B019 STATISTICAL MOMENT EQUATIONS FOR FLOW IN
COMPOSITE HETEROGENEOUS POROUS MEDIA
L. Li1 and H. A. TCHELEPI2
(1) ChevronTexaco Energy Technology Co., (2) Standard U.
Abstract
We describe a conditional statistical moment equations (CSME) method for quantifying the uncertainty in flow related quantities (pressure and velocity) due to incomplete knowledge (uncertainty)
about the permeability field in a composite system. We identify multiple permeability regions in
the flow domain. For each region, the mean, variance, and covariance of permeability are specified.
We focus on the flow problem; specifically, we study the first two statistical moments of pressure
for incompressible systems. The applicability of the method for composite media is demonstrated
using several examples. We present and analyze the statistical moments of the pressure for several
composite systems with various combinations of the mean, variance and correlation lengths in the
different regions. We focus on the predication uncertainty for systems where the overall variance
of the log-permeability is (much) larger than unity.
Introduction
The permeability of natural porous formations, such as aquifers and oil reservoirs, often displays
highly complex spatial correlation structures and large levels of variability[1]. However, we typically have measurements at a few locations only. Incomplete knowledge about formation properties leads to uncertainty in the computed predictions of flow and transport in natural heterogeneous
systems. To deal with having limited information about highly heterogeneous systems, a probabilistic framework is usually employed. In this framework, we treat reservoir properties, e.g.
porosity and permeability, as random space functions. Usually, a Monte Carlo Simulation (MCS)
approach is used to quantify the uncertainty in predictions of flow and transport in heterogeneous
reservoirs. In the MCS approach, first a large ensemble of realizations of the permeability field,
each honoring the available measurements and statistical information, is generated using various
geostatistical techniques[2]. Then, a flow simulator is run, one simulation per realization, for the
displacement process under study. Statistical moments for the quantities of interest can then be
computed and analyzed.
Here, we use a direct method to quantify the prediction uncertainty, where equations governing the
statistical moments of flow-related quantities (pressure and velocity) are derived from a stochastic
statement of flow in the heterogeneous porous medium. Low-order approximations of the governing equations are then solved for the statical moments of flow-related quantities, e.g., pressure, velocity, production rate. This is referred to as the Statistical Moment Equations(SME) approach[3].
In all the computations presented here, we assume that the only source of uncertainty is our limited
knowledge of the permeability field.
It is now well documented that SME methods are applicable for a wide range of variability levels
2
and correlation length scales[4, 5]. However, as the variance of log-permeability, σln
k , increases
significantly beyond two or three, the accuracy of the computed statistical moments, especially the
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
second moments, decreases[5]. In this paper, we extend the Conditional SME method (CSME)
[6, 7] to composite heterogeneous systems. A composite, or multi-region, porous medium is one
where multiple permeability regions are delineated. In addition to available measurements, the
permeability field within each region is described by its mean, variance and covariance. We assume that the boundary separating the regions is known with certainty. Such a description is
possible when available data allow for accurate delineation of large-scale features, but where only
a weaker form of information (statistical moments) is available about property variation within
these large-scale features. Examples include the decomposition of the reservoir region into fault
blocks or the identification of stratigraphic layers within a thick formation. In practice, it is often
the case that large-scale reservoir characterization models are actually constructed as a composite of complex, yet individually identifiable, geologic units. While the global (log)-permeability
variance of a composite system may be quite high, the ability to make reliable predictions of the
statistical moments of the flow field using the SME method depends only on the local variability level within each region. As a result, a computationally efficient multi-region SME method
extends the applicability to more complex domains that are amenable to a composite description.
We focus here on the first two statistical moments of pressure in the quarter-five-spot geometry.
Recently, Winter and Tartakovsky[8] presented a theoretical treatment using an integral SME approach for flow in composite systems. In their analysis, Winter and Tratakovsky[8] also discuss
the case where the specific location of the boundary is subject to uncertainty. Numerical solutions
for two-dimensional (2D) composite systems have been presented by Winter et al.[9].
