Georg.hi.is Files Talk GEORG Seminar 0

8/12/2019 Georg.hi.is Files Talk GEORG Seminar 0

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Mathematical modeling of energy

flow in a geothermal reservoirHalldr Plsson

University of Iceland

Mathematical modeling of energy flow in a geothermal reservoir p. 1


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The topic

The project involves the following:

Calculation of fluid flow in porous media, according toDarcys law.

qg = rg

g(p+mg) ql =

rl

l(p+mg)

m

t + (gqg+lql) = 0

Calculation of energy balance

mh

t + (1 )rcr

Tr

t +q h= (krTr)

Estimation of model parameters, based on data.Mathematical modeling of energy flow in a geothermal reservoir p. 2


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Basic assumptions

The reservoir consists of different porous rockformations. The pores are filled with fluid, primarilywater, which exists in liquid phase, gas phase, or as a

two phase mixture. The solid rock material has elasticproperties but does not move as the fluid itself.

The reservoir is bounded by Earths surface as well as

outer boundaries within the crust. Conditions must bespecified for selected variables everywhere on theboundary.

All variables and parameters are functions of position ina three dimensional space, and in some casesparameters depend on some of the variables.

The state of the reservoir generally varies with time, butin some cases a steady state solution can be found.Mathematical modeling of energy flow in a geothermal reservoir p. 3


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Motivation for the project

Various models exist, notably iTOUGH2, developed atBerkeley Labs. But, . . . the purpose of this work is to:

Develop an Icelandic reservoir model in cooperationbetween University of Iceland, Reykjavk University andIcelandic GeoSurvey.

Use the model or different parts of it for teaching and

training students and specialist at the mentionedinstitutions.

Involve people from different fields in the work such as

geophysicists, engineers and mathematicians.Promote new ideas and methods in the modeling work,using state of the art methods from the literature.



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The research group

Assoc. prof. Elnborg I. lafsdttir, UI

Dept. man. Guni Axelsson, IGS

Ass. prof. Gurn Svarsdttir, RUAssoc. prof. Halldr Plsson, UI

Assoc. prof. Jan Valdman, UI

Prof. emer. Jnas Elasson, UI

PhD. stud. Lrus orvaldsson

Prof. Ragnar Sigursson, UIRes. spec. Stefn I. Valdimarsson, UI



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Software framework

Modular software will well defined interfaces.

Focus on numerical mathematics and solvers.



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Mesh generation

Several high quality programs are available:

GMSH (http://www.geuz.org/gmsh/), two and threedimensional, mainly trinangles and tetrahedra elements.

Triangle (http://www.cs.cmu.edu/ quake/triangle.html),two dimensional Delaunay triangulator. Works well forFEM and FVM.

TetGen (http://tetgen.berlios.de/), three dimensionaltetrahedra generator. File interface similar to Triangle.



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Solution strategies

Discretization results in a large system of equations,possibly non-linear. Solution strategies involve

Direct solution of linear equations.

Iterative methods for linear equations, enhanced bypreconditioners:

Incomplete factorization.Fast methods based on FFT.Multigrid methods, notably AMG.

Non-linear iterations, e.g. Newton-Raphson with trustregion or linesearch, or more specialized methods.

Time stepping methods for unsteady problems.



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Interpretation of results

Results are stored in a consistent manner. Postprocessinginvolves:

Calculation of combined values, using e.g. Matlab orOctave.Visualization of field variables with:

the extensive graphical abilities of Matlab.available programs, like ParaView(http://www.paraview.org/).known commercial programs, e.g. ANSYS or Fluent.

Comparison of results with other calculations.



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Parameter estimation

Motivation:

It is necessary to be able to estimate reservoirparameters, based on known data.

Known asinverse modelingas an alternative toparameter estimation.

A replication of the abilities of iTOUGH2.Methods available for estimation:

Standard methods, e.g. Levenberg-Marquardt.

Robust methods with stochastic gradient search.Evolutionary programming for very difficult problems.



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A simple case: Two dimensional flow

The Darcy equation for pressure variation

(Ra z) = 0

coupled with the advective-diffusive energy balanceequation

=

(

Ra z) +

where

Ra=2cg(T1 T0)L

k



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Results for Ra = 60

0.5 1 1.5 2 2.5 3 3.5

0

0.20.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5

0

0.2

0.4

0.60.8

1

0.2

0.4

0.6

0.8

1

2

1

0

1

2



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Conclusions

An initiative has been started by UI, RU and IGS todevelop their own geothermal reservoir model.

The people involved are applied mathematicians,geophysicists and engineers.The purpose is to educate students and scientists andinvolve them into the inner workings and development

of state of the art reservoir models.An emphasis will be put on using the newest findings inimplementing solution methods, e.g. for solving large

sets of equations and computation on clusters.Finally, the group hopes that the developed models willbe used by research institutions and industry involved in

utilization of geothermal energy.Mathematical modeling of energy flow in a geothermal reservoir p. 14

Georg.hi.is Files Talk GEORG Seminar 0

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