Table Of Content.. ... " ... ~"
"
" \
"
~
s k
on o
egi o
m r b
u
........ . .; ~ p"Lrolc the
OJJin in
'~ ; : ",'nIL" """11,1 ... : II ....... n~'oI.,u. ,1/ = or Boundaries Ih" he ex-USSR in I estigated
...... nv
i
,
R
S
S
U
-
x
e
, " he
..... / OW .... . ... i .... wtJnNl ' Olllf,N1UJlU: e ""'I,ICVI ~'4."-IIt:ItN '" Sf:'I,nU \ MU ...... ltt~( G~··· ~I.~,·~·! .,.~;:,~:.~) "" .... ', nd Gas Fields of t
.. MOSC l a
i
O
Petroleum Engineering and
Development Studies
Volume 4
Mathematical Theory of
Oil and Gas Recovery
With Applications to
ex-USSR Oil and Gas Fields
PETROLEUM ENGINEERING AND
DEVELOPMENT STUDIES
Advisory Editor John S. Archer
Volume 1 Fundamentals of Casing Design
Hussain Rabia ISBN 0-86010-863-5
Volume 2 Directional Drilling
T. A. Inglis ISBN 0-86010-716-7
Volume 3 Quality Assurance in the Offshore Oil and Gas Industry
J. H. Rogerson (editor) ISBN 0-8601 0-947-X
Volume 4 Mathematical Theory of Oil and Gas Recovery. With Applica
tions to ex-USSR Oil and Gas Fields
P. Bedrikovetsky ISBN 0-7923-2381-5
Petroleum Engineering and
Development Studies
Volume 4
Mathematical Theory of
Oil and Gas Recovery
With Applications to
ex-USSR Oil and Gas Fields
by
Pavel Bedrikovetsky
Moscow State Oil and Gas Academy, Russia
Scientific Editor
eren Rowan
Consultant, formerly with British Petroleum
Translator
Ruth Loshak
M. Phil., MITJ
SPRINGER-SCIENCE+BUSINESS MEDIA, B.v.
Library of Congress Cataloging-in-Publication Data
Bedrikovetsky, cavei.
Mathematical theorv of oii and gas recovery with applications to
ex-USSR oii and gas fields 'by Pavel Bedrikovetsky.
p. cm. -- <Petroleum engineer1ng and development studies v.
4>
Includes bibliographlcal references <p. > and index.
ISBN 978-90-481-4300-9 ISBN 978-94-017-2205-6 (eBook)
DOI 10.1007/978-94-017-2205-6
1. Oii fields--Product1on methods--Mathematical models. 2. Gas
fields7·Production methods--Mathematical models. 3. Oii fields
·Former Soviet republics--Production methods. 4. Gas fields--Former
Sov1et republics--Production methods. I. Title. II. Series.
TN870.8415 1993
622' .33B--dc20 93-14525
ISBN 978-90-481-4300-9
Printed on acid-free paper
Ali Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
With love to my mother, Rita Klinger
FOREWORD
It is a pleasure to be asked to write the foreword to this interesting new book. When
Professor Bedrikovetsky first accepted my invitation to spend an extended sabbatical
period in the Department of Mineral Resources Engineering at Imperial College of
Science, Technology and Medicine, I hoped it would be a period of fruitful collaboration.
This book, a short course and a variety of technical papers are tangible evidence of a
successful stay in the UK. I am also pleased that Professor Bedrikovetsky acted on my
suggestion to publish this book with Kluwer as part of the petroleum publications for
which I am Series Editor.
The book derives much of its origin from the unpublished Doctor of Science thesis
which Professor Bedrikovetsky prepared in Russian while at the Gubkin Institute. The
original DSc contained a number of discrete publications unified by an analytical
mathematics approach to fluid flow in petroleum reservoirs. During his sabbatical stay at
Imperial College, Professor Bedrikovetsky has refined and extended many of the chapters
and has discussed each one with internationally recognised experts in the field. He
received great encouragement and editorial advice from Dr Gren Rowan, who pioneered
analytical methods in reservoir modelling at BP for many years.
To the eyes of western petroleum reservoir engineers, used to numerical methods and
reservoir simulation, this book is relatively unusual. It is, however, very timely and
provides an opportunity to revisit the mathematical basis of analytical continua problems
for which there are exact solutions. Of necessity, the reservoir description is relatively
simplistic and may be something of a shock to numerical modelling colleagues immersed
in permeability characterisation of heterogeneous reservoirs with ten thousand grid cells.
Although the source and reliability of charactrerisation data are quite properly a matter of
concern, it is not the objective of this book to comment on where these data come from or
how effective values may be obtained. Similarly, the appropriateness of particular
representations of relative permeability relationships and partition coefficients are not the
primary concern. Rather, it is the way such information could be used, if it were
available, that is of importance in this analytical mathematics approach.
