_{1}

In this paper, we discuss the assumptions, the balances, and the constitutive relationships in order to provide a set of tools for the mathematical modeling of a geothermal system. In particular, we present a model for pressure and saturation supposing that: 1) the geothermal fluid flows in a porous medium, 2) it is composed of pure water, 3) the simultaneous presence of the gaseous (vapor) and liquid phases occurs, and 4) the effects of capillarity action can be introduced.

Geothermal energy is the energy in the form of heat contained within the Earth that can be extracted using manpower [_{2} saving from geothermal electricity can be in around 1000 million tons per year” [

A challenge with which we strive is trying to achieve this goal with description, understanding, and prediction of geothermal system behavior. Such a system is been defined schematically as “convecting water in the upper crust of the Earth, which, in a confined space, transfers heat from a heat source to a heat sink, usually the free surface” [

Nevertheless, this amount of information and concepts presented in literature may leave, especially at first glance, the researcher with an unclear perception of the main underlying and characterizing elements. The purpose of this work is to provide a short presentation of a general mathematical model that allows in a simple (but not simplistic) way to identify which balances and assumptions can be considered to model a geothermal system. The aim is therefore to introduce, albeit under simplified hypotheses (fluid composed only of pure water and isotropic system), a not trivial starter kit for modeling a geothermal field. To facilitate the reading of the work, we display in

The basic method for mathematical model development and physics modeling of a geothermal reservoir is represented by the following equation balance equations. An elementary balance equation for the generic quantity M can be written in the following form [

d d t ∫ V M d V = − ∫ Γ F ⋅ n d Γ + ∫ V q d V . (1)

The integration takes place on an arbitrary V sub-domain enclosed by a Γ surface. F denotes the flow and q the source and absorption terms. Finally, is the normal versus outer to surface. The local form of Equation (1) is given by the arbitrary of the V volume, using the divergence theorem. The result can be obtained as:

∂ ∂ t M + ∇ ⋅ F = q . (2)

Symbol | Physical quantity | Physical dimension |
---|---|---|

P | Pressure | [M∙L^{−}^{1}∙t^{−}^{2}] |

T | Temperature | [T] |

M | Mass | [M] |

ρ | Density | [ML^{−}^{3}] |

V | Volume | [L^{3}] |

Γ | Surface | [L^{2}] |

Λ | Latent heat | [L^{2}∙t^{−}^{2}] |

r | Specific gas constant | [L^{2}∙T^{−}^{1}∙t^{−}^{2}] |

ϕ | Porosity | [-] |

S | Saturation | [-] |

μ | Viscosity | [M∙L^{−}^{2}∙t^{−}^{1}] |

u | Fluid flow rate | [L∙t^{−}^{1}] |

v | Fluid velocity | [L∙t^{−}^{1}] |

K | Absolute permeability | [L^{2}] |

k r β | Relative permeability of β | [-] |

T | Stress tensor | [M∙L^{−}^{1}∙t^{−}^{2}] |

g | Acceleration of gravity | [L∙t^{−}^{2}] |

γ | Surface tension | [M∙t^{−}^{2}] |

〈 κ 〉 | Mean curvature | [-] |

Notation | ||

β | Phase | |

v | Vapor | |

l | Liquid | |

H_{2}O | Pure water | |

* | Clapeyron | |

mol | Molar | |

r | Relative | |

c | Capillary | |

w | Wetting | |

nw | Non-wetting |

However, the fluid is not free in our case, but it is under dynamics conditions within the rock fractures that form the reservoir of the geothermal system. Therefore, in the following system, a non-deformable porous medium is introduced. Let us introduce how we can describe the balances in a porous medium which are filled with pure water in gaseous and liquid state.

A porous medium is defined as any material in which a solid matrix and an empty space are present; or rather, a part where the fluid can penetrate [_{2}O. The solid matrix is distributed throughout the porous medium (see

In the porous medium, the H_{2}O flows through a complex network of pores structure which constitutes the space of the material. This flow is therefore delimited by the microscopic interface between solid and fluid. This configuration can be essentially analyzed in two different ways:

• Microscopic Level: Conceptually, the H_{2}O flowing in a porous medium could be studied at a microscopic level, that is by focusing attention on what happens in the single pore. In this case, the fluid is modeled as a continuous means subjected to specific boundary conditions. These conditions are defined on the interface that delimits the domain under consideration. However, this type of approach is generally theoretical because of the complex geometry of the interface. Besides, the mathematical model solutions from measurements at these levels can hardly be checked.

