reservoir geo mechanics

8/10/2019 Reservoir Geo Mechanics

1/29

Reservoir Geomechanics

Stefan Hergarten

Institut fur Geo- und UmweltnaturwissenschaftenAlbert-Ludwigs-Universitat Freiburg


2/29

Processes and Interactions in Reservoirs

Components of a Reservoir Model

Hydrocarbon reservoirs Geothermal reservoirstwo/three-phase ow single-phase owFluid ow

(oil/gas/water) (water)Heat transport no yes

Deformation optional optional

Parameters and Variables

Which of the parameters and variables in the list below belong to the

processes uid ow, heat transport, and deformation? Which are (input)parameters, and which are (output) variables?

Displacement, heat capacity, heat ux density, permeability, porosity,pressure, shear modulus, strain, stress, temperature, thermal conductivity,velocity, viscosity, Youngs modulus


3/29

Processes and Interactions in Reservoirs

Parameters and Variables

Parameters Variables

Fluid ow porosity pressure

permeabilty velocity (ow rate)

viscosity

Heat transport thermal conductivity temperature

heat capacity heat ux density

velocity (ow rate)Deformation Youngs modulus displacement

shear modulus strain

pressure stress


4/29

Fluid Flow in Porous Media

Darcys Law

Empirically found by Henry Darcy (1856)Describes the average ow through a porous medium on macroscopicscalesCan be theoretically derived from the Navier-Stokes equations (which

describe the small-scale ow in pore space) under some simplicationsSimplest form (without gravity):

v = k

p l

wherev = ow rate (volume per time) per cross section areak = hydraulic permeability = dynamic viscosity of the uid

p l = pressure drop per length


5/29


The Hydraulic Permeability

Units:SI unit: m2

Widely used unit: Darcy (D)1 D = 9.869 10 13 m2 10 12 m2 = 1 m2

k = 1 D results in a ow rate of 1 cms at a pressure drop of 1 atmcm in

water at 20

C ( = 10 3

Pas).Permeability of oil reservoirs should be at least about 0.1 D.Unconsolidated sand: k 5 D


6/29


Example: Hydraulic Permeability of a Tubular Medium

Hagen-Poiseuille law for an individual pipe:

q = r 4

8 p l

Total ow per cross section area through a cubic block of length l with nparallel pipes:

v = nq

l 2 =

nl 2

r 4

8 p l

= r 2

8 p l

with the porosity = n

r 2

l 2

k = r 2

8 =

d 2

32


7/29


Darcys Law in Three-Dimensional Space

Without gravity:

v ( x ) = k p ( x )

where

p ( x ) = grad p ( x ) =

x 1 p ( x )

x 2

p ( x )

x 3p ( x )

is the gradient of the pressure eld.

Including gravity:

v ( x ) = k

p ( x ) + g 001


8/29


Hydraulic Head (piezometric head, hydraulic potential)

Dene h( x ) in such a way that

h( x ) = p ( x )g

+001

v ( x ) = gk

h( x )


9/29


10/29


The Water Balance v ( x ) is the mass ux (mass per time and cross section area).

div( v ( x )) = x 1

(v 1( x )) + x 2

(v 2( x )) + x 3

(v 3( x ))

is the (negative) change of water mass per time and bulk volume (rockand pore space) due to

changes in uid saturation in pore space,changes in uid density (compression), andcompression of the matrix.


11/29


The Water BalanceCompressibility of water: 5 10 10 Pa 1

Compressibility of most rocks is even smaller.

div( v ( x )) = 0

in case of completely saturated pore spaceand moderate changes in temperature, salinity, etc.

This is not true for the gas component in hydrocarbon reservoirs.


12/29


The Water Balance

If sources or sinks are present:

div( v ( x )) = s ( x )

wheres = source term (supplied uid volume per total volume and time)

Differential equation for pressure or hydraulic head:

divk p ( x ) = s ( x ) or div

gk

h( x ) = s ( x )


13/29


Pressure Field around a Point-Like (line, area) Injection

3D (point source)p (r ) =

4 k

i 1r + const

where r = distance from the injection point, i = injection rate [ m3

s ]

2D (line source in 3D space)

p (r ) = 2 k

i lnr + const

where i = injection rate per length [ m2

s ]

1D (area source in 3D space)

p (r ) = k

i r + const

where i = injection rate per area [ ms ]


14/29

Heat Transport

The Heat Equation (conduction, advection, production)

c t

T ( x , t ) = d iv( T ( x , t ) cT ( x , t ) v ) + Q

where = density [ kgm3 ]

c = specic heat capacity [ Jkg K ]

= thermal conductivity [ Wm K

] v = velocity [ms ]

Q = heat production rate [ Wm3 ]


15/29

Heat Transport

Typical Values of c

Water: c = 4180 Jkg KRocks: c = 8001000 Jkg K

Typical Values of Material [ Wm K ] Rocks [

Wm K ]

diamond 2300 granite 2.8iron 80 basalt 2

quartz 1.4 dolomite 2.5sand 0.6 limestone 2.5

expanded polystyrene 0.033 sandstone 2.5water 0.6 shale 2

air 0.026 widely used value 2.5


16/29

Heat Transport

The Heat Equation for a Porous Medium

(m c m + f c f ) t

T ( x , t ) = div (( m + f )T ( x , t )

f c f T ( x , t ) v ) + Q

where

f , c f , f = parameters of the uidm , c m , m = parameters of the dry matrix (not the solid!)

v = ow rate (Darcy velocity)


17/29

Stress and Strain

The Stress Tensor

Stress = force per areaUnit: Pa = Nm2

Force is a vector.At each point, the force acting on (hypothetic) surfaces of arbitraryorientations can be considered. The orientation can be characterizedby a normal unit vector n (perpendicular to the surface, | n | = 1).

