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 or ROI can be manual or can be automated with SensorCast or SensorPCast workflows (requires Tieto's ECIAM^® third-party workflow integration and optimization software).

For assumed 1d radial flow, it may be referred to as the outer radius to the no-flow 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 permeability-thickness" 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 "semi-infinite" well model such that there are no effects on results (gridblock pressure (temperature), composition, and saturations) at the outer no-flow 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.E-10 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 semi-infinite 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. Datta-Gupta) 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, multi-stage 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 well-documented 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 = (1-phi) Crock + phi (Sw Cw + Sg Cg + So Co)

Crock is defined as the solid, non-porous 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 pore-volume "compressibility" Cf, where Cf = Crock (1-phi) / phi, and is used as Cf= (1/phi) dphi/dp. Some authors refer to Cf as a “porosity-defined compressibility".  It is not a true compressibility, as physics relates the elastic pressure-volume 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 full-physics 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 10-day 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 x-direction 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 rate-limited 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 pressure-limited case because the oil remains undersaturated, and oil mobility is less than water or gas, giving less production at reservoir conditions.  The constant-rate 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 full-physics 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 full-physics 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 2-component 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 matrix-fracture 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.