Governing Equations
The equation governing the pressure is given
∂Y (x)
∂ 2 p(x) ∂Y (x) ∂p(x)
+
+ ρg
= −e−Y (x) q(x),
2
∂xi
∂xi ∂xi
∂x3
(1)
where p is pressure and Y (x) = ln k(x) is the log-transform of the permeability, k. The source
term, density and acceleration due to gravity are denoted by q, ρ and g, respectively. Because we
assume that Y (x) is a stochastic process, the pressure p(x) is a stochastic process, and thus Eq. (1)
is a stochastic partial differential equation. We assume that ρ and q(x) are deterministic variables.
We use a perturbation expansion of the unknowns, in terms of log-permeability variance, σY2 , and
we construct low-order (first- and second-order ) partial differential equations (PDEs) for the mean
and covariance of pressure and for the cross covariance of log-permeability and pressure[6, 5].
Another approach is based on deriving integral equations with random Green’s functions[9].
To deal with composite systems, one can divide the problem into several sub-problems and prescribe continuity conditions at the boundary between any two regions, namely, that both the pressure and flux are continuous at the boundary[8]. Detailed treatment of composite heterogeneous
domains based on solving a collection of boundary value problems, one per region, using integral
equations with random Green’s functions is given by Winter and Tartakovsky[8] and Winter et
al.[9]
We employ a conservative, three-dimensional (3D), finite-volume discretization scheme for our
system of PDEs. The scheme ensures continuity of both pressure and flux at the interface of a
computational cell, or gridblock. As a result, dealing with multiple permeability regions using
our numerical finite-volume approach turns from an elaborate scheme of handling sub-problems
with complex boundary conditions to a simple matter. Since the continuity conditions are satisfied
everywhere by construction, we simply need a screening function that returns a zero for the permeability covariance between two points that lie in different regions. The details of the numerical
Table 1: Case Description
Case
Description
Statistics
Global Statistics
1
One block
Y = 0.000
σY2 = 1.000
Y = −0.008
σY2 = 0.964
2
One block
Y = 0.000
σY2 = 2.250
Y = −0.011
σY2 = 2.169
3
Two
horizontal
blocks separated at
x = 0.5
Y1 = 0.500, Y2 = −0.500
σY21 = 0.500, σY22 = 0.500
Y = 0.019
σY2 = 0.774
7
Two
horizontal
blocks separated at
x = 0.5
Y1 = 2.00,Y2 = −2.00
σY21 = 1.000, σY22 = 1.000
λY1 = 0.2, λY2 = 0.2
Y = 0.027
σY2 ≈ 5.0
8
Two
horizontal
blocks separated at
x = 0.5
Y1 = 2.000, Y2 = −2.000
σY21 = 1.000, σY22 = 1.00
λY1 = 0.4 , λY2 = 0.1
Y = 0.025
σY2 ≈ 5.0
treatment will be published later; our emphasis here is on reporting results for composite systems
using our low-order SME method.
Results
We present results using our multi-region CSME (conditional statistical moment equation) simulator. We study the first two statistical moments of the pressure field for incompressible flow in a
quarter-five-spot geometry. The boundary conditions are assumed known. Specifically, pressure is
specified at the injection and production wells. The reservoir is divided into regions. Within each
region, the permeability field is defined in terms of the mean, variance and covariance. For all the
computations presented here we used the exponential covariance function. Table 1 summarizes the
cases we present here. In the Table < Y >, σY2 denote the mean and variance of log-permeability.
The computational grid for the SME computations was 21 by 21 by 11 for the 3D results. For
2D computations, a 41 by 41 grid is used. We compare our CSME results with high-resolution
Monte Carlo Simulation (MCS). The MCS computations were performed on a 41 by 41 by 21
grid. In the 2D MCS computations, a grid of 81 by 81 was used. In order to ensure statistical
convergence of the MCS results, we used 5000 realizations of the permeability field for each of
the cases reported here. The permeability realizations were generated using SGSIM from Deutsch
and Journel’s GSLIB[10].
