Numerical modelling of geothermal systems A short ... - Geo.X

Numerical modelling of geothermal
systems
A short introduction
Mauro Cacace
Björn Onno Kaiser
Yvonne Cherubini

Mathematical model
Description (approximated) of a system using mathematical concepts
Mathematical representation
o
o
o
o
Variables
Inputs
Outputs
Internal state
Internal processes
Mathematical equations
Description of the
interactions among these
variables
Conceptual model
Analytical model
Numerical model
Homogeneous and simple (linear) processes Heterogenous, coupled (non-linear) processes
Geothermal Reservoirs
Modelling

Conceptual model
Numerical model
Continuos
PDEs
∆f x, y, z = 0
Discretization
Linear Algebraic
Systems
A ij ∙ f i = 0
Geothermal Reservoirs
Modelling

Finite Differences vs Finite Element
o Linear approximations
o Regular or pseudo-irregular
supprting meshes (grids)
o Simple geometries
o Low computational efforts
o Polynomial approximations
(Weak solution)
o Fully irregular supporting
meshes
o Complex geometries
o High Computational efforts
Geothermal Reservoirs
Modelling

Regionally localized geological setting where heat flow from the Earth‘s interior is
transported close enough to the surface by circulating steam or hot water to be readily
harnesed for use
Heat
Water
Energy
Working Fluid
© 2000 Geothermal Education Office
Requirements
Reservoir Permeable and porous rock bounded by impermeable formations
Geothermal Reservoirs
Introduction

Reservoir systems
Engineering perspective
Courtesy of Guido Blöcher GFZ Potsdam
Geothermal Reservoirs
Introduction

Reservoir systems
Engineering perspective
Geothermal reservoir is …
Heat
and
! PERMEABILITY, PERMEABILITY, PERMEABILITY !
Geothermal Reservoirs
Introduction

Multiphysics systems
Reservoir systems
Modelling perspective
Coupled Processes
H
T
M
C
Hydraulic processes
Fluid and two phase flow, heat and
mass transport, fluid pressure changes
Thermal
Conductive and convective heat flux,
internal heat production
Mechanical
Rock deformation, fracture initiation and
propagation, thermo and poroelastic
effects
Chemical
Chemical reactions, corrosion and
scaling
Geothermal Reservoirs
Introduction

H T M
Processes
Fluid or Two Phase
Flow
Permeability
Heat flow
Fracture & Fault
Mechanics
Poroelastics
C
Thermoelastics
Chemical Reactions
Corrosion & Scaling
Natural
Engineered
Reservoir
Components
Geology
Fault Systems
Natural Fractures
Completed wells
Induced Fractures
Geothermal fluid
loop
Enhanced
Geothermal
System
Courtesy of Guido Blöcher GFZ Potsdam
Geothermal Reservoirs
Introduction

H T M
Processes
Fluid or Two Phase
Flow
Permeability
Heat flow
Fracture & Fault
Mechanics
Poroelastics
C
Thermoelastics
Chemical Reactions
Corrosion & Scaling
Natural
Engineered
Reservoir
Components
Geology
Fault Systems
Natural Fractures
Completed wells
Induced Fractures
Geothermal fluid
loop
Enhanced
Geothermal
System
Courtesy of Guido Blöcher GFZ Potsdam
Geothermal Reservoirs
Introduction

Groß Schönebeck at a glance
Hamburg
Groß
Schönebeck
o
Location
approx. 40 km north of Berlin
Hannover
Berlin
o
Doublet system
January 2007
Köln
o
o
o
o
Low-enthalpy Reservoir
-3850 m to -4260 m
Reservoir rocks Lower Permian
Siliclastics and Volcanics
Natural major fault zones
NW-, and NNE-striking
Hydraulically Induced fractures
3 stimulation treatments
Moeck et al. (2009)
Groß Schönebeck Geothermal Facility

Groß Schönebeck Geothermal Facility

H T M
Processes
Fluid or Two Phase
Flow
Permeability
Heat flow
Fracture & Fault
Mechanics
Poroelastics
C
Thermoelastics
Chemical Reactions
Corrosion & Scaling
Natural
Engineered
Reservoir
Components
Geology
Fault Systems
Natural Fractures
Completed wells
Induced Fractures
Geothermal fluid
loop
Enhanced
Geothermal
System
Courtesy of Guido Blöcher GFZ Potsdam
Geothermal Reservoirs Modelling
Components

• Geological Units
Geothermal Reservoirs Modelling
Components

• Geological Units
• Induced Fractures / Natural fault Systems
Geothermal Reservoirs Modelling
Components

• Geological Units
• Induced Fractures / Natural fault Systems
• Injection & Production Well
Geothermal Reservoirs Modelling
Components

