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# Fluid flow in horizontal wells

Productivity estimates in horizontal wells are subject to more uncertainty than comparable estimates in vertical wells. Further, it is much more difficult to interpret well test data because of 3D flow geometry. The radial symmetry usually present in a vertical well does not exist. Several flow regimes can potentially occur and need to be considered in analyzing test data from horizontal wells. Wellbore storage effects can be much more significant and partial penetration and end effects commonly complicate interpretation.

In vertical wells, variables such as average permeability, net vertical thickness, and skin are used. Horizontal wells need more detail. Not only is vertical thickness important, but the horizontal dimensions of the reservoir, relative to the horizontal wellbore, need to be known.

## Steps in evaluating horizontal well-test data

There are three basic steps in evaluating pressure-transient data from a horizontal well:

- Identify the specific flow regimes in the test data
- Apply the proper analytical and graphical procedures to the data
- Evaluate the uniqueness and sensitivity of the results to properties derived from analysis or simply assumed

### Identify flow regimes

Evaluation of data from a vertical wellbore will generally center on a single flow regime, such as infinite-acting radial flow, known as the MTR. However, a pressure-transient test in a horizontal well can involve as many as five major and distinct regimes that need to be identified. These regimes may or may not occur in a given test and may or may not be obscured by wellbore storage effects.

### Apply the proper procedures

Each flow regime can be modeled by an equation that can be used to estimate important reservoir properties. At best, only groups of analytical parameters can be determined directly from equations. It is imperative that the proper analytical and graphical procedures be applied to the data. In many cases, when solving for specific parameters, the application of these analytical expressions may involve a complex iterative procedure.

### Evaluate uniqueness and sensitivity

Experience indicates that results of horizontal well test analysis are seldom unique, so it is important that the uniqueness and sensitivity of the results to assumed properties be evaluated. Simulation of the test using properties that have been determined from the test can confirm that at least the analysis is consistent with the test data. A simulator can also determine whether other sets of formation properties will also lead to a fit of the data.

## Horizontal well flow regimes

Different formation properties can be calculated from the data in each of the five different flow regimes. Any flow regime may be absent from a plot of test data because of geometry, wellbore storage, or other factors. Nor does the fact that they can appear mean that they do appear. The five different flow regimes that can occur are:

- Early radial
- Hemi-radial
- Early linear
- Late pseudoradial
- Late linear

**Fig. 1** shows a horizontal well with length, *L*_{w}, within a reservoir that is assumed to be a rectangular parallelepiped or a "box reservoir" drainage area. In this discussion, it is assumed that the axes of the coordinate system coincide with the direction of principal permeability and the well produces over its entire length, *L*_{w}.

The axes for this box are the usual *x*-, *y*-, and *z*-axes. Notice that the x-axis is measured along the bottom edge of the reservoir, going from left to right in the direction perpendicular to the well. The *y*-axis lies along the axis from front to back of the reservoir, parallel to the wellbore. The *z*-axis is oriented in the direction of reservoir thickness.

The total width of the reservoir perpendicular to the wellbore is *a*_{H}, the total length in a direction parallel to the wellbore is *b*_{H}, and the total height of the reservoir is the net pay thickness, *h*. Notice the parameters for the distance from the well to the various borders. Along the axis of the well, the shortest distance from the end of the well to a boundary is *d*_{y}, and the longest distance from the other end of the well to the boundary is *D*_{y}. In the vertical direction, the shortest distance to a vertical boundary is *d*_{z}, and the longest distance to a vertical boundary is *D*_{z}.

### Characteristics of flow regimes

Consider a well producing at a constant rate. The early radial flow regime occurs before the area drained or the pressure transient caused by this production encounters either of the vertical boundaries of the reservoir. **Fig. 2** shows a radial flow pattern penetrating out into the reservoir. Actually, however, this flow pattern is likely to be elliptical, moving further into the reservoir at a given time in the higher-permeability *x*-direction than in the lower-permeability *z*-direction. This phenomenon causes no significant complications in our analysis.

