformed (current) and its undeformed (reference) configurations are given by the position vectors x and X, respectively. The displacement vector of the material point is introduced as the difference between x and X, i.e., u D x X: (2) Calculating the first and second time derivatives of the position vector x, we obtain the velocity vector v D Px and the acceleration a D Rx. Here, we indicate the material or substantial time derivative with the dot operator. By applying the chain rule, the material or substantial time derivative for an arbitrary scalar field function.x;t/depending on the current position x reads d.x; t/ d t D P.x; t/d @ x.x; t/ v C @ t.x; t/: (3) Page 3 of 55
Throughout this chapter we employ the frequently used short notation for partial derivatives indicating the variable(s) with respect to which to differentiate as index on @, i.e., @ x.x; t/wd @.x; t/=@x. 2.1.2 Velocity Gradient and Linear Strains The spatial velocity gradient is obtained from the partial time derivative of the velocity with respect to the position vector in the current configuration, i.e., L D @ x v D grad v. Note that L D L ij e i e j is a general second-order tensor which can be additively split into a symmetric and a skewsymmetric part. We apply the abbreviated summation convention according to Einstein if an index appears twice in an equation. The symmetric and skew-symmetric parts of the velocity gradient are given by L D D C W with D D 1 2 L C L T and W D 1 2 L L T ; (4) where D represents the strain-rate tensor and W the spin or rate of rotation tensor. To interpret simple homogeneous experiments (simple shear, pure compression, tension, torsion tests), we split the strain-rate tensor, i.e., the symmetric part of the velocity gradient into a traceless part, the deviator, and an additional term characterizing the volumetric deformation, i.e., D D dev.d/ C vol.d/ with vol.d/ D 1 tr.d/ D div v: (5) 3 The second-order unity tensor is denoted as I D ı ij e i e j, with the Kronecker symbol ı ij D 1 for i D j and ı ij D 0 for i j. From the gradient of the displacement vector grad u, we define linear strain measures D 1 2 grad u C grad T u : (6) Again, the second-order (symmetric) linear strain tensor can be split into a volumetric and a deviatoric part, i.e., D dev./ C vol./ with vol./ D 1 tr./: (7) 3 2.2 Conservation Laws We introduce mechanical and thermal balance relations of a deformable material body B. Inthe present discussion, we restrict ourselves to classical continua, i.e., the Cauchy stress tensor is axiomatically introduced as a symmetric tensor. Therefore, the balance of moment of momentum is a priori fulfilled. Furthermore, we do not intend to discuss the formulation of thermodynamically consistent constitutive relations since we consider a discussion of the balance of entropy out of the scope of this handbook contribution. Theoretical details of modern material theory related to constitutive modeling of solids, fluids, and multiphase materials can be found, e.g., in Haupt (2000), Hutter and Jöhnk (2004), and Bower (2010). Page 4 of 55
2.2.1 Balance of Mass Axiom: The total mass M.B;t/of a material body B given as Z Z M D dm D dv (8) B B is constant during its deformation d.m / D const. (9) dt The density is given as the ratio of mass dm and volume dv in a certain representative volume element (RVE) of size dv. 2.2.2 Balance of Linear Momentum Axiom: Linear momentum J.B;t/ D R Px B dm D R Px B dv of a deformable material body B is Rchanged by the sum F D F.@B;t/C F.B;t/ D F @B C F B of near-field or contact forces F @B D @B t da and far-field or body forces F B D R B b dv, i.e., d.j/ D F: (10) dt Surface tractions t are related to the Cauchy stress tensor by Cauchy s theorem t D n. The vector n is the outward normal vector of @B. 2.2.3 Balance of Energy Axiom: The sum E D E kin C E int of kinetic energy E kin D R 1 B Px Px dv and internal energy 2 E int D R B dv ( denotes the specific internal energy) of a deformable material body B is changed by the sum P D P mech CP therm of mechanical power P mech D R Pxt @B da CR Px B b dv and thermal power P therm D R @B q da C R B r dv, i.e., d dt.e/ D P D P mech C P therm : (11) Here, q denotes the heat flux across the body s surface, and r is the specific heat production. Furthermore, the heat flux is related to the heat flux vector and the outward surface normal vector by Cauchy s heat-flux theorem q D q n. 2.3 Constitutive Equations Constitutive relations have to be formulated to close the system of equations, i.e., the number of unknown field variables has to match the number of kinematic, balance, and constitutive equations. In particular, kinematic and dynamic quantities have to be related as an expression of substance s Page 5 of 55