To present our results in a concise manner, we plot the mean and variance of pressure along the
diagonal line connecting the two wells. Figure 1 is a comparison between SME and MCS for
2
Case 1, in which there is one permeability region with mild variability, namely σlnk
= 1. The
figure shows the mean (left) and variance (right) of the pressure in the top layer along the diagonal
line connecting the two wells. The profile is nearly the same for all the layers. Examination of
the results in Figure 1 indicates that the agreement between the SME method and MCS is quite
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
0
0.6
MC
SME
MC
SME
0.5
Pressure Variance
Pressure Mean
-1
-2
-3
-4
0.4
0.3
0.2
0.1
-5
0
0
1
2
3
4
5
6
Distance from Well 1 to Well 2
7
8
0
1
2
3
4
5
6
7
8
Distance from Well 1 to Well 2
Figure 1: Comparison between SME and MCS computations of the pressure mean (left) and variance (right) in the top layer along the diagonal line connecting the two wells for Case 1.
good for both the mean and variance of pressure. The MCS profile of the pressure variance is not
exactly symmetric about the center point. This indicates that perhaps an even larger ensemble of
realizations may be necessary for MCS. Figure 1 indicates that steep gradients in the mean pressure
take place around the wells. Away from the wells, the gradient of the mean pressure is smaller and
nearly uniform. This is expected given the converging/diverging character of the flow field in the
quarter-five-spot configuration. Figure 1 indicates that the pressure variance, which is an indication
of prediction uncertainty, is maximum near the center of the domain. Note that in Figure 1, the
pressure variance at the wells is zero. This is due to our specification of deterministic Dirichlet
boundary conditions. Analysis of the first two statistical moments of pressure in the quarter-fivespot geometry for problems similar to Case 1, i.e. a single permeability region with relatively mild
permeability variability, is given elsewhere [6, 7]. Figure 2 is similar to Figure 1, but for Case 2,
2
in which the permeability variability is relatively high, (σln
k = 2.25). Notice that the agreement
between SME and MCS for the mean pressure is excellent. The results for the pressure variance
reflect a different behavior, however. As the permeability heterogeneity increases, the differences
between the SME predictions of the pressure variance and those obtained from MCS, which we
take as reference, increase. A detailed study of the statistical moments of pressure and velocity for
high-variance cases is treated by Li et al.[5].
Now, we study the behavior of composite systems. Figure 3 shows the profiles of the mean and
variance of pressure along the diagonal line in the top layer for Case 3. Results from both MCS
and SME are shown in the figure. Note that in Case 3, we have two different regions separated
by a vertical plane, where the mean of log-permeability in region 1, < Y1 >, is 1/2, while that
in region 2, < Y2 > is -1/2. In Case 3, the variance is the same in the two regions, namely, σY2
= 1/2. The overall agreement between our multi-region SME simulations and MCS for Case 3
is excellent as shown clearly in Figure 3. The mean-pressure gradient in Figure 3 is smaller in
region 1 compared to region 2, which is due to the larger mean of log-permeability in region 1.
The obtained predictions of the pressure variance in Figure 3 are quite interesting. The variance is
zero at the well locations, but now the profile of the pressure variance in the SME computations is
not symmetric about the center point. The pressure variance is higher in the region with the lower
mean of log-permeability. Consider two points along the diagonal line connecting the two wells
that are equally distant from center of the domain. The pressure variance at the point in region 1,
0
1.4
MC
SME
MC
SME
1.2
-1
Pressure Variance
Pressure Mean
1
-2
-3
0.8
0.6
0.4
-4
0.2
-5
0
0
1
2
3
4
5
6
7
Distance from Well 1 to Well 2
8
0
1
2
3
4
5
6
7
Distance from Well 1 to Well 2
8
Figure 2: Comparison between SME and MCS computations of the pressure mean (left) and variance (right) in the top layer along the diagonal line connecting the two wells for Case 2.