When the wellbore is much nearer one vertical boundary than the other, another flow regime, called half-radial or hemiradial flow (**Fig. 3**) may exist. Hemiradial flow can occur immediately following the early radial flow regime, if the well is much nearer one of the vertical boundaries than the other. Eventually, the area affected by the production will include the entire thickness of the reservoir. When that happens, a linear flow pattern may develop, as **Fig. 4** shows.

Eventually, flow will begin to come into the wellbore from beyond the ends of the well. Until these end effects become important, early linear flow continues. Once end effects become important a transition period is followed by a later pseudoradial flow regime, as **Fig. 5** illustrates.

This flow regime continues until the area affected by the production reaches one of the sides of the reservoir. Once the area affected is the entire width of the reservoir (that is, the pressure transient has reached both sides of the reservoir), then the late-linear flow regime begins (**Fig. 6**).

## Identifying flow regimes in horizontal wells

All of these flow regimes in a test can be identified on a diagnostic log-log plot of the pressure change, Δ*p*, and pressure derivative, *p*′, against the logarithm of time (**Fig. 7**).

A unit-slope line appears during wellbore storage; a horizontal derivative during early radial flow, and then, later, in pseudoradial flow; and a half-slope line in early-linear flow and then in late-linear flow. (These half-slope lines appear on the derivative but not on the pressure-change curves.) This does not imply that all these flow regimes will appear in any given test; in fact, that would be rare. But these are the shapes that identify the flow regimes that may appear in the test being analyzed.

The shapes that may appear in a drawdown test (which is the basis of **Fig. 7**) may not appear in a buildup test because of the complex superposition of flow regimes. For example, a test would have to be in linear flow both at time (*t*_{p} + Δ*t*) and at time Δ t to ensure appearance of a derivative with half-slope; this is highly unlikely. The best way to solve the problem is to ensure that a buildup test on a horizontal well is run with a producing time, *t*_{p}, much greater than the maximum shut-in time in the test (that is, *t*_{p} > 10 Δ*t* max ).

**Table 1** summarizes the working equations for permeability, skin, and start and end of each of the recognized flow regimes. Different investigators have found different equations for start and end of various flow regimes, especially linear flow regimes. This is partly because of a difference of assumptions about flow into the wellbore. Uniform flux or infinite conductivity models are common; neither is rigorously applicable in practice. ^{[1]} In this section, the equations for duration of flow regimes derived by Odeh and Babu^{[2]} are used. This model assumes uniform flux into the wellbore.

### Early-radial flow

Early-radial flow is similar to the radial flow period in a vertical well (**Fig. 2**). The governing equation for this flow regime is

Data for this period may be masked by wellbore storage effects, but, when present, they may be analyzed on a semilog plot.

The early-radial flow regime may in theory start at time zero, in absence of wellbore storage effects. The end of the early-radial flow regime may occur when the transient reaches a vertical boundary or when flow comes from beyond the end of the wellbore. The end of the period is the smaller of these two values.

**Eq. 2**^{[2]} says that the period must end when the transient reaches the nearest boundary, *d*_{z}, from the well. This equation includes the permeability in the vertical direction:

The radial flow regime may also end when flow from beyond the end of the wellbore becomes important. **Eq. 3** gives the time by

*L*_{w} is the completed length of the well, and k y is the permeability in the direction parallel to the wellbore. The actual end is the lesser of the two times calculated from **Eqs. 2** and **3**. It is helpful to check the expected duration of the early-radial flow regime after estimating the parameters necessary to make these calculations.

**Eq. 1** suggests that possible radial flow on the diagnostic plot be identified and then bottomhole flowing pressure be plotted against time during the appropriate time range on semilog coordinates. The slope of the straight line that results is

The group can be found from the slope, *m*_{erf}:

Effective completed length of the well must be known to make this calculation. This is not necessarily the same as the perforated or completed length of the well. Some sections of the well may not produce at all.

The equation for calculating the altered permeability skin, *s*_{d}, for early-radial flow is

When analyzing a buildup test rather than a constant-rate flow test, plot the HTR or equivalent time on the horizontal axis of the semilog plot, and then plot shut-in or equivalent time on the vertical axis. Note that this plotting is correct only if (*t*_{p} + Δ*t*) and Δ*t* appear in this time period simultaneously; that is, radial flow must exist at both time (*t*_{p} + Δ*t*) and time Δ*t*. This is unlikely because radial-flow regime may exist at time Δ*t*, but a different flow regime is likely at time (*t*_{p} + Δ*t*).

#### Example 1: Well Erf-1

For drawdown test data from Well Erf-1, ^{[2]} the diagnostic plot indicates the data from approximately 0.24 to 24 hours may be in early-radial flow. The following information is available for this well: *q* = 800 STB/D, *μ* = 1 cp, *B* = 1.25 RB/STB, *r*_{w} = 0.25 ft, *ϕ* = 0.2, *c*_{t} = 15×10 –6 psi^{–1}, centered in box-shaped drainage area, *h* = 200 ft, *b*_{H} = 4,000 ft, and *a*_{H} = 2,000 ft, *L*_{w} =1,000 ft, and, from analysis of data in early linear flow regime, *k*_{x} = 200 md. **Table 2** shows the pressure change data for 0.24 to 24 hours.

Plot (*p*_{i} − *p*_{wf} ) = Δ*p* vs. *t* on semilog coordinates (**Fig. 8**). The plot results in a straight line with a slope of 8 psi/cycle. In **Fig. 8**, at *t* = 2.4 hours, the points begin to deviate from the straight line, as expected from calculations for flow regime duration that follow. The pressure change at 1 hour is 39 psia. Using the slope of 8 and **Eq. 5**,

Thus, because *k*_{x} = 200 md, *k*_{z} = 2 md. Using the value of 39 for Δ*p*_{1hr} from **Fig. 8**, skin from **Eq. 6** is

The start of the early-radial flow regime is controlled by wellbore storage, which appears to have vanished at times earlier than 0.24 hours in this example. The end of the early radial flow regime is expected at the lesser of the two values derived from **Eqs. 2** and **3**. For a centered well, *d*_{z} = *h*/2 = 100 ft, and **Eq. 2** gives

Assuming *k*_{y} = *k*_{x} = 200 md, **Eq. 3** gives

Thus, expect the early-radial flow regime to end at approximately 1.875 hours, which is the smaller value and is consistent with observed test data.

### Hemiradial flow

The hemiradial flow period (**Fig. 3**) will occur only when the well is close to one of the vertical boundaries (either the upper or the lower boundaries) and is analogous to a vertical well near a fault. The governing equation is^{[1]}

A horizontal derivative on the diagnostic plot identifies hemiradial flow. If data appear to fall into this flow regime, a straight line on a semilog plot would provide more confidence that radial flow has been identified. Consistency checks in the analysis coupled with a well survey will be required to distinguish hemiradial flow from early radial flow.

The time range in which the analysis for hemiradial flow is valid begins after the closest vertical boundary, *d*_{z}, affects the data and before the farthest boundary, *D*_{z}, affects them. In the absence of wellbore storage, the start of hemiradial flow is given by

Note that the start of the hemiradial flow regime involves the shortest distance to a vertical boundary and the permeability in the vertical direction. However, wellbore storage will most likely determine the actual start of hemiradial flow.

The end of hemiradial flow occurs when pressure is affected by the farther vertical boundary or flow from beyond the ends of the wellbore, whichever occurs first. It is the smaller of the times calculated using **Eqs. 9** and **10**. If the hemiradial flow regime ends when pressure reaches the farthest vertical boundary, it depends on the distance, *D*_{z}, and the vertical permeability, *k*_{z}:

When the appearance of end effects—flow from beyond the ends of the wellbore—causes the end of the hemiradial flow regime to appear, the end of the flow regime occurs when

The completed length of the well, *L*_{w}, and the permeability *k*_{y}, parallel to the wellbore appear in this equation. These parameters determine when enough flow has come from beyond the ends of the wellbore to distort the radial flow pattern that appeared earlier.

gives the slope of the semilog straight line for semiradial flow, *m*_{hrf}. The multiplier, 325.2, is twice the multiplier for early-radial flow. The equation to estimate the damage skin factor is also similar to that for radial flow but has a multiplier that differs by a factