#### Modeling of Tight Fractured Reservoirs
Dual porosity and dual permeability (DP) fractured
reservoir models were developed decades ago by upscaling orthogonal
discrete fracture systems. They were developed by making the least
possible number of simplifying assumptions to obtain a discrete numerical model
that can be practically and robustly solved. The purposes of DP
models are
1. To improve efficiency by combining (upscaling) a
group of discrete matrix and surrounding fracture blocks into a
reservoir block containing both a single matrix block and a single
fracture block. The upscaling results in an expression for the DP
exchange transmissibility **Tex** between matrix and fracture blocks, as
given in the Sensor Manual and originally by Khazemi. The rate of
transfer between each matrix-fracture block pair is independent of fracture
permeability, and depends on fracture spacings and matrix permeability
and the phase potential gradients representing the difference in capillary and
gravity forces.
**T****ex
**= 4
(0.0011274) ΔxΔyΔznetm (1-фf) (kx/Lx2+ky/Ly2+kz/Lz2) rb-cp/d-psi
The flow rate of phase j from matrix to fracture is
qj
= TexλjΔpj
rb/d
where j = w, o, g, for water, oil, and gas, respectively, and
λj
= phase j upstream
mobility, krj/μj
, 1/cp
krj
= phase j upstream relative permeability
Δpj
= pjm
- pjf,
psi
pjm
= phase j pressure in
matrix
Matrix-fracture transfer can also occur due to diffusion, with
similar rate expressions and transfer coefficients (see Sensor Manual).
2. To eliminate the need to conform the grid to any
details such as fractures. A sugarcube approximation specified by
gridblock fracture spacing arrays Lx, Ly, Lz allows us to obtain a
robust and efficient model solution and to robustly treat the fracture
system description as uncertain distributions, the shapes of which are
determined by the model, but are strongly guided by experience and
expectation, such as extents of the fracture systems that may be
detected by microseismic measurements. This allows the use of a single
grid in our attempts to optimize fracture spacings, orientations,
distributions, treatments, operating conditions, and production.
If a reservoir has no significant natural fractures, the
DP model may not gain any efficiency over the discrete model because the
hydraulically fractured blocks are sparse, but it still provides a robust
framework for the addition of an empirical geomechanical model representing
the effects of fracturing on model parameters. This allows the
fracturing and production processes to be represented in a single model, and
eliminates the impossibility of accurate specification of initial conditions in a discrete model
after the well has been fractured. Much more extensive use of available fracture
treatment
and production data (especially BHP data during fracturing and production)
greatly increases our abilities and decreases uncertainty in
characterization, history matching, and optimization.
Dual porosity / dual permeability (DP) models have
enabled efficient field-scale modeling of naturally fractured reservoirs.
They have historically been validated by comparing DP and discrete model
results for immersion tests for one or few-cell systems in which the
fractures are flooded with oil or gas and the matrix originally containing
hydrocarbon is allowed to equilibrate towards its equilibrium saturations
Swe, Sge, from its initial saturations Swi, Sgi. In conventional
fractured or highly heterogeneous systems, the transient time is usually on
the order of days to months and is not nearly as important as are the
equilibrium matrix saturation distributions Swe and Sge, which (for a
water or gas flood) are mainly a function of capillary pressure.
Capillary-gravity equilibrium is achieved between adjacent high-perm
(fractures) and low-perm (matrix) regions in the well-swept zones, which are
determined by boundary conditions, geology, and sweep
efficiency. This capillary crossflow is described in detail on our
Capillary Pressure
page
with simple conventional examples.
An example comparison of single porosity (SP) and dual
porosity (DP) results for an immersion test using Sensor for conventional
systems is given at Dual Porosity Immersion Test Examples.. We
continue here mainly to focus on issues of applicability to very tight and
hydraulically fractured systems.
A common assumption made in modeling naturally fractured
reservoirs is that of infinite fracture conductivity, or equivalently
infinite fracture permeability. Pressure drop in the fractures is
assumed negligible. Corresponding "infinite" permeability values are
determined as those for which increased values have no effect on results
(use of fracture permeabilities that are higher than necessary can cause
numerical difficulties, due to difficulty in determining flow directions
between small blocks). If the assumption is valid in a naturally
fractured reservoir, it is certainly valid for hydraulic fractures, at least
close to the well.
The same principles apply to tight unconventional
fractured systems, with matrix permeabilities on the order of nanodarcies
and with extremely high capillary pressures due to the much smaller flow
paths, or "pore sizes". In naturally fractured systems, where
hydraulic fractures are created, the effects on gridblock Tex values are
purely due to changes in the orthogonal model fracture spacings (the
sugarcube DP model gridblocks can be locally orthogonal approximations of
input corner-point reservoir gridblocks). The differences in behavior
of conventional and tight systems are mainly due to the fact that the bulk of
the matrix in the reservoir will be in transient (changing in saturations
towards Swe, Sge but never reaching them) at all times during production.
This places greater demands on the accuracy of the matrix-fracture transfer
function Tex in the two-point upscaled DP representation of the discrete
matrix-fracture system.
There are many questions and complications regarding the
modeling of extremely tight systems, such as modified flow behavior (Klinkenberg
effect) and pvt ("bubble point suppression" due to high capillary pressure)
in very small pores. But we will start with the simplest possible
assumptions and reservoir and fluid systems in our investigation of the
capabilities of reservoir models to optimize well completions, spacings,
treatments, operations, and production in very tight systems. Complexity will be added until we
determine the limits of the capabilities of our tools and methods in
determining reliable answers to our questions.
