Unconventional Reservoir Theory

1/30/2014 Unconventional Reservoir Theory

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Production Behaviour of Unconventional Reservoirs

The term 'unconventional' refers to production from very low-permeability formations. This often includes shale and very tight sands. When the

permeability of a formation is very low, an enormous amount of surface area is required between the well completion and the reservoir to be able to

produce at economic rates. This required area is achieved through multiple stages of massive fracture stimulations. The flow regime most often

observed in these wells is linear flow. Linear flow may last several years and is often the only flow regime observed in analysis. In addition to fractures

created in the reservoir, linear flow can be caused by natural fracturing or enhancements to the natural fractures. For more information see Fracture

Properties.

The effective drainage boundaries for these wells often coincide with the fracture length. In other words, drainage beyond the stimulated region is not

significant and in the analysis method presented, this contribution is neglected. The base reservoir geometry used to develop this method is that of a

single fracture centered in a rectangular reservoir, where the fracture width extends to the reservoir boundary. In this geometry, linear flow is observed

until the reservoir width (Ye) boundary is reached, and the well enters boundary-dominated flow. The reservoir geometry is shown below:

Constant Flowing Pressure

Current analysis techniques in the industry use material balance time. Since material balance time is actually boundary-dominated flow superposition

time, these analyses may appear to show boundary-dominated flow even when the reservoir is still exhibiting transient flow. In the method presented,

no superposition functions are used, which avoids a bias towards any flow regimes. The production data is analyzed using a plot of reciprocal rate

versus square-root time. In this plot, linear flow appears as a straight line trend. The basic assumption is of infinite conductivity in the fracture; finite

conductivity manifests as a positive intercept on the plot. The equations presented are based on the assumption of a constant flowing pressure at the

well. This is a reasonable simplification for tight gas and shale production, in which wells are typically produced under high drawdown.

Based on the straight-line behaviour of the square root time plot, the simplest form of the linear flow equation is:

(Equation 1)

In this equation, the intercept captures a number of near well effects such as skin and finite fracture conductivity, and the slope is given by:

For oil:

(Equation 2a)



For gas:

(Equation 2b)

From the slope of the equation, fracture half-length and permeability are determined as a single product. To determine either explicitly, the other

parameter must be known.

To determine the skin effect, the intercept of the line is used with the following:

For oil:

For gas:

Duration of Linear Flow

When a well is producing under constant flowing pressure, the distance of investigation can be obtained from the following equation during the linear

flow period:

(Equation 3)

Combining this equation with the reservoir geometry detailed in the diagram presented previously, the end of linear flow is given by the following:

(Equation 4)

Equation 4 can be rearranged as follows:

(Equation 5)

This is not a practical form of the equation, since permeability (k) and reservoir width (Ye) are usually not explicitly known. The permeability is tied

into the (xfk) term and reservoir width is related to the drainage area (A); however, both of these are related to fracture half-length (xf). Using the

definition of drainage area (A = 2 * xf * Ye), equation 5 becomes:



(Equation 6)

The (xfk) term is determined from the slope of the square-root time plot using a rearranged form of equation 2. For unconventional gas reservoirs,

xfk departs from its analytical value as the pressure drawdown, DD, becomes higher. To correct for this drawdown effect, a correction factor, fcp, is

implemented in the gas constant pressure module, allowing for a more accurate determination of xfk.

DD and fcp are defined as follows:

xfk can be determined by using following equations:

For oil:

(Equation 7a)

For gas:

(Equation 7b)

Substituting equation 7 into equation 6, the duration of linear flow (and hence the beginning of boundary-dominated flow) is determined using the

following equation:

For oil:

(Equation 8a)

For gas:

(Equation 8b)

An approximation of the gas equation can be obtained by replacing pseudo-pressure terms with:

and cti by the inverse of pi

The major unknown in determining the end of linear flow is the effective drainage area. Analog wells can be used to determine an appropriate range of

drainage areas for the subject well. Well spacing can provide an upper bound on drainage area in high-density developments. Interpretation of flowing

material balance can be used to determine a minimum area based on the current production data.



Modifications in RTA

The above derivations for the linear flow period use (xfk). In the module interface, this term is replaced with (Ac * k). This area is the total fracture

area, or (4 * xf * h).

Boundary-Dominated Flow

Given the geometry of the reservoir considered, linear flow is followed directly by boundary-dominated flow. There are two ways to represent this flow

regime for purposes of forecasting: (a) pseudo-steady state equations, material balance time, and pseudo-time; (b) traditional (Arps) hyperbolic

decline. In the interest of keeping the method simple and practical, the hyperbolic decline method is used. Hyperbolic decline is defined in the

following equation:

(Equation 9)

In this equation, qi and ai are flow rate and decline rate (respectively) at the start of the forecast period; and t is the time that has elapsed since the

start of the forecast. Since the hyperbolic decline forecast starts at the end of linear flow, the flow rate, the decline rate, and time will be with respect

to time at the end of linear flow (telf). The decline exponent (b) is selected to be between 0 and 0.5, which are typical values for boundary-dominated

flow in gas. Equation 9 is then re-written to represent this:

(Equation 10)

Equation (1) is re-written to show how rate at the end of linear flow (qelf) is determined based on the interpretation of the square-root time plot:

(Equation 11)

Decline rate from the hyperbolic decline equation is defined as:

(Equation 12)

In order to determine decline rate at the end of linear flow (aelf), an expression is needed for decline rate during linear flow. This is obtained by

combining Equation 1 and Equation 12:

(Equation 13)

This equation is now written to reflect the decline rate at the end of linear flow:

(Equation 14)

From Equation 14, it is also apparent that if b' is negligible, the equation reduces to:

(Equation 15)

Once a decline exponent (b) is selected, Equation 10 can be used for any duration of forecast and even to determine production forecast based on an

abandonment rate.

Variable Flowing Pressure



In the initial development of this interpretation technique, constant flowing pressure has been assumed. An additional case for variable flowing

pressure may also be considered when interpreting the square-root time plot. The standard y-axis of reciprocal rate is simply replaced with

normalized pressure to account for changes in flowing pressure during the linear flow portion of the data. Transient linear flow appears to be a straight

line in the normalized pressure vs. square-root of time plot. The slope of the straight line, m, is used to calculate xfk with the following:

For oil:

For gas:

As stated earlier, the observation has been that most unconventional wells are produced at very high drawdown to maximize recovery. Because of

this common practice, the forecasting approach in the variable pressure module uses hyperbolic decline with the assumption of constant pressure for

the duration of the forecast.

Superposition Time

Superposition is a mathematical tool that allows us to use simple solutions (such as constant rates) to produce complex ones (such as variable

rates). Superposition uses the theory that a rate that changes from q1 at time t to a new rate q2 is equivalent to q1 continuing forever, superposed, or

added on (q2 - q1) starting at time t and continuing forever.

For unconventional reservoirs, two superposition time functions are commonly used in production analysis: linear superposition time and material

balance time.

Linear superposition time is given by:

Material balance time is given by:

Both two superposition time functions can effectively convert variable rates to their equivalent constant rates solutions. Therefore, the plot of

versus linear superposition time (or square root of material balance time) on a Cartesian graph results in a straight line. The slope of this line (m) can



be used to calculate :

For oil:

For gas:

Application to Horizontal Wells

The majority of wells in unconventional gas production are horizontal wells with multiple stage fracturing. The approach first considered with the

development of this method was to treat the horizontal wellbore as a fracture and incorporate fracture stimulation in this well as part of the effective

permeability interpreted on the square-root time plot. An additional method has been incorporated using the same process as that developed for a

single fracture. If the horizontal well is cased or contributes very little production compared to the fractures, the well can be treated as a series of

fractures in the reservoir. In a well with n equally spaced and identically sized fractures, the single fracture model can be multiplied by n fractures to

represent the whole well. The linear flow into the fractures is followed by boundary-dominated flow when no-flow boundaries are formed between

adjacent fractures. This layout is displayed below with its relationship to the single-fracture model.

Most Likely Model

The following is a new methodology for identifying the possible range of stimulated reservoir volume (SRV) configurations based on specified well

performance data, assumed system geometry, and known external system constraints.

The method is based on providing single values for known completion and petrophysical properties, and lower/upper limits for unknown parameters.

The method is suitable for early well performance data, where transient linear flow is dominant.

The method is graphical and involves a plot of exposed area of the fractures (Ax) versus drainage area of the reservoir (Ad). Based on user-defined

constraints and limits, this graph defines a region which contains the full set of possible SRV configurations.



The definition of the allowable range of Ax versus Ad is accomplished in two parts, as follows:

Part 1: Identify the region that defines minimum and maximum values for Ax and Ad

Limiting values for Ax are determined as follows:

Where Ak is extracted from the slope of the square root straight line on the specialized plot, and km(min) and km(max) define the acceptable range of

the unknown parameter Stimulated Region Permeability. The parameter km is considered to be an effective permeability inclusive of any natural or

induced fractures that have been activated (or are naturally active) and are connected to the primary hydraulic fracture network, but it does not include

the primary fracture permeability. In other words, it is the permeability that acts directly on the primary hydraulic fracture system. Ax is defined as the

total matrix area exposed to fractures. For a fracture of half-length (xf), Ax is defined as:

Ax = 4xfhf

Limiting values for Ad are determined as follows:

Ad(min) = Ad(tp)

if Ad is in acres

Where Ad(tp) equals the currently contacted drainage area, which is determinable from a combined analysis of the square root straight line and the

flowing material balance (Anderson et al. 2010). Y equals the projected average distance between adjacent laterals once the field has been fully

developed.

Based on the above, a box-shaped region can be defined on a graph of Ax versus Ad.

Part 2: Identify the functional relationship between Ax and Ad



Assuming parallel transverse fractures (slab model) of uniform distribution and length, Ad is a linear function of Ax as follows:

If the fracture network is assumed to be 2D (matchstick geometry) the coefficient in the above equation becomes 4. If the fracture network is

assumed to be 3D (sugar cube geometry), the coefficient becomes 6.

Since hf and nf are unknowns, we can define minimum and maximum functional relationship between Ax and Ad, assuming that we can provide

allowable ranges for hf and nf.

The above equations plot as straight lines through the origin on the Ax versus Ad plot. The intersection of the previously defined rectangular region

with these straight lines yields a region that contains all the possible Ad and Ax combinations.

Formulae for calculating reservoir/completion properties, given a sample configuration (Ad1, Ax1) from the Ax-Ad region, are shown as follows:

The mass center of the Allowable Ax vs Ad region is used to calculate the Most Likely values.

Copyright 2013 IHS Inc.

Unconventional Reservoir Theory

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