H T M
Processes
Fluid or Two Phase
Flow
Permeability
Heat flow
Fracture & Fault
Mechanics
Poroelastics
C
Thermoelastics
Chemical Reactions
Corrosion & Scaling
Natural
Engineered
Reservoir
Components
Geology
Fault Systems
Natural Fractures
Completed wells
Induced Fractures
Geothermal fluid
loop
Enhanced
Geothermal
System
Courtesy of Guido Blöcher GFZ Potsdam
Geothermal Reservoirs Modelling
Processes

Fluid flow
H T M
C
Darcy‘s
Law
v f = − k μ ∇ (p + ρgze z)
v f = Darcy velocity m ∙ s −1
k = permeability [m 2 ]
μ = dynamic viscosity [Pa ∙ s]
p = pressure [Pa]
ρ = fluid density [kg ∙ m −3 ]
g = acceleration coefficient [m ∙ s −2 ]
z = reference height [m]
Geothermal Reservoirs Modelling
Processes

Fluid flow
H T M
C
Piezometric head [h]
h ≝ z + p
ρg
Hydraulic conductivity [K]
K ≝ (ρg) k μ
v f = −K ∇ ∙ h = −K grad(h)
Geothermal Reservoirs Modelling
Processes

Permeability – porous medium
H T M
C
o Permeability in an intrinsic property of the porous medium
o Measure the easiness of flow
o Magnitude is controlled by the grain size (pore size)
Hornberger et al. (1998)
Geothermal Reservoirs
Introduction

Permeability – fractures
H T M
Formulas are analytical solutions and valid for laminar (Hagen Poiseuille or
Couette) flow between two ideal plates without roughness
C
Fracture permeability
Fracture conductivity
k = a2
12
K = a2 ∙ ρ ∙ g
12 ∙ μ
Geothermal Reservoirs
Introduction

Fluid flow
H T M
C
Balance equation
S ∂h
∂t = −∇ ρK∇h + Q = −∇ ρv f
+ Q
Geothermal Reservoirs Modelling
Processes

Fluid flow
H T M
C
Balance equation
S ∂h
∂t = −∇ ρK∇h + Q = −∇ ρv f
+ Q
Specific Storativity
Mass source term
S = γ α + φ ∙ β
γ = ρg = Specific weight of water [Nm −3 ]
φ = Effective porosity [−]
α = Bulk compressibility [m 2 N −1 ]
β = Fluid compressibility [m 2 N −1 ]
Darcy velocity
v f = −K ∇ ∙ h
Geothermal Reservoirs Modelling
Processes

Heat flow
Conductive and convective
H T M
C
[φ ρc p f
+(1 − φ) ρc p S
] ∂T
∂t + ρc p f
∇ ∙ v f T − ∇ ∙ φλ f + 1 − φ λ S ∇T = H Sf
Geothermal Reservoirs Modelling
Processes

Heat flow
Conductive and convective
H T M
C
[φ ρc p f
+(1 − φ) ρc p S
] ∂T
∂t + ρc p f
∇ ∙ v f T − ∇ ∙ φλ f + 1 − φ λ S ∇T = H Sf
Fourier‘s Law
q = −λ ∙ ∇T
q = heat flow density [W ∙ m −2 ]
λ = heat conductivity W ∙ (m ∙ K) −1
∇T = thermal gradient [K ∙ m −1 ]
Geothermal Reservoirs Modelling
Processes

Heat flow
Conductive and convective
H T M
C
[φ ρc p f
+(1 − φ) ρc p S
] ∂T
∂t + ρc p f
∇ ∙ v f T − ∇ ∙ φλ f + 1 − φ λ S ∇T = H Sf
Darcy‘s Law
v f = −K ∇ ∙ h
Geothermal Reservoirs Modelling
Processes

Fluid and Heat flow
H T M
C
S ∂h
∂t = −∇ ρv f
+ Q
[φ ρc p f
+(1 − φ) ρc p S
] ∂T
∂t + ρc p f
∇ ∙ v f T − ∇ ∙ φλ f + 1 − φ λ S ∇T = H Sf
Geothermal Reservoirs Modelling
Processes

Fluid and Heat flow
H T M
C
S ∂h
∂t = −∇ ρv f
+ Q
v f = −K ∇ ∙ h
[φ ρc p f
+(1 − φ) ρc p S
] ∂T
∂t + ρc p f
∇ ∙ v f T − ∇ ∙ φλ f + 1 − φ λ S ∇T = H Sf
Geothermal Reservoirs Modelling
Processes

Boundary conditions
1 st type (Dirichlet):
• pressure (head), temperature,
concentration
2 nd type (Neumann):
• flux
3 rd type (Cauchy):
• transfer
4 th type (FeFlow):
• well

Results – Step 1
Natural flow field

Results – Step 2
Doublet System (hydraulic)

Results – Step 3
Doublet System (hydro-thermal)

Results – Step 4
Doublet System with induced fractures