Type-Curve Models Available in PIE

The following is a summary of the type-curve models available in the PIE well-test analysis software. All of these models use a full analytic solution using a variety of techniques (e.g. Laplace or Fourier transforms, Bessel functions, etc.). All models include well-bore storage and skin effects, and represent a finite well-bore radius (i.e. no line-source solutions are used).

1) Homogeneous Reservoir

This model represents radial flow for a finite radius, vertical well-bore with storage and skin in a homogeneous reservoir. Solution obtained by Laplace inversion of the equation presented by Mavor and Cinco for a well in an infinite homogeneous reservoir with storage and skin.

This is the most basic of models and should be used to obtain some idea of how the test-data deviates from simple radial flow. This will assist in selecting more complex models by identifying characteristic behaviour relative to radial-flow.

Parameters:             Permeability, skin factor, and well-bore storage coefficient.

Static Data:             None.

Specialised:              None.

Reservoir:                All reservoir boundary geometry's supported.

References:              Gringarten, A. C.; "Interpretation of Tests in Fissured Reservoirs and Multi-layered Reservoirs with Double-Porosity Behaviour: Theory and Practice"; SPE Paper 10044.

Mavor, M. J. and Cinco, H.; "Transient Pressure Behaviour of Naturally Fractured Reservoirs", SPE Paper 7977, Presented at the 1979 California Regional Meeting of the SPE.

          

2) Double-Porosity Reservoir

This model represents radial flow for a finite radius, vertical well-bore with storage and skin in a fissured reservoir. There is exchange of fluids between the matrix and fissures with flow to the well-bore occurring only in the fissures. Flow between the matrix and fissures can either be pseudo-steady-state (PSS), 1-dimensional transient (1DT), or 3-dimensional transient (3DT) behaviour. The solution is obtained by Laplace inversion of the same equation as the homogeneous model, but including an inter-porosity flow term. Note that all double-porosity models include well-bore storage and skin.

Parameters:             Permeability, storage coefficient, skin, omega, and lambda. Omega is the ratio of fissure storivity to total system storivity, and lambda is the inter-porosity flow coefficient.

Static Data:             None.

Specialised:              On the derivative plot the LAMBDA key can be used to define the pressure level at which the inter-porosity flow transition occurs by selection of the derivative minimum. This will define the value of Lambda provided that the total-system stabilisation has been set by the STABIL key. Using this key a value of Omega will also be estimated.

Radial-Flow plots can be used to determine Omega if two parallel straight-lines can be identified. After the radial-flow line has been identified, the OMEGA key allows a second, parallel line to be drawn through the selected early time data and Omega to be calculated.

Lambda can be estimated on radial flow plots using the LAMBDA key which defines the pressure level at which the inter-porosity flow occurs by setting the point at the inflection of the transition between the two straight-lines.

Reservoir:                All reservoir boundary geometry's supported.

References:              Gringarten, A. C.; "Interpretation of Tests in Fissured Reservoirs and Multi-layered Reservoirs with Double-Porosity Behaviour: Theory and Practice"; SPE Paper 10044.

Mavor, M. J. and Cinco, H.; "Transient Pressure Behaviour of Naturally Fractured Reservoirs", SPE Paper 7977, Presented at the 1979 California Regional Meeting of the SPE.

          

3) Infinite Conductivity Hydraulic Fracture

The infinite conductivity fracture model describes a well with a vertical hydraulic fracture in a homogeneous reservoir. The hydraulic fracture has a very high conductivity relative to the reservoir permeability-fracture length product i.e. the fracture can move fluid to the well-bore with negligible pressure drop. This model represents the linear and radial flow for a line source well with a vertical hydraulic fracture. The solution is obtained in real space with well-bore storage added by Laplace inversion.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.3_files\image001.gif

Parameters:             Permeability, storage coefficient, hydraulic fracture half-length.

Static Data:             None.

Specialised:               The HALFSL key on the derivative plot will define a half-slope which represents the linear flow into the fracture. This line will allow the fracture half-length to be calculated if the radial flow stabilisation has been set with the STABIL key.

Linear-flow plots can be used calculate the fracture half-length by selecting an early time straight-line with AUTOSL. The estimate of fracture half-length from this line and the radial-flow permeability will set a time marker on the plot indicating the end of linear flow. A correct analysis is achieved when the data for the AUTOSL line lies before the end of linear flow time marker.

Reservoir:                All reservoir boundary geometry's supported. Note that these are added by de-superposition of Ei-functions. If boundary effects occur during linear flow, this solution is not accurate.

References:              Gringarten, A. C., Ramey, H. J., and Raghavan, R.; "Unsteady-State Pressure Distributions Created by a Well with a Single Infinite-Conductivity Vertical Fracture"; SPEJ (Aug. 1974) 347-360; Trans. AIME, 257.

Earlougher, R. C.; "Advances in Well-Test Analysis"; SPE Monograph Volume 5, Equation C.9.

          

4) Finite Conductivity Hydraulic Fracture

The finite conductivity fracture model describes a well with a vertical hydraulic fracture in a homogeneous reservoir. The hydraulic fracture has a low to moderate conductivity relative to the reservoir permeability-fracture length product i.e. there is significant pressure drop along the fracture as fluid enters from the reservoir and moves to the well-bore.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.4_files\image001.gif

This model is the same as the infinite conductivity hydraulic fracture, except that the flow capacity of the vertical fracture is finite instead of infinite. The solution is by table look-up using Specialised interpolation algorithms and analytic extrapolation past the table bounds. Table values were generated by the "semi- analytic" solution presented by Cinco et al.

Parameters:             Permeability, storage coefficient, fracture half-length, fracture conductivity.

Static Data:             None.

Specialised:               The 1/4SLP key on the derivative plot will define the quarter-slope which represents fracture bi-linear flow. This line allows the fracture conductivity to be calculated if the radial-flow stabilisation has been set with the STABIL key and the linear-flow 1/2 slope line has been set with the HALFSL key.

The HALFSL key on the derivative plot will define a half-slope which represents the linear flow into the fracture. This line will allow the fracture half-length to be calculated if the radial flow stabilisation has been set with the STABIL key.

A Bi-linear flow plot can be used to calculate the fracture conductivity. This analysis should be done after the radial-flow and linear-flow analyses. Use the AUTOSL function to draw an early-time straight-line. The fracture conductivity is calculated using the fracture half-length and permeability of the linear and radial flow plots. A time marker to indicate the end of bi-linear flow is draw. A correct analysis is achieved when the data for the AUTOSL line lies before this marker.

Reservoir:                All reservoir boundary geometry's supported. Note that these are added by de-superposition of Ei-functions. If boundary effects occur during linear flow, this solution is not accurate.

References:              Cinco, H. L. et al; "Transient Pressure Behaviour for a Well with a Finite Conductivity Vertical           Fracture"; SPE paper 6014; New Orleans Fall       Technical Conference of the SPE of AIME, Oct. 3-6,        1976.

          

5) Partial-Penetration Model

This model represents a well with a small perforated interval within a thick homogeneous reservoir. There is a transition between flow over the perforated thickness to flow over the effective total reservoir thickness. This transition is controlled by the ratio of perforated thickness to effective thickness and the ratio of vertical to horizontal permeability. The flow during the transition behaves like spherical flow, which is related to the -1/2 power of time. The following schematic shows the layout of this well and reservoir geometry

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.5_files\image001.gif

Note that this model can also be used with a double-porosity reservoir and all three types of double-porosity behaviour can be used. For the sake of simplicity, the homogeneous reservoir is used for this example. The solution is obtained by numerical integration of a Green's function equation. Well-bore storage and double-porosity behaviour is added by Laplace inversion

Parameters:             For a homogeneous reservoir; permeability, storage coefficient, perforation skin, global skin, Kv/Kh ratio, and effective reservoir thickness.

For a double porosity reservoir; the same as homogeneous plus the double-porosity parameters, Omega, and Lambda.

Static Data:             The perforation thickness and the distance between the perforation centre and the reservoir top.

Specialised:              Derivative plots can set an early-time stabilisation representing flow over the perforation thickness with the PPNSTB key. This sets the effective reservoir thickness if the main analysis stabilisation has been selected via STABIL.

The spherical flow transition can be analysed on the Derivative plot using the SPHERE key. This will define a slope of -1/2 on the derivative data from which the value of Kv/Kh can be calculated if the total thickness stabilisation has been set by the STABIL key.

Radial-Flow plots can be used to set an early-time straight-line representing flow over the perforation thickness by using either the PPNSLP or PPNSKN function key. The PPNSLP key will set the permeability, perforation skin factor, and, if the main analysis line is present, the value of effective reservoir thickness. The PPNSKN key will only set the value of perforation skin.

A Spherical-flow plot can be used to analyse the transition between radial-flow over the perforation thickness to radial-flow over the effective thickness. This period will have a spherical flow straight-line line from which Kv/Kh can be calculated provided that a radial-flow plot has been used to set the permeability.

Reservoir:                All reservoir boundary geometry's are supported by de-superposition of Ei function image wells. This is not correct when boundary effects occur during the early-time period when flow is over the perforation thickness.

References:              Gringarten, A. C., Ramey, H. J.; "Unsteady-State Pressure Distributions Created by a Well with a Single Horizontal Fracture, Partial Penetration, or Restricted Entry"; SPEJ, August, 1974.

          

6) Two-Layer Reservoir With Cross-Flow

This model represents radial flow for a two-layer reservoir with the well completed in both layers and with flow between the layers in the reservoir (This model is also called a dual-permeability system, or bi-layer reservoir). The solution is obtained in Laplace space.

This model has quite complex behaviour; so much so that there are no Specialised analysis methods available. As the ratio of the layer permeability's goes towards zero or infinity, the behaviour of this model is like a double-porosity reservoir (see Section 8.2). The late time behaviour is governed by the thickness average permeability of the two layers. The flow coefficient, Lambda, is used to control the amount of cross-flow between layers.       For the case where a thin, low-permeability 'wall' separates the two layers, use:

           Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.6_files\image001.gif    (1)

Where dW is the thickness of the 'wall' and KV is the low permeability of the 'wall'. For the case where the vertical flow is controlled by a uniform reservoir vertical permeability, Kv, use the equation:

           Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.6_files\image002.gif      (2)

For the above equations (1) and (2); rw is the well-bore radius, KH(total) is the sum of the layer permeability-thickness values, and H is the total thickness of both the layers.

Parameters:             Storage coefficient, permeability for layer 1, permeability for layer 2, skin factor for layer 1, skin factor for layer 2, Omega, and Lambda.

Static Data:             Thickness, porosity, water saturation, and fluid compressibility of layers 1 and 2 required to set the default value of Omega. The thickness, porosity, etc. entered are updated to be consistent with the total reservoir thickness average values of the layer data.

Specialised:              Same as for the double-porosity model (see Section 8.2). However, note that even for ideal data the values of Omega and Lambda are estimates.

Reservoir:                No reservoir boundaries are supported.

References:              Bourdet, D; "Pressure Behaviour of Layered Reservoirs with Crossflow"; SPE Paper 13628.

          

7) Inclined-Well Model

This model represents the behaviour of a well drilled on a slant, (i.e., an inclined well),through a homogeneous or double-porosity reservoir. It is characterised by a transition between radial flow perpendicular to the inclined well-bore and horizontal radial flow over the total reservoir thickness. The geometry and parameters for this model are shown in the following cross-section:

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Cross-Section

Due to the fact that the well-bore is inclined, the initial radial flow is governed by an effective permeability that has a value between the vertical and horizontal permeability. There are no Specialised analysis methods available for analysis of this transition period.

The inclined well model also includes well-bore storage and perforation skin factor. The solution is found using Green's functions for the well and reservoir without storage and skin, and a Laplace transform adds the effect of storage and skin. The model also includes reservoir boundaries as shown in the following plan-view of this model:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.7_files\image002.gif
Plan-View

The Azimuth of the well is requested only when reservoir boundaries are included in the model selection.

Note also that this model can also be used with a double-porosity reservoir. For the sake of simplicity, the homogeneous reservoir is used for this example.

Parameters:             For a homogeneous reservoir; permeability, storage coefficient, perforation skin, Kv/Kh, Global skin, and Geometric skin.

For a double-porosity reservoir; As for homogeneous reservoir plus      lambda and omega.

Static Data:             Length of the inclined "drain" (i.e. the perforated length), position of the inclined "drain" from the reservoir base, and the angle from the vertical of the inclined "drain".

Specialised:              None. However, as the angle of deviation becomes close to 90 degrees, the horizontal well   Specialised analysis can be used to obtain        estimates of the parameters.

Reservoir:                All rectangular boundaries are supported. Boundaries can be no-flow (fog-factor=1), constant pressure (fog-factor=-1), or not present (fog-factor=0).

References:              Cinco, H., Miller, F. G., Ramey, H. J.; "Unsteady-State Pressure Distribution Created By a Directionally Drilled Well"; Journal of Petroleum Technology (JPT), Nov. 1975, pg. 1392.

          

8) Horizontal Well Model

This model describes a horizontal well in a homogeneous or double porosity reservoir. The well can have well-bore storage and skin.

The flow behaviour for this model is characterised by a transition between vertical radial flow over the length of the horizontal drain, perpendicular to the axis of the horizontal well, to pseudo-radial flow over the reservoir thickness. The two radial flow regimes result in two straight-lines on a semi-log plot, or two stabilisation's on a derivative plot. If these two straight-lines or stabilisation's can be identified, then the vertical permeability can be determined along with the flow efficiency of the completion (the global skin factor).

Solution is by Green's functions with well-bore storage and skin added by a transform in Laplace space.

Whilst this well model can also be used with a double-porosity reservoir, for the sake of simplicity, the homogeneous reservoir is used in this example. The cross-section geometry and the parameters for this model are shown in the following figure:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.8_files\image001.gif
Cross-Section

The distances to reservoir boundaries from the horizontal-well are shown in the following plan-view of the this model:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.8_files\image002.gif
Plan-View

Note that the +x and -x distances are measured from the centre of the horizontal well-bore.

Parameters:             Permeability, storage coefficient, Kv/Kh, well-bore perforation skin, Global skin (pseudo- radial flow skin factor), and geometric skin. For double-porosity models, lambda and omega are results. The square-root of Kv*Kh is also calculated.

Static Data:             Length of the horizontal "drain" (i.e. the perforated horizontal length), and the position of the horizontal well-bore from the base of the reservoir.

Specialised:              On the derivative plot, the HORSLP key may be used to estimate the drain length. A second stabilisation can then be set using the HORSTB key. If the permeability for pseudo-radial flow has been estimated previously using the STABIL key then value of perforation skin, Geometric skin, and Kv/Kh are calculated. If only the HORSLP line is drawn, the perforation skin and square-root of Kv*Kh are set.

On a radial-flow analysis plot, a second line can be drawn through the data in the vertical radial-flow-period using the HORSLP key. This is the analogous function to the HORSLP key on the derivative analysis plot and the same calculations are performed.

Reservoir:                A limited subset of reservoir boundary geometry's are supported since the image wells must be horizontal wells for this model. All boundaries must have the same boundary fog-factor. Only fog-factors of -1, 0, or 1 (constant pressure, no boundary, or no-flow) are accepted. Any other input is silently transformed to meet these limitations.

The geometry's supported are single linear, parallel, parallel closed and one end, and closed rectangular. Note that the image well solution using horizontal wells is a large calculation problem. Use boundaries with this model only if you are working on a computer with substantial capacity (e.g. a high-speed 386-PC, a   NIX workstation, etc).

References:              Raghavan, R., Ozkan, E., and Joshi, S. D.; "Horizontal Well Pressure Analysis"; SPE paper 16378, SPE California Regional Meeting, Ventura California., April 8-10, 1987.

          

9) Radial Composite Reservoir

This model is used to describe a vertical well in a reservoir that has two radial zones, each with different reservoir properties, to form a radial composite geometry. The reservoir can be homogeneous or double-porosity. The inner zone closest to the well has different rock and fluid properties than the outer-zone. The most common case where this model is used is for injection wells where the inner-zone represents a flooded region and the outer zone represents the unaltered reservoir. The layout and parameters for this model are shown in the following figure:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.9_files\image001.gif        

The model is characterised by a transition between radial flow in the inner zone governed by the inner zone total compressibility and mobility, to flow in the outer zone, (unaltered reservoir), governed by it's reservoir total compressibility and mobility. Note that the Mobility, Storativity, and Diffusivity ratios are all defined as the outer zone properties divided by the inner-zone properties. Solution is found by inversion of an equation in Laplace space.

Note that this well model can also be used with a double- porosity reservoir. For the sake of simplicity, the homogeneous reservoir is used for this example. Also note that the static data (volume factor, viscosity, etc.) entered for the basic test information are assumed to refer to the inner-zone. The static data for the outer zone are entered when this model is selected. Note that the mobility and storativity ratio used by this model are based on the ratio of the outer-zone properties divided by the inner-zone properties.