rheology, the way a specific material deforms and flows. The first fundamental characterization of fluids concerns their density fr D dm f =dv f, i.e., the ratio between the mass element of the fluid dm f and the volume dv f that it occupies. The effective weight of the fluid calculates as fr D fr g, where g denotes gravitational acceleration. The density fr of an incompressible fluid remains constant during the flow process (or equivalently tr D D div v D 0, seeeq.5). Compressible fluids change their density in flows; their pressure is calculated from an equation of state, i.e., p D p. fr ;/. Prominent examples for equations of state are Boyle-Mariotte s law (ideal gas), van der Waals law, or the versatile Muskat s law (e.g., Bear 1972): fr D fr 0 p n expœˇf.p p 0 / : (12) p 0 For incompressible liquids, it is proposed n D 0, ˇf D 0 and for compressible liquids n D 0, ˇf 0. For gases, Muskat distinguishes between isothermal cases (n D 1, ˇf D 0) and adiabatic processes (n D c v =c p, ˇf D 0), where c v and c p denote specific heat capacity at constant volume and pressure, respectively. The fluid pressure has unit [ N/m 2 D Œ Pa D 10 5 Œ bar D 10 1 Œ dynes / cm 2, but it is also frequently expressed by the height of a continuous water column 1ŒmH 2 0 D 0:09807 Œ bar 0:1 Œ bar. Throughout this presentation, we follow a notation according to which ˇ D @ p ln R D @ p ln v f denotes a compressibility [Pa 1 ]andk D 1=ˇ denotes a bulk modulus [Pa]. Superscripts to these properties indicate the particular material of interest, for example, ˇf and K f represent fluid properties. Subscripts in contrast indicate that these parameters express the pressure sensitivity of a physical property other than volume, for example, ˇ D @ p ln and K denote relative changes in porosity. In many cases, it is warranted to treat the fluid as barotropic, i.e., fluid density is a function of pressure alone, i.e., fr D fr.p/,cf.(12). 2.3.1 Viscous Fluids Flow of real fluids is a dissipative process and requires continuous operation of forces. A velocity imposed on a fluid s surface, e.g., by the action of a tangential or shear force, is only partly transmitted into the bulk of the fluid by momentum interaction between neighboring fluid particles. The velocity perturbation decays with distance from the surface. The penetration depth depends on the specific fluid and is quantified by its viscosity. As a result of the balance of entropy, the total stress tensor T f is split into an equilibrium and into a nonequilibrium part: T f D T f eq C Tf neq D Tf E p I: (13) The deviatoric part of the equilibrium stresses is zero (dev.t f eq / D 0), and the volumetric part of the nonequilibrium stresses is also zero (vol.t f neq / D 0). The nonequilibrium or extra stress part is given by T f E T f E D 2fR dev.d/ (14) introducing formally dynamic viscosity fr Œ kg=ms or Œ Ns/m 2. Thus, dynamic viscosity constitutes the proportionality constant between shear stress, i.e., the off-diagonal elements of the stress (or the extra) tensor, and the component of the velocity gradient normal to the direction Page 6 of 55
of the shear stress, i.e., in the direction of the normal vector of the surface under consideration. Performing a rheological shear test, for example, in a plate-plate viscosimeter, we could obtain the relation between the shear stress component and the shear rate P of a linear viscous fluid as D fr P. In a three-dimensional setting, this linear relation is expressed by Eq. (14). Linear viscous behavior, i.e., a constant viscosity, is commonly addressed as Newtonian behavior. The dynamic viscosity of water at room temperature D 293 Kisgivenby fr D 1:00 10 3 Ns/m 2. The dynamic viscosity is also still given in centi-poise Πcp with Πcp D 10 3 ΠPa s D 10 3 ΠNs/m 2.The fr D fr = fr 0 is expressed in Πm 2 =s or in the unit Stoke, with Πm 2 =s D 10 4 ΠStoke. Kinematic viscosity can be considered a diffusivity of momentum linking characteristic length and time scales of momentum transfer between fluid particles. 2.3.2 Linear Elastic Solids For a linear elastic solid, the extra stress part consists of a volumetric and a deviatoric part given by T s E D vol.ts E / C dev.ts E / D 3Kvol.s / C 2Gdev. s / (15) with bulk modulus K and shear modulus G. For this presentation, we adopt the sign convention from continuum mechanics. Thus, compressive (total and effective) stresses (leading to a decrease in volume) are negative. Pore pressure and confining pressure are positive. Note that in many fields in geosciences, compressive stresses are treated positive. 2.4 Field Equations 2.4.1 Navier-Stokes Equations To derive the Navier-Stokes equations, i.e., the set of field equations or partial differential equations (PDEs) expressed in partial derivatives of the primary variable v, we insert the introduced kinematic relation (5) and the constitutive expression for a Newtonian fluid (14) into the balance of momentum in local form. This approach leads to @ t. fr v/ C fr grad v v fr div grad v C grad p D fr b; (16) a set of three partial differential equations commonly addressed as the Navier-Stokes equations. This set of PDEs has to be supplemented by an equation of state for the pressure if the fluid is compressible, e.g., (12), or a kinematic constraint if the fluid is incompressible (17), i.e., div v D 0; (17) known as the continuity equation. The constraint fr.x;t/d fr.x;t 0 / WD fr 0 is a consequence of mass conservation (8)and(9). In dimensionless form, the Navier-Stokes equations read St @ t?.v? / 1 Re div? grad? v? C grad? v? v? C grad? p? D b? ; (18) Page 7 of 55
where we introduced the Reynolds number Re and the Strouhal number St Re D Q l Qv fr fr and St D Q l Qt Qv (19) describing the ratio of inertia to viscous forces and the ratio of local to convective (or advective) acceleration, respectively (e.g., Truckenbrodt 1996, p. 128). In the definition of Re and St (19), we introduced characteristic quantities of the problem (indicated with a tilde), like length Q l, velocity Qv, and time Qt. Only very few exact analytic solutions of the Navier-Stokes equations are known today for rather specific cases. Among these are the ones for stationary flow in simple conduits that we present in the next section. With the help of computational methods (finite volumes, finite elements, finite differences, or smoothed particle hydrodynamics), numerical solutions of the Navier-Stokes equations can be obtained for various problems. Especially for laminar flow conditions, computational fluid dynamics (CFD) has extended the theoretical knowledge about the Navier-Stokes equations within the last decades. 2.4.2 Lamé-Navier Equations Another set of fundamental field equations are the Lamé-Navier equations that describe the linear propagation of elastic waves in solids or, if inertia terms vanish ( sr Ru D 0), the deformation behavior of elastic solids under the influence of near-field and far-field forces. The derivation of these field equations follows the technique described for the Navier-Stokes equations. We insert the kinematic relations for the linear strains (6) and(7) and the generalized Hooke s law (15) into the local form of the balance of momentum to get sr Ru D div grad u C. C /grad div u C sr.1 div u/ b: (20) From Eq. (20), we find the velocity c P of a longitudinal wave mode called the compressional wave or P-wave and the velocity c S of a transversal wave mode called the shear wave or S-wave. The standard notation uses the subscript P for primary wave, the faster of the two waves, and the subscript S for secondary wave. In the Lamé-Navier equations (20), we used the properties of linear elastic isotropic materials. As introduced in Eq. (15), the stress tensor depends on two material parameters, bulk modulus K and shear modulus G, that relate to the two Lamé parameters and according to sr c 2 P D K C 4 3 G D C 2; D K 2 3 ; and sr c 2 S D D G: (21) 3 Stationary Flow Processes: Transport Stationary flow processes are entirely dominated by the geometry of the pathway provided to the moving fluid, the conduit(s), and the fluid s intrinsic resistance to flow. The solid constituting the conduit can safely be treated as non-deformable in many applications since the conduit geometry remains fixed at steady state. In principle, the fluid is heated or cooled during compression or dilation, and thus flow rate depends on the heat exchange between the fluid and its environment. Page 8 of 55
We neglect this type of hydrothermal coupling in the following. In geothermal applications, the volume changes of the involved fluids are typically moderate, and heat flow is dominated by the thermal characteristics of the reservoir. The temperature differences between the pumped fluid and the reservoir will typically largely exceed any temperature variations due to volume changes of the fluid. 3.1 Hydraulic Conduits with Simple Geometries 3.1.1 Basic Equations I: Stokes Equation The classic flow problems to be presented next consider a fluid volume and its movement within a conduit of simple geometry in response to an imposed stationary pressure difference. The solid comprising the conduit is initially treated as rigid, i.e., undeformable. This constitutive assumption ensures that the boundaries for the fluid flow remain fixed in space and time. According to fundamental observations, fluids flow from regions where they experience high pressure to regions where they experience low pressure. Spatial pressure variations drive fluid flow in a direction toward pressure equilibration. Given the flow is sufficiently slow, i.e., inertia forces can be neglected, an incompressible fluid volume experiences essentially normal or pressure forces due to pressure differences between its opposing surfaces and tangential or (viscous) shear forces due to velocity differences of its surface relative to neighboring fluid volumes. At steady state, the fluid volume moves with constant velocity, and thus the total force is zero. Evaluating the force balance and performing the transition to infinitesimal volumes leads to the Stokes equation for creeping flow in the absence of body forces: fr div grad v C grad p D 0; (22) where the gradient of pressure represents the imbalance of normal forces across the fluid volume and the imbalance of tangential forces is converted to an expression for velocity using the constitutive relation (14). Equation (22) is supplemented by an equation of state for compressible fluids, e.g., (12), or the continuity equation in case of incompressible liquids (17). This partial differential equation allows for analytic solutions in case of specific conduit geometries and boundary conditions. Stokes equation is the specific form of the full Navier-Stokes equation (16) for stationary conditions (@ t. fr v/ D 0) in the absence of body forces (b D 0) when the nonlinear term fr grad v v is negligible. 3.1.2 Law of Hagen-Poiseuille: Tubes The flow of fluids through tubes is of obvious fundamental technical interest given that the transport of fluids as well as their local provisionand discharge are overly realized by employing man-made tube networks. However, tubes also constitute one of the fundamental idealized pore geometries of the pore space of rocks (e.g., Bernabé et al. 1982) or more generally porous materials. Exploiting the cylindrical symmetry for a tube with its axis oriented along the direction of e z at z D 0 (Fig. 1) simplifies Stokes equation (22)to 1 fr @ zp D 1 r @ r.r@ r v/: (23) Page 9 of 55
