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B011 STOCHASTIC RESERVOIR – THE BUCKLEYLEVERETT MODEL
PEPPINO TERPOLILLI
ELF EP TOTAL AV LARRIBAU 64018 PAU CEDEX
Abstract
The stochastic approach has found broad acceptance in weather forecasting, global climate
modelling or hydrology. In the oil business, already in the sixties such tools were used to
represent the complex and intricate structure of a porous media. Scheidegger, Matheron and
Beran were among the people who used such approach to deduce Darcy law at a macroscopic
scale, the flow being modelized at the microscopic scale by Stokes equation. A modern revival
of such an approach, more mathematically founded, could be the homogenisation theory which
was developed since the eighties.
In this talk we first present the challenge faced by the reservoir engineer who is supposed to
obtain production profiles useful to ascertain the economy of the project under consideration.
Most of the time prediction of field performance is based on specific knowledge which is in
general a mixture of hard and soft data. Hard data such as logs data collected along the wells, are
known with minimal uncertainty, while soft data have a broader spectrum of uncertainties. This
uncertainty is managed using geostatistical tools to represent for example the porosity or the
permeability field between the wells: in this way we obtain stochastic fields which are the
coefficients of the equations modelizing the flows in the field. Each realization of these
stochastic processes gives a possible geological model as an input for the reservoir simulator.
But such a model could be very large (million of cells) to conform the hard data at the well.
Ensemble-based prediction is then hindered by the computational cost of running a flow
simulation on a single geological realization.
This talk will then present a simplified flow model known as the Buckley-Leverett model,
introduced in the thirties, to face the previous challenge : in that case the cost of a flow
simulation is affordable. We will present some prediction results obtained recently by Cho and
Lindquist, we present the Integral approach to uncertaity and then we assess an homogenisation
problem for the Buckley-Leverett model extending previous work by T Hontans and P Terpolilli
to multiphasic flows.
Introduction
The basic law used to model flow in porous media is the so called Darcy law discovered in the
mid eighties. This law is remarkable for several reasons: first because it allows a simple
macroscopic description of a very complex situation mainly due to the inner geometry of porous
rocks . Another distinctive feature, which was recognised later, is the fact that Darcy law is a
scale law in the sense that it allows a description of phenomena at various scales, with an
effective permeability which fits at each chosen scale. A third striking feature is the local relation
between the filtration velocity and the gradient of pressure, in the spirit of non-equilibrium
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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thermodynamic and in contrast with the Navier equation for the modelling of free flows. This
fact is the fundamental reason why single phase flow in porous media is modelled with linear
PDEs.
These features have always been linked with the issues faced by reservoir engineers on the
processes of evaluation of reserves and forecasting of production. But in recent years, many
reservoirs with complicated physical and geological properties have entered into development as
the extraction industry of oil and gas has reached more mature time. Fundamental problems of
enhancing oil and gas recovery from rocks has been intensively analysed: among them
uncertainty assessment and upscaling issues need a greater understanding .
Uncertainty assessment is becoming a crucial step for the appraisal of a new field and/or for the
decision of development. In the last decades we witnessed a big change in the practise of
reservoir engineering. Actually, a characteristic point for the applications of subterranean
hydrodynamics is that the information concerning the underground is always rather poor and
limited. It includes geophysical information coming from interpretation surveys, geological data
from core analysis, logs along the wells, petrophysic results and well testing. When the wells are
in production, we also collect the production history of the fields. However, even if the total
amount of such information were available, which is by no means always the case, it is still
insufficient to enable an unambiguous construction of an adequate reservoir model. Indeed, any
model requires the interpolation over the whole field of the data measured in majority in the
wells and the near vicinity. Under these circumstances, the basic goal of the reservoir engineer
until recently was to establish qualitative features of the field under study, stable trends as well
as certain quantitative predictions, stable with respect to the variation of the input data, poorly
known. The purpose of a reservoir study was not an illusive precise prediction of all flow
properties, but rather an enlarging of the amount of information taken into account. It is the
problem statement and the analysis of the results of its solution that plays the governing role and
allows the engineers to come to some conclusions.