Parameters:             Storage coefficient, skin factor, permeability (outer), permeability (inner), storativity-ratio and injection radius. For double-porosity models, lambda and omega are results.

Static Data:             Fluid viscosity, fluid compressibility, porosity, and water saturation of the Outer Zone. This defines the default value of the storativity ratio and the data for setting the mobility ratio.

Specialised:              On the derivative plot, the INJSTB key is used to define inner-zone permeability, radius, and perforation skin factor by allowing a stabilisation representing radial flow in the inner-zone to be set. If the INJSLP key is used without having used the STABIL key to define the outer-zone radial-flow stabilisation then only the inner-zone permeability and perforation skin are calculated.

On a radial-flow analysis plot, use the AUTOSL key to define the outer-zone radial flow to obtain the permeability and Global Skin factor. Use of the INJSLP key is analogous to the Derivative function and defines the inner-zone permeability, radius, and perforation skin factor. If the INJSLP key is used without the AUTOSL key, then only the inner-zone permeability and perforation skin are calculated.

Reservoir:                All boundary geometry's are supported by de-superposition of Ei-function image well solutions in Laplace space. However, it should be noted that this is correct only if the effect of the boundaries occurs in the outer-zone radial flow-period.

References:              Abbaszadeh, M.; Medhat K.; "Pressure Transient Testing of Water-Injection Wells"; SPE Reservoir Engineering Journal, Feb. 1989, pp. 115-124.

Sutman, A., Eggenschwiler, M., and Ramey, H. J.; "Interpretation of Injection Well Pressure Transient Data in Thermal Oil Recovery"; SPE paper 8909, California Regional Meeting of the SPE of AIME, Los Angeles California, April 8-11, 1980.

          

10) Reservoir Boundaries (Linear, Channel, Closed-Channel, Closed-Rectangle, Intersecting)

All of the models that support reservoir boundaries have the same basic late-time behaviour. With no reservoir boundaries, the late-time behaviour for these models is "radial flow" i.e. either a late-time stabilisation on a derivative plot, or a late-time straight-line on a radial-flow plot (M.D.H, Horner, or Superposition).

When there is a single linear boundary close enough to have been "seen" by the test, then there is a deviation from that late-time radial flow behaviour. When the linear boundary is sealing (no-flow), the deviation is a doubling of the slope on a radial-flow plot or a doubling of the derivative plot stabilisation. The "Specialized" analysis described in the following sections is used to determine the distance to a linear boundary based on the time at which the deviation from the late-time radial flow occurs.

The type-curve models in PIE implement reservoir boundaries according to the solution method used for the model. These implementations fall into two broad categories:

·    The type-curve solution method itself includes reservoir boundaries. For example, models based on "Green's functions" will use the appropriate instantaneous source functions for a bounded reservoir. The boundaries can be either "no-flow" (i.e. a sealing boundary) or "constant-pressure". The user input of the reservoir boundary parameters is customised for each model.      

·   The type-curve solution method allows boundaries to be added using an "image-well" technique (see Appendix B.4 of the SPE Monograph "Advances in Well-Test Analysis" by R. Earlougher). The user-interface for the reservoir boundary parameters will request distances and "fog-factors" (described below).

For the image-well solution, various reservoir boundary shapes can be implemented. The shapes available in PIE are linear, parallel, parallel with one end closed, intersecting, and closed rectangle boundaries.

In addition to the published image-well technique, PIE also includes a boundary "fog-factor" to approximately represent a linear discontinuity in reservoir properties. The name "fog-factor" derives from the optical analogy of an image in a "foggy mirror". A fog-factor of "1.0" represents a no-flow boundary, "0.0" represents no boundary, and "-1.0" represents a constant pressure boundary.

Fractional fog-factor values between -1 and 1 represent a change in reservoir transmissibility for a constant diffusivity. Assuming a constant reservoir thickness and analysis-constants, a fog-factor corresponds to a change in permeability across the boundary interface according to the following equation:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.10_files\image001.gif

where 'k' is the permeability of the reservoir containing the well, and 'ko' is the permeability on the outside of the boundary. Note that ko = 0.0 represents a no-flow boundary, and ko = infinity represents a constant pressure boundary.

Single Linear Boundary

When the linear boundary is sealing (no-flow), then the deviation is a doubling of the slope on a radial-flow plot or a doubling of the derivative plot stabilisation. The Specialised analysis described in this section determines the distance to a linear boundary based on the presence of the double slope, or the time at which the deviation from the late-time radial flow occurs.

Parameters:             Distances in the current units for length, and angles if these are part of the boundary geometry.

Static Data:             None.

Specialized:             There are three keys on the derivative plot which allow analysis of the distance to the nearest boundary, (the FAULT key), the width of a channel, (the CHANEL key), and the angle between intersecting boundaries, (the ANGLE key). When used together these function keys allow you to specify the location of a well between single, multiple or intersecting boundaries.

Radial-Flow plots can be used to estimate the distance to the nearest boundary by using the BND-ON key. This defines a line of double-slope and is specifically for a single linear no-flow boundary. If a doubling of slope is not obvious the BND-DV function can be used, after selecting the middle-time, (radial-flow), straight-line with the AUTOSL key. This sets the time where boundary effects begin to be 'felt' from which a distance can be calculated. Note that the BND-DV key is useful for all boundary geometry's. It will accurately report the distance to the nearest boundary.

References:              Van Poolen, H. K., et al; "Effect of Linear Discontinuities on Pressure Build-up and Draw-down Behaviour"; Journal of Petroleum Technology (JPT) August 1963.

          

Parallel Boundaries

Parallel no-flow boundaries are characterised by a continuous increase in slope on a radial-flow analysis plot once the boundaries affect the late-time radial flow straight-line. On the derivative plot, parallel no-flow boundaries have a derivative slope of 1/2 after the radial flow stabilisation. The derivative 1/2 slope means that a square-root power of time can be used to analyse linear-flow which will give the width between the parallel boundaries and the well location within the channel formed by the boundaries.

Parameters:             Distances in the current units for length, and angles if these are part of the boundary geometry.

Static Data:             None.

Specialised:              There are three keys on the derivative plot which allow analysis of the distance to the nearest boundary, (the FAULT key), the width of a channel, (the CHANEL key), and the angle between intersecting boundaries, (the ANGLE key). When used together these function keys allow you to specify the location of a well between single, multiple or intersecting boundaries. Radial-Flow plots can be used to estimate the distance to the nearest boundary by using the BND-DV key, after selecting the middle-time, (radial-flow), straight-line with the AUTOSL key. This sets the time where boundary effects begin to be 'felt' from which a distance can be calculated. Note that the BND-DV key is useful for all boundary geometry's. It will accurately report the distance to the nearest boundary.

For a two-sided channel parallel boundary geometry, using the Linear-flow plot will compute the distance from the well to each side of the channel. Note that a radial-flow analysis should be done first to provide the channel width calculations with a value of permeability.

For Parallel boundary geometry's (either 2-sided, or 3-sided channels), the Linear-flow analysis plot can be used to compute extrapolated pressure and channel half-width. Note that selection of a hydraulic-fracture model will over-ride these calculations.

References:              Van Poolen, H. K., et al; "Effect of Linear Discontinuities on Pressure Build-up and Draw-down Behaviour"; Journal of Petroleum Technology (JPT) August 1963.

          

Closed-Channel (Three-sided Channel)

Parallel no-flow boundaries are characterised by a continuous increase in radial-flow plot slope when the boundaries influence the pseudo-radial flow straight-line. On the derivative plot, parallel no-flow boundaries have a derivative slope of 1/2 after the pseudo-radial flow stabilisation. The derivative 1/2 slope means that a linear flow plot can be used to analyse the data for width between the parallel boundaries and the well location within the channel. The third side of this model closes the end of the channel formed by the parallel boundaries and adds an extra transition to the basic linear flow behaviour of a channel. This extra transition depends on how close to the well the third side is relative to the other two sides of the channel.

Parameters:             Distances in the current units for length, and angles if these are part of the boundary geometry.

Static Data:             None.

Specialised:              The 3-sided channel does not have a specific derivative plot analysis. The distance to the nearest boundary can be estimated with the FAULT key. When used together with the CHANEL and ANGLE keys it is possible to specify the location of a well between single, multiple or intersecting boundaries.

