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# Fluid flow in hydraulically fractured wells

Many wells, particularly gas wells in low-permeability formations require hydraulic fracturing to be commercially viable. Interpretation of pressure-transient data in hydraulically fractured wells is important for evaluating the success of fracture treatments and predicting the future performance of fractured wells. This page includes graphical techniques for analyzing post-fracture pressure transient tests after identifying several flow patterns that are characteristic of hydraulically fractured wells. Often, identification of specific flow patterns can aid in well test analysis.

## Flow patterns in hydraulically fractured wells

Five distinct flow patterns (**Fig. 1**) occur in the fracture and formation around a hydraulically fractured well.^{[1]} Successive flow patterns, which often are separated by transition periods, include fracture linear, bilinear, formation linear, elliptical, and pseudoradial flow. Fracture linear flow (**Fig. 1a**) is very short-lived and may be masked by wellbore-storage effects. During this flow period, most of the fluid entering the wellbore comes from fluid expansion in the fracture, and the flow pattern is essentially linear.

Because of its extremely short duration, the fracture linear flow period often is of no practical use in well test analysis. The duration of the fracture linear flow period is estimated by^{[1]}

where *t*_{LfD} is dimensionless time in terms of fracture half-length,

The dimensionless fracture conductivity, *C*_{r}, is

and *η*_{fD} is dimensionless hydraulic diffusivity defined by

Bilinear flow (**Fig. 1b**) evolves only in finite-conductivity fractures as fluid in the surrounding formation flows linearly into the fracture and before fracture tip effects begin to influence well behavior. Fractures are considered to be finite conductivity when *C*_{r} < 100. Most of the fluid entering the wellbore during this flow period comes from the formation. During the bilinear flow period, BHP, *p*_{wf}, is a linear function of t^{1/4} on Cartesian coordinates.

A log-log plot of (*p*_{i} – *p*_{wf}) as a function of time exhibits a slope of 1/4 unless the fracture is damaged. The pressure derivative also has a slope of 1/4 during this same time period. The duration of bilinear flow depends on dimensionless fracture conductivity and is given by **Eqs. 5a** through **5c**^{[1]} for a range of dimensionless times and fracture conductivities:

Formation linear flow (**Fig. 1c**) occurs only in high-conductivity (*C*_{r} ≥ 100) fractures. This period continues to a dimensionless time of *t*_{LfD} ≅ 0.016. The transition from fracture linear flow to formation linear flow is complete by a time of *t*_{LfD} = 10^{–4} . On Cartesian coordinates, p wf is a linear function of t ^{1/2} , and a log-log plot of (*p*_{i} – *p*_{wf}) has a slope of 1/2 unless the fracture is damaged. The pressure derivative plot exhibits a slope of 1/2. Elliptical flow (**Fig. 1d**) is a transitional flow period that occurs between a linear or near-linear flow pattern at early times and a radial or near—radial flow pattern at late times.

Pseudoradial flow (**Fig. 1e**) occurs with fractures of all conductivities. After a sufficiently long flow period, the fracture appears to the reservoir as an expanded wellbore (consistent with the effective wellbore radius concept suggested by Prats *et al.*^{[2]}). At this time, the drainage pattern can be considered as a circle for practical purposes. (The larger the fracture conductivity, the later the development of an essentially radial drainage pattern.) If the fracture length is large relative to the drainage area, then boundary effects distort or entirely mask the pseudoradial flow regime. Pseudoradial flow begins at *t*_{LfD} ≅ 3 for high-conductivity fractures (*C*_{r} ≥ 100) and at slightly smaller values of *t*_{LfD} for lower values of *C*_{r}.

These flow patterns also appear in pressure-buildup tests and occur at approximately the same dimensionless times as in flow tests. The physical interpretation is that the pressure has built up to an essentially uniform value throughout a particular region at a given time during a buildup test. For example, at a given time during bilinear or formation linear flow, pressure has built up to a uniform level throughout an approximately rectangular region around the fracture. At a later time during elliptical flow, pressure has built up to a uniform level throughout an approximately elliptical region centered at the wellbore. At a given time during pseudoradial flow, pressure has built up to a uniform level throughout an approximately circular region centered at the wellbore. The area of the region and the pressure level within that area increase with increasing shut-in time. **Example 1** illustrates how to estimate the duration of flow periods for hydraulically fractured wells.