r v(r) p out p in R z L Fig. 1 Schematic of the radial velocity distribution for flow in a cylindrical tube Accounting for two dynamic boundary conditions, specifying the pressures at the inlet and outlet (p in, p out ), and two kinematic boundary conditions, v.r D R/ D 0, i.e., no-slip condition at the tube s inner wall, and @ r vj rd0 D 0 owing to axial symmetry, yields the well-known quadratic velocity profile v.r/ D 1 4 fr @ zp R 2 r 2 D (4p WD p in p out ) and upon integration the law of Hagen-Poiseuille 4p R 2 r 2 (24) 4 fr L Q z D @ t V f D R4 8 fr 4p L : (25) The minus sign emphasizes that the fluid yield is downhill relative to the direction of the pressure gradient that indicates the direction and magnitude of the steepest increase in fluid pressure. For given boundary conditions, i.e., prescribed pressures at the opposing ends, the conduit geometry and the fluid s properties expressed by tube radius and viscosity, respectively, determine jointly the volume-flow rate. The fourth-power dependence of volume-flow rate on tube radius bears eminent consequences for hydraulically preferential paths in networks composed of tubes with varying geometrical characteristics (e.g., Bernabé and Bruderer 1998). The inverse dependence on fluid viscosity is of equal importance considering that viscosities of geoscientifically relevant fluids differ by orders of magnitude from 10 5 Pa s for gases to 10 3 Pa s for water, to 10 3 (light) and up to 10 0 Pa s (heavy) for oil, and to 10 8 Pa s for magmas. The viscosity dependence of flow rate accounts for the necessity to operate turbines with gases and for the enhancement of productivity of heavy-oil reservoirs upon heating. 3.1.3 Cubic Law: Slits The flow of fluids through slits is also of obvious fundamental technical interest, for example, in lubrication. Slits also constitute one of the idealized pore geometries used to model pore networks of rocks (e.g., Bernabé et al. 1982). Employinga Cartesian coordinate system and analyzing Stokes equation (22) for flow in direction of e z in a slit of width (aperture) w between parallel plates with a normal vector in the direction of e x and located at x D w=2 (Fig. 2) yields 1 fr @ zp D @ xx v: (26) Page 10 of 55
p in x v(x) z w p out B y L Fig. 2 Schematic of the velocity distribution for flow between parallel plates Solving the partial differential equation for dynamic (prescribed in- and outlet pressure as above) and kinematic boundary conditions (the no-slip condition at the plates surfaces v.x D w=2/ D 0 and the requirement from symmetry @ x vj xd0 D 0) leads to a quadratic velocity profile v.x/ D 1 w 2 2 fr x2 @ z p D 1 w 2 4p 4 2 fr x2 (27) 4 L closely resembling the one for a tube (24). Upon integration over the slit s cross section, one arrives at the famous cubic law Q z D @ t V f D B w3 4p 12 fr L : (28) The slit s extension in the direction of e y is denoted B. As the radius for tubes, the aperture is the critical geometrical parameter due to its third-power relation to flow rate. Even creeping flow between plates becomes immediately more complicated than presented here when altering the boundary conditions only slightly. For example, a steady-state solution does not exist for a point source between the plates (a zero-order model for a fracture or joint intersected by a borehole) unless a constant-pressure boundary condition is enforced at some finite distance from the source. 3.1.4 Extended Cubic Law: Rock Fractures Real rock fractures deviate from the simple model of parallel plates by their surface roughness that also allows the opposing surfaces to be in physical contact at a finite number of asperities without hindering fluid transport per se. Surface roughness perturbs the simple geometry of flow lines along the slit. Flow is redirected around local protrusions, and yield is thus reduced for given boundary conditions. Yet, the simple cubic law maintains its validity when using an effective hydraulic aperture for rough fractures and fractures in mechanical contact as demonstrated by the seminal work of Witherspoon et al. (1980). However, ignoring the out-of-plane flow components apparently may lead to overestimated transport properties of fractures. At large separations, the topography of the opposing surfaces has little effect. The larger the protrusion of individual asperities relative to the average distance between the surfaces, the larger the disagreement between the real flow rate and the prediction by the parallel-plate model. A significant number of experimental and numerical investigations aimed at determining empirical factors, so-called friction factors f > 1, that account for roughness and multiply the geometrical factor of 12 in the denominator of (28)(e.g.,Brown1987; Renshaw1995). Specifically, Renshaw Page 11 of 55