Radial-Flow plots can be used to estimate the distance to the nearest boundary by using the BND-DV key, after selecting the middle-time, (radial-flow), straight-line with the AUTOSL key. This sets the time where boundary effects begin to be 'felt' from which a distance can be calculated. Note that the BND-DV key is useful for all boundary geometry's. It will accurately report the distance to the nearest boundary.

For Parallel boundary geometry's (either 2-sided, or 3-sided channels), the Linear-flow analysis plot can be used to compute extrapolated pressure and channel half-width. Note, however, that a radial-flow analysis should be performed initially to provide the channel width calculations with a value of permeability.

For a three-sided channel geometry, there are two analysis possibilities. If the boundary at the channel end is closer than the two sides of the channel, the data should show only one straight-line on the linear-flow plot. In this case, use the 3-SIDE key to obtain an estimate of the distances to the boundaries.

If the channel end is significantly further away than the two sides of the channel, the data should show two straight-lines with a doubling of slope. In this case, use the AUTOSL key for the first straight-line and the 3-SIDE key for the second to estimate the distances to the boundaries.

References:              Van Poolen, H. K., et al; "Effect of Linear Discontinuities on Pressure Build-up and Draw-down Behaviour"; Journal of Petroleum Technology (JPT) August 1963.

          

Closed Rectangle Boundary

The closed rectangle boundary geometry with no-flow boundaries has pseudo-steady-state late-time behaviour. For build-up tests, this results in the pressure reaching a constant value equal to the average pressure. For a draw-down test, the late-time pressure decreases linearly. These responses have characteristic shapes on the radial-flow and derivative analysis plots. For a build-up test, the radial-flow plot has a final horizontal portion at the average pressure and the derivative plot has a derivative that decreases rapidly to very small values. For a draw-down test, the radial-flow plot shows the data continually curving upward whilst the derivative plot shows a late-time unit slope.

A reservoir geometry that has a much longer 'length' than width' can have a linear-flow period prior to the onset of pseudo-steady-state flow. A linear flow analysis can be used in this period to define the distances to three of the four boundaries (see Section 8.12 for an example). With this specialised analysis, a good guess for the final fourth boundary is a value larger than the other three distances.

If the test is a single-rate draw-down, (as in this example), then the calculations for reservoir drainage volume and area are the same as for the calculation of well-bore storage coefficient. You can obtain these parameters by doing a well-bore storage analysis, but selecting the pseudo-steady-state data for the analysis straight-line instead of the data near the plot origin.

Note that for models using the image-well technique, a closed rectangle has a slightly different definition of the fog-factor:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.13_files\image001.gif

Parameters:             Distances in the current units for length, and angles if these are part of the boundary geometry.

Static Data:             None.

Specialised:              The closed rectangle does not have a specific derivative plot analysis. The distance to the nearest boundary can be estimated with the FAULT key. For a single rate draw-down test in a closed boundary the derivative plot will exhibit a late-time unit slope.

Radial-Flow plots can be used to estimate the distance to the nearest boundary by using the BND-DV key after selecting the middle-time, (radial-flow), straight-line with the AUTOSL key. This sets the time where boundary effects begin to be 'felt' from which a distance can be calculated. The remaining three sides must be at distances larger than this value.

If the length of the rectangle is longer than its width then the Linear-flow analysis plot can be used to compute extrapolated pressure and channel half-width. Note, however, that a radial-flow analysis should be performed initially to provide the channel width calculations with a value of permeability.

For a single-rate draw-down (i.e., a reservoir limits test), a Cartesian well-bore storage analysis plot can also be used to estimate the reservoir drainage volume and area, (reported as 'volume' and 'area'), using the AUTOSL key to draw a line through the late-time linear portion of the data.

References:              Van Poolen, H. K., et al; "Effect of Linear Discontinuities on Pressure Build-up and Draw-down Behaviour"; Journal of Petroleum Technology (JPT) August 1963.

          

Intersecting Boundaries

The intersecting no-flow boundary model has late-time radial-flow behaviour that is related to the angle of intersection of the boundaries. As the angle between the boundaries becomes small, the behaviour of this model tends to the parallel boundary model. Hence, there is a linear flow transition between infinite-acting and intersecting boundary radial flow. On the derivative plot, these flow regimes result in a first stabilisation for infinite acting flow, a 1/2 slope transition due to linear flow, and a final stabilisation for the intersecting boundaries radial flow. On a semi-log plot, these flow regimes are shown by an initial straight-line for infinite acting behaviour and a transition to a final straight-line for intersecting boundary radial flow.

Parameters:             Distances in the current units for length, and angles if these are part of the boundary geometry.

Static Data:             None.

Specialised:              There are three keys on the derivative plot which allow analysis of distance to the nearest boundary (the FAULT key), width of a channel (the CHANEL key), and the angle between intersecting boundaries (the ANGLE key). These function keys together allow you to specify the location of a well between intersecting boundaries. Note also that the transition between the two derivative radial-flow-periods will tend towards linear-flow.

If there is a clear transition towards linear flow as indicated by the presence of a late-time 1/2 slope on the derivative plot a linear-flow plot will show a straight-line related to the width between the boundaries at the well location.

Radial-Flow plots can be used to estimate the distance to the nearest boundary by using the BND-DV key after selecting the middle-time, (radial-flow), straight-line with the AUTOSL key. This sets the time where boundary effects begin to be 'felt' from which a distance can be calculated. The remaining three sides must be at distances larger than this value.

References:              Van Poolen, H. K., et al; "Effect of Linear Discontinuities on Pressure Build-up and Draw-down Behaviour"; Journal of Petroleum Technology (JPT) August 1963.

          

11) Multi-Layer Reservoir With Cross-Flow

This model represents radial flow in a layered reservoir where there is vertical communication between layers. The solution is solved in Laplace space. The fluid flow between layers is controlled by the inter-layer flow coefficient, Lambda. The value of Lambda can be related to the vertical permeability and the thickness of the layers by the following equation:

           Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.15_files\image001.gif         (1)

For a uniform reservoir vertical permeability, Kz(i) is the same for all layers. Replacing the Kz(i) values in equation (1) with a constant value, Kv, results in:

           Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.15_files\image002.gif  (2)

For the above equations; rw is the well-bore radius, KH(total) is the sum of the layer permeability-thickness values, and H(i) is the thickness of the i'th layer. Note that for an N layer system, only N-1 values of Lambda are defined.

These models also use the parameters Omega and Kappa where Omega is the fraction of the total porosity-thickness-compressibility product for a layer and Kappa is the fraction of the total permeability-thickness for a layer. For an N layer system, there are only N-1 values of Omega and Kappa required since the sum of the Omega's and Kappa's must be equal to one.

Note that these models have a large number of unknowns and are virtually impossible to use to obtain a unique answer. It is recommended that other information (e.g. log, core, etc.) be used to fix as many parameters as possible, in particular, try to fix the porosity distribution so that the layer values of Omega can be fixed. The best use for these models is the representation of complex well completion problems such as partial penetration under a gas-cap.

Parameters:             Storage coefficient, skin factors for each layer, permeability's for each layer, Omega and Lambda for each layer except the last.

Static Data:             A table of thickness, porosity, water saturation, fluid compressibility, and rock compressibility for each layer is entered to define default values of Omega. The information entered here will UPDATE the global static data entered from the Data Manipulation menu for the total reservoir thickness, compressibility, and porosity.

Specialised:              None.

Reservoir:                No reservoir boundaries are supported.

References:              Economides, C. A.; "A New Test for Determination of Individual Layer Properties in a Multi-layered Reservoir"; SPE paper 14167, presented at the 60th Technical Conference, 1985.

          

12) Finite Conductivity Fracture With Skin

This model is the same as the finite conductivity hydraulic fracture, except the flow from the reservoir to the vertical fracture is restricted by a skin on the fracture face. The solution is by the "semi-analytic" method presented by Cinco, but solved in Laplace space.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.16_files\image001.gif

Parameters:       Permeability, storage coefficient, fracture half-length, fracture conductivity, fracture skin.

Static Data:        None.

Specialized:        Same as the finite-conductivity hydraulic-fracture model. However, the fracture skin can obscure the early-time bi-linear flow regime.

Reservoir:          No reservoir boundaries are supported.

Reference:          Cinco, H. L. and Samaniego, V.; "Effect of Wellbore Storage and Damage on the Transient Pressure Behaviour of Vertically Fractured Wells"; SPE paper 6752; Denver Fall Technical Conference of the SPE of AIME, Oct. 9-12, 1977.

See the Finite-Conductivity Hydraulic-Fracture example.

