BiotBiot--Barenblatt double Barenblatt double porosity theoryporosity theory

Dmitriy SilinDmitriy SilinUniversity of California, BerkeleyUniversity of California, Berkeley

University of Houston, December 05 2005

ObjectivesObjectives

►► Theoretical interpretation of the frequency Theoretical interpretation of the frequency dependent reflection from the hydrocarbondependent reflection from the hydrocarbon--bearing reservoirbearing reservoir

►► Asymptotic analysis of attenuation, reflection, Asymptotic analysis of attenuation, reflection, transmission coefficients in order to obtain transmission coefficients in order to obtain relatively simple expressionsrelatively simple expressions

“All that is simple is false and all that is complex is useless” // P.Valery

►► Investigate the role of local formation permeability Investigate the role of local formation permeability heterogeneitiesheterogeneities

LayoutLayout►►PoroelasticityPoroelasticity + fluid flow = + fluid flow = Biot’sBiot’s mediummedium►►Fluid flow + local heterogeneities = Fluid flow + local heterogeneities =

Barenblatt’sBarenblatt’s double porosity mediumdouble porosity medium►►PoroelasticityPoroelasticity + fluid flow + + fluid flow +

local heterogeneities = local heterogeneities = BiotBiot--BarenblattBarenblattdouble porosity modeldouble porosity model

►►Asymptotic analysis of reflection and Asymptotic analysis of reflection and transmission coefficientstransmission coefficients

►►Future workFuture work

Fractured rockFractured rock

Local heterogeneities: Local heterogeneities: fractures and matrixfractures and matrix

Source: Markus Tuller and Dani OrVadose Zone Journal 1:14–37 (2002)

MatrixMatrix

2 micron2 micronDiatomite

Matrix permeability is small due to the complexity of the pore space geometryBerea sandstone

Images: Liviu Tomutsa LBNL (ALS and FIB)

DoubleDouble--porosity modelporosity model

T. D. van Golf-Racht, Fundamentals of fractured reservoir engineering, Elsevier, Amsterdam, 1982.

DoubleDouble--porosity model: some historyporosity model: some history

► Barenblatt, G. I., Zheltov, Yu. P., and Kochina, I. N. (1960), basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Applied Mathematics and Mechanics (PMM), 24, No. 5, pp. 1286-1303 (English Translation from Russian).

► J. E. Warren and Root, P. J.: “The Behavior of Naturally Fractured Reservoirs,” SPEJ, (Sept. 1963) pp. 245-255

► S. R. Pride, J. G. Berryman. Linear dynamics of double-porosity dual-permeability materials. I - II. Phys. Rev. E 68, (2003)

DualDual--Medium ApproachMedium Approach

• In every REV, two types of media are presented simultaneously:

– Matrix: stores the fluid but only allows for fluid exchange with the surrounding fractures

– Connected system of fractures: practically zero storage capacity, but fluid flow due to the simple geometry

–Two fluid pressures are associated with each point in the medium

• The rate of fluid exchange is proportional to the difference between the fluid pressures in the two media

DoubleDouble--porosity mediumporosity medium

Compression Dilation

MatrixFr

actu

re

Frac

ture

Matrix

Low matrix pressure

High fracture pressureHigh matrix pressure

Low fracture pressure

Fluid flow and compressibilityFluid flow and compressibility

2

2f

f

pW uWt x t

κτ ρη

∂ ∂ ∂+ = − + ∂ ∂ ∂

Fractures: dynamic Darcy’s law

( )fm f mq Aq p pt

τη

∂+ = − −

∂Matrix-Fractures flow: A -> shape factor

fd dVdp

Vρ βρ

= = −

Fluid compressibility: slightly compressible fluid

ux

βσ∂= −

∂Drained solid: linear elasticity

Mass BalanceMass Balance

Wqx

∂=

∂Fractures: no fluid accumulation

2m m

m f mp u qt t t x

φφ β φ∂ ∂ ∂+ = − −

∂ ∂ ∂ ∂

Matrix: fluid accumulation exclusively due to compression and deformation

( )2

1mm

ut t xφ φ∂ ∂

= −∂ ∂ ∂

Solid: deformation results in porosity variation

Momentum balanceMomentum balance2

2 2 2

2

2

1 1

b f

fmm f

u Wv t v t

ppux x x

α β α β

∂ ∂+

∂ ∂

∂∂∂= − −∂ ∂ ∂

Acceleration of fluid-solid system

Total stress

2 1b

b

vβρ

= 2 1f

f

vβρ

= Two modified speeds of sound

Harmonic wave solution:Harmonic wave solution:asymptotic formasymptotic form

( ) ( )i t k x i t k xs Fs Fu U e U eω ω− −= + Slow + fast wave

Small (dimensionless) parameter i fρ κω

εη

=

( )1F F

f

Oa

vς

εε ω=

+

Example: attenuation factors ( )1S S

f

Oa

vς

εω

ε=

+

Reflection from impermeable top of Reflection from impermeable top of the reservoirthe reservoir

Incident P-wave

Reflected P-wave

TransmittedFast & Slow waves

Boundary conditionsBoundary conditions::Material and total stress balance

ImpermeableImpermeableelastic mediumelastic medium

FluidFluid--saturated saturated fractured reservoirfractured reservoir

Reflection from impermeable top of Reflection from impermeable top of the reservoirthe reservoir

( )0 1

R R R Oε ε= + + Reflection coefficient

Transmission coefficient (fast) ( )0 1

F F FT T T Oε ε= + +

Transmission coefficient (slow) ( )ST O ε=

10

1

2

2

F

F

Z ZR

Z Z

−=

+1

1

02

2FF

Z

Z ZT =

+Z1 – impedance of the impermeable medium

1

21 1 1

2

SF F

F

Z

Z ZR T Tς==

+– fast and slow

impedances of the reservoir formation

2 2,F SZ Z

ConclusionsConclusions

►►LowLow--frequency asymptotic analysis of elastic frequency asymptotic analysis of elastic wave propagation and reflection in fractured wave propagation and reflection in fractured fluidfluid--saturated reservoir implies:saturated reservoir implies:

The fast wave attenuation factor is small of The fast wave attenuation factor is small of higher order relative to that of slow wavehigher order relative to that of slow waveThe main frequencyThe main frequency--dependent components of dependent components of reflection and fast transmission coefficients are reflection and fast transmission coefficients are proportional to the square root of frequencyproportional to the square root of frequencyThe frequencyThe frequency--independent component of the independent component of the slow transmission coefficient is zeroslow transmission coefficient is zero

Future plansFuture plans

►►Analyze reflection from a finiteAnalyze reflection from a finite--thickness reservoirthickness reservoir

►►Include internal reservoir Include internal reservoir heterogeneitiesheterogeneities

►►Extend analysis to not normal incident Extend analysis to not normal incident wave (frequencywave (frequency--dependent AVO)dependent AVO)

AcknowledgmentsAcknowledgments

This work has been performed at the University of California, Berkeley, University of Houston, and Lawrence Berkeley National Laboratory

Grant No. DE-FC26-04NT15503, U.S. Department of Energy