1
P004 SIMULATION OF NATURALLY FRACTURED
RESERVOIRS UTILIZING AN EFFECTIVE TIMESPLITTING AND UPSCALING METHODS
SUTOPO
Dept. of Petroleum Eng., Institut Teknologi Bandung Jl. Ganeca 10 Bandung 40132, Indonesia
Abstract
The single-continuum approach employing effective permeability is one of the practical methods
for simulating naturally fractured reservoirs. The MFVE method for full-tensor model was used
to deal with permeability tensors resulted from upscaling of fractured systems.
The governing equations can effectively be formulated in fractional flow equations; i.e. in terms
of global pressure and saturation equations. An algorithm is implemented applying a highly
accurate numerical approach based on the mixed finite volume element (MFVE) method for
discretizing the pressure equation, and a combination of the MFVE and the finite volume
element (FVE) methods for the saturation equation. The saturation equation is discretized by
means of a time-splitting algorithm that allows an explicit time stepping for FVE applied to the
advection equation, and an implicit time stepping for MFVE applied to the corrective equation of
capillary diffusion.
Four fracture systems comprise discrete non-crossing fractures distributed randomly. Numerical
experiments of 2-D flow are presented to demonstrate the efficiency and robustness of the timesplitting technique and upscaling for simulation of fractured reservoirs.
Introduction
Two main approaches are currently available for simulating the behavior of naturally fractured
reservoirs, the dual porosity or dual permeability model [1], and the single continuum model
with effective permeability [2−5]. In the dual continuum models, matrix blocks are divided by
regular fracture patterns. However, field characterization studies have shown that fracture
systems are very irregular, often disconnected, and occur in swarms [6]. In the single continuum
models, the flow through both the fracture and matrix systems as well as the interference flow is
represented by the effective permeability [4, 5].
The effective permeability is estimated by solving the flow equations reflecting the coupled
matrix and fracture flow. In our previous study [7], we applied the complex variable boundary
element method [8] (CVBEM) to semi-analytically solve the coupled potential problem for
matrix and fracture flow under periodic boundary conditions. Effective permeability resulting
from flow calculations is generally of tensor form.
In this research we examine the equivalent single-continuum system for flow performance
prediction of naturally fractured reservoirs. Using Euler time-stepping, the convective term is
discretized by an upstream weighting FVE scheme, while the diffusive flux is discretized using a
MFVE technique [9]. The choice of these two schemes is dictated, on the one hand, by their
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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accuracy, robustness and efficiency in handling highly heterogeneity of the porous media. On the
other hand, both FVE and MFVE are based on the weak formulation of the governing equation
and use similar functional spaces for the approximation of the dependent variable, making them
ideally suited for combination in a time-splitting approach. More precisely, the time splitting
scheme employs explicit and implicit Euler time-stepping for FVE and MFVE, respectively.
Both MFVE and FVE are locally (at the grid-block level) conservative and monotone. The
combination of the two methods in the time splitting approach should maintain these two
properties as long as stability requirements are met, as confirmed also by numerical results.
Effective Permeability
Simulation study of stochastic fracture distributions was performed to evaluate how the effective
permeability behaves as the fracture length is changed. Fracture systems comprise discrete noncrossing fractures distributed randomly. Lengths li and orientations θi of individual fractures are
normally distributed with mean length ml and standard deviation σl, and with mean angle mθ and
standard deviation σθ, respectively. Four fracture systems were generated for simulation study
with parameters ml = 0.02, 0.06, and 0.2 m, σl = 0.4ml, mθ = 20° with σθ = 0°, and mθ = 45° with
σθ = 5° as shown in Fig. 1.
Effective Permeability Computation by CVBEM
The effective permeability tensor is calculated by solving the two-dimensional, single-phase,
incompressible fluid flow equation subject to the periodic boundary conditions (PBC). Flow
within a grid-block is described by the following equation assuming unit viscosity:
∇ ⋅ [k∇p ] = 0 .
(1)
The effective permeability k* is defined with the average velocity u and the average pressure
gradient G as follows:
u = −k * G ,
where u = ( u x , u x
)T
(2)
and G = (1, 0)T or (0, 1)T.
Eq. (1) needs to be solved twice by applying the unit pressure drop across the grid-block in one
direction only. The solution of Eq. (1) with the unit pressure drop in the x direction provides a
pressure distribution in the grid-block, from which the velocity distribution u is computed. The
average velocity through the grid-block can be obtained as follows:
ux = −
∫Γ u ⋅ n dy ,
3
3
and
uy = −
∫Γ u ⋅ n dx ,
1
(3)
1
where n1 and n3 are unit normal vectors on the bottom side Γ1 and left side Γ3 of the grid-block,
respectively. These average velocities and the pressure drop G = (1, 0)T are substituted into Eq.