          

13) Interference Test Model

This model calculates the pressure response at an observation well some distance away from a flowing well. The reservoir can be homogeneous or double porosity and there is well-bore storage and skin at both the observation well and the flowing well. The solution is based on two line-source wells with each well having storage and skin. The equations are solved in Laplace space.

Parameters:       Permeability, Skin factor(pulser), Storage Coefficient(pulser), Skin factor(obs), Storage coefficient(obs), Storativity, x-location, y-location.

For double-porosity reservoirs, results include omega and lambda.

Static Data         None.

Specialised:        For a simple pulser and observer problem, treat the test like a normal well-test. Use the Optimizer to find the best-fit solution based on regression over all flow-periods in the test. Problems with more than two wells need to use the specialized "Multi-Well Interference Test" option.

Reservoir:          All reservoir boundary geometry's supported.

Reference:          D. O. Ogbe and W. E. Brigham, "A Model for Interference Testing with Wellbore Storage and Skin Effects at Both Wells", SPE paper 13253 presented at the SPE 59th Technical Conference, Sept. 16-19, 1984.

No example is given for this model.

The geometry for this model is shown in the following figure:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.17_files\image002.gif

Note that the origin of the co-ordinate system is at the "active" or "flowing" well. This origin is used for both the distances to the reservoir boundaries and to the observation point in the reservoir. The observation point is at "(x-location, y-location)" and is the location where pressure measurements have been made.

14) Multi-Layer Reservoir Without Cross-Flow

This model represents radial-flow for a multi-layer reservoir with the well completed in all layers. There is no cross-flow between layers in the reservoir, but there is cross-flow between layers in the well-bore. Each layer can be infinite acting or reservoir boundaries of constant pressure, partially transmissible, or no-flow type.     This model is available for 2, 3, or 4 reservoir layers.

Each reservoir layer can have a different initial reservoir pressure. The initial pressure distribution is for a well just prior to perforating i.e. the initial pressure is uniform and constant within each layer. The pressure distribution data is entered as a pressure differential between layer 1 and the layer of interest.    All dimensionless quantities are based on the maximum initial pressure implied by these differential pressures.

Note that this model tends to behave like the equivalent homogeneous system. It is recommended that this model be used only for deterministic problems i.e. you know what most of the layer properties are.

Results:              Storage coefficient, layer permeability's, layer skin factors, Omega's, pressure differentials between layers, layer drainage radii, and layer fog-factors

Static Data         Thickness, porosity, water saturation, and fluid compressibility of the layers. This sets default values of Omega. The thickness, porosity, etc. entered for the global well static data are updated to be consistent with the total thickness average values of the layer data entered for this model.

Specialised:        A fog-factor not equal to 1 or -1 indicates a radial permeability discontinuity. The outer permeability, 'ko' can be found from:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.18_files\image001.gif  (single, or channel)

           Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.18_files\image002.gif           (rectangular)

          

Reservoir:          Closed circular boundary only. Fog-factors determine the boundary type (1 = no flow, 0 = no boundary, -1 = constant pressure).

Reference:          Kuchuk, F. J. and Wilkinson, D. J.; "Transient Pressure Behaviour of Commingled Reservoirs"; SPE Paper 18125, SPE Houston Technical Conference, Oct. 1988.

          

          

15) Linear Composite Reservoir

This model describes an homogeneous reservoir with linear discontinuities in the reservoir properties. The geometry and parameters for a three zone model with boundaries is shown in the following figure. Up to five zones can be specified with or without reservoir boundaries.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.19_files\image001.gif

Note that the distances to the interfaces are signed directions i.e. a positive value lies to the right of the well and a negative value lies to the left.

Select this model by choosing a "vertical well" in a "homogeneous" reservoir with a "linear-composite" geometry. Options will be presented to select the number of zones and boundaries to use with the model.

          

16) Vertical Pulse-Test

This model represents radial flow in a homogeneous or double-porosity reservoir where there is a transition between flow over the perforation thickness to flow over the effective reservoir thickness. The pressure measurement location can be defined as a point vertically above or below the perforated interval.

The solution is obtained by numerical integration of a Green's function equation. Well-bore storage is added by Laplace transform.

Parameters:       Permeability, storage coefficient, perforation skin, Kv/Kh ratio, and effective reservoir thickness.

Static Data:        The perforation thickness, the distance between the perforation centre and the reservoir top, and the location of the pressure measurements.

Specialised:        Use the Optimiser to match data in all flow-periods. This will give a pulse test Cartesian plot of delta-p vs. delta-t with the reference point as the first pressure data in the vertical pulse test.

Reservoir:          All reservoir boundary geometry's are supported by de-superposition of Ei function image wells. This is not correct when boundary effects occur during the early-time period when flow is over the perforation thickness.

Reference:          Gringarten, A.C., Ramey, H.J.; "Unsteady-State Pressure Distributions Created by a Well with a Single Horizontal Fracture, Partial Penetration, or Restricted Entry"; SPEJ, August, 1974.

          

          

17) Fetkovich And Arps Decline-Curves

Decline curve analysis uses the dimensionless Fetkovich decline curves based on a closed circular reservoir production decline (exponential) and the empirical curves due to Arps. This model is used EXCLUSIVELY for analysis of well production decline given a pressure history. They cannot be used for analysis of pressure data. If you select this model for pressure interpretation, no type-curve is calculated.

Parameters:       Permeability, skin factor, drainage radius, and Arps "B" coefficient.

Static Data:        None.

Specialized:        This model is only applicable to the well-deliverability analysis and production prediction options.

Reservoir:          The implied geometry is a closed circle.

Reference:          Fetkovich, M. J.; "Decline Curve Analysis using Type-Curves", J. Pet. Tech. (June 1980); pg 1065-1077.

          

          

18) User-Defined Model

This "model" allows PIE to work with a tabulated type-curve function. The data for these tables are specified in an input data file using keywords and arrays of td and pd values. This model allows one parameter in addition to time match and pressure match, hence, a family of curves can be input. PIE will interpolate between curves and extrapolate a particular curve for large and small values of dimensionless time. Extrapolation for large Td is done on a semi-log basis, while extrapolation for small Td is done using log-log.

The input data file is a plain text file. This file has the following format:

LINE 1:        An 80 character string specifying the name of the type-curve.

LINE 2:        A 14 character string specifying the name of the parameter.

LINE 3:        Four numbers specifying the default log-log type-curve axis scales. These values are the log values i.e. -1 means 0.1, 1 means 10, etc. The first two values specify the x-axis minimum and maximum, and the last two values specify the y-axis minimum and maximum.

LINE 4:        Specify one of the following time match calculation keywords:

TDSK:  Time match is based on a skin factor

TDXF:  Time match is based on a length

TDCD:  Time match is based on a storage coefficient.

In addition, you can use the +LOG keyword to specify that the interpolation between parameter values is performed using logs.

$TD n:          This keyword denotes the start of an array of dimensionless time values. It must be on a line after LINE 4, and before any $CURVE keywords. The value of "n" that is on the $TD keyword line specifies the number of points in each type-curve. The maximum value is 1500.

ALWAYS ENSURE THAT THERE ARE SUFFICIENT POINTS TO MAKE A SMOOTH CURVE. IT IS RECOMMENDED THAT THIS BE AT LEAST 100 POINTS AND/OR 9 POINTS PER LOG CYCLE OF TIME. FAILURE TO DO THIS WILL RESULT DERIVATIVE CURVES THAT ARE NOT SMOOTH AND POOR OPTIMIZER OPERATION.

On the line immediately after this keyword, enter an array of dimensionless time values going from the smallest to largest value. There must be "n" values entered. You can enter more than one value on one line. Note that these are the values of dimensionless time and not the log values.

$CURVE p:   This keyword denotes the start of an array of the dimensionless pressure values to specify one curve for the model.

This keyword must be after a $TD keyword and there can be more than one $CURVE keyword in the file. The value of "p" specifies the type-curve parameter value for this curve. Each value in this array corresponds to the dimensionless time array read with the $TD curve i.e. the first Pd value corresponds to the first Td value of the $TD keyword.

On the line immediately after this keyword, enter an array of dimensionless pressure values going from the value at the smallest Td value to the value at the largest Td value. There must be "n" values as specified by the $TD keyword. You can enter more than one value on a line.