##### Gridding Requirements (upscaling)
The first step is to determine the gridding requirements
for given simple fracture geometries and distributions in order to obtain an
accurate numerical description of flow behavior using either discrete or DP
reservoir models. We will find that these are highly dependent on
rock/fluid properties, phase behavior, and initial and boundary conditions, as are the answers to all of our other questions
of reservoir behavior. Some
have reported that for real fluids, very fine gridding is required in the
matrix adjacent to the fractures in their discrete fracture models. We
will first determine the fine uniform grid solution and then the coarsest
sufficient uniform grid for a number of simple problems, and then attempt to find
non-uniform grids with sufficient accuracy and better efficiency (fewer
number of cells) for some problems of interest.
We start with horizontal 1D flow of a single phase
incompressible fluid through 1000 ft of homogeneous slightly compressible matrix into a single
fracture on the left end (i=1), with a matrix-fracture area of 10,000
square ft (dy = dz = 100 ft), in a discrete fracture model. This is a symmetrical
pattern element model of a 1/2 of a single planar fracture within
semi-infinite matrix. This case can be solved analytically. Data are:
single phase incompressible unit mobility fluid (water,
Bwi=1, denw=62.4 lb/ft3, visw=1 cp)
arbitrary hydrocarbon pvt
formation compressibility Cf=3.e-6 1/psia, pref=5000 psia
matrix perm Km=30 nd, porosity=.05
i=1 is the fracture block with width delx=.0002 ft,
fracture perm = 1D, poros=1 (perm and porosity are irrelevant)
Pinit=5000 psia, well in block 1 (fracture), bhp=14.7
psia, produce for 10
years
df2w.dat, df11w.dat, df21w.dat, df41w.dat, df51w.dat, df101w.dat, and df1001w.dat
are 1D discrete fracture models (data files) dfn.dat having n total blocks,
the fracture block at i=1 and the n-1 matrix blocks with delx about
equal to 1000/(n-1).
Figure 1 indicates that for Km=30 nd, use of uniform
matrix blocks as large as 10 ft gives accurate results. Cumulative
water production at 10 years (from df101w.out or df1001w.out) is 101.8 stb.
Figure 1
Figure 2 shows results for Cf changed from 3.e-6 to 5.e-6
(data files df2wa.dat, df11wa.dat, df21wa.dat, df41wa.dat, df51wa.dat,
df101wa.dat, df1001wa.dat). The maximum block size for good accuracy
remains about 10 ft over this range of (small) total compressibility.
Cumulative production at t=10 years increases by about 29% (to 131.3 stb)
as a result of the increased formation compressibility.
Figure 2
Figure 3 shows matrix depletion results for slightly
compressible water and formation (Cw=5.e-6 1/psia, Cf=5.e-6 1/psia).
Data files are df2wb.dat df11wb.dat df21wb.dat df41wb.dat, df51wb.dat,
df101wb.dat, and df1001wb.dat. Again, uniform 10 ft matrix blocks give
sufficiently accurate results.
Figure 3
Figure 4 shows 1d matrix depletion results for a dry gas
rather than water (Swc=.4 in matrix). Data files are df51wc.dat,
df101wc.dat, df501wc.dat, and df1001wc.dat, and df10001wc.dat. Runs
using matrix block sizes of 0.01 ft and 0.1 ft. give virtually identical
results. Use of uniform delx=1 ft matrix blocks gives about 2% error
in cumulative production at end of run. High gas compressibility
results in much less propagation of pressure decline into the matrix and in
a requirement for much finer gridding than in the all-water case.
Figure 4
Figure 5 shows 1D depletion results for an initially
undersaturated oil, using spe1 blackoil pvt data in uniform grids. Data files are
df51d.dat, df101d.dat, df501d.dat, df1001d.dat, df10001d.dat, df100001d.dat.
Again, use of uniform 1 ft grid in the matrix gives about 2% error in
cumulative production, and results using 0.1 and 0.01 ft gridblocks are
virtually identical.
Figure 5
Figure 6 shows spe1 depletion results for geometrically
spaced grids in the matrix, using a geometric factor g of 1.5 (delx(i+1) =
delx(i) * g). The cases using 20 or more matrix blocks give excellent
matches to the fine-grid uniform case (delxm=.01). Variable grid
spacing can maximize performance while preserving accuracy. It is
significantly more accurate than the uniform cell case using 50 times more
gridblocks (df1000d.dat), DELXM=1). The geometric matrix block
spacings in the case with 20 matrix blocks are:
0.1504096E+00 0.2256143E+00
0.3384215E+00 0.5076323E+00 0.7614484E+00 0.1142173E+01 0.1713259E+01
0.2569888E+01 0.3854833E+01 0.5782249E+01 0.8673373E+01 0.1301006E+02
0.1951509E+02 0.2927263E+02 0.4390895E+02 0.6586343E+02 0.9879514E+02
0.1481927E+03 0.2222891E+03 0.3334336E+03
Figure 6
The gridding limitations shown so
far apply to matrix-fracture flow for discrete fracture single porosity
systems. In dual porosity and dual permeability models, there is no
direct flow between a fracture and matrix blocks in adjacent cells
(matrix-fracture transfer is instead represented in each cell by the m-f
exchange transmissibility Tex). Dispersion is controlled the same as any other
case, by using the largest physical gridblocks for which there is no change
in results (over cases using smaller blocks). There is no lower limit
on physical gridblock size in dual systems. A single fracture can be
represented by specifying the minimum fracture spacings which are equal to
gridbblock dimensions (for example Lx=dx, indicating 2 fractures, one on
each side of the matrix block in the yz plane) and then multiplying the
computed Tex value by 0.5. Cells containing no fracture are treated as
single porosity (having fracture porosity or pv equal to zero). |