Now the situation has drastically changed with the appearance of geostatistical tools enabling
the construction of detailed geologic models constrained by the data at the wells. In his famous
book “ Estimer et choisir” G Matheron gives deep insight about the use of stochastic tools for the
modelling of geological bodies. The level of details of these stochastic models conforms to the
resolution of the data collected at the wells. Geostatistic offers efficient tools to interpolate
between the wells but each such models is by no means the reality but a possibility, linked with
the probabilistic framework inferred from data. This very fact leads to the development of
uncertainty assessment studies in the field of reservoir engineering trying to figure the
propagation of uncertainty from the data- like porosity and permeability fields- to the solutions
like production rates and so on. Several methodologies are now available as alternatives to the
ensemble averaging, hindered by the cost of a single run for a possible model. We have at our
disposal experimental design methodology, with which were already developed industrial tools
in reservoir engineering. A more prospective way is the solution by stochastic partial differential
equations (SPDE) governing the different moments of quantity of interest . In this paper we
briefly review SPDE approaches and give a new proposal to tackle with the uncertainty issue.
Upscaling has appeared as a necessary step for the development of the theory of subterranean
flows since the early sixties in the task to deduce Darcy Law at a macroscopic scale when the
flow at the microscopic scale is modelled by Stokes equation. We refer to the works by
Matheron, Scheidegger, Schvidler… More recently upscaling was used to obtain affordable input
for the simulation of oil fields: for example it became a decisive step to transfer porosity and
permeability fields from the geological model to the dynamical model. Upscaling is also
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critically used to define the geological model directly at a macroscopic scale if we have a
stochastic description at the microscopic scale. It allows the determination of the variogram and
other statistical parameters at the larger scale.
On the upscaling issue we discuss the possibility to extend the work by Terpolilli-Hontans [7 ]
to the case of multiphase flows.
Uncertainty Assessment
We will not discuss here the experimental design approach as it is already a mature technique
routinely used in operational study. We first focus on methods using SPDE developed since a
decade in reservoir engineering. These methods are designed to take into account the variability
of parameters as permeability, in each cell of the reservoir model. This fact is very satisfactory in
a theoretical point of view even if in many situations not necessary at all to obtain the answers
we look for in a reservoir study.
We first mention the works of Zhang [ 8 ] for single phase flow, extended to 2-phase flow in
collaboration with Tchelepi [ 8 ]. A related approach has been discussed in his PhD thesis by
Jarman [5 ]. Most of these works deals with Buckley-Leverett equation. The main goal of these
approaches was to obtain PDEs to describe the evolution of the moments of quantities like
production of oil or other parameters with value for operational purposes. These moment
equations are obtained using different techniques such as closure approximation method or
perturbative expansion method. To be more complete we may mention Cvetkovic-Dagan [ 3 ]
and Langlo-Espedal [ 6 ] for related works and assumptions of importance for the operational
interest of the results obtained.
These works are very ambitious and give interesting results which stimulate research and
discussions on uncertainty assessment in reservoir field. Different points have been made
concerning the methods used and the assumptions made.
As in turbulence theory the closure problem is always critical and lead to a-priori assumptions
impossible to justify, only checking if it works on some cases. For the perturbation technique we
need to be not too far away from a reference situation and this could be a limitation. Artus [ 1 ]
in his PhD thesis has raised more physical questions, emphasizing for example the importance of
viscous coupling in 2 phase flows, which is not taken into account in the Lagrangian approach
taken in Zhang [8 ] nor in the Eulerian approach taken in Langlo-Espedal[6 ] or Jarman[ 5]. To
our knowledge there is no systematic comparison of the results obtained and this could have
great interest for our community.
We will now recall some results given in Cho-Lindquist [ 2 ] about an ensemble average
approach. This paper presents a study in predictability in stochastic reservoirs. For a reservoir
model consisting of purely stochastic, a priori, small scale geological information, the a
posteriori distribution of reservoir oil production behaviour is established. In particular the
inherent limits for ensemble based prediction are emphasized. The model considered is
sufficiently simple to allow rapid solution and thus enable deeper analysis of the mapping from
geology to production. The model geometry is a 2D cross section with non-interacting 1D
stratum, homogeneous layer with constant porosity and permeability. The porosity and
permeability are modelled by a stochastic process with correlation between the stratum. The
relative permeability curves are of the Corey type with exponent equal 2. Buckley-Leverett
equation to model 2 phase flows subject to a drop of pressure is clever used in each stratum .
The results obtained show that production uncertainty has a strong dependence on both the
heterogeneity strength and the spatial correlation length of the geology. The number of
realizations needed to achieve a given level of accuracy of ensemble based means is established.