The following is an   input file for a horizontal fracture model. Note that this example has too few points per curve and an insufficient number of curves to define the range of Pd between curves. Use the above purely as an example of the data input format.:

Horizontal Fracture
Hd
-3          2           -2          2
TDXF +LOG
$TD 9
0.001 0.004 0.010 0.040 0.100 0.400 1.00 10.0 100.0
$CURVE 100.0
3.50 7.00 11.0 22.0 35.0 56.0 70.0 90.0 95.0
$CURVE 1.0
0.035 0.070 0.110 0.220 0.380 0.750 1.10 2.20 3.35
$CURVE .10
0.0037 0.0095 0.021 0.080 0.195 0.600 1.00 2.20 3.35

The following is a description of the basic model information as it applies to a user-defined model.

Parameters:       Time match result as per the TDSK, TDXF, or TDCD keyword, permeability, and the user defined parameter.

Static Data:        None.

Specialised:        None

Reservoir:          No reservoir boundaries supported; user is responsible for the addition of these behaviour to the data.

Reference:          None. It's all up to you.

          

19) Multi-Layer Horizontal-Well With Cross-Flow

This model represents a single horizontal well-bore in a multi-layer reservoir with cross-flow between the different reservoir layers. The flow between layers is fully transient i.e. this model computes the vertical transient flow through the different reservoir layers. A maximum of four layers can be defined in this model with the horizontal-well located in any of these layers. The following cross-section schematic shows the layout and parameters for this model.

NOTE: in the following, the thickness "H" shown as a denominator refers to the total reservoir thickness i.e. the sum of all the layer thickness values. The value "Zw" refers to the distance from the top of the reservoir to the mid-point of the horizontal drain.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.23_files\image001.gif

Note that the thickness for each layer is entered as static-data when this model is selected. The parameters for this model also include a "perforation skin" and a "well-bore storage" coefficient.

Select this model by choosing a "Horizontal-Well" in a "Multi-layer, Cross-Flow" reservoir type with a "Radial" geometry. This model does not include reservoir boundaries.

Reference:     Kuchuk, F.J.; "Pressure Behaviour of Horizontal Wells in Multilayer Reservoirs with Crossflow"; SPE 22731 presented at the 1991 SPE Annual Technical Conference in Dallas, October 6-9, 1991.

          

20) General Heterogeneity Radial/Linear Composite

This model can be used to represent a very complex set of reservoir heterogeneity's in three zones; a radial zone around the well, a linear zone to the right of the well, and another linear zone to the left of the well. The following plan-view schematic shows the layout and parameters for this model.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.24_files\image001.gif

The "R_Zone", "+X_Zone", and "-X_Zone" can contain up to 9 discontinuities in the reservoir properties. The discontinuities can be either piece-wise linear and/or step-wise changes in the properties. The variation of properties in each zone are specified in a table of normalised properties verses a normalised distance from the well-bore. The properties are normalised with respect to the reservoir at the well-bore and the distances are normalised with respect to the zone length.

There are several restriction on the use of this model:

a) The sum R_Zone start and length must be less than 1/2 the sum of the +y boundary distance and the -y boundary distance.

b) The distance to the +X_Zone or -X_Zone must be greater than the sum of the +y and -y boundary distances.

These restrictions come the de-superposition of a linear-flow solution for the behaviour in the -X_Zone and +X_Zone.

Select this model by choosing a "Vertical" well in a "General Heterogeneity" reservoir type with a "Linear-Composite" geometry.

          

21) General Heterogeneity Radial Composite

This model can be used to represent a complex radial-composite system with step-wise or piece-wise linear changes in reservoir properties. This model can be used to refine a match obtained using the radial-composite model where there is a complex transition in reservoir properties between the inner and outer zones (e.g. a water-flood). This model can also be used for interference-tests to compute the pressure response at an arbitrary point in the reservoir. The following plan-view schematic shows the layout and parameters for this model.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.25_files\image001.gif

The "R_Zone" can contain up to 9 discontinuities in the reservoir properties. The discontinuities can be either piece-wise linear and/or step-wise changes in the properties. The variation of properties in this zone are specified in a table of normalised values verses a normalised distance from the well-bore. The properties are normalised with respect to the reservoir at the well-bore and the distances are normalised with respect to the zone length.

Select this model by choosing a "Vertical" well in a "General Heterogeneity" reservoir type with a "Radial-Composite" geometry. Note that this model can be used for an interference-test by selecting an "Interference-Test" geometry.

This model is complex, therefore, the following example demonstrates how to setup this model and understand the behaviour of the model. The following figure is a "not to scale" schematic of a three-zone, infinite-acting (unbounded) reservoir i.e. the third zone extends to "infinity". The reservoir around the well has a permeability of 30 md and a porosity of 10%.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.25_files\image002.gif

The following reservoir properties are also defined foreach zone:

Zone 1: Permeability = 3 x 30 = 90 md
Porosity = 10%
Starts at Radius= 10 feet
(Properties are constant throughout the zone)

Zone 2: Permeability = 9 x 30 = 270 md
Porosity = 10 %
Starts at Radius = 100 feet
(Properties are constant throughout the zone)

Zone 3: Permeability = 27 * 30 = 810md
Porosity = 20%
Starts at Radius = 1000 feet
(extends to "infinity")

The total compressibility is 1e-4 psi-1 and the reservoir thickness is 82.1 feet.

After select this model by choosing a "Vertical" well in a "General Heterogeneity" reservoir type with a "Radial-Composite" geometry, a table is displayed where the heterogeneity data are entered to define each radial zone. For the above reservoir description in this example, the table should be filled in as shown below:

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.25_files\image003.jpg

Each zone is entered in a column in the table. The locations are in real distances, but are "nominal" values that can be scaled later by the "R-zone Start" and "R-zone length" parameters (this is discussed later).

The changes in permeability are entered as ratios relative to the value of permeability at the well (30 md in this case). In this example, the properties in each zone are constant throughout the zone, hence, the table needs to be set-up for a step-wise change in permeability. Note there are two permeability ratios for each location in the table. The first is the ratio that applies "up to" that location, and the second is the ratio that "begins" to be applied at that location. The values that begin at a location are tagged with a "[+]" symbol.

To help visualize the heterogeneity data, the permeability distribution is shown in the above table is plotted in the following figure highlighting the "Perm Ratio" and "Perm Ratio [+]" values.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.25_files\image004.gif

The porosity distribution is set-up in the same way as the permeability, with the "Porosity Ratio [+]" values set at the location where the step-change occurs at 1000 feet from the well.

The above table and figure can be read as follows:

Shown in the following figure is the derivative-plot of the response for the above example 3-zone model together with the model parameters "R_zone Start" and "R_zone Length". Note that the sum of these two parameters equals the 1000 foot distance to the location of zone-3. The derivative curve begins with a stabilisation corresponding to a 30 md reservoir with a 10% porosity. Once the radius-of-investigation (RI) reaches the start of zone-1, the response makes a transition over to a stabilization corresponding to a 90 md reservoir. When the RI reaches start of zone-2, a transition to a 270 md reservoir begins. Finally, when the RI reaches 1000 feet, the derivative response makes a transition to the zone-3 permeability of 810 md and a porosity of 20%.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.25_files\image005.gif

Shown in the next figure is a comparison of the three-zone model to a simple two-zone radial-composite model (plotted as data points), where the inner-zone permeability and porosity are 30 md and 10%, and the outer-zone properties are 810 md and 20%. Also shown on the figure are two curves. The first is for the 3-zone model shown above, and the second is for a smaller "R_zone length" value with the "R_zone start" adjusted so zone-3 starts at the same location 1000 feet from the well. The difference between the two cases of the 3-zone model are to "squeeze" the heterogeneity into a smaller distance while keeping the outer-edge of the heterogeneity constant at 1000 feet.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.25_files\image006.gif

As the "R_zone Length" becomes smaller, the effect of the 3-zone heterogeneities "collapses" to become a simple two-zone reservoir i.e. the net effect of the three-zones is just a step-change from the well-bore properties of 30 md and 10% porosity to the zone-3 properties of 810 md and 20%. In the limit as the zone-length becomes very small, the general heterogeneity model will become a simple radial-composite system.

Note that both 3-zone models reach the final stabilization at an earlier time than the simple radial-composite model. This is caused by the higher average permeability within the reservoir out to the start of zone-3, which means the RI reaches 1000 feet faster. As the zone length decreases, the RI takes longer to move through the larger volume of 30 md reservoir. Hence, stretching or shrinking the value of the "R_zone length" will also affect the time at which the heterogeneity effects show up in the response.