### Example 1: Estimating duration of flow periods in a hydraulically fractured well

For each case, estimate the end of the linear flow period and the time at which pseudoradial flow period begins. Assume that pseudoradial flow begins when *t*_{LfD} = 3. **Table 1** gives the data for each case.

*Solution*. The end of the linear flow regime occurs at a dimensionless time of *t*_{LfD} ≅ 0.016 or, using **Eq. 2**,

Similarly, the time to reach pseudoradial flow is *t*_{LfD} ≅ 3, or

**Table 1** summarizes the results.

## Flow geometry and depth of investigation of a vertically fractured well

Fluid flow in a vertically fractured well has been described using elliptical geometry. ^{[3]} The equation for an ellipse with its major axis along the *x*-axis and minor axis along the *y*-axis is

where the endpoints of the major and minor axes are (±*a*_{f}, 0) and (0, ±*b*_{f}), respectively. The foci of the ellipse are ±*c*_{f} where *c*_{f}^{2} = *a*_{f}^{2} – *b*_{f}^{2}. In terms of a well with a single vertical fracture with two wings of equal length, *L*_{f}, the relation becomes *L*_{f}^{2} = *a*_{f}^{2} – *b*_{f}^{2}, where *L*_{f} is the focal length of the ellipse. Fig. 8.76 shows the elliptical geometry of a vertically fractured well.

Hale and Evers^{[3]} defined a depth of investigation for a vertically fractured well. Their definition is based on a definition of dimensionless time at a distance *b*_{f}, the length of the minor axis:

Solving for the length of the minor axis,

Assuming that pseudosteady-state flow exists out to distance, *b*_{f}, at dimensionless time *t*_{bD} = 1/*π* as in linear systems, **Eq. 8** becomes

which represents the depth of investigation in a direction perpendicular to the fracture at time, *t*, for a vertically fractured well. In gas wells, the terms *μ* and *c*_{t} should be and , evaluated at average drainage-area pressure, .

The elliptical pattern of the propagating pressure transient can be fully described in terms of the lengths of the major axis, *a*_{f}, the minor axis, *b*_{f}, and the focus, *L*_{f}. Using the estimate of *b*_{f} from **Eq. 9** and an estimate of *L*_{f} obtained by one of the methods described in sections that follow, the length of the major axis can be estimated from

Given values of *a*_{f} and *b*_{f}, the depth of investigation at a particular time, *t*, in any direction from the fracture can be calculated using **Eq. 6**. Furthermore, the area, *A*, enclosed by the ellipse at time, t (the area of the reservoir sampled by the pressure transient), is given by

The coefficient 0.0002878 in **Eq. 9** is strictly correct only for highly conductive fractures (*C*_{r} ≥ 100). As *C*_{r} becomes smaller, the ratio *a*_{f}/*b*_{f} also becomes smaller. The lower bound of *a*_{f}/*b*_{f} is 1 (a circle) as *C*_{r} approaches 0.

## Fracture damage

Two major types of fracture damage are frequent: choked fracture damage and fracture-face damage. The choked-fracture damage means that the fracture has a reduced permeability in the immediate vicinity of the wellbore (**Fig. 3**). In this case, *k*_{f} is used for the permeability in the propped portion of the fracture farther along the wellbore, and *k*_{fs} for reduced permeability near the wellbore, out to a length, *L*_{s}, in the fracture.