The improvement of accuracy obtained from history matched ensemble is quantified. It is shown
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a Markovian property for the prediction of reservoir production. This implies that no single
realization, even one that is history matched, is likely to retain the same level of accuracy for an
extended period of time.
The model considered in [ 2 ] contains fluid and geometrical complexity representative of real
reservoir behaviour. At the same time it is our feeling that these result are not subjected to the
general critic one can read about Monte Carlo approach, for example in Zhang [ 8 ]. This gives a
great relevance to the results obtained. Nevertheless we cannot expect to extend the Monte
Carlo approach for sufficiently general situations.
Finally we will give some insight on some works done at the end of the nineties which could be
of some help in defining a new approach for uncertainty assessment. For convenience these
works will be referred to as the Integral approach for reasons to be made clear in the sequel.
The previous works using SPDE were really in the spirit to evaluate the propagation of
uncertainties: actually these works were looking for PDEs describing the moments of different
parameters of importance. In the Integral approach we directly obtain the desired moment as a
functional integral. In fact supposing that we have the characteristic function of the stochastic
process describing the data, for example the permeability random field, we obtain the
characteristic function of the pressure field (and more generally for general function of the
pressure field as the velocity field) as a functional integral involving the characteristic field of
the permeability field. This was done for single phase flows. The techniques used to obtain such
representation were taken from quantum field theory and the justifications are at the level of
physical standard of rigor. Let introduce some mathematical and physical formalism to present
briefly the Integral approach.
Let u denote the field of interest, for example the pressure field; the actual value u of this field
is obtained solving some equations:
g j (u , A) = 0;
(1 )
where A denote, for example, the permeability field which is supposed to be a random field (
physicist say that A is a disordered field and we will use this terminology in the sequel ) with a
known characteristic function given by:
Ψ (ϕ ) = ln ⟨ exp i ∑ ϕ n . An ⟩
(2)
n
where ⟨. ⟩ denote the average taken over the disorder and ϕ is a dual variable ( similar to the
wave-number for the Fourier transform). The value f (u ) is random through the probability
distribution of A .
We wish to express the expectation value exp F = exp if (u ) in terms of the functional Ψ .
For single phase flow in a random porous media equation ( 1 ) is the pressure equation wich
depend of the permeability random field A .
We give now the functional integral representation obtained for exp F :
⎧⎪
⎫⎪
⎡
⎤
exp F = ∫ D ( X , Y ) exp ⎨i ∫ dθ ⎢ θ f ( X ) + ∑ Y j g j ( X ) ⎥ Ψ (ϕ ) ⎬
( 3 ).
j
⎣
⎦
⎩⎪
⎭⎪
The expression ( 3 ) formally achieves our goal. At the price of introducing auxiliary variables
( physicist called them bosonic and fermionic variables as they obey different algebraic rules) we
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have integrated over the random field A and expressed the result in terms of the characteristic
functional Ψ . The result has the standard form of a functional integral over various fields
similar to the so-called partition function in quantum field theory. For single phase flow we can
give a more explicit form to equation ( 3 ) . This integral representation could then be evaluated
using techniques of field theory such as explained in Glimm-Jaffe [4].
A crucial assumption to obtain ( 3 ) is the fact that equation ( 1 ) is linear with respect to field A .
This is indeed the case for single phase flow, but not in the multiphase case. It is an interesting
challenge to extend the Integral approach to such case, for example for flows modelled with
Buckley-Leverett equation.
Upscaling
We begin this section recalling the work done in Terpolilli-Hontans [ 7 ] for single phase flow.
In this paper a new framework was introduced to compute upscaled permeability for general
boundary conditions, without restriction, including then the usual no-flux, linear or periodic
boundary conditions. We introduce the framework and give the necessary results to be able to
present our proposal for a new approach to the upscaling of relative perm for 2 phase flows.
Solving auxiliary problems, often called local problems, is a basic step for most of the existing
methods for the computation of upscaled quantities. We introduce the local problem we consider.