• Macroscopic Level: To overcome the difficulties mentioned above, another level of description is needed, namely the macroscopic one. On this scale, the quantities involved can be easily measured. To obtain a description of the flow at this level, the so-called continuous approach is adopted. According to this approach, the real porous medium is replaced by a fictitious model in which each phase (liquid and gaseous) is seen as a continuous medium that fills the entire AEV. Let us proceed with the concept of macroscopic quantity in the porous medium. Suppose, for instance, we aim to define ρ quantity (the quantity may be the volume fraction occupied by grains, the mass of liquid water, etc.) of the medium at the point x ∈ ℝ 3 (if the domain is considered three-dimensional). The procedure is given as follows: We consider an ideal spherical ball centered in the point x and we measure the average value of ρ on such a sphere varying the radius length. It is expected to obtain an oscillating behavior of the measurement values; precisely, due to the random nature of the medium (being that as more or less solid parts can be included in the sphere). Nevertheless, by observing the graph of the measurements (a diagram of which is shown in

The theory exposed in this paper refers to the macroscopic level, and the following quantities are considered as average quantities. In Section 3, we describe the balance equations which are used in continuum mechanics for these average quantities of the medium.

Let us first consider the mass and momentum balance equations for a system in which the geothermal fluid is composed of pure liquid and gaseous H_{2}O. The mass of H_{2}O presented in the volume unit of the porous medium is given by:

M H 2 O = ∑ β S β ρ β ϕ , (3)

where

• β is the index representing the phase ( β = v for vapour and β = l for liquid water);

• ϕ is the porosity of the medium, that is, the fraction of volume not occupied by the solid matrix;

• S β is the saturation of phase β , that is the volume fraction occupied inside the pores by the water of the corresponding state. The saturation variables comply with the constraint

S v + S l = 1 ;

• ρ β is the density of phase β , that is the mass of β phase per volume unit occupied by the β phase. For example, ρ l = 1 gr / cm 3 .

The mass flow F is the sum of all the phases

F = ∑ β F β , (4)

whereas each F β flow of the phase β is given by

F β = ρ β S β u β , (5)

by the product between the density and the volumetric flux u β of the fluid in the β phase.

The fluids move in response to forces in which the pressure forces, gravity force, and resistance forces are most important due to the viscosity and friction within the grains of the porous medium. The fundamental law of the cause of fluid fluxes in porous media was established by Henri Darcy yet in 1856 [

u = − K μ ∇ P , (6)

being P the pressure of H_{2}O. The sign minus is needed because the fluid flows from more high-pressure areas to low-pressure areas. In Equation (6), K is called permeability tensor and measures the ease of the fluid to move inside the porous medium (depending on the porosity and the microscopic geometric structure). Finally, μ denotes the viscosity, i.e., the resistance of a fluid to a change in shape. We can also consider the contribution of gravitational force [

u = − K μ ( ∇ P − ρ g ) . (7)

We remark that u represents the volume of fluid that crosses a unit area section per unit time. The volumetric flow u is also called Darcy’s velocity; but, the volumetric flow is not the average velocity with which fluid particles are moving. This quantity is known as pore velocity v and is defined according to the following equation:

u = ϕ v . (8)

This definition implies that in a medium with low porosity, the velocity v may be greater as compared to the volumetric flow u .

Let us now proceed to specify the meaning of pressures of the H_{2}O and its phases. As specified, P denotes the pressure of H_{2}O while with P v and P l the pressures of H_{2}O in vaporous state and liquid phase, respectively. By way of definition it follows that

S l = 1 ⇔ P l = P , and P v = 0 , S l = 0 ⇔ P l = 0 , and P v = P .

If 0 < S l < 1 holds, P l and P v have to be specified in a constitutive manner. In that case, the liquid and vapor are considered as a two-component mixture with P representing mixture pressure in its totality. In particular, the partial stresses of the components in the fluid mixture (see [

T ˜ l = S l P l I and T ˜ v = S v P v I ,

while the total stress T exerted by the mixture fluid is

T = T ˜ l + T ˜ v = S l P l I + S v P v I , (9)

where T = P I .