The choked-fracture skin factor, *s*_{f}, is^{[4]}

Fracture face damage in a hydraulically fractured well (**Fig. 4**) is a permeability reduction around the edges of the fracture, usually caused by invasion of the fracture fluid into the formation or an adverse reaction with the fracturing fluid. The equation for fracture face skin is^{[4]}

## Specialized methods for post-fracture well-test analysis

Generally, the objectives of post-fracture pressure-transient test analysis are to assess the success of the fracture treatment and to estimate:

- The fracture half-length
- Fracture conductivity
- Formation permeability

Three specialized methods of analyzing these post-fracture transient tests are included here:

- Pseudoradial flow
- Bilinear flow
- Linear flow

### Bilinear flow method

The bilinear flow method^{[5]} applies to test data obtained during the bilinear flow regime in wells with finite-conductivity vertical fractures. Bilinear flow is indicated by a quarter-slope line on a log-log graph of pressure derivative vs. *t* or Δ*t*_{e}.

During bilinear flow,

The following procedure is recommended for analyzing test data obtained in the bilinear flow regime (that is, data in the time range with quarter slope on the diagnostic plot). In Step 1, note the use of "bilinear equivalent time," Δ*t*_{Be}. Radial equivalent time is rigorously correct as a plotting function only for infinite-acting radial flow.

- For a constant-rate flow test, plot
*p*_{wf}vs.*t*^{1/4}on Cartesian coordinates. For a buildup test, plot*p*_{ws}vs. Δ*t*_{Be}^{1/4}, where - Determine the slope,
*m*_{B}, of the straight line region of the plot. - Determine the pressure extrapolated to time zero,
*p*_{o}, and the fracture skin,*s*_{f}, from - From independent knowledge of
*k*(for example, from a prefracture well test), estimate the fracture conductivity,*w*_{f}*k*_{f}, using*m*_{B}and the relationship

**Fig. 5** is an example of bilinear flow analysis. The bilinear flow analysis method has the following important limitations.

- No estimate of fracture half-length,
*L*_{f}. - In wells with low-conductivity fractures, wellbore storage frequently distorts early test data for a sufficient length of time so that the quarter-slope line characteristic of bilinear flow may not appear on a log-log plot of test data.
- An independent estimate of
*k*is required. This suggests that prefracture well tests should be conducted before fracturing the well, thus obtaining independent estimates of formation properties.

### Linear flow method

The linear flow method^{[5]} applies to test data obtained during formation linear flow in wells with high-conductivity fractures (*C*_{r} ≥ 100). After wellbore storage effects have ended, formation linear flow occurs up to a dimensionless time of *t*_{LfD} = 0.016, which means that a log-log plot of pressure derivative against time will have a slope of one-half. The plot of pressure change vs. time, however, will have a half-slope only if the fracture skin is zero. The pressure and pressure derivative are

so that

which indicates that a log-log plot of the derivative against time will have a slope of one-half. Radial equivalent time applies rigorously only for radial flow in an infinite-acting reservoir. When linear flow is the flow pattern occurring at both times (*t*_{p} + Δ*t*) and Δ*t*, a more useful equivalent time function is the linear equivalent time, Δ*t*_{eL}.

Test conditions in which linear flow occurs at both (*t*_{p} + Δ*t*) and Δ*t* are rare, and, consequently, **Eq. 22** is not necessarily rigorously correct for well-test analysis. Fortunately, when *t*_{p} >> Δ*t*_{max}, Δ*t*_{eL} ≈ Δ*t*. **Fig. 6** is an example of a plot used in linear flow analysis.

The linear flow analysis method also has limitations.

- The method applies only for fractures with high conductivities. Strictly speaking, linear flow occurs for the condition of uniform flux into a fracture (same flow rate from the formation per unit cross-sectional area of the fracture at all points along the fracture) rather than for infinite fracture conductivity. Therefore, only very early test data (
*t*_{Lf D}≤ 0.016) exhibit linear flow in a high-conductivity fracture. - Some or all of these early data may be distorted by wellbore storage, further limiting the amount of linear-flow data available for analysis.
- Estimating fracture half-length requires an independent estimate of permeability,
*k*, which suggests the need for a prefracture well test.

