What is Drainage Radius?*
Also see:

Reserves Definitions

Uncertainty
Quantification  P10, P50, P90

The Failure
of Decline Curve Analysis for Real 3D Multiphase Systems

Sensor's HWELEMENT and
ECONOMICS options for the simplest and fastest method to determine
theoretically optimal fracture and well spacings and landings (and
optionally optimal operating strategy) for
simultaneous depletion or flooding with idealized sets of hydraulically
fractured parallel and stacked horizontal wells in reservoirs with or
without natural fractures, both asking and answering the question "what
is drainage radius" in meaningful terms, and much more, for idealized
but heterogeneous and multiphase 3D systems. HWELEMENT is a hybrid
discrete hydraulic fracture / single porosity or dual porosity or dual
permeability naturally fractured reservoir pattern element model.
Model size and the number of optimization variables are minimized to
achieve robust and efficient automatic (or manual) predictive
optimization. See SensorFrac_HWE.dat (confidential, requires a
license agreement). Generalized predictive optimizations performed
by iterative adjustment of input variables to maximize NPV can be manual
or can be automated with SensorCast or SensorPCast workflows (requires
thirdparty optimization software ). Following optimization of the
set of desired variables to maximize deterministic or probabilistic NPV,
sensitivities to any other input data are easily determined.
For assumed 1d radial flow, it may
be referred to as the outer radius to the noflow boundary of a cylindrical
system. There is no such thing as a
meaningful value of "drainage radius (or area or volume)" for real
wells, except as may be determined by experience or optimization studies.
No meaningful value can be determined by well testing or by pressure
response in reservoir modeling. It is mostly a result of
misconceptions regarding fluid flow and recovery.
In real reservoirs and models of them, flow and
recovery, and "effective well permeabilitythickness" kh as measured in a
well test along with any other effective or overall well or reservoir
properties, are determined by porosity and permeability distributions, rock
type distributions and properties (relative permeability and Pc), fracture
distributions and orientations, initial and boundary conditions (including
well locations/completions and operating constraints vs. time), and multiphase flow
considerations and pvt (fluid phase behavoir and initial capillary/gravity
and local thermodynamic equilibrium). Also see
Fracture Conductivity
and Capillary Pressure. Reservoir
simulators are designed to determine or predict how production and injection will behave
with time, along with pressure (and temperature in thermal models), composition** and saturations in the
reservoir as a function of space and time,
given those inputs, assumptions, and considerations. While effective kh may be a useful comparative measure of overall well productivity, there
are no effective well or reservoir properties that can be derived from well
tests that provide any useful descriptive information in modeling
heterogeneous or fractured systems, or any useful information regarding
optimal well placement. Well tests are generally useful in
modeling for determining initial conditions and for obtaining fluid samples,
for determining individual well rates when production is normally combined
with that of other wells in surface facilities, for testing communication between wells, and in providing valuable data for
history matching that can be used to reduce uncertainty (see
Uncertainty Quantification  P10, P50, P90).
What some mean by "drainage radius or area or
volume" or "radius
or area of investigation" is the extent of the reservoir that must be
included in an areally "semiinfinite" well model such that there are no
effects on results (gridblock pressure (temperature), composition, and
saturations) at the outer noflow boundaries. In general that can only
be determined by examining pressure response at the end of a simulation run.
If a significant response is seen, the reservoir extent is
increased.
There are only 2 other ways to calculate a drainage radius
or area that has any significance whatsoever  1) operational trial and
error optimization of spacings and completions within a field with little
variation in geology, fluids, and initial and operating conditions, or 2) computed from the results of a simulation study
of optimal well spacings/completions and boundary conditions. Drainage area may be defined as some estimate
of the optimal number of wells divided by lease or field surface area.
Some define it simply as a given land surface
area divided by the number of wells drilled in it, regardless of how that number may have been determined.
In well tests or depletion of isolated wells, the
closest meaningful quantity to drainage radius might be affected volume,
where affected volume would be determined by some cutoff value(s) of
pressure change, or of fractional recovery. But that affected volume
would be highly dependent on the cutoffs selected, so its value even in
homogeneous systems is highly questionable. It
is the rates of oil and gas recovery and injection/production as predicted
and output by reservoir simulators as a function of the input data that are
important and meaningful.
Consider the following questions when asking the question, "what is drainage
radius?".