Let us consider a finite piece (larger cell) ω with boundary Γ , of a bounded reservoir Ω ⊆ IR d , 1 ≤ d ≤ 3 . We suppose to be given the permeability field K ( x ) over ω . Our goal is to
find a constant permeability tensor K in order to affect it to ω . For this purpose, we introduce
the following pairs of local problems:
let m>0 be a given integer and i an index such that 1 ≤ i ≤ m
⎧− div( K ( x ) ∇Pi ) = f i in ω
(1.1.i) ⎨
Pi = g i
⎩
⎧− div( H ∇ U i ) = f i
(1.2.i) ⎨
U i = gi
⎩
on Γ
in ω
on Γ
Problems (1.1.i) model a monophasic flow through ω with permeability field K ( x ) ; f i is a
source term and gi the condition imposed on the boundary Γ . We denote by Pi the unique
→
solution of (1.1.i) and q i = − K ( x ) ∇Pi the corresponding filtration velocity. The same applies to
problems (1.2.i) where a constant field H is substituted to the heterogeneous given field K ( x ) .
→
Let U i denote the unique solution of (1.2.i) and vi = − H ∇ U i the corresponding filtration
velocity. We introduce the set M (h, a , ω ) of constant uniformely positive definite symmetric
⎛ 1
tensor on ω , where h denotes the harmonic average i.e. h = ⎜⎜
⎝ω
arithmetic average defined by: a =
1
ω
⎞
K ( x )dx⎟⎟
ω
⎠
∫
∫ K( x)dx : the constant fields
ω
−1
−1
and a is the
H in M (h, a , ω ) satisfy:
hξ . ξ ≤ Hξ . ξ ≤ aξ . ξ ∀ξ ∈ IR d . This set is a bounded closed convex set in the vector space
generated by symmetric tensors of order 3.
Dissipated energy
The basic idea is to look for the tensor H which minimizes the discrepancy between the
dissipated energy in problems (1.1.i) and (1.2.i).
For each index i, j such that 1 ≤ i , j ≤ m we consider the following quantities:
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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E ij ( K ( x )) =
E ij ( H ) =
1
2
1
2
∫
ω
∫
ω
K ( x )∇Pi . ∇Pj dx −
H ∇U i . ∇U j dx −
∫
ω
∫
ω
f i Pj dx
f i U j dx
For i = j these quantities correspond exactly to the energy dissipated for the corresponding
flows. To compute the upscaled permeability we consider the following optimization problem:
m
( P)
min
H ∈M ( h ,a ,ω )
I (H) =
∑[ E
ij ( H )
− E ij ( K ( x ))
i , j =1
]
2
This optimization problem is finite dimensional, this is the main point since the existence of a
minimizer is then obvious.
Definition 1: We say that H * is an upscaled tensor for the energy if H * is solution of
problem ( P) .
Averaged velocity field
We consider now for each local problem the averaged velocity fields defined by:
→
Vi ( K ( x )) =
and
→
Vi ( H ) =
1
ω
1
∫
ω
∫
→
qi dx
ω
→
vi dx .
ω
We consider then the following optimization problem:
'
(P )
m
min
H ∈M ( h ,a ,ω )
J (H) =
→
→
i
i
∑ V ( H ) − V ( K ( x))
2
i =1
where |.| is the Euclidien norm.
Definition 2: We say that H * is an upscaled tensor for the velocity if H * is solution of the
problem ( P ' ) .
This new framework encompasses the classical approaches and has close connection with
homogenization theory. The point of interest for us herein, is that for the classical cases, i.e. with
no-flux or linear or periodic boundary conditions, definition 1 and 2 define the same upscaled
permeability as the classical approaches.
Upscaling 2 phase flows.
A possible way to extend the approach in [7] is to work with the velocity fields, trying to extend
Definition 2 to the multiphase flows. Actually a generic extension is straightforward, working
with the velocity fields of each phase at the same time, trying to enforce each of these fields at
the large scale to be close to the same field at the fine scale. Choosing a cost function to measure
the discrepancy between fields an optimization scheme could be devised, using standard
techniques coming from control theory to compute the desired gradients.
But it is obvious that such a blind strategy has little chance to be successful at the first stroke.
We feel the need to gain first experience with simple cases where the physic is well understood.
To be more precise we mention ongoing work with simple model of two phase flow, including
Buckley-Leverett model, for which we have a good knowledge of velocity fields and of the
saturation fields which allows for an easy implementation of the velocity field strategy. Let be
more precise: we consider the case of linear displacement in a thin reservoir of porous material
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of length L and section A. We assume a constant porosity Φ and a constant permeability K, but
a repartition of rock-types with the associated rel perm kri i ∈ {1,..., m} . We consider the
reservoir filled with oil at time 0, when we begin the injection of water at rate qw at the left
q
hand. We denote by q the total flow of water and oil, and by f w = w the water fractional flow
q
and we suppose negligible capillary and gravity effects. We suppose known the saturation field
S ( x, t ) at each point and each time.