(2) to determine two elements of the effective permeability, kxx and kyx. Calculations are repeated
with G = (0, 1)T to obtain kxy and kyy.
The pressure equation subject to the PBC is solved semi-analytically by the CVBEM. The
complex potential is expressed as Ω = Φ + iΨ where Φ (= (k/µ)p) and Ψ are potential and
stream functions, respectively. The Laplace equation ∇ 2 Ω = 0 is solved under the PBC, where the
outside boundary is divided into segments by placing nodes. The constraints of the PBC require
the relationships in Φ ’s and Ψ ’s, i.e. Φ ’s at the opposite nodes to be the same in one direction,
and to differ by 1 in the other direction. Ψ ’s are also assumed to have the same flow rate through
the opposite boundary segments.
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The internal boundary condition along the fracture is based on continuity of flow such that the
flow rate at any point in the fracture is equal to the net inflow at the both sides. The complex
potential at an arbitrary point in the grid-block is expressed as the summation of three
components, i.e. non-singular, fracture, and fracture crossing complex potentials,
nf
nc
i =1
i =1
Ω = Ω ns + ∑ Ω fi + ∑ Ω ci ,
(4)
where nf and nc are numbers of fractures and crossings, respectively. Ωns is given by the
Cauchy’s integral. Ωfi and Ωci are given in Reference [5].
In this paper, fractures are assumed to be discrete and distributed regularly or randomly in two
dimensions in an otherwise isotropic and homogeneous matrix rock of permeability km =
9.869×10-4 µm2 (1 mD). Individual fractures are assumed to be infinitely thin and infinitely
conductive, to be perpendicular to the bedding, and to terminate at the bedding surfaces.
Model Formulation
The usual equations describing two-phase immiscible flow through porous media, such as oil
and water, are given by the mass balance equation and Darcy's law for each fluid phase, α.
∂ (φρα Sα )
+ ∇ ⋅ (ρα uα ) = ρα qα ,
∂t
and
uα = − K
krα (Sα )
µα
(∇pα − ρα g∇Z )
(5)
For phase α = w, o (i.e., water and oil), let Sα , uα , and pα be the phase saturations, Darcy
velocities, and pressures. Let Sw = 1 − So, φ be the porosity, K the absolute permeability, g the
gravitational constant, and Z the depth. The mobilities are related to the relative permeabilities
and fluid viscosities as λα(Sw) = k α (Sw)/µα and λ(Sw) =λw(Sw) + λo(Sw), and Pc(Sw) = po − pw is the
capillary pressure. Conservation of mass of each phase gives the governing equations. After
reformulation, we obtain the following (see, e.g., [10]).
r
Pressure Equation
∇ ⋅ u = qt ,
u = −Kλ (S w )(∇p − G (S w )∇Z ) ,
(6)
(7)
where qt = qw + qo and u = uw + uo. Define the global pressure [11] to be
Sw ⎛ λ
dPc ⎞
⎟⎟(ξ )dξ ,
p = p o − ∫ ⎜⎜ w
0
⎝ λ dS w ⎠
(8)
and G(Sw) = [fw(Sw)ρw + fo(Sw)ρo]g. Since λ(Sw) > 0 and K is a symmetric definite tensor, this is
an elliptic equation. For compressible flow, the pressure equation is parabolic.
Saturation Equation
φ
∂S w
+ ∇ ⋅ ( f w (S w )u ) + ∇ ⋅ (Kf w λo [∇Pc (S w ) + G s ∇Z ]) = q w (S w ) ,
∂t
(9)
where Gs(Sw) = (ρw − ρo)g.
Eqs. (6) and (7) are referred to as the pressure equations, while the saturation equation is
described by Eq. (9). They determine the main unknowns p, u, and Sw.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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To calculate the saturation using the MFVE method, we decouple Eq. (9) as
∂S w
+ ∇ ⋅ ( f w (S w )u ) + ∇ ⋅ V = q w (S w ) ,
∂t
(10)
⎛
⎞
Gs
∇Z ⎟⎟ ,
V = −KD(S w )⎜⎜ ∇S w +
∂Pc( S w ) / ∂S w
⎝
⎠
(11)
φ
and
where D(Sw) is the capillary diffusion coefficient defined by
D (S w ) = − f w λ o
∂Pc (S w )
.
∂S w
(12)
The second term of the left side of Eq. (10) is the convective flux, while Eq. (11) expresses the
diffusive flux. This parabolic equation is degenerate in the sense that D(Sw) can be zero at certain
Sw. This is due to the behavior of the relative permeability and the capillary pressure functions.
We will approximate the solution of Eqs. (10) and (11) using the operator splitting procedure
stated in the next section.