If it is defined as the radius
to which pressure is affected, by what degree is pressure affected at
the outer boundary? Is it equal to the real double precision of
our computers? Is it the accuracy of our downhole pressure
measurements? Is it 1.E10 psia? Or 1 psia? Even in
homogeneous systems, any such radius is defined by and highly dependent
on the cutoff used.

From the gridblock pressure
changes or oil and gas gridblock
recoveries computed and reported by reservoir simulation models, one
could compute an affected or "drainage" volume, based on some
chosen cutoff
values (gridblock oil and gas recoveries =
1  fraction of original oil and gas in place, respectively). For depletion of a single well in an areally
semiinfinite homogeneous reservoir, pressure
change and recovery*** will
decrease exponentially to 0 with increasing distance from the well. What would be
the value of any such measurement, that would greatly depend on the
cutoffs selected?

In a heterogeneous or fractured
system with arbitrary boundaries and rock and fluid property
distributions, how is drainage radius defined? How exactly is it
entered or calculated or represented in our model inputs or outputs?
The answers to those questions are
the reasons why there is no such thing as a drainage radius in real systems,
and why any effective value calculated from well testing or from pressure
changes in modeling results is generally a meaningless quantity
in reservoir description and modeling.
Whenever permeability or any reservoir or well
descriptive property is given as a constant, one knows that the theory does
not apply to heterogeneous or fractured systems. Except as noted,
drainage radius does not even apply to homogeneous systems. This is
proven by the simple reproducible examples and results given below.
Fekete says (or said):
"Radius of investigation represents how far into
the reservoir the transient effects have traveled. A pressure transient is
created when a disturbance such as a change in rate occurs at a well. As
time progresses, the pressure transient advances further and further into
the reservoir. This concept is not theoretically rigorous, but ..."
"The radius of investigation is calculated using the
following equation:
When analyzing a pressure transient
test, radius of investigation can be used to estimate drainage area as
follows:
Halliburton says (or said)
"Drainage Radius: The radius of the approximate
circular shape around a single wellbore from which the hydrocarbon flows
into the wellbore. The drainage radius of a single well will help determine
how many wells will be needed (and where they should go) to most efficiently
drain the reservoir."
Schlumberger says
"transient drainage radius
1. n. [Well Testing]
The calculated maximum radius in a
formation in which pressure has been affected during the flow period of a
transient well test. While not absolutely accurate, the value has meaning in
relation to the total volume of reservoir that is represented by calculated
reservoir parameters, such as kh, the permeability thickness. This may also
be termed radius of investigation.
Synonyms: radius of investigation
See: permeability thickness"
"drainage area
1. n. [Well Testing]
The reservoir area or volume drained by the well. The terms drainage
area, reservoir area and drainage volume are often incorrectly used
interchangeably. When several wells drain the same reservoir, each
drains its own drainage area, a subset of the reservoir area.
See: drainage volume"
"drainage volume
1. n. [Well Testing]
The portion of the volume of a reservoir drained by a well. In a
reservoir drained by multiple wells, the volume ultimately drained by
any given well is proportional to that well's production rate: Vi = Vt x
qi/qt, where Vi is the drainage volume of Well i, Vt is the entire
drainage volume of the reservoir, qi is the production rate from Well i,
and qt is the total production rate from the reservoir.
See: drainage area, well production rate"
SPE 120515 (F. Kuchuk, Schlumberger) says
"Although it is often used in pressure transient
testing, radius of investigation still is an ambiguous concept, and there is
no standard definition in the petroleum literature."
SPE 180149 (M. King, Z. Wang, and A. DattaGupta)
says
"Understanding how pressure
fronts propagate (diffuse) in a reservoir formation is fundamental to
welltest analysis and reservoir drainage volume estimation"
"The method is especially well suited to the
interpretation of the drainage volume, which is of great help in well
spacing calculations and in the context of unconventional reservoirs,
multistage fracture spacing optimization."
It is the purpose of reservoir simulators to represent
how pressure and composition and saturations and production and injection
change as a function of time and space, and how they do this is
welldocumented and understood (it is the reservoir description and boundary