If, as in a geothermal reservoir occurs, the H_{2}O is present in several phases then it is proved that each phase flows moved by its own pressure gradient and by its own weight [

The Darcy’s expression for the single β phase becomes

u β = − K k r β μ β ( ∇ P β − ρ β g ) , (10)

with, resuming:

• K the absolute permeability tensor of the medium ( [ K ] = [ L 2 ] = m 2 );

• k r β the effective permeability of the single phase β , [ k r β ] = [ − ] , 0 ≤ k r β ≤ 1 . It is a reduction factor of the permeability because only a part of the space available in the pores is occupied by the β phase. It is indeed true that

k r β ( S β = 1 ) = 1 and k r β ( S β = 0 ) = 0 ,

while, for intermediate saturation values, these functions are defined experimentally. It also applies

∑ β k r β = 1 ;

• μ β the viscosity of the β phase, usually [ μ β ] = Pa ⋅ s .

From Equation (4) and Equation (5) the total mass flow is obtained as F of H_{2}O, given by

F = − ∑ β K k r β μ β ρ β ( ∇ P β − ρ β g ) . (11)

It can be stated that the local mass conservation equation [

∂ ∂ t { ϕ [ S l ρ l + ( 1 − S l ) ρ v ] } + ∇ ⋅ ( ρ l S l u l + ρ v S v u v ) = 0 , (12)

or as an alternative

∂ ∂ t { ϕ [ S l ρ l + ( 1 − S l ) ρ v ] } + ∇ ⋅ ( ρ l S l ϕ v l + ρ v S v ϕ v v ) = 0 , (13)

where any type of wells or sources is not considered. There are three unknown variables in this equation, which are S l , P l and P v . Note that the vapor density ρ v , albeit is variable, does not introduce an unknown variable because it depends on the vapor pressure P v . We can assume to apply the ideal gas law

P v ρ v = r T , (14)

where T is temperature in Kelvin degrees (K) and r is the specific gas constant, in H_{2}O case equal to r = 4.6 × 10 2 J / kg ⋅ K .

We provide three remarks concerning Equation (12), according to three different cases that can occur.

Remark 1. First, if the whole water is in the liquid state ( S l = 1 and S g = 0 ), then the Equation (12) is reduced to

∇ ⋅ u l = 0, ⇒ ∇ ⋅ ( ϕ K μ l ( ∇ P l − ρ l g ) ) = 0 ,

that is an elliptic equation in the unknown P l : the classic equation of porous media. Resolving it and specifying appropriate boundary conditions, the equation provides the pressure field in the domain, and using this Equation (10) we can evaluate the water flow.

Remark 2. Let us suppose now, in this second case, that S l = 0 while S v = 1 . We consider a system in which all the water is in the gaseous state. We obtain

∂ ∂ t ( ϕ ρ v ) + ∇ ⋅ ( ρ v u v ) = 0 , ⇒ ∂ ∂ t ( ϕ ρ v ) − ∇ ⋅ ( ρ v ϕ K μ v ( ∇ P v − ρ v g ) ) = 0 ,

with ρ v and P v linked by Equation (14). A not-uniform parabolic equation in the unknown P v is formulated here. If it is assumed that T ( x ) is known, the solution of this equation provides the vapor pressure field in the domain.

Remark 3. The third case occurs when the water is present in both the gaseous and liquid phase. According to this configuration, we must reduce the number of unknowns, which for now are: S l (or S v ), P l , and P v . Through constitutive equations, we can introduce a mutual relationship between the two pressures and a second relationship that binds one of them to the saturation degree.

In Sections 4 and 5, we present therefore a constitutive model depending on whether the capillarity effects are or are not neglected, respectively.

The state of the H_{2}O is determined via the Clapeyron diagram [

d P d T = Λ T ( V β m o l − V α m o l ) , (15)

being Λ the latent heat (per unit of mass) of transition from one phase to another, and V m o l is the molar volume of the two phases. An approximate solution of this equation in the case of the liquid-vapor transition of pure H_{2}O is given after fitting of the experimental data [

P * ( T ) = 961.7 exp { 17.35 T − 273 T } Pa . (16)

If, at a fixed temperature, the pressure P is such that P > P * , then the H_{2}O is in the liquid phase and we have P = P l . Conversely, if P < P * then the H_{2}O is in the gaseous phase (vapor), and the result is P = P v . Unlike again, if P = P * then there is the coexistence of the liquid and gaseous phase. In such a situation, the pressure of water (now viewed as a mixture of two phases) exactly corresponds to P * . Under this condition, P l and P v must be specified. By treating the fluid as a liquid-vapour mixture, from Equation (9) we have

( 1 − S l ) P v + S l P l = P * . (17)

Therefore, if we neglect the capillary effects, the simplest constitutive assumption, encountered in literature [

P v = P * , (18)

P l = P * . (19)

In this case, the variable that describes the system behavior is therefore the saturation of one of these phases. For instance, we can choose S l ; the density of the phases follows accordingly. The liquid phase is calculated with constant density while the vapor density is given by the following Equation (14) in this way,

ρ v = ρ v * ( T ) with ρ v * ( T ) = P * ( T ) r T .

