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MODELLING OF TRANSIENT HYDROCARBON RESERVOIRS
The present invention relates to systems and methods for modelling of hydrocarbon 5 well and reservoir systems, and in particular to modelling of highly transient well and reservoir systems.
Tight oil/gas reservoirs exhibit low permeability, which means that the pressure distribution within the reservoir is constantly transient during production and the 10 productivity of an un-stimulated well is very low. For this reason multiple-fractured horizontal wells are required subject to large transient changes in effective productivity. Systems with up to 20,000 wells with 100 wells being drilled per month are common and the time to create and successfully run numerical simulations capturing all the complexities of gridding, anisotropy etc. is prohibitive.
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These types of reservoirs often have complex well geometries where gas adsorption, diffusion, directional stress dependent permeability variations require complex and lengthy numerical simulation. Such complex numerical simulation is often not possible within the timescales required, particularly with so large a number of wells, 20 each of which would need separate modelling.
Issues arise from application of existing analytical methods to highly transient reservoirs. Traditional analytical approaches based on standard transient analytical solutions to the diffusivity equation or material balance exhibit limitations or no 25 solution when applied to highly transient reservoirs.
It is desirable to be able to perform rapid calculations for short and medium term forecasting for tight (i.e. transient) reservoirs with complex well geometries.
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SUMMARY OF INVENTION
In a first aspect of the invention there is provided a computer apparatus for monitoring of a reservoir system, said a reservoir system comprising one or more 5 wells, said computer apparatus comprising:
means operable to create a simulation of a first well comprised within said reservoir system based upon the geometry of the first well;
means operable to create a dimensionless analytical analogue to the geometry of the first well based upon the simulation;
10 means operable to use said dimensionless analytical analogue to monitor said reservoir system during reservoir operations.
Said dimensionless analytical analogue may capture transient geometrical effects of the well and reservoir using dimensionless variables.
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Said means operable to create a dimensionless analytical analogue may be operable to run the simulation so as to determine a response curve for dimensionless pressure over dimensionless time according to the geometry of the well.
20 Said dimensionless time and dimensionless pressure may each be calculated from, respectively, time and flowing bottom hole pressure response of said simulation. Each of said dimensionless time and dimensionless pressure may correspond to a solution to the diffusivity equation for flowing bottom hole pressure of the well.
25 Said computer apparatus may comprise means operable to match said response curve to a production history of the first well in analytical space.
Said computer apparatus may be operable to match the response curve to a production history of a well other than said first well, comprised within the 30 reservoir system, said other well comprising a similar geometry to said first well. Two wells may be considered to have similar geometry if the shape of at least
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portions of the determined response curves for dimensionless pressure over dimensionless time for each well is similar.
Said matching of the response curve to a production history of a well may comprise 5 applying multipliers to one or more terms of the equations characterising the response curve so that portions of said response curve are matched with corresponding portions of a response curve characterising the production history, which have a similar shape. Said corresponding portions of said response curves may correspond to periods of near infinite radial or near infinite linear response. 10 Said terms may comprise one or more of permeability, porosity, cleat compressibility factor or terms related to pressure, volume and temperature.
Periods of time where near infinite radial linear response exist may be where: dpD „ 1 ain {tD) 2
15 where pd is dimensionless pressure and tD is dimensionless time.
Periods of time where near infinite linear flow response exist may be where:
^ln(/^L 1
01n (tD) 2
where pd is dimensionless pressure and tD is dimensionless time.
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Said apparatus may be operable to calculate flowing bottom hole pressure of the well from the matched response curve.
Said apparatus may be operable to calculate transient Inflow Performance from the 25 matched response curve.
Said reservoir operations may comprise extraction of hydrocarbons from said reservoir.
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Said means operable to create said dimensionless analytical analogue may comprise means for predicting reservoir behaviour during hydrocarbon extraction processes from said reservoir.
5 Said apparatus may comprise means operable to capture the geometry of the first well.
In a further aspect of the invention there is provided a method for monitoring of a reservoir system, said reservoir system comprising one or more wells, said method 10 comprising: creating a simulation of a first well comprised within said reservoir system based upon the geometry of the first well; creating a dimensionless analytical analogue to the geometry of the first well based upon the simulation; and using said dimensionless analytical analogue to monitor said reservoir system during reservoir operations.
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In a yet further aspect of the invention there is provided a computer program or computer program product comprising computer readable instructions for running on suitable computer apparatus, said computer readable instructions causing said computer apparatus to perform the method of the above further aspect. Said 20 computer program product may comprise any program storage device, such as a computer disk (magnetic, optical or other) or computer memory. Alternatively it may not necessarily be a physical product but instead may be a program product downloadable over a network.
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BRIEF DESCRIPTION OF THE DRAWINGS
Embodiments of the invention will now be described, by way of example only, by reference to the accompanying drawings, in which:
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Figure 1 is a flowchart illustrating steps according to an embodiment of the invention; and
Figure 2 illustrates matching of a curve of dimensionless pseudo-pressure against 10 dimensionless pseudo-time to historical data in log space.
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DETAILED DESCRIPTION OF THE EMBODIMENTS
Described herein is a methodology for generating a generic dimensionless transient reservoir response (poto) capturing the well, fracture and reservoir geometry and 5 matching this potD against historical data in analytical space. Also described is the concept of transferability of the generic dimensionless potD across wells with similar geometries. These concepts are applicable (but not limited) to oil, gas and retrograde condensate reservoirs characterized by low permeability below lmD and often below l^iD. This includes shale oil and gas as well coal bed methane 10 reservoirs. These concepts are scalable to large fields and can be used as a basis for surveillance monitoring, prediction optimization, shut-in scheduling etc.
The numerical methods disclosed are applicable for any well fracture geometry whether low/high angle, and include complex reservoir anisotropy, faulting 15 heterogeneity, directional stress dependent permeability changes, gas desorption and diffusion etc.
It is desirable to provide a fast and reliable method to calculate transient productivity Inflow Performance Relationships (IPR). These may be used for short 20 and long term modelling objectives for reservoirs with tens of thousands of wells. For example, they may be used in:
> Vertical lift performance (VLP)/IPR rate estimation workflows.
> Shut-in stimulation to estimate average reservoir pressures for reservoir 25 reserve estimation.
> Field optimisation and scenario management such as cycling well production.
Full numerical modelling of transient wells is not feasible in the timescales required, 30 due to the complexities of the models. While analytical solutions exist (for example flowing p/z method for Gas-Variable rate) where the pressure drop due to flow is not constant, such techniques tend to fail for wells exhibiting transient behaviour.
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An issue with present analytical solutions is that they exist only for simple geometries, e.g. the radial and linear. For slightly more complex geometries, there are analytical methods to obtain solutions, but these methods (Green's functions or methods of images) are difficult to solve, and usually contain slowly converging 5 infinite series. Furthermore, they do not extend as the geometry increases in complexity (e.g. inhomogeneous reservoir, faults, non-parallel reservoir boundaries) and are limited to what it is practical to formulate and solve.
In general drawdown curves do not match expected 'Concave' type variable rate 10 response i.e. there is large drawdown/noise and other effects including permeability variations with depletion. In these and other related scenarios it is not possible to use traditional analytical methods for mean reservoir pressure, original gas in place (OGIP) etc. estimates where pseudo steady state (PSS) is never reached or where very dynamic changes to transient/rates exist over the production history.
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Figure 1 is a flowchart indicating the general methodology of the concepts disclosed herein. This approach uses a numerical model for single well systems where the transient reservoir response of low permeability reservoirs cannot be adequately modelled using standard transient analytical solutions to the diffusivity equation or 20 material balance. Effectively, the numerical model is being used to create a dimensionless analytical analogue, as it is impractical to create and solve analytical equations directly.
The approach does not require, but may include, a complex reservoir description 25 including anisotropy, faulting, heterogeneity, multi-phase or complex fluid behaviour, directional stress dependent permeability changes, gas desorption and diffusion etc.. The purpose of this methodology is to capture the geometrical response of the highly transient system with sufficient detail of the well and fracture system without the requirement to build complex simulation models, and then tune 30 this geometrical response to match historical data.
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The transient nature of the productivity means that the normal concepts of Productivity Index (PI) and estimates of mean reservoir pressure do not apply. The current production rate for a given bottom hole pressure (Pwf) depends on the entire history of production.
The diffusivity D of the reservoir gives a good indication of the scale of the transient effects and can be calculated by:
D = a (m2/s) Equation 1
(f)fJCT
10 where <p is porosity (dimensionless), k is permeability (m2), n is viscosity (Pa.s), Ct is compressibility (Pa1) and a is a constant. The diffusivity D represents the signal that a pressure signal can sweep in a given time.
The following are often characteristics of transient reservoirs.
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> The wells can usually be considered independently since the time for the pressure communication is very long.
> Permeability may be initially increased around the region of the fracture simulation.
20 > Permeability may decrease significantly in the fractured region during production where large drawn down is present.
> The transient productivity (IPR) response changes continually and is largely dependent on the geometry of the well configuration and previous production history.
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In a first step 105, a well/fracture is chosen and the geometry of the fractured well system and reservoir is captured. Following this, a simulation is created 110. The simulation may be created using a software package, based upon inputted reservoir property data. The reservoir property data may include some or all of: the reservoir
30 dimensions (e.g. area, length, thickness) etc.; well length, position and orientation within the reservoir; the number of fractures, their dimensions and orientations;
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and other properties such as permeability, porosity etc. The software determines the required gridding and simulation properties to reflect the inputted data.
The simulation is then run at a constant production rate 112, and a bespoke 5 dimensionless type curve (dimensionless pseudo time versus dimensionless pseudo pressure or poto) is created 115. The potD captures the transient geometrical effects of the well and reservoir using dimensionless variables and can be used where it is impractical to create and solve analytical equations directly.
10 The definitions in field units of the dimensionless pseudo-time (tD] and dimensionless pseudo-pressure (pD) are given below:
tn = b-^-r f Equation 2
<t>rlJo /JCT
iTikh rP° dp „
15 PD=a — Equation 3
Q h iaB
where t is time, <p is porosity, k is permeability, h is well length, rw is the well flowing radius, Q is the constant well production rate, P° is the initial reservoir pressure, p is the current bottom hole pressure, n is viscosity, ct is the total compressibility 20 (rock+fluid), B is the formation volume fraction (FVF), which is the volume of fluid at reservoir pressure/volume of fluid at standard (surface) conditions and a and b are constants.
The dimensionless pseudo-time tD can therefore be calculated from time using 25 Equation 2, by solving the equation at a changing bottom hole pressure Pwf (viscosity and compressibility are fluid properties that are a function of pressure). In a similar manner, dimensionless pseudo-pressure (pd) can be calculated from bottom hole pressure Pwf using Equation 3 (1 /|aB is function of pressure).
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Equations 2 and 3 are solutions to the diffusivity equation for the flowing bottom hole pressure (P) and depend on the geometry and boundary conditions of the system.
The solution to the equation is linear in pd and allows for superposition. This means that for any two solutions of pd as a function of tD, their sum is also a solution. Consequently, multiple solutions can be used for different rates and superposed to 10 obtain a solution for a varying rate response. Therefore, obtaining the potD time function enables calculation (using superposition) of the Pwf response to any history of production.
At step 130 the calculated potD match is matched to historical well data (120 or 125) 15 in analytical space. The applicability of superposition means it is possible to vary parameters such as permeability, porosity, PVT by applying multipliers to the k, 0 and PVT terms (or any other constant) in the equations above, so as to match the historical data, since it will remain a solution to the diffusivity equation. This is as a consequence of the potD being a dimensionless curve capturing the geometry of the 20 transient response, and that its shape is independent of the rate, permeability etc.
Matching parameters may include:
Equation 4
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> The permeability multiplier affects the transient response and can be used to match the magnitude of the transient spikes.
> The porosity multiplier relates to well productivity (as it appears alongside rw in the equations above) and can be used to match later (nearer steady state) response.
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> PVT term multipliers scale the pressure between current pwf(l) and the reservoir pressure(O) calculated from a shut-in.
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Figure 2 is a log graph of dimensionless pseudo-pressure against dimensionless pseudo-time illustrating the potD response 200 being matched to the evolving production history 210. A match may be more readily visualized in log and 'log of derivative' space where it is possible to identify periods of near infinite linear and 5 radial response.
A fully developed infinite linear flow response has the form:
5 In {pD)_ 1
and therefore ^n(^o) ^ Equation 5
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A fully developed infinite radial flow response has the form:
, , 0.80907 + ln(<„) <K '
Pd\1dJ ~ fj\r\(t 1 2
2 and therefore ^D ' Equation 6
15 Such regions are identified by the ringed areas 220 and 220' on the graph.
The application of appropriate multipliers to the permeability (k), porosity (cf)), and PVT terms in the pd and tD expressions characterising the potD curve 200 effectively moves the curve 200 on the graph to the position labelled 200' where the region 20 representing infinite linear/radial flow 220 is matched with a corresponding region 220' of the production data 210.