The author has been ambitious in the breadth of the field that he has taken on. He has
divided the book into seven parts that broadly reflect his experience in tackling problems
in the petroleum reservoirs of the oil USSR.
In Part I he deals with the mathematics of waterflooding, and describes two phase
incompressible and compressible flow in I-D in porous media of various complexity.
vii
viii FOREWORD
In Part II he deals with chemical flooding and is particularly interested in the
Riemann problem. Continuous injection and slug injection of one or two chemicals are
treated, and the 1-0 models are extended to 2-D for the case of 2-phase displacement.
The author tantalisingly speculates on a 3-D analytical model for chemical flooding in a
stratified reservoir.
Thermal waterflooding is treated in Part III and the approaches presented are utilised
in an example from the Kharyagi field.
In Part IV we find a treatment of gas and solvent injection into oil and gas condensate
reservoirs. In oil fields the injection process is considered both as continuous and with
slugs, and the experience with the Vuktyl field is used as an example. An exact analytical
solution for the WAG process is developed and utilised in an example from the Kharyagi
field. Gas condensate initially in place is estimated from phase equilibrium assumptions
and an interesting case of displacement of retrograde condensate by a LPG slug is
discussed.
A single chapter comprises Part V, and ideas of in situ chemical refining as an
example of chemical reactions during flow in porous media are explored. The
development of gas fields containing hydrogen sulphide provide the background, and
examples are drawn from Gugurtly and Sovetabad.
In Part VI the effects of temperature, gravity and capillarity on immiscible fluid
stratification in porous reservoirs are developed.
Finally, in Part VII, the author develops relationships for gravity stabilised gas
injection into thick gas condensate reservoirs, and provides an example with the
Karachaganak field in Kazakhstan.
The book should be of interest to many petroleum reservoir engineers, and it will
provide an opportunity to validate a range of numerical reservoir simulation models by
comparison with the exact results of simple analytical methods. It also provides an
insight into the ways in which petroleum reservoir engineering has been considered in the
ex-USSR and it offers, through its examples, a glimpse of some of the reservoirs which
may become the subject of future East-West collaborative developments.
Professor John Archer, F.Eng
Pro Rector and Professor of Petroleum Engineering
Imperial College of Science, Technology and Medicine
London SW7 2AZ
5th April 1993
Table of Contents
Foreword by Professor John Archer vii
xvii
Preface
PART I. WATER·FLOODING
CHAPTER 1. ONE-DIMENSIONAL MOTION OF A TWO-PHASE SYSTEM OF
IMMISCIBLE LIQUIDS IN A POROUS MEDIUM 3
1.1. Equations of two-phase flow through a porous medium 3
1.2. Formulation of the displacement problem. Discontinuous solutions 7
1.3. Method of characteristics 9
1.4. Discontinuities 12
1.5. The problem of stabilized zones 15
1.6. Solution of tbe Riemann problem (decay of a discontinuity) 16
1.7. Contour integration metbod. Motion of discontinuity 20
1.8. Graphical-anal ytical calculation of displacement coefficient. Welge' s method 22
1.9. Inverse problems of two-phase displacement 23
l.l0. Features of water-drive displacement of a rbeologically anomalous oil 25
CHAPTER 2. PERCOLATION MODELS OF FLOW THROUGH A POROUS MEDIUM 27
2.1. Basic relations of percolation theory 27
2.2. Percolation model of a porous medium 31
2.3. Conduction parameters of a porous medium 33
CHAPTER 3. ANALYTICAL MODELS OF WATER-FLOODING OF STRATIFIED
RESERVOIRS
3.1. Asymptotic analysis of tbe effect of capillary and gravitational forces on the
two-dimensional transport of two-phase systems through porous media 40
3.2. Water-flooding of a stratified reservoir in the viscous dominated case 48
3.3. Displacement from stratified reservoirs in the case of capillary-gravitatiooal
equilibrium 57
CHAPTER 4. EFFECTS OF COMPRESSIBILITY ON TWO-PHASE DISPLACEMENT 60
4.1. Statement of the one-dimensional problem 60
4.2. Conditions at discontinuities 61
4.3. Self-similar solutions of one-dimensional problems 63
4.4. Solutions for displacement of a liquid by gas and a gas-liquid mixture 65
4.5. Decay of a discontinuity 67
4.6. Equations of the displacement of gas by water from stratified reservoirs 71
4.7. Derivation of the equations for the displacement of oil (water) by gas from
stratified reservoirs 71
4.8. 'Quasi-one-dimensional' displacement from a heterogeneous reservoir 73
IX