### Pseudoradial flow method

The pseudoradial flow method applies when a short, highly conductive fracture is created in a high-permeability formation, so that pseudoradial flow develops in a short time. The time required to achieve pseudoradial flow for an infinitely conductive fracture (*C*_{r} ≥ 100) in either a flow test or a pressure buildup test is estimated by

The beginning of pseudoradial flow is characterized by the flattening of the pressure derivative on a log-log plot and by the start of a straight line on a semilog plot. Hence, when the pseudoradial flow regime is reached, conventional semilog analysis can be used to calculate permeability and skin factor. For a highly conductive fracture, skin factor is related to fracture half-length by^{[2]}

**Fig. 7** shows an example.

A recommended procedure for analyzing test data from the pseudoradial flow regime is as follows.

- For a drawdown test, plot
*p*_{wf}vs. log*t*. For a buildup test, plot*p*_{ws}vs. the Horner time ratio (HTR). - Determine the position and slope,
*m*, of the semilog straight line and the intercept,*p*_{1hr}on the line. - Using
*m*, calculate values of*k*and*s*(or*s*′ for a gas well). - Calculate the fracture half-length,
*L*_{f}, using**Eq. 24**.

The pseudoradial flow method has the following limitations that seldom make it applicable in practice. ^{[5]}

- The conditions that are most favorable for the occurrence of pseudoradial flow are short, highly conductive fractures in high-permeability formations. These formations, however, are rarely fractured. The most common application of hydraulic fractures—wells with long fractures in low-permeability formations—require impractically long test times to reach pseudoradial flow.
- For gas wells, the apparent skin factor,
*s*′, calculated from test data is often affected by non-Darcy flow. - The pseudoradial method applies only to highly conductive (
*C*_{r}≥ 100) fractures. For lower conductivity fractures, fracture lengths calculated using the skin factor (**Eq. 24**) will be too low.

## Using type curves for hydraulically fractured wells

Type curves are the most common method of analyzing hydraulically fractured wells. The independent variable for most type curves for analyzing hydraulically fractured wells is the dimensionless time based on hydraulic fracture half-length, *t*_{Lf D}. The dependent variable is usually the dimensionless pressure, *p*_{D}.

For type curves used for manual type-curve matching, most vary only one parameter. The Cinco type curve^{[1]} is obtained for zero *C*_{Lf D} and *s*_{f} ; the only parameter is dimensionless fracture conductivity, *C*_{r} or *F*_{cD} (where *F*_{cD} = *πc*_{r}). The choked-fracture skin is analyzed by assuming *C*_{Lf D} and infinite *C*_{r} with single parameter *s*_{f}. The wellbore-storage type curve^{[6]} sets *s*_{f} to 0 and *C*_{r} (*F*_{cD}) to infinity and varies the coefficient *C*_{Lf D}.

When using type curves in commercial software, the computer can set any two of the three parameters to fixed values (other than their limiting values) and vary the third parameter to obtain the matching stems.

### Procedures for analyzing fractured wells with type curves

The following steps outline the procedure for analyzing fractured wells with type curves.

- Graph field data pressure change and pressure derivatives.
- Match field data to the appropriate type curve.
- Find the match point and matching stem.
- Calculate the formation permeability from the pressure match point.
- Calculate
*L*_{f}from the time match point. - Interpret the matching stem value appropriate for a given type curve. For one type curve, this can be
*w*_{f}*k*_{f}, which will provide an estimate of fracture conductivity. For another, it can be*s*_{f}, the choked-fracture skin, or, for a third, it can be*C*, the wellbore-storage coefficient.

To interpret the match points for a test with unknown permeability, use **Eqs. 25** and **26**. The formation permeability, *k*, is determined from the pressure match point; that is, the relationship between the pressure derivative and pressure change found at a match point given by

From the time match point, calculate the fracture half-length:

Matching can be ambiguous for hydraulically fractured wells; the data can appear to match equally well in several different positions. The ambiguity can be reduced or eliminated if a prefracture permeability is determined, and the post-fracture test data forced to match the permeability.