Therefore, the only state variable is precisely the saturation of one of the phases.

We aim to generalize these results, by introducing the following constitutive relationship for S l in terms of P. We describe S l with the graph S ^ l ( P ) on the pressure, i.e.

S l ∈ S ^ l ( P ) , (20)

being

S ^ l ( P ) = { 1 if P > P * , ( 0,1 ) if P = P * , 0 if P > P * , (21)

as shown in

P v = ( 1 − H ( P − P * ) ) P , (22)

P l = H ( P − P * ) P , (23)

where H is the Heaviside function

H ( x ) = { 1 if x ≥ 0 , 0 if x < 0 . (24)

Summing up, if capillary effects are neglected, the H_{2}O flow in the porous medium is described by the Equation (12), read as an equation in P, with S l given by Equation (21), and with P v and P l given by Equations (22) and (23), respectively.

When the liquid and gaseous phases coexist and phenomena of capillarity are unable to neglect, a notion of capillary pressure [

P c = P w − P n w , (25)

where P w is the wetting fluid pressure which tends to adhere with the pour walls while P n w is the fluid pressure which tends to minimize the surface contacting the porous medium. In the case of the H_{2}O in the form of vapor and liquid in the soil, the wetting phase is the liquid phase. Therefore

P c = P l − P v , (26)

with P c described by the following so-called Laplace formula

P c = γ 〈 κ 〉 cos α , (27)

with

• γ = γ ( T ) the surface tension;

• 〈 κ 〉 the average curvature of the liquid-vapour interface throughout the REV;

• cos α the contact angle between the liquid and the pore wall.

Let us now consider an isotherm line in the Clapeyron plane, according to Equation (15). At the phase transition (assuming as always, the equilibrium between the phases), we observe that

V v m o l d P v = V l m o l d P l , (28)

being V β m o l , β = l , v the molar H_{2}O volume of the respective phases. By differentiating Equation (26) we have

d P c = d P l − d P v = ( V v m o l V l m o l − 1 ) d P g = V v m o l V l m o l ( 1 − V l m o l V v m o l ) d P v .

Now, if we are “sufficiently” far from the critical point, it is legitimate to consider V l m o l ≪ V v m o l , and then we can state that

d P c = V v m o l V l m o l d P v .

Let us now assume that vapor is an ideal gas; it can be considered the equation of state as,

P v V g m o l = R T , (29)

and we get accordingly

d P c = R T V l m o l d P v P v . (30)

Now we integrate Equation (30) remarking that P c = 0 corresponds to P v = P * ,

∫ 0 P c d P c = R T V l m o l ∫ P * P v d P v P ′ v ,

in order to arrive at

P v = P * e x p { V l m o l R T P c } , (31)

that is well-known Kelvin equation. In particular, being

M H 2 O m o l = ρ l V l m o l ,

the Equation (31) can be written as

P v = P * exp { ρ l r T P c } . (32)

The model so defined has an additional variable: P c . It is necessary to introduce a constitutive law that links capillary pressure to some other variable. Following classical literature [

P c = P c ( S l ) , con { P c ( S l ) < 0 , se 0 < S l < 1 , P c ( S l ) = 0 , se S l = 1 , (33)

recalling that it is required to specify P c ( S l ) for 0 < S l < 1 .

An important estimation of capillary pressure is obtained from Equation (27). Using the experimental tabulations which reproduce this relationship and the typical values of 〈 κ 〉 and γ , it is estimated that the P c inside a capillary and can be obtained as a comparison parameter which confutes or not the approximation of capillary absence. For example, the Young-Laplace formula [

P c = 2 γ cos α η , (34)

where η is the radius of the capillary section, to evaluate some reference values. In _{2}O is taken from [

Thus, in the case of capillary not negligible, the state of the system is described by only two variables: P and S l , with the liquid phase pressure P l linked to P and to S l from this relationship