22) Multi-Layer, No-crossflow hydraulic-fracture

This model represents a layered reservoir with no cross-flow between layers in the reservoir i.e. commingled production from reservoirs separated by shale intervals. Each layer is completed with a hydraulic-fracture. The following cross-section schematic shows the layout and parameters for this model.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.26_files\image001.gif

Each layer can also include reservoir boundaries i.e. each layer can be a reservoir of a different size and shape. The boundary distances include "fog-factors" to control the type of boundary in each layer (e.g. -1 is a constant pressure boundary, 0 is no boundary, and 1 is a no-flow boundary).

Note that this model includes cross-flow through the well-bore. This model will correctly represent differential depletion between layers and will compute the pressure response caused by flow between layers with the well shut-in at the surface.

Select this model by choosing a "Hydraulic Fracture" well and a "Layered, No Cross-Flow" reservoir type with a "Radial" geometry.

Reference:     Kuchuk, F. J. and Wilkinson, D. J.; "Transient Pressure Behaviour of Commingled Reservoirs"; SPE Paper 18125, SPE Houston Technical Conference, Oct. 1988.

          

23) Multi-Layer, No-crossflow Horizontal-well

This model represents a layered reservoir with no cross-flow between layers in the reservoir i.e. commingled production from reservoirs separated by shale intervals. Each layer is completed with a horizontal-well. The following cross-section schematic shows the layout and parameters for this model.

NOTE: in the following, the thickness "H" shown as a denominator refers to the reservoir thickness of a particular layer. The value "Zw" refers to the distance from the top of a particular layer to the mid-point of a perforated interval within that layer.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.27_files\image001.gif   

Note that each horizontal-well is connected to a common well-bore as indicated by the dashed line on the figure. Because each layer is separated by a no-flow shale barrier, only the vertical position of the horizontal-wells in each layer needs to be specified.

Each layer can also include reservoir boundaries i.e. each layer can be a reservoir of a different size and shape. The boundary distances include "fog-factors" to control the type of boundary in each layer (e.g. -1 is a constant pressure boundary, 0 is no boundary, and 1 is a no-flow boundary). Note that the distances to the boundaries for each layer are measured from the centre of the horizontal-well in that layer.

Note that this model includes cross-flow through the well-bore. This model will correctly represent differential depletion between layers and will compute the pressure response caused by flow between layers with the well shut-in at the surface.

Select this model by choosing a "Horizontal-Well" and a "Layered, No Cross-Flow" reservoir type with a "Radial" geometry.

Reference:     Kuchuk, F. J. and Wilkinson, D. J.; "Transient Pressure Behaviour of Commingled Reservoirs"; SPE Paper 18125, SPE Houston Technical Conference, Oct. 1988.

          

24) Inclined-well in a multi-layer reservoir with cross-flow

This model represents a layered reservoir with cross-flow and an inclined-well penetrating one or more of the reservoir layers. The following cross-section schematic shows the layout and parameters for this model.

NOTE: in the following, the thickness "H" shown as a denominator refers to the total reservoir thickness i.e. the sum of all the layer thickness values. The value "Zw" refers to the distance from the top of the reservoir to the mid-point of a perforated interval.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.28_files\image001.gif

There are no restrictions on the number of layers the well-bore penetrates. A deviation angle of zero yields the behaviour for a vertical well and a deviation angle of 90 degrees yields the behaviour for a horizontal-well.

Note that this model uses very complex calculations (a combination of Laplace and Fourier transforms), hence, it is necessary to exercise some patience while waiting for the type-curve to be displayed.

Select this model by choosing an "Inclined Well" in a "Layered, Cross-Flow" reservoir type with a "Radial" geometry. This model does not include reservoir boundaries.

Reference:     Kuchuk, F.J.; "Pressure Behaviour of Horizontal Wells in Multilayer Reservoirs with Crossflow"; SPE 22731 presented at the 1991 SPE Annual Technical Conference in Dallas, October 6-9, 1991.

          

25) Multi-Lateral Well In A Multi-Layer Reservoir With Cross-Flow

This model represents a layered reservoir with cross-flow and multiple horizontal well-bores penetrating one or more of the reservoir layers. The following cross-section schematic shows the layout and parameters for this model.

NOTE: in the following, the thickness "H" shown as a denominator refers to the total reservoir thickness i.e. the sum of all the layer thickness values. The value "Zw" refers to the distance from the top of the reservoir to the mid-point of a perforated interval.

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There are two options for the plan-view layout of these lateral well-bores. The first is a 'star' formation where each horizontal segment is aligned back to a common origin:

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The second option is to arrange the lateral well-bores parallel to each other, but displaced to arbitrary (x,y) locations:

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Note that in both cases, the location of the origin is arbitrary and not fixed to any particular lateral segment. Also note that the offset distances are always to the centre of the lateral segment.

Note that this model uses very complex calculations (a combination of Laplace and Fourier transforms), hence, it is necessary to exercise some patience while waiting for the type-curve to be displayed.

Select this model by choosing an "Multi-Lateral Well" in a "Layered, Cross-Flow" reservoir type with a "Radial" geometry. This model does not include reservoir boundaries.

Reference:     Kuchuk, F.J.; "Pressure Behaviour of Horizontal Wells in Multilayer Reservoirs with Crossflow"; SPE 22731 presented at the 1991 SPE Annual Technical Conference in Dallas, October 6-9, 1991.

          

26) Multi-Layer Multi-Perforation

This model represents a multi-layer reservoir with cross-flow between the reservoir layers with one to four perforated intervals set at arbitrary depths and locations. This model can be used to analyse wells with selective perforations or deviated wells in a reservoir with a low vertical-to-horizontal permeability anisotropy ratio.

NOTE: in the following, the thickness "H" shown as a denominator refers to the total reservoir thickness i.e. the sum of all the layer thickness values.   The value "Perf.H" refers to the vertical length of the perforated interval. The value "Zw" refers to the distance from the top of the reservoir to the mid-point of a perforated interval.

There are two variations of this model. The first is to keep all of the perforated intervals aligned in a single vertical column. The following cross-section schematic shows the layout and parameters for this option.

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Cross-Section

The second variation of this model allows the perforated intervals to be positioned at an arbitrary location in the reservoir as shown in the following cross-section and plan-view schematic. Note that the origin is arbitrary and not fixed to any one perforated interval.

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Cross-Section

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Plan-View

Note that the perforated intervals can cut across the reservoir layer interfaces and all perforated intervals are connected to a common well-bore. The transient pressure response that travels vertically through the reservoir layers is calculated exactly (i.e. there are no "lambda" terms that assume pseudo-steady-state vertical flow between layers).

Select this model by choosing a "Partial-Penetration" well in a "Layered, Cross-Flow" reservoir-type with a "Radial" geometry. This model does not include reservoir boundaries.

Reference:     Kuchuk, F.J.; "Pressure Behaviour of Horizontal Wells in Multilayer Reservoirs with Crossflow"; SPE 22731 presented at the 1991 SPE Annual Technical Conference in Dallas, October 6-9, 1991.

          

27) Multi-Layer Horizontal-Well Interference-Test

This model computes the pressure response between two horizontal-wells in a multi-layered reservoir. One of the horizontal-wells is on production and the other is shut-in. The pressure response is computed at the shut-in horizontal-well. The following cross-section schematic shows the layout and parameters for this model.

NOTE: in the following, the thickness "H" shown as a denominator refers to the total reservoir thickness i.e. the sum of all the layer thickness values. The value "Zw" refers to the distance from the top of the reservoir to the mid-point of a perforated interval.

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The producing well can be positioned at an arbitrary depth and location from the observation well. The observation horizontal-well can be oriented along a particular azimuth relative to the producing well. The following plan-view schematic shows this layout.

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This model is selected by choosing a "Horizontal-Well" in a "Layered, Cross-Flow" reservoir type with an "Interference-Test" geometry. This model does not include reservoir boundaries.

Reference:     Kuchuk, F.J.; "Pressure Behaviour of Horizontal Wells in Multilayer Reservoirs with Crossflow"; SPE 22731 presented at the 1991 SPE Annual Technical Conference in Dallas, October 6-9, 1991.

          

28) Multi-Layer Multi-Perforation Interference-Test

This model computes the pressure response between up to three perforated intervals and an observation perforated interval in a multi-layer reservoir. The following cross-section schematic shows the layout and parameters for this model.

NOTE: in the following, the thickness "H" shown as a denominator refers to the total reservoir thickness i.e. the sum of all the layer thickness values.   The value "Perf.H" refers to the vertical length of the perforated interval. The value "Zw" refers to the distance from the top of the reservoir to the mid-point of a perforated interval.

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.32_files\image001.gif     

Note that producing well interval is the origin of the co-ordinate system. The producing perforated intervals are constrained to a single (x,y) location, but can be positioned at an arbitrary vertical position in the reservoir. This layout is shown in the following schematic.

          

Description: Description: Description: Description: Description: C:\Users\Mike Wilson\smcd\pie\manual\htmlhelp\html\chapter_8.32_files\image002.gif     

This model is selected by choosing a "Partial Penetration" well in a "Layered, Cross-Flow" reservoir with an "Interference-Test" geometry. This model does not include reservoir boundaries.

Reference:     Kuchuk, F.J.; "Pressure Behaviour of Horizontal Wells in Multilayer Reservoirs with Crossflow"; SPE 22731 presented at the 1991 SPE Annual Technical Conference in Dallas, October 6-9, 1991.

          

29) Inclined-Well Interference-Test

This model computes the pressure response between two inclined-wells in a homogeneous or double-porosity reservoir. One of the inclined-wells is flowing and the other is shut-in. The pressure response is computed at the shut-in inclined-well. The following cross-section schematic shows the layout and parameters for this model.

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Note that this model supports reservoir boundaries, and the distances to each boundary are measured from the origin at the producing well. Each inclined-well has an azimuth angle relative to the reservoir boundaries as shown in the following plan-view.

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Note that the x-location and y-location distances are measured from the mid-point of the producing inclined-well to the mid-point of the observation well.

Select this model by choosing an "Inclined-Well" in a "Homogeneous" or "Double-Porosity" reservoir with an "Interference-Test" geometry.

References: Cinco, H., Miller, F. G., Ramey, H. J.; "Unsteady-State Pressure Distribution Created By a Directionally Drilled Well"; Journal of Petroleum Technology (JPT), Nov. 1975, pg. 1392.

                   Kuchuk, F. J. and Wilkinson, D. J.; "Transient Pressure Behaviour of Commingled Reservoirs"; SPE Paper 18125, SPE Houston Technical Conference, Oct. 1988.

          

30) Partial-Penetration Hydraulic-Fracture

This model computes the pressure response for an infinite-conductivity hydraulic-fracture which has a vertical height less than the reservoir thickness (partial penetration). This model includes a "fracture skin" which represents a zone of reduced formation permeability on the face of the hydraulic fracture. The following cross-section schematic shows the layout and parameters for this model.

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Select this model by choosing an "Hydraulic-Fracture" well in a "Homogeneous" or "Double-Porosity" reservoir with an "Radial" reservoir geometry. From the list of different hydraulic-fracture models, select the "partial-penetration" model.

References: "The Use of Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs", A. C. Gringarten and H. J. Ramey., SPE paper 3818, 1973

          

          

31) Linear-Composite Interference-Test

This model computes the interference response at a shut-in observation well caused by flow at an active well. Up to five zones can be specified for the linear-composite reservoir and the observation well can be positioned at any point in the reservoir. The model can include various reservoir boundary geometry's. The following schematic shows a three-zone reservoir:

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Note that the active well is the origin for the "x-location, y-location" point in the reservoir, the distances to the interfaces, and the distances to the reservoir boundaries.

Select this model by choosing a "vertical well" in a "homogeneous" reservoir with a "linear-composite interference" geometry.

          

32) Linear-Composite Multi-Lateral Well

This model computes the pressure response for a multi-lateral well in a linear-composite reservoir. Each of the producing segments are represented by a vertical well positioned at an arbitrary location in the linear-composite reservoir. Up to four producing segments and five reservoir zones can be specified. This model also includes various reservoir boundary geometry's. The following figure show a three zone reservoir with two producing segments.

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Note that the first producing segment is always located at the origin. Select this model by choosing a "multi-lateral" well in a "homogeneous" reservoir with a "Linear-composite" geometry.

          

34) Linear-Composite Multi-Lateral Interference-test

This model computes the interference pressure response at an observation well caused by production at a multi-lateral well in a linear-composite reservoir. Each of the producing segments are represented as a vertical well positioned at an arbitrary location in the linear-composite reservoir. Up to four producing segments and five reservoir zones can be specified. This model also includes various reservoir boundary geometry's. The geometry and layout of this model is identical to the linear-composite multi-lateral well.

Note that the first producing segment is always located at the origin. Select this model by choosing a "multi-lateral" well, "homogeneous" reservoir, and a "linear-composite interference" reservoir geometry.

          

35) General Multi-Layer No Cross-Flow Model

This model represents a multi-layer system where there is no cross-flow between layers in the reservoir. There is cross-flow between layers through the well-bore. Each layer can be one of a number of possible well/reservoir types ranging from a simple vertical well in a homogeneous reservoir to a horizontal-well or linear-composite reservoir.

Select this model from the "Type-Curve Model" option under the "ANALYSIS" pull-down menu. Then select a "General" well-type in a "Layered, no cross-flow" reservoir with a "Radial" geometry. This selection will result in a prompt for the number of layers, a table to define the "static" properties for each layer, and an option to include reservoir boundaries in the model. The static properties for the layered system are entered on a table where each layer has a "Set Model" and a "Set Prop." button.

The "Set Model" button will display a list of well and reservoir models to apply to a particular layer. By default, a homogeneous model is used for each layer.

The "Set Prop." button displays a small table where the thickness, porosity, fluid compressibility, etc. values are entered for a layer. These values are used to define initial "omega" values for the multi-layer system where "Omega" is defined as:

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These values are used in the type-curve simulation for this model and can be altered as a model-parameter in the analysis.

A model and static properties must be entered for each layer. Once these data are defined, the analysis constants are updated to reflect the layer-average static properties.

The option to include or not include reservoir boundaries applies the model in all layers i.e. each layer can have an individual reservoir geometry and size. When boundaries are included, the default boundary geometry for each layer is a closed rectangle. Each boundary in each layer has an associated "fog-factor" (i.e. transmissibility factor) where -1 represents a constant pressure boundary, 0 is no boundary, and 1 is a no-flow (sealing) boundary.

The "fog factors" are set along with the other model parameters in the analysis after selecting the model. Use these "fog-factors" to derive other boundary geometry's from the default closed rectangle. The boundary distances and the "fog factors" are set during an analysis using the "Model Parameters" option under the "ANALYSIS" pull-down menu.

Reference: Kuchuk, F. J. and Wilkinson, D. J.; "Transient Pressure Behaviour of Commingled Reservoirs"; SPE Paper 18125, SPE Houston Technical Conference, Oct. 1988.

          

          

36) Interference-Test in a Multi-Layer Reservoir with Cross-Flow

This model represents a layered reservoir with pseudo-steady-state cross-flow between the layers controlled by "lambda" coefficients at each layer interface. After selecting a "vertical well" in a "layered with cross-flow" reservoir using a "interference-test" geometry, a table of properties for each layer are entered in order to define the value of "Omega" (the fraction of porosity-compressibility-thickness in each layer). This table of layer values is used to update the analysis constants to represent the thickness-averaged values for the whole system.

The pressure at an arbitrary (x,y) location in the reservoir is reported for a particular layer. This model can include reservoir boundaries. The areal layout for this model is shown in the following figure.

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Each layer is given values to define the flow into the active well-bore at the origin of the co-ordinate system. The pressure response at the (x,y) location is reported for a specified layer number.

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37) Radial-Composite Interference-Test

This model computes the pressure response at an arbitrary (x,y) location within a radial-composite reservoir. After selecting a "vertical well" in a "homogeneous" reservoir with a "Radial Comp. Interf." geometry, there is an option to add reservoir boundaries to this model. The radial-composite reservoir contains two zones; the inner-zone containing the active well and an outer-zone. The mobility ratio is defined as the outer-zone permeability-thickness divided by the inner-zone permeability-thickness. Similarly, the storivity ratio is the outer-zone porosity-compressibility-thickness product divided by the inner-zone value. The areal layout of this model is shown in the following figure. Note that the active well defines the origin of the co-ordinate system and the distances to the reservoir boundaries.

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38) Water Injection Model

This model computes the displacement of reservoir fluid by the injection of water. Note that this is a semi-analytic solution to the general two-phase flow problem and uses a very complicated algorithm to solve this non-linear problem. The resulting solution is specific to the injection history specified for the test. It is important to note that this model does NOT have a simple "draw-down" type-curve solution and is NOT used with convolution to simulate a pressure response. After selecting an "Injection" well in a "Homogeneous" reservoir with a "Radial" geometry, a number of extra reservoir and fluid parameters are specified for this model. Follow the instructions carefully.

Note that the rate-history for the test MUST specify injection or shut-in flow periods (i.e. all rates are either negative or zero). The analysis constants MUST specify a non-zero water compressibility.