(1995) proposed a modification that employs standard deviation of aperture and average aperture as statistical measures of the roughness. In numerical simulations of rough fractures, the local cubic law approach is commonly employed since results correspond closely to solving the full flow problem (e.g., Brush and Thomson 2003). At small separations and for fracture halves in contact, the flow is tortuous, tending to be channeled through high-aperture regions (Witherspoon et al. 1980). As the fractional contact area increases, the friction factor increases (e.g., Zimmerman et al. 1992; Piggott and Elsworth 1992). Eventually, the channel network along the slit becomes subject of percolation issues (e.g., Nolte et al. 1992). Fracture geometry is sensitive to loading. Normal loads prominently change average aperture. The resistance to normal displacement is typically expressed by normal stiffness. The nonlinear asperity deformation (Hertzian contacts) often causes a load dependence of stiffness. Increasing normal load reduces the yield of fluid volume for given boundary conditions in pressure. The correlation between the mechanical and hydraulic properties of fractures was investigated in some detail by Pyrak-Nolte and Morris (2000). Sisavath et al. (2003) presented a simple model for deviations from the cubic law for a fracture undergoing dilation or closure. Tangential loads may cause dilatant shear displacements that increase average aperture (e.g., Yeo et al. 1998). Furthermore, shear displacements may give rise to anisotropy in flow beyond intrinsic structures of the surfaces (e.g., Auradou et al. 2006; Giacomini et al. 2008). Until today, the critical issue for quantitative predictions remains to include the results of surface roughness measurements in micromechanical concepts from which hydraulic and mechanical properties can be derived at the same time (e.g., Olsson and Brown 1993). 3.2 Complex Geometries: From Conduit Networks to Porous Media The governing equations of (macroscale) single-phase fluid flow in porous media can be developed on the basis of (a) averaging procedures of the microscale flow processes in individual conduits (pores) described by the Navier-Stokes field equations (Howes and Whitaker 1985); (b) empirical (transport) equations on the macroscale disregarding the morphology of the porous network, cf. classic Darcy s law; and (c) the continuum-mixture theory that may be regarded as an extension of the classic continuum theory (Hassanizadeh and Gray 1979a, b;bowen1982; Coussy 1995;Ehlers and Bluhm 2002). We will focus on continuum-mixture theory after presenting some basic aspects of microscale approaches to hydraulic properties. Treating pore space as a three-dimensional network composed of individual hydraulic conduits varying in shape and size is conceptionally appealing but leaves one with the incredible task to find a finite number of simple descriptors: 1. of the statistics of the geometrical characteristics of the hydraulic conduits that in the best of cases are accessible to direct determination, and 2. of the network design or topology. One such approach often associated with the keyword hydraulic radius is also subsumed under Kozeny-Carman models in honor of early contributors (e.g., Schopper 1982). Hydraulic radius is considered the central geometrical characteristic of the pore space. Typically, it is quantified from the ratio between pore volume and pore surface. Pore volume is readily measured in the laboratory (by either fluid saturation or determination of bulk and component density), while pore surface Page 12 of 55
is accessible only by indirect methods such as nitrogen adsorption. In Kozeny-Carman models, tortuosity, a measure of flow path length relative to sample length, quantifies the topology of the pore space and is closely related also to measures of connectivity between pores. The controlled design of networks is a basic step in the quest for representative parameters. While numerical approaches have dominated research in this direction with the advent of fast personal computers, analogue models composed of electrical resistors were employed in early work (Rink and Schopper 1968; Greenberg and Brace 1969), and synthetic porous media, for example, prepared from well-sorted powders, are used until today (e.g., Blair et al. 1996; Revil and Cathles 1999; Mok et al. 2002). Network modeling relying on tubes has shown to be very valuable for understanding the role of heterogeneity, for example, (e.g., Madden 1976; Bernabé and Bruderer 1998). Heterogeneity of pore space actually has two aspects, the variability in geometry of conduits and the variability of connectivity of conduits in the network. The range of networks associated with a certain number and arrangement of a certain ensemble of pores is specifically a subject of percolation theory (Berkowitz and Ewing 1998). A continuous line of work on numerical network modeling has been presented by Bernabé and coworkers (Bernabé 1995; Bernabé and Bruderer 1998). Recently, coordination number was suggested to suffice as a complementary parameter to hydraulic radius for permeability modeling (Bernabé et al. 2010, 2011). Extended micromechanical modeling suggests that three geometrical parameters rather than a single characteristic length scale are required for substantial models of hydraulic parameters, pore radius, and length and separation (e.g., Gavrilenko and Gueguen 1989). These models invoke penny-shaped cracks or cylindrical tubes as basic geometries for pores (e.g., Matthewset al. 1993). Many of the theoretical developments happened in parallel for the two closely linked transport properties, hydraulic and electric (e.g., Madden 1976; Schopper 1982; Avellaneda and Torquato 1991). Investigations into the length scale relating hydraulic and electric transport properties revealed that the subtle differences between electrical and fluid fields are affected by spatial randomness of the pore space that may not be easily realized in numerical models that, for example, employperiodic structures (Martys and Garboczi 1992). Analytic models complemented the experimental and numerical work (e.g., Torquato 2002). A peculiarity of hydraulic properties of natural rock volumes appears to be a distinct scale dependence related to some hierarchical organization of the hydraulic conduits (e.g., Neuman 2008). This notion rests on direct experimental observations (e.g., Tidwell and Wilson 1999) but also on the indirect comparison of results from experiments involving flow on different spatial scales (e.g., Brace 1980; Hünges et al. 1997; Ingebritsen and Manning 1999). The variability of hydraulic transport properties with spatial scale can be explained in the context of percolation theory (e.g., Hunt 2005). A range of further methods (bounding, heuristic methods, deterministic methods, stochastic methods, etc.) have been invoked for the calculation of equivalent hydraulic properties of heterogeneous porous media (see, e.g., review by Renard and de Marsily 1997). Specifically, the scale dependence was associated with the systematic spatial variation of local percolation probabilities in a statistical treatment employing local porosity theory (Hilfer 2002). Geometrical details of the pore space like pore size and specific surface area and their distributions are not accounted for in mixture theory. Continuum-mixture theory is motivated by the idea that various (miscible or even nonmiscible) constituents coexist in a representative volume element (Fig. 3). Neglecting small-scale details (e.g., morphology of pore space) and upscaling physical properties of the observed constituents and their interaction, one reaches a model of Page 13 of 55
effective geological model (macro-scale) discrete grain scale (micro-scale) L λ l heterogeneous meso-scale Fig. 3 Representative volume element (RVE) and related scales. Quantities like effective densities R are defined on mesoscale superimposed continua. Note that each constituent follows its own motion; therefore kinematic quantities and balance relations have to be individually formulated for each constituent. In contrast to classic continua, interaction mechanisms between the constituents (phases) have to be accounted for in balance relations. 3.2.1 Basic Equations II: Biphasic Mixtures In mixture theory, the description of flow processes of a viscous pore fluid (phase ' f ) in a porous skeleton (phase ' s ) with a complex pore geometry requires the usage of effective, i.e., smeared out, flow properties of the RVE with given representative volume dv. The effective flow properties are based on homogenized, coarse-grained morphological quantities, like porosity WD dv f =dv or (more general) volume fractions n WD dv =dv of the solid and the fluid constituent 2fs; fg. The mass dm of a constituent in the RVE related to the total volume of the RVE defines the partial density dm =dv DW. Consequently, the effective (or true) density R follows from the definition of the volume fractions n and the definition of the partial density, i.e., R D dm =dv D =n. Note that porosity constitutes the only quantity of the pore space preserved in classic macroscopic continuum-mixture theories. Morphological details, like specific surface areas, and geometrical details of the pore channels like averaged diameter or distribution of diameters or grain size distribution are not taken into account. To extend the classic mixture theory to multiphase flow in porous media, it was attempted to account for specific surface area as additional microscopic information (Hassanizadeh and Gray 1979a, b). As a consequence of the consideration of two constituents, i.e., the porous solid skeleton and the viscous pore fluid, we have to extend the conservation laws. Neglecting mass exchange, i.e., reactions by which fluid mass can be transformed into solid mass or vice versa, the extended formulation of the partial conservation laws for phase ' should account for an exchange of momentum and also an exchange of energy. Page 14 of 55