Algorithm of Operator Splitting
We employ a time-descritization procedure based on a two-stage operator splitting technique. In
the first stage of the splitting the pressure and saturation equations are solved sequentially that is
described in our previous work [10]. In the second stage, the convection and diffusion associated
with the saturation equation are treated sequentially.
Since the pressure changes less rapidly in time, a larger time step is taken for pressure than that
for the saturation. Because the stability of the convective time step is determined by the CFL
constraint, while the diffusive time step is not subject to stability restriction, we employ a finer
convection time step together with a coarser diffusion time step. Define the partition of three
different time steps:
∆t p = n sd ∆t sd = n sd n sc ∆t sc ,
(13)
where nsd and nsc are the number of diffusion and convection time steps, respectively; ∆tp is the
time step for the pressure calculation, ∆tsd is the time step for the diffusion stage saturation
calculation, and ∆tsc is the time step for the convection stage saturation calculation. The discrete
times are given by
t m = m∆t p , t n = t m + n∆t sd , t n ,k = t n + k∆t sc ,
(14)
where the normal range for m = 0,…, np, n = 0,…, nsd, and k= 0,…, nsc.
Primary Operator Splitting
The general algorithm for the primary operator splitting is as follows:
1.
Given Swm, determine pm and um by solving the pressure equations of Eqs. (6) and (7),
( )(∇p
u m = − Kλ S w
m
m
( ) )
m
− G S w ∇Z ,
m
∇ ⋅ u m = qt .
2.
For [tm, tm+1], solve the saturation equations of Eqs. (10) and (11)
(15)
(16)
5
⎛
⎞
Gs
∇Z ⎟⎟ ,
V = −KD(S w )⎜⎜ ∇S w +
∂Pc( S w ) / ∂S w
⎝
⎠
φ
∂S w
+ ∇ ⋅ ( f w (S w )u ) + ∇ ⋅ V = qw (S w ) .
∂t
(17)
(18)
Secondary Operator Splitting
The secondary operator splitting is procedure for carrying out the solution of the saturation
equations of Eqs. (17) and (18). This technique can be viewed as a predictor-corrector approach
and can be described by the following algorithm:
1.
Let tn1 = tm and assume known p, u, and Sw for t ≤ tn1.
2.
do n = n1,..., n2 = n1 + nsd -1
• do k = 0,..., nsc-1
Compute the predictor Swn, k+1 over subinterval time [tn,k, tn, k+1] by solving explicitly the
system
φ
( (
n ,k
∂S w
∂t
+ ∇ ⋅ fw Sw
n,k
)u ) = q (S ).
n ,k
w
w
(19)
•
end do
•
Set Swn = Swn, nsc -1(tn, nsc -1) = Swn, nsc - 1(tn+1).
•
Compute the diffusion effect over subinterval time [tn, tn+1] by solving implicitly the
system
⎛
⎞
Gs
∇Z ⎟⎟ ,
V = −KD(S w )⎜⎜ ∇S w +
∂Pc( S w ) / ∂S w
⎝
⎠
φ
∂S w
+∇⋅V = 0.
∂t
(20)
(21)
end do
Set Swm+1 = Swn2(tn2+1) = Swn2(tm).
3.
Numerical Discretization
The two-phase flow formulation is solved numerically by using a MFVE method for the pressure
equation and FVE-MFVE methods based on time-splitting technique for the saturation equation.
MFVE Discretization for Pressure Equations
The MFVE method is applied to the discretization of the pressure equations of Eqs. (15) and (16),
following the derivation developed in our previous works[10]. Assume that Sw0 = Swinit for m = 0
or assume that Swm has been computed through the saturation time step procedure of tm = tn2+1.
Now, the MFVE scheme to solve Eqs. (15) and (16) for pressure and velocity is defined as
follows:
∫ (Kλ (S ))
m
w
Ω
9th
−1
( )
u m ⋅ vdx + ∫ ∇p m ⋅ vdx = ∫ G S w ∇Z ⋅ vdx ,
m
Ω
(22)
Ω
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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∫∇ ⋅u
Ω
m
wdx = ∫ qt dx .
m
(23)
Ω
FVE-MFVE Discretization for Saturation Equations
The time-splitting technique applied to the saturation equations combines FVE and MFVE
methods. The MFVE method is used in the descritization of the diffusive flux because on one
hand, it is cell-centered, and thus intrinsically compatible with the FVE method employed. On
the other hand, it allows for the harmonic-weighting type of the diffusion-tensor coefficient
(KD(Sw)) across the grid blocks.
FVE Discretization for Convective Step
In the convective step, Eq. (19) can be written explicitly for subinterval time [tn,k, tn,k+1]. The
discretization of Eq. (19) is obtained by means of the FVE method. Integrating against a scalar
test function w and applying the divergence theorem, then the predictor Swn,k+1 at the new time
level k+1 is determined by
Sw
n , k +1
fwup(Swn,k)
= Sw
n ,k
+
(
)
∆t sc
∆t sc
n ,k
q
S
wdx
−
w
w
φVi , j Ω∫
φVi , j
∫ ∇ ⋅ ( f (S
n ,k
up
w
w
)u )⋅ nwdΓ .
(24)
∂Ω
fw(Swn,k)i+1/2,j,
Since
=
the last term of right-side of Eq. (24) requires fw(Swn,k) to be
evaluated on the edges. Because Sw is piecewise constant on any element, fw(Swn,k)i+1/2,j cannot be
directly computed. One-point upstream weighting is used to determine this value.
MFVE Discretization for Diffusive Step
In the diffusive step, the MFVE method will be implemented to the discretization of Eqs. (20)
and (21). As we mentioned in previous section D(Sw) can be zero at certain Sw, however on the
other hand for the application of the MFVE method, D(Sw) needs to be inverted. To treat this
difficulty, we employ a regularized version of this equation obtained by perturbing D(Sw) with
small constant, and let this constant go to zero in computations [12].
Consider the solution of the diffusive time step for Swn+1, as the result of the convective step, we
have the function Swn = Swn,nsc-1(tn+1) that will serve as the initial condition at time tn for the
diffusive time step. With similar operations performed to the saturation equations in our previous
work,[10] Eqs. (20) and (21) become
∫ (KD(S
))
n +1 −1
w
Ω
Ω
φ
∫ ∆t
Ω
V n +1 ⋅ vdx + ∫ ∇S w
sd
Sw
n +1
n +1
⋅ vdx = − ∫
wdx + ∫ ∇ ⋅ V n +1wdx = ∫
Ω
G
s
∂Pc S w n +1
Ω
∂S w
Ω
φ
∆t sd
(
n
) ∇Z ⋅ vdx ,
S w wdx .
(25)
(26)
The MFVE discretization for the diffusive step of Eqs. (25) and (26) is nonlinear. The Picard
iteration method is used for solution of the nonlinearly coupled system of Eqs. (25) and (26).
Numerical Experiments
An efficient approach for simulating behavior of a naturally fractured reservoir is to represent the
fracture-matrix system by effective permeability and use its distribution in a single continuum
simulator. Effective permeability of a grid block integrates flow through the fracture network
and flow coupling between the fractures and matrix. We examined effects of length of the
fracture distribution on accuracy of the MFVE method. To this end, four types of fracture
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configurations were employed. To generate the effective gridblock permeability data, we
descretized each domain by use of a uniform mesh of 40 × 40 gridblocks. Figs. 1 show the
impact of variations in the effective permeability on the movement of the fluid through the
domain of four fracture systems, respectively. In these figures we superimpose the fracture
systems onto plots of the water saturation counters. The fluid movement in the fracture system
can be reproduced well using single continuum model. Distinct features of fingering caused by
the fractures were commonly generated by this method for four types of fracture length. Because
of the fracture length and density difference, the water saturations front are faster in Figs. 1(c)
and 1(d) than those in Figs. 1(a) and 1(b).
350
350
300
300
250
250
200
200
Y
400
Y
400
150
150
100
100
50
50
0
0
100
200
X
300
0
400
0
(a) ml = 0.02, mθ = 20°
200
X
300
400
(b) ml = 0.06, mθ = 20°
350
350
300
300
250
250
200
200
Y
400
Y
400
150
150
100
100
50
50
0
100
0
100
200
X
300
(c) ml = 0.20, mθ = 20°
400
0
0
100
200
X
300
400
(d) ml = 0.20, mθ = 45°
Fig. 1: Simulated water saturation contours of fracture system at t = 2000 days.
Conclusions
A time-splitting method for numerical solution of the two-dimensional immiscible and
incompressible problem was developed. This technique is based on the separate discretization of
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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the diffusive and convective terms of the governing equation by means of the MFVE and FVE
methods, respectively. The final solution of the saturation equations are obtained by means of a
two-step time-spitting procedure employing explicit time stepping for the FVE method and
implicit time stepping for the MFVE method. Different time-stepping strategies can be used for
two steps. In the diffusive step of the most computational demanding, the time-step size is not
limited by stability constraints and its choice is dictated by accuracy consideration. On the other
hand, the convective step has to satisfy a CFL stability condition. To minimize the differences in
the computational costs, several convective time steps can be performed in a single diffusive
time step. Modeling naturally fractured reservoir can be effectively simulated by full tensors.
Fluid motion is primarily determined by the orientation, intensity, and length of the fractures.
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