We search for an effective medium with one rock-type which under the same conditions gives
the same flow. Hereafter we will make precise the meaning we give to our last sentence. We
recall some known facts about Buckley-Leverett equation in a homogeneous media: the water
saturation satisfy the following equation
∂S w ⎡ q df w ⎤ ∂S w
+⎢
=0
⎥
∂t ⎣ ΦA dS w ⎦ ∂x
(4)
which is the Buckley-Leverett equation and if x = x(t ) is chosen to coincide with a surface of
fixed S w we have the relation
q df w ( S w )
⎛ dx ⎞
⎜ ⎟ =
⎝ dt ⎠ S w ΦA dS w
(5 )
which gives the velocity of the section of saturation S w .
As we have supposed to know S w ( x, t ) for each point and each time for the experiment with
several rock-types, we can deduce from this knowledge the average velocity for different section
of saturation S wi for i∈ (1,......k ) . We denote those velocities by Vi . Using then equation (5 ) a
genuine approach to obtain an effective medium could be to match each such velocity
considering the minimization of the following cost function:
i
q df w ( S w ) 2
J (k ro , k rw ) = ∑ ( Vi −
) .
ΦA dS w
i
We need then to choose a parameterization of the rel perm to reduce our search to an
optimization problem.
This approach take into account the viscous coupling [1], but it implicitly assume the validity of
the Buckley-Leverett equation at the large scale. We have the feeling, comforted by the single
phase case, that the velocity field strategy is a good approach for the up scaling of 2 phase flow.
Conclusion
In this work we have tried to give a perspective on two important issues of reservoir engineering
and have made two modest proposals as alternative approaches to each of them. A common
point has been the use of Buckley-Leverett formulation for the modelling of two phases flows in
porous medium. The validity of this model for reservoir purposes is possible if one can discard
the capillary and gravitational effects. For laboratory measurements Buckley-Leverett equation is
of standard use. Actually me make ours the following words of John von Neumann :‘ It is that
the sciences do not try to explain, they hardly even try to interpret, they mainly make models. By
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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a model is meant a mathematical construct which [……] describes observed phenomena’ with
sufficient accuracy . In fact Buckley-Leverett equation turns the Monte Carlo approach
affordable for the uncertainty issue. This nonlinear equation retains many features of more
complex model such as black-oil or compositional models even if it could not tell us the full
story. It is also the simplest model to deal with if we want to get a better understanding of the up
scaling of relative permeability for multiphase flows.
A last word for this conclusion to emphasize a dual aspect of modern reservoir engineering: very
often we shift from stochastic model to deterministic one back and forth. Having a stochastic
description of a complex porous media we look for an average description of the flow and obtain
eventually a deterministic model for the averaged parameters: this is the spirit of upscaling.
Another time we have a real, deterministic oil field but with poorly described physical
parameters: we then use stochastic tools to figure out what could be the all possibilities. I have
the feeling that our field need for a better integration of these two aspects.
References
[1 ] Artus, Mise à l’echelle des ecoulements diphasiques dans les milieux poreux heterogenes
PhD thesis. November 2003
[2 ] Cho-Lindquist, Predictability in stochastic reservoirs. SUNYSB-AMS-01-21
Department of Applied Mathematics and Statistics. Stony Brook
[3 ] Cvetkovic-Dagan, Reactive transport and immiscible flow in geological media. II
applications . Proc. R. Soc. London, 452 :303-328, 1996
[4 ] Glimm-Jaffe, Quantum Physics. A functional point of view. Springer Verlag . 1987.
[5 ] Jarman. Stochastic immiscible flow with moment equations. PhD thesis, Graduate School of
the University of Colorado, 2000
[6 ] Langlo-Espedal, Macrodispersion for two-phase, immiscible flow in porous media.
Advances in Water Resources, 17:297-316, 1994
[7 ] Terpolilli-Hontans, Boundary effects in the upscaling of absolute permeability-A new
approach. Proceedings of 7th ECMOR, Baveno, Lago Maggiore, Italy, 5-8 Sept. 2000.
[8 ] Zhang, Stochastic methods for flow in porous media. Academic Press, 2001.