Kinematics The relative (seepage) velocity is introduced as the difference between the velocity of the fluid and the velocity of the solid w f D v f v s. Furthermore, the filter velocity or Darcy s velocity is given by q f D w f. Partial Balance of Momentum of a Biphasic Mixture The momentum of each phase ' J D Z Z Px dm D Px dv; (29) B B is now complemented by a momentum interaction (compared to 33). The momentum interaction terms (e.g., drag forces) account for the exchange of momentum between the solid and the fluid phase. One important example of momentum interaction is related to flow of a viscous fluid through pores. Assuming no-slip boundary conditions, velocity is zero at the pore wall. In accord to this Dirichlet condition, the reaction forces associated with viscous flow constitute the interaction terms, i.e., Z X OP D Op dv; with Op D Op s C Op f D 0; (30) @B where Op denotes local momentum interaction. Finally, the momentum of a constituent is changed by the sum of body forces and contact forces, F, and interaction forces, OP,.J / 0 WD @ tj C v grad J D F C OP : (31) Here, we represent the material or substantial derivative with respect to the partial velocities by terms in parentheses that are primed and subscripted to indicate the partial velocity to be used for the advection term. Near-field and far-field forces F D F @B C F B are defined in analogy to single-phase continua (see presentation of Eq. 10): F @B D Z @B t da; F B D Z B b dv: (32) The constraint in Eq. (30) links the partial balance of momentum to the balance of momentum of the mixture when Eq. (31) is summed up for all constituents '. Constitutive Equations The extended momentum balance (31) requires formulating a further constitutive equation for the momentum interaction Op f. Based on thermodynamic considerations (cf. Ehlers and Bluhm 2002; Hutter and Schneider 2009), we split the momentum interaction into an equilibrium and nonequilibrium contribution, i.e., Op f D Op f eq C Opf neq. Regarding the entropy principle at thermodynamic equilibrium, we obtain Op f eq D p grad. Thus, RVEs with conical pore shapes exhibit a static (no-flow) solid-fluid interaction term. Close to thermodynamic equilibrium, the nonequilibrium momentum exchange is proportional to the seepage velocity, i.e., Op f neq / w f or fr Op f neq D 2 w f D 2 fr k f k s w f : (33) Page 15 of 55
The proportionality factor is determined from experimental observations. Here, k s and k f denote intrinsic permeability (unit [ m 2 ]ordarcy[d] 10 12 [m 2 ]) and Darcy s permeability or the hydraulic conductivity (unit [ m=s ]), respectively. These two transport measures are related by k s D fr fr g kf D fr fr kf : (34) 3.2.2 Linear Darcy s Flow Next, we discuss a mixture which consists of a rigid (undeformable) porous skeleton that is fully saturated with a viscous pore fluid. Chemical reactions, e.g., dissolution and precipitation, are disregarded. Furthermore, the system is assumed to be under isothermal conditions. Localizing the partial balance of momentum of the fluid phase D f (Eqs. 30 31) yields Rx f div T f D f b C Op f : (35) Disregarding inertia forces ( Rx f 0) and viscous shear stresses of the fluid (T f pi, cf. Hassanizadeh and Gray 1987) and combining the resulting equation with the constitutive equation for the exchange of momentum (33), we obtain Darcy s law 1 q f D k f fr g grad p b : (36) g Regarding one-dimensional flow processes in the direction of e z, with b D g D ge z,we specifically yield 1 q f D k f fr g @ zp C 1 D k f i (37) where we introduced the hydraulic gradient i. Taking a closer look at Eq. (36), we observe that the directions of Darcy s velocity q f and of the driving force (right-hand side of (36)) are identical. For porous media with an anisotropic pore space, this equation can be generalized by replacing the scalar permeability k f with a second-order permeability tensor K f D K f ij e i e j q f D K f 1 fr g grad p b : (38) g Thus, the second-order permeability tensor transforms, in the sense of an affine mapping, the driving forces to Darcy s velocity. Let us emphasize here that we have introduced Darcy s law from the balance of momentum and a (linear) constitutive relation for the nonequilibrium momentum exchange between the solid and fluid phase. Thus, Darcy s law does not constitute a linear constitutive law but rather a result of a linear constitutive assumption. Comparing Darcy s law with Hagen-Poiseuille s law (25) and the cubic law (28), we discover that both relations, though obtained in quite different ways, represent linear relationships between a total discharge in a pipe (Q z ) or a local seepage velocity (q f ) and a driving force given by a pressure difference between two spatial positions (4p or grad p). Page 16 of 55
3.2.3 Nonlinear Extensions While formally often treated alike, the nonlinear extensions of the transport laws for simple conduits, like straight tubes with circular cross section or perfectly smooth and parallel slits, on the one hand, and real fractures with rough surfaces and porous media with complex pore structure, on the other hand, actually have to account for two different phenomena. Hagen-Poiseuille s law and the cubic law are exact linear equations that break down suddenly when inertia terms become relevant at high flow rates. Then, flow lines exhibit substantial complexity due to the occurrence of spontaneous eddies even in simple conduits (e.g., Costa et al. 1999). In real fractures and porous media, the presence of obstacles and the intrinsically winding pores enforce curved flow lines at all rates, but changes in dominating fluid path may occur with increasing flow velocity and the increasing role of inertia effects (e.g., Andrade et al. 1997). A linear effective transport law thus requires nonlinear convective terms in the Navier-Stokes equations (16) to be negligible. The transition to turbulent flow has enormous consequences for the efficiency of fluid-volume