T (˚C) | γ ( T ) | P c [ Pa ] | P c [ Pa ] | P c [ Pa ] |
---|---|---|---|---|

η = 0.1 mm | η = 1 mm | η = 10 mm | ||

250 | 0.026 | 5.2 × 10^{3} | 5.2 × 10^{2} | 5.2 × 10^{1} |

280 | 0.020 | 4.0 × 10^{3} | 4.0 × 10^{2} | 4.0 × 10^{1} |

310 | 0.014 | 2.8 × 10^{3} | 2.8 × 10^{2} | 2.8 × 10^{1} |

340 | 0.008 | 1.6 × 10^{3} | 1.6 × 10^{2} | 1.6 × 10^{1} |

P l = { P if P > P * , P c ( S l ) + P * ( T ) e x p { ρ l r T P c ( S l ) } if P = P * , (35)

being P c ( S l ) given by Equation (33).

Considering the capillarity action, the mass and momentum balance problem thus arrives at the following mathematical model:

{ ∂ ∂ t { ϕ [ S l ρ l + ( 1 − S l ) ρ v ] } + ∇ ⋅ ( ρ l S l u l + ρ v ( 1 − S l ) u v ) = 0 , u l = − K k r l μ l ( ∇ P l − ρ l g ) , u v = − K k r v μ v ( ∇ P v − ρ v g ) , P v = P * exp { ρ l r T P c ( S l ) } , P l = P c + P v , P c = P c ( S l ) , S l + S v = 1 , (36)

or, expressed in function of the pore velocity

{ ∂ ∂ t { ϕ [ S l ρ l + ( 1 − S l ) ρ v ] } + ∇ ⋅ ϕ ( ρ l S l v l + ρ v ( 1 − S l ) v v ) = 0 , v l = − K k r l ϕ μ l ( ∇ P l − ρ l g ) , v v = − K k r v ϕ μ v ( ∇ P v − ρ v g ) , P v = P * exp { ρ l r T P c ( S l ) } , P l = P c + P v , P c = P c ( S l ) , S l + S v = 1. (37)

The model must be closed with the relative permeability curves specification

k r l = k r l ( S l ) ,

k r v = k r v ( S v ) = k r v ( 1 − S l ) ,

while the permeability dependence of the structure is given by the definition of the permeability tensor K.

The main purpose of the paper was not limited to reach a generic model of a geothermal system that we presented as a result. The significance of this work is recognized in the approach used in building the model. We think the aim of this work lies in how we can obtain such model. The paper was written in dissertation style. It treats how to work with physical quantities in porous media, how to consider the consequences of the various hypotheses, and how to insert the constitutive relationships. Inter alia, we gave attention to the clarity of the steps and the reproducibility of the deductions.

The main result of our findings is to propose a generalised mathematical model for pressure and saturation of a two-component H_{2}O mixture flow. This finding is consistent with the sub-problems in which the fluid consists only of a single phase. In the case of the presence of pure water in the liquid state only, the problem is reduced to the classic elliptic PDEs problem of a liquid fluid in a porous medium. When the fluid is completely gaseous, we obtain a parabolic PDEs problem that can be solved if the system temperature is known. Now, in the case of simultaneous coexistence of the two phases, the addition of a constitutive model is necessary. Under the hypothesis of neglecting the effects due to capillarity, we adopted the application of the Clapeyron pressure and the law of perfect gases, to obtain a constitutive law for liquid saturation that depends on pressure, and using the Heaviside function. Finally, assuming to consider the effects of capillarity, we took advantage of the fact that capillary pressure can be expressed as a function of saturation. Therefore, we have expressed a relationship between the pressure of the liquid phase and the saturation, with the introduction of the measurement estimates of the capillary pressure, according to the Young-Laplace formula.

The various equations and the described methods can be used as a set of tools, a not-trivial starter kit, to understand and operate with models for geothermal basins, even by a readership who does not have strictly physical and mathematical skills. For these reasons, we believe this work can contribute to potential future directions [

The author would like to express his gratitude to Professor Angiolo Farina (Università Degli Studi di Firenze, Italy) for having introduced him to the study of porous media and for the very important contribution to the realization of this work. The author also acknowledges with appreciation the foundation CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) of the Ministry of Education of the Federal Republic of Brazil for receiving economic support within the funding program (Financial code PROEX-9740044/D) of academic excellence PROEX (Programa de Excelência Acadêmica).

The author declares no conflict of interest.

Meacci, L. (2020) Modeling a Geothermal Field: A Not-Trivial Starter Kit. Open Access Library Journal, 7: e6836. https://doi.org/10.4236/oalib.1106836