9 degrees of freedom at each point


18/29

Stress and Strain

The Stress TensorCauchys stress theorem: State of stress at each point can be

characterized by a 3 3 matrix, the stress tensor

=11 12 1321 22 2331 32 33

so that n is the force per area acting on a surface with orientation n.

Symmetry: is symmetric: T

= , ij = ji

Only 6 independent stress components at each point

d


19/29

Stress and Strain

Normal Stresses, Shear Stresses, Principal Stresses, and Mean Stress

Normal stresses: The component of n parallel to n (i. e., perpendicular tothe surface) is called normal stress on the surface dened by n.

Shear stresses: The absolute value of the component of n perpendicularto n (i. e., parallel to the surface) is called shear stress on the surface

dened by n.Principal stresses: Due to the symmetry of , there are always 3

perpendicular directions where n is parallel to n (i. e., where no shearstresses occur):

n(i ) = i n(i )

with real numbers i . The values 1, 2, and 3 are called principalstresses, the vectors n(i ) dene the principal stress directions.

Mean stress: =

11 + 22 + 33

3 =

1 + 2 + 3

3

S d S i


20/29

Stress and Strain

The Navier-Cauchy Equations

2

t 2 u ( x , t ) = div( x , t )

x 111 +

x 2

12 + x 3

13

x 1 21 + x 2 22 + x 3 23

x 131 +

x 2

32 + x 3

33

+ F

where

= density u ( x , t ) = displacement of the point x

F = body force per volume (e. g., gravity)

S d S i


21/29

Stress and Strain

The Navier-Cauchy Equations

The Navier-Cauchy equations reect Newtons law and relate theforce acting on a small volume to its acceleration.The acceleration term is only important in uids and for thepropagation of seismic waves in solids.Steady-state equation:

div( x , t ) + F = 0

St d St i


22/29

Stress and Strain

Displacement Gradient Tensor and Strain Tensor

Describe the state of deformation and are not directly related to forces.Displacement gradient tensor (arguments x and t omitted):

u =

x 1

u 1

x 2u 1

x 3

u 1

x 1u 2

x 2

u 2

x 3u 2

x 1

u 3

x 2u 3

x 3

u 3

Removes the translational component of u and contains rotation anddeformation.

St d St i


23/29

Stress and Strain

Displacement Gradient Tensor and Strain Tensor

Strain tensor (arguments x and t omitted):

= 12 u + u T + u T u

Removes the rotational component of u and contains onlydeformation, i. e., changes in the distances between points.

Linear approximation for small deformations:

= 12 u + u

T

In components:

ij = 12

x j

u i + x i

u j

Stress and Strain


24/29

Stress and Strain

Normal Strain, Shear Strain, Principal Strains, and Volumetric Strain

Basically the same as for the stress tensorNormal strain: The component of n parallel to n is called normal strain in

direction of n.Shear strain: The absolute value of the component of n perpendicular to

n is called shear strain in direction of n.Principal strains: Due to the symmetry of , there are always 3

perpendicular directions where n is parallel to n:

n(i ) = i n(i )

with real numbers i . 1, 2, and 3 are called principal strains.Volumetric strain = relative change in volume:

V = 11

+ 22

+ 33

= 1 +

2 +

3

Stress and Strain


25/29

Stress and Strain

The Signs of Stress and Strain

Physics and engineering: As dened here

positive normal/principal/mean stress = tensile stressnegative normal/principal/mean stress = compressive stress

positive normal/principal/volumetric strain = expansionnegative normal/principal/volumetric strain = compression

Geology: Just opposite

positive normal/principal/mean stress = compressive stressnegative normal/principal/mean stress = tensile stress

positive normal/principal/volumetric strain = compression

negative normal/principal/volumetric strain = expansion

Elastic Behavior


26/29

Elastic Behavior

Hookes Law

Linear relationship between stress and strainHookes law for an isotropic elastic medium:

= V 1 + 2

with

1 = identity matrix

, = Lame parameters of the medium

Elastic Behavior


27/29

Elastic Behavior

Hookes Law

Hookes law for an isotropic elastic medium written the other wayround:

= 1

2

2( + 23 )

1

= 1

E ((1 + ) 3 1 )

with

E = (3 + 2 )2( + )

= Youngs modulus

= 2( + )

= Poissons ratio

Poroelasticity


28/29

Poroelasticity

Basic Ideas

If remains constant, a positive pore uid pressure p causes an isotropicexpansion of the porous medium:

= 12

2( + 23 ) 1

p 3H

1

H is some kind of bulk modulus with respect to the uid pressure.

Written the other way round:

p 1 = V 1 + 2

with

= K

H = Biots coupling parameter

K = + 23 = bulk modulus of the porous medium

Poroelasticity


29/29

Poroelasticity

The Concept of Effective Stress

p 1 = V 1 + 2

and p together cause the same deformation as the so-called effectivestress

eff = p 1

would cause without uid pressure.

It is often assumed that rock failure etc. depends on eff instead of .

reservoir geo mechanics

Documents