transport and advective heat transport. The onset of turbulence and thus the range of validity of Darcy s law are conventionally characterized by the magnitude of the dimensionless Reynolds number Re, the ratio of inertia, and viscous forces. This approach stems from the stability analysis for straight tubes and slits, though real tortuous conduits require estimating the magnitude of convective terms, too. Explicit formulations of Re lead to Re D q f l Q fr = fr D q f l= Q f where l Q denotes a characteristic length scale of the flow problem. For porous media, the hydraulic radius r hyd D p k s = is often employed as a measure of mean pore size in the calculation of Re. Fractures are characterized by their effective aperture. In turbulent flow in tubes, momentum exchange in the transverse direction causes the radial velocity distribution to be significantly more uniform than the parabolic distribution characteristic for laminar flow. The profile is almost flat, pluglike, with steep flanks. Despite the locally chaotic nature of the flow, analytic velocity profiles can be derived for steady turbulent flow in tubes with circular cross section (e.g., Munson et al. 2006) since the eddies reach a statistical steady state when the flow is sufficiently developed (e.g., Smits and Marusic 2013). The boundary layer remains devoid of eddies due to the local dominance of viscous forces. Relatively recent work related the characteristics of turbulent flow in tubes to nonlinear traveling waves (Hof et al. 2004). In irregular tubes and slits (rough fractures) or granular media, deterministic eddies occur at predictable sites associated with obstacles, protrusions, and the like that may actually cause a sensitivity of transport to flow direction (Cardenas et al. 2009). The influence of inertial forces on the bulk flow rate across fractures however remains small when Re < 1 is met among other kinematic criteria (Brush and Thomson 2003). Numerical simulations of high-velocity flow in a self-affine channel indicated that the effective permeability is dominated by the narrowest constrictions at low velocity (Skjetne et al. 1999); the effective tube thickness decreases with increasing Reynolds number. Similarly, accounting for a reduction in effective aperture, extended parallel-plate models comprise a Re dependence of the friction factor f (e.g., Phipps 1981; Nazridoust et al. 2006). Numerical work that focused on modeling the results gained from analogue models of rock fractures yields a transitional regime for Re between 1 and 10 (in which the non- Darcy pressure drop varies with the cube of the flow rate) and a fully turbulent regime for Re larger than 20, in which the non-darcy pressure drop is quadratic in the flow rate (Zimmerman et al. 2005). Such quadratic extensions of Darcy s law are commonly addressed as Forchheimer equation (Skjetne and Auriault 1999) with the additional proportionality constant for the quadratic term in flow velocity termed inertial resistance, inertial permeability, or turbulence factor (Sen 1987). Page 17 of 55
4 Fundamental Laws: A Second Step The set of governing partial differential equations for a biphasic porous medium (solid skeleton ' s and pore fluid ' f ) is described in closed form by the balance of momentum of a mixture and the balance of mass/volume of the mixture. This set of equations is of great practical value since the controllable conditions in physical, i.e., laboratory and field, experiments can be applied in numerical analysis as boundary conditions and initial conditions. 4.1 Balance of Momentum of the Mixture Neglecting inertia forces, the balance of momentum of a biphasic mixture (31) reads in local form as div T D b: (39) The total stress of the mixture contains a solid and a fluid contribution T D T s C T f. The balance of momentum is unaffected by the compressibility of the individual constituents '. 4.2 Balance of Mass of the Mixture The partial balances of mass for both constituents ' are given as. / 0 C div v D 0: (40) Recalling the definition of relative velocity, w f D v f v s, the partial mass balances are reformulated in the seepage velocity and the velocity of the skeleton (cf. Ehlers 2002). f / 0 s C div.f w f / C f div v s D 0;. s / 0 s C s div v s D 0: (41) This formulation of the basic mass balances in primary kinematic variables fw f ; v s g can easily be linked to experimentally motivated boundary conditions and can also be used to formulate numerical methods, e.g., finite element methods (cf. Zienkiewicz et al. 1999). Equation (41) can be formally linearized around a reference state B 0 expressed by a set of state variables x.t 0 / DW x 0 D. w f;0 ; v s;0 ; fr 0 ; 0 / T where initial porosity 0 and initial effective density of the fluid fr 0 represent given material properties. In most linearized poro-elastic applications, this set can be simplified to x 0 D. 0; 0; fr 0 ; 0 / T. While hardly explicitly stated, this simplification also implicitly underlies linear poro-elasticity. Thus, we will also use the latter set of state variables x 0 to obtain the linearized balance of mass 0 @ t fr C fr 0 @ t C 0 fr 0 div w f C 0 fr 0 div v s D 0 (42) that forms the basis for the derivation of the diffusion equation (storage equation) in linear poro-elasticity. Page 18 of 55