Permeability variation with depletion due to pressure dependent compressibility may also be captured within the generic dimensionless potD curve. Exponential 25 permeability decline with compressibility factor can be used to regenerate PVT parameters and then applied to permeability decline in the dimensionless pseudo-time (to) and dimensionless pseudo-pressure (pd) solutions. The compressibility factor may be the exponential cleat compressibility exponent term C in the expression K = k e 3C(po p5, where K is the modified pressure dependent permeability, 30 k is the unmodified (input or match parameter permeability), C is the cleat
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compressibility, PO is the initial reservoir pressure and P is the current pressure (this may be Pwf, reservoir pressure or something in between depending on the PVT matching method).
5 Following matching, the Pwf response can be calculated (step 135) from the matched potD and superposition of the historical varying rate. This can then be used to make predictions of future reservoir behaviour. Specifically the Pwf response can be used to calculate future and historical transient IPR 140 and to simulate analytical solutions estimating the mean reservoir pressure (e.g. for P/Z plots) 145.
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Of particular note is that the dimensionless generic nature of the potD curve means that it is transferrable between wells of similar geometries. Similar well geometries may match to the same potD curve even if the permeability, number of fractures, well length etc. are significantly different.
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Consequently, step 130 may entail matching the potD to historic data from another well 125, provided that the geometry of the other well is sufficiently similar to the well for which the potD was calculated. In this way, the same potD curve may be used to make predictions for a large number of different wells.
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It is not generally possible to match-historical data using a potD curve that does not represent the geometry of the well. However, the method is quite discriminatory and the geometries may show significant differences. Similar wells are those that exhibit a similar geometric response; for example those which have similar potD
25 curves, or at least have similar regions within their potD curves. Similar regions may be those that exhibit similar periods of time where near infinite radial or infinite linear response exists. Matching the curves may comprise matching those similar regions.
30 Steps 130 and 135 can be further broken down into the following steps, so as to obtain the Pwf:
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> Break the history into a series of steps;
> For each step calculate the rate change with respect to the previous rate and apply this rate change from where it occurred to the current time;
> The tD for this period is calculated (in a series of steps applying the current PVT [Pressure, Volume, Temperature] dependencies for fluid properties);
> The pd is updated from the potD definition, and the rate change applied and added to the current time value for pd (superposition)
> Finally the is pd inverted to obtain the Pwf.
The superposition in the penultimate step above is performed to back calculate a Pwf from a changing rate history. Superposition is used because the inversion of the potD to Pwf from pd is for a fixed production rate Q: superposition is used when the rate is not constant.
Pwf is calculated from production rate during the initial matching process to test that the calculated value matches the observed value. Superposition is also used to calculate Pwf from production rate for future predictions and transient IPR etc using the matched potD.
For example after the first step the dimensionless time (to) is calculated using the viscosity and compressibility at initial conditions (from Equation 2).
bk r dt tDl <t>rl ! r)o
The dimensionless pseudo pressure at this tD is taken from the PD,tD curve and (from Equation 3):
P
, a x 2nkh r dp Therefore pseudo pressure I(Pwf) can be calculated as follows.
P Pwf In PO r let I{P) = [ —- - Therefore: I(Pwf)= f — = I(P0)~ f
J0 jiB I ijB lf!iB
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I(Pwf),=I(P°)-(Q' _Qf>xPD\!m\
a x 2 nkn
Using the PVT from the beginning of the step a value of Pwf is obtained.
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For a series of rate changes the following equation can be used to calculate the pseudo pressure at the wellbore:
HPwf), = /(PO) - x pD
i=o a
10 k,(|),rw,h,Q,|j,,B,CT,t,p and pO are as previously defined, a and p are the constants in the dimensionless pseudo pressure and time equations respectively.
One or more steps of the methods and concepts described herein may be embodied in the form of computer readable instructions for running on suitable computer 15 apparatus, or in the form of a computer system comprising at least a storage means for storing program instructions embodying the concepts described herein and a processing unit for performing the instructions. As is conventional, the storage means may comprise a computer memory (of any sort), and/or disk drive or similar. Such a computer system may also comprise a display unit and one or more 20 input/output devices.
The concepts described herein find utility in all aspects of surveillance, monitoring, optimisation and prediction of hydrocarbon reservoir and well systems, and may aid in, and form part of, methods for extracting hydrocarbons from such 25 hydrocarbon reservoir and well systems.
It should be appreciated that the above description is for illustration only and other embodiments and variations may be envisaged without departing from the spirit and scope of the invention.
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t■ —t-,
7 7-1
(pcT)j-i
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