conditions that are usually uncertain). We can easily construct a
heterogeneous fractured case with some specified distributions of fracture
orientations and spacings and permeability, say as a function of distance
from the well. It is absolutely impossible for any well test result to
give any valuable descriptive information of the fracture network. Any
drainage volume estimated from pressure response is useless in "fracture
spacing optimization". None of the single or effective (homogeneous)
well or reservoir properties claimed to be determined in well testing have
any value in reservoir characterization or modeling of heterogeneous or
fractured systems. The examples below show that to be true for
"drainage" radius, area, or volume, even in homogeneous reservoirs.
Propagation of pressure change in reservoirs is
generally due to
convection and compressibility (and all the inputs and considerations in our
simulation models)****. In the limit of 0 total system compressibility,
pressure change propagates instantaneously to an infinite extent. In
the limit of infinite system compressibility, pressure change does not
propagate at all. Total compressibility Ct at any point depends on phase compressibilities and saturations (that are
functions of pressure and composition) and rock compressibility at that point (or at the center of a
discrete numerical
model gridblock), and is given by
Ct = (1phi) Crock + phi (Sw Cw + Sg Cg + So
Co) Crock is defined as the
solid, nonporous rock compressibility, at zero porosity. The
above equation reflects that total compressibility must be equal to Crock
for porosity phi=0 and to fluid average compressibility for phi=1. The phase compressibilities are
functions of local thermodynamic equilibrium and the gridblock variables
P,(T),Xi,Yi,do,dg,dw. Compressibilities are represented in reservoir
simulators by the pressure derivatives of phase densities do dg dw, and formation
porevolume "compressibility" Cf, where Cf = Crock (1phi) / phi, and is
used as Cf= (1/phi) dphi/dp. Some authors refer to Cf as a “porositydefined
compressibility". It is not a true compressibility, as physics relates
the elastic pressurevolume behavior of a (single phase, fixed composition,
homogeneous) substance using compressibility
defined by c = 1/V dV/dp, which is the inverse of bulk modulus.
Accurate modeling of production and gridblock
variables and properties as a function of time requires a fullphysics
reservoir simulator. Streamline models are based on incompressible and
immiscible flow neglecting capillary pressure and gravity, and are not
generally applicable to multiphase flow. Published claims to the
contrary are unsubstantiated by results for any reproducible example
problems where those multiphase effects are important.
Examples
1D simulation of 10day drawdown test in a homogeneous
reservoir. Estimate drainage radius Rd, defined as the distance from
the well to the point at which pressure change becomes less than the cutoff
tolerance of 0.05 psia, at the end of the test.
Data are based on spe1, with the following changes:
Permeability = 1 md, , Pinit=4800.0 at center
Depth=8375.0 ft
1000 xdirection blocks, dx=10, dy=dz=100, producer in (1,1,1), Q=200 rb/d,.
Rd = distance from center of (1,1,1) to center of
first block for which printed pressure in .out file printed map at 10.0 days
is shown exactly equal to Pinit = 4800.0.
The producing well is ratelimited at 200 rb/d out to
10 days time in all these runs:
spe1_1dw.dat
spe1_1dw.out
all water, Map Table at 10.0 days shows
Rd=2790 ft
spe1_1dg.dat
spe1_1dg.out
all gas,
Rd=1020 ft
spe1_1dgw.dat
spe1_1dgw.out
gas with connate water, Swc=.12, Rd = 1090 ft
spe1_1dgow.dat
spe1_1dgow.out
oil reservoir, Swc=.12, Rd=1150 ft
spe1_1dgows.dat
spe1_1dgows.out
oil
reservoir, Swc=.25, Rd=1210 ft
If we use a cutoff for pressure change of 1.0 psia rather than 0.05 psia,
then the Rd values for the above 5 cases are seen to be equal to
2050, 640, 700, 860, and 890 ft,
respectively.
If we use a cutoff of 0.05 psia for affected pressure
and we change the well boundary conditions to impose a constant bottomhole
pressure of 4500 psia rather than the constant reservoir volumetric rate,
then we get Rd values of 2300, 1150, 1230, 1060, and 1110 ft
for the 5
cases, respectively. The values are less for the oil reservoir cases
here in the pressurelimited case because the oil remains undersaturated,
and oil mobility is less than water or gas, giving less production at
reservoir conditions. The constantrate runs are more reflective of
the effects of total compressibility on pressure change
propagation.
If we use a cutoff of 1.0 psia for pressure change and
a BHP=4500, we get Rd values equal to
1600, 800, 850, 730, and 770 ft,
respectively. If we use
radial instead of Cartesian 1d grids, we get similar behavior, but with
smaller Rd values, because of increasing gridblock pore volume with
increasing distance from center..
Conclusions:

Drainage radius, area, or
volume, as may be determined by pressure response in a well test or in reservoir modeling has no significance in heterogeneous or fractured
systems, or even in homogeneous systems. Any estimate of any
affected radius, area, or volume is a function of time and is highly
dependent on the cutoff used to define it. The measure can not
possibly provide a
robust estimate of optimal well spacing, completion, or operation.

Pressure response to any change
in boundary conditions as a function of time and space is a function
of all the inputs and considerations of our reservoir simulation models,
just like all its other outputs. It is highly dependent on initial
and boundary conditions and fluid distributions and compositions and
fluid and rock properties (particularly phase and rock compressibilities and phase saturations determined by local
thermodynamic equilibrium).

Since any "affected volume" can
only be computed from simulation results, then production and injection
vs. time are already known from those results, and it makes no sense to
even attempt to compute any affected volume. Any such computation
is not useful since it cannot provide any valid estimate of optimal well spacings or placement. Those determinations can be made only by
experience in regions with little or no variation in geology, fluids, and
initial and operating conditions, or through probabilistic analysis of
optimal well placements and completions and boundary conditions using a
fullphysics reservoir simulator. Probabilistic analysis is
required due to the large numbers of uncertain variables (generally
equal to many times the numbers of gridblocks and wells).

Given a pressure change
tolerance, all simulation models could easily compute and output the
corresponding affected pore volume as a function of time and space
(in the examples, the total affected pore volume Vd at 10 days is equal
to Rd * phi * dy * dz ft3). Their developers have chosen not to do
so because it is not a useful or desirable result or
feature. Any
claims of value or ability related to computation of an affected or
drainage area or volume, such as in SPE 120515 and SPE 180149 and in
various publications on streamline models are unsubstantiated and false.

Streamline models do not
generally apply to simulation or well testing or any optimizations in
hydrocarbon (oil/gas) reservoirs. Their assumptions (incompressible
and immiscible, no capillary or gravity effects) are almost never valid
in gas/oil(/water) systems. They are generally incapable of robustly
predicting the pressure response, or any response, to any event or
process in real hydrocarbon reservoirs. To make any predictions or
optimizations for real reservoirs and fluids, robust solution of
multiphase flow is required and that requires a fullphysics
reservoir simulator along with consideration of uncertainties in
reservoir descriptive and/or operational variables.
We could move on to heterogeneous and multidimensional
systems and look at the effects of rock compressibility and fluid
distributions (aquifers have a large effect) to further investigate the
significance of any drainage radius, area, or volume that may be estimated
from pressure changes observed in well testing or modeling, but that is
obviously not
necessary or productive.
* See LinkedIn discussion of
Drainage Area Calculation
** Black oil is a 2component compositional fluid
representation
*** In depletion, gridblock pressure change and
recovery are directly related and their rates of change at any given time
are proportional
**** Where convection exists, flow due to diffusion is
so small that it is negligible. But where there may be no potential
for convection, pressure can propagate by fluid flow due to molecular
diffusion, such as in matrixfracture transfer in models of fractured
reservoirs. Diffusion is not generally included in interblock flow in
reservoir simulation models because it has no discernable effect on results.