0
0.3
MC
SME
MC
SME
0.25
Pressure Variance
Pressure Mean
-1
-2
-3
-4
0.2
0.15
0.1
0.05
-5
0
0
1
2
3
4
5
6
Distance from Well 1 to Well 2
7
8
0
1
2
3
4
5
6
7
8
Distance from Well 1 to Well 2
Figure 3: Comparison between SME and MCS computations of the pressure mean (left) and variance (right) in the top layer along the diagonal line connecting the two wells for Case 3.
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
0
-1
-2
-3
MC
SME
-4
-5
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
3
2.5
2
1.5
1
0.5
MC
SME
0
0
0.2
0.4
Figure 4: Comparison between SME and MCS computations of the pressure mean (upper) and
variance (lower) obtained for Case 7 modelled as a single region with global statistics of
2
< Y > = 0 and σln
K =5. The profile is along the diagonal line connecting the two wells.
where < Y1 > = 1/2, is smaller than that in region 2, where < Y2 > = -1/2. Large values of the
pressure variance appear in region 2, the one with the lower mean of log-permeability, close to the
boundary between the two regions.
Case 7 in Table 1 represents a two-region case where the global variance of log-permeability is
quite large, σY2 = 5. The large global variance is due to a large contrast in the mean permeability
between the two regions. If we use SME with a single characteristic structure of permeability to
predict the pressure moments, we may significantly over predict the pressure variance compared
with MCS as shown in Figure 4. However, if we model Case 7 using our multi-region SME
approach, the results are quite different as shown in Figure 5. The excellent agreement in the
predicted statistical moments of pressure between the multi-region SME computations and MCS
suggests that the extended SME method is well suited for quantifying the uncertainty associated
with predictions of the flow field in composite heterogeneous domains.
The results of Figure 5 indicate that the profiles of both the mean and variance of pressure are
quite small in the left region, where < Y1 >= 2. In the second region, where < Y2 >= −2, large
mean-pressure gradients and relatively high variances are observed. The location of the boundary
between the two regions can be inferred from examination of the two plots in the figure. Notice
that the maximum variance in the pressure profile is obtained deep into the second region. That is,
the uncertainty in the pressure prediction is greatest in the second region some distance away from
the production well.
Figure 6 shows the two statistical moment of pressure for Case 8, where the two regions have
different means and different correlation lengths, but the variance is the same in both. The behavior
is qualitatively quite similar to that observed for Case 7. The variance level in the second region is
slightly higher in Case 8 compared to Case 7, and this is due to the larger correlation length in the
second region.
0
-1
-2
-3
MC
SMEY
-4
-5
0
0.2
0.4
0.6
0.8
1
0.8
1
0.35
0.3
MC
SMEY
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
Figure 5: Comparison between SME and MCS computations of the pressure mean (upper) and
variance (lower) obtained for Case 7. The profile is along the diagonal line connecting
the two wells.
0
-1
-2
-3
MC
SMEY
-4
-5
0
0.2
0.4
0.6
0.8
1
0.8
1
0.45
0.4
MC
SMEY
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
Figure 6: Comparison between SME and MCS computations of the pressure mean (upper) and
variance (lower) obtained for Case 8. The profile is along the diagonal line connecting
the two wells.
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
We found that the high level of accuracy in the computed moments shown in Figures 5 and 6 is
typical of composite systems where the highest regional variance, σY2 is less than about two.
Summary
We extend our SME approach to composite systems. This multi-region SME approach produces
accurate predictions of the statistical moments of the flow field in heterogeneous systems where
multiple permeability regions, each with relatively mild variability, say (σY2 ≤ 2, can be delineated.
In such systems, the global variance may be quite large due to the large contrasts in the regional
means and variances.
Acknowledgements
We thank Hui Cao, who implemented the first 3D statistical moment equation (SME) reservoir
simulator, and we thank Victor Ginting, who made the first attempts at extending that work to
composite heterogeneous media.
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