報 告 者：林 建 文指導教授：陳 瑞 昇 博士2011/03/24

Analytical solution for coupled multi-species reactive transport of N-member radionuclide sequential decay chains in

finite geological media

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OUTLINE INTRODUCTION OBJECTIVES METHODS PRELINRY FUTURE WORK

how to deal with …?

examples of common international

INTRODUCTION

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α

α

β

γ

(USEPA, 2010)

Pu-238

Th-230

Ra-226

U-234

+DAY

Radioactive decay often involves a sequence of steps (decay chain). For example, Pu-238 decays to U-234 which decays to Th-230 which decays, and so on, to Ra-226.

Decay products are important in understanding radioactive decay and the management of radioactive waste.

INTRODUCTION

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Analytical solutions for transport problems involving sequential decay reactions have been developed mostly for steady-state boundary conditions and for infinite or semi-infinite spatial domains.

Relatively very little literature is available about analytical solutions for multispecies

transport problems for either finite media or time-dependent boundary conditions.

INTRODUCTION

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OBJECTIVES

In this project we will develop analytical solutions for the two-dimensional couple multi-species reactive transport of radionuclide sequential decay chains through a finite-length geological media.

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METHODSEstablish the initial and boundary conditions,

derivation of two-dimensional advection dispersion equation

Finite Fourier cosine transform

General integral transform technique (GITT)

Decouple technique in combination

Solve the particular solution for differential equation

Inverse transform, Analytical solution obtained

PRELINRY

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Establish the initial and boundary conditions, derivation of two-dimensional

advection dispersion equation

1 1 12 2

0

2 2

( 1, 2,3, , ; 0)

i i iL T i i i i i i

C C C CiR D D V λ RC λ R Ci t x y x

i N λ

0,

1,

initial conditions:

, , 0 ,

boundary conditions:

- (for constant source) 0, ,0, ,

0 - &

( ) 0, ,0, ,

i i

iiL i

y y

iiL i

C x y t G x y

VC B y BC x y tD VC x y t

L y B B y Lx

VC tC x y tD VC x y t

x

- (for time-varying source)

0 - &

( , , ) ( , - , )( , , )0, 0, 0

y y

i y i yi x

B y B

L y B B y L

C x y L t C x y L tC x L y t

x y y

PRELINRY

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00

0 0

2 22

1 12 2

, , , , , , , ,

1

i i x i i x xTD D D i i L i

x Y x L y

i i i ii i i i i

D L D D L D

C C VL λ R L LDx y Vtx y t c c Pe r X β

L L L C C D V D L

c c c cXβR rc r ct Pe x x Pe y

2 2 2 2, , ,

1 12

,

1

,

0

1

Finite Fourier cosine transform ( , , ) ( , , )

( , , ) ( , , ) cos( )

F i F i F ii i i i i

D L D D L

i D D D F i D D

F i D D i D D D D D

c c c Xβ n πR r c r c

t Pe x x Pe

c x y t c x n t

c x n t c x y t n π y dy

Finite Fourier cosine transform

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FUTURE WORKEstablish the initial and boundary conditions,

derivation of two-dimensional advection dispersion equation

Finite Fourier cosine transform

General integral transform technique (GITT)

Decouple technique in combination

Solve the particular solution for differential equation

Inverse transform, Analytical solution obtained

General integral transform technique (GITT)

Decouple technique in combination

Solve the particular solution for differential equation

Inverse transform, Analytical solution obtained

2 2 2 2

1 12

10i i

i i i iL D D L

F F Xβ n πr F r F

Pe x x Pe

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2 2 2 21

Ωi

mk

k m km i

L

rx F x

Xβ n πr r

Pe

Analytical solutions

HYDROGEOCHEM

4.0

Transport safety assessment of nuclear substances

Risk assess-ment

Geochemical transfer mode (HYDROGEOCHEM)

Biogeo chemica

l Transfer

Groundwater Flow

Numerical solutions

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FUTURE WORK

Thanks for your attention

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