11. SPE-18172-MS OK

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    SP

    Society of PetroIeun

    Engineers

    SPE 8 72

    Pressure

    Transient

    nalysis of Wells With Finite

    Conductivity

    Vertical

    Fractures

    in

    Double Porosity

    Reservoirs

    by

    H

    Cinco-Ley,· PEMEX and University of Mexico and H.-Z. Meng,· Dowell Schlumberger

    MemberSPE

    Copyright 1988. Society of Petroleum Engineers

    This paper was prepared for presentation at the 63rd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in

    Houston. TX. October 2-5. 1988.

    This paper was selected for presentation by an SPE Program Committee following review of information contained in

    an

    abstract submitted by the

    author s). Contents of the paper. as presented. have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the

    author s). The material. as presented. does not necessarily reflect any position of the Society of Petroleum Engineers. its officers. or members. Papers

    presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is

    restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of

    where and by whom the paper is presented. Write Publications Manager. SPE. P.O. Box 833836. Richardson. TX 75083·3836. Telex. 730989 SPEDAL.

    ABSTRACT

    This paper

    presents

    the

    results

    obtained in

    the

    study

    of the

    transient behavior of a well intersected by a finite conductivity

    vertical fracture in a double porosity reservoir. Two models are

    considered to

    take

    into account the fluid

    transfer

    between matrix

    blocks and fractures: the pseudo-steady-state matrix flow model

    and the transient

    matrix

    flow model.

    A general semianalytical model and simplified fully

    analyt

    ical models are presented. t is demonstrated that these systems

    exhibit the basic behavior of a well with a finite conductivity

    fracture: that is bilinear flow, pseudolinear flow and pseudoradial

    flow

    in addition

    to

    the transition

    flow periods.

    Each of

    thes e flow

    periods is

    under

    the influence of the different states of

    the

    fluid

    transfer between matrix and fractures;

    that

    is

    fracture dominated

    period, transition period

    and total

    system dominated period.

    t

    is shown that correlating

    parameters are the

    dimension

    less

    fracture

    conductivity kfbf

    )v,

    the

    fracture

    storativity coeffi

    cient wand the interporosity flow parameter Af

    or

    the dimension

    less matrix hydraulic diffusivity 1)maD).

    t

    was found, for the

    transient

    matrix flow model,

    that the

    pressure behavior exhibits 1/8 slope in a log-log graph during the

    bilinear flow dominated by the trans ition period of the fluid trans

    fer. Hence a graph of pressure versus t

    l

    /

    S

    yields a straight line

    passing

    through the

    origin.

    During

    the pseudolinear flow, and if the fluid transfer is in

    the

    transition period, a log-log graph of

    the

    prerssure versus

    time

    exhibits 1/4 slope straight line. This means that a graph of p

    versus t

    l

    /

    4

    yields a

    straight

    line. Hence it is concluded that bilin

    ear flow is not the only type of flow that exhibits the one quarter

    slope type of behavior.

    References and illustrations at

    end

    of

    paper.

    645

    Type

    curves

    are

    presented

    to

    analyze data falling in

    the

    bi

    linear - pseudolinea r flow regions. The effect of wellbore storage

    are also included. The general semianalytical models yields simul

    taneous the

    constant

    flow rate and

    the

    constant pressure solutions

    as well as the pressure derivative function for the constant rate

    case.

    INTRODUCTION

    In recent years interest has been growing for the evalua

    tion

    of hydraulically

    fractured

    wells producing in double porosity

    reservoirs.

    Although

    considerable efforts have been dedicated

    to

    study

    either fractured

    wells in homogeneous reservoirs

    or

    wells

    producing in double porosity

    reservoirs,there is not

    a complete

    study that includes fractures in double porosity reservoirs.

    t has been shown in the

    pastl

    that well intersected by

    fi-

    nite conductivity vertical fracture in a homogeneous reservoir can

    exhibit several flow periods: bilinear, pseudolinear and pseudo

    radial

    in

    addition to

    the

    transition

    between

    them.

    For each of

    these flow periods there is a specific graph of interpretation that

    produces a

    straight

    line

    portion

    for

    the

    pressure

    data. The

    pres

    sure data

    within the

    bilinear flow, the pseudolinear flow and

    the

    pseudoradial flow exhibit a straight line in a graph of pressure

    versus tl/4,

    t

    l

    /

    2

    and log t respectively. The complete behavior

    of the system

    can

    be

    correlated by a

    parameter

    called dimen

    sionless

    fracture

    conductivity kfbf)v. In order to identify the

    different flow regimes type curves have been presented in

    terms

    of

    pressure

    l

    ,2

    or

    pressure derivative

    3

    4

    On the other hand, the studies on the behavior of wells

    in double

    porosity

    reservoirs have considered two kinds of models

    for the matrix-fracture fluid transfe r: the pseudo- steady-s tate flow

    model Warren and Root

    5

    ) and the

    transient

    matrix flow model

    de Swaan

    6

    and

    KazemF). t

    has been established that both

    models exhibit

    three

    well-defined

    flow

    periods: at early time,

    the

    behavior is dominated by the storativity o f the natural fractures,

    at

    intermediate time

    the fluid

    transfer

    from matrix to fractures

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    2

    PRESSURE

    TRANSIENT

    ANALYSIS OF ,WELLS WITH FINITE

    CONDUCTIVITY

    VERTICAL

    FRACTURES IN DOUBLE POROSITY RESERVOIRS SPE 18172

    becomes more

    important and

    the pressure in the network of natu

    ral fractures

    tends

    to stabilize originating the so called transition

    period

    and

    at long time the system is dominated by the storage

    capacity of the total system. The behavior of a double

    porosity

    system can be correlated by

    the

    fracture storage capacity param

    eter w and the

    interporosity

    flow parameter

    or

    dimensionless

    matrix hydraulic diffusivity l1maD).

    For the case of an

    unfractured

    well in a double porosity

    reservoir a semilog

    graph

    shows two parallel

    straight

    lines repre

    senting the

    natural

    fracture dominated

    flow

    period

    and

    the

    total

    system dominated

    flow period. In between these parallel

    straight

    lines there is a transition period sometimes called stabilization

    period; here the pressure curve is represented by a rather flat line

    for the

    pseudo-steady-state

    matrix flow model and by a straight

    line whose slope is

    half

    the slope of the parallel straight lines for

    the transient

    matrix

    flow model.

    Several type curves have been presented to interpret tran

    sient pressure data of naturally fractured reservoirs

    8

     9 .

    These

    type curves are expressed in terms of pressure or pressure deriva

    tive. Recently Houze et al

    10

    developed a model to study the

    behavior of wells intersected by infinite conductivity vertical frac

    tures in

    double porosity reservoirs.

    They

    presented

    type

    curves

    that can be used to estimate w, ~ f x J and the formation flow

    capacity kh if a large span of pressure data is available. Their

    models are semianalytical and consider

    the

    pseudo-steady-state

    type

    of

    matrix-fracture fluid

    transfer

    and

    both

    the uniform flux

    fracture and the infinite conductivity fracture.

    Later Lancaster

    and

    Gatens

    11 presented a methodology for

    analyzing pressure data for fractured wells in double porosity

    reser