### Type curves used for analysis in fractured wells

The Cinco type curve (**Fig. 8**), ^{[1]} assumes that *C*_{Lf D} = 0 and *s*_{f} = 0. The type-curve stems on this curve are obtained by varying values of *C*_{r} or *F*_{cD}. With the Cinco type curve, the fracture conductivity, *w*_{f}*k*_{f}, can be determined from the matching parameter:

#### Choked-fracture type curve

**Fig. 9** shows the choked-fracture type curve. ^{[4]} The choked-fracture type curve is generated with wellbore-storage coefficient, *C*_{Lf D}, of zero and infinite fracture conductivity, *C*_{r}. On this type curve, the stems represent different values of the fracture skin, *s*_{f}. The fracture skin, *s*_{f}, can be used to find the additional pressure drop from

#### Wellbore-storage type curve

The wellbore-storage type curve (**Fig. 10**) takes into account the possibility of wellbore storage. The wellbore-storage type assumes *s*_{f} = 0 and *C*_{r} = ∞. To interpret a best-fitting stem for this type curve, use the following:

## Limitations of type-curve analysis

Although it is the most common methodology for analyzing hydraulically fractured well, type-curve analysis still has some limitations.

First, type-curves for analysis of hydraulically fractured wells are usually based on solutions for constant-rate drawdown tests. For buildup tests, shut-in time itself may possibly be used as a plotting function in those cases in which producing time is much greater than the shut-in time. Equivalent time can be used in some cases, but equivalent time has different definitions depending on the flow regime: radial, linear, and bilinear flow. Another possibility is to use a "superposition" type curve, which depends on the specific durations of flow and buildup periods. Superposition type curves can be readily generated with computer software.

Another problem with type curves is that they may ignore important behavior. The type curve that takes into account wellbore storage does not consider a variable wellbore storage coefficient. This can be caused by phase redistribution in the wellbore, for example. The widely available type curves that have been discussed do not include boundary effects. With gas wells, the probability of non-Darcy flow is high, but available type curves don’t take this into account.

An independent estimate of permeability may also be needed. A number of different type curves or a variety of stems on a given type curve may seem to match test data equally well. To remove this ambiguity, the best solution is to have an independent estimate of permeability.

## Nomenclature

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}Cinco-Ley, H. and Samaniego-V., F. 1981. Transient Pressure Analysis for Fractured Wells. J Pet Technol 33 (9): 1749-1766. SPE-7490-PA. http://dx.doi.org/10.2118/7490-PA - ↑
^{2.0}^{2.1}Prats, M., Hazebroek, P., and Strickler, W.R. 1962. Effect of Vertical Fractures on Reservoir Behavior--Compressible-Fluid Case. SPE J. 2 (2): 87-94. http://dx.doi.org/10.2118/98-PA - ↑
^{3.0}^{3.1}Hale, B.W. and Evers, J.F. 1981. Elliptical Flow Equations for Vertically Fractured Gas Wells. J Pet Technol 33 (12): 2489–2497. SPE-8943-PA. http://dx.doi.org/10.2118/8943-PA - ↑
^{4.0}^{4.1}^{4.2}Cinco-Ley, H. and Samaniego-V., F. 1981. Transient Pressure Analysis: Finite Conductivity Fracture Versus Damaged Fracture Case. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 4–7 October. SPE-10179-MS. http://dx.doi.org/10.2118/10179-MS - ↑
^{5.0}^{5.1}^{5.2}Lee, W.J. 1989. Postfracture Formation Evaluation. In Recent Advances in Hydraulic Fracturing, J.L. Gidley, S.A. Holditch, D.E. Nierode, and R.W. Veatch Jr. eds., Vol. 12. Richardson, Texas: Monograph Series, SPE. - ↑ Ramey, H.J. Jr. and Gringarten, A.C. 1975. Effect of High Volume Vertical Fractures on Geothermal Steam Well Behavior. Proc., Second United Nations Symposium on the Use and Development of Geothermal Energy, San Francisco.

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## See also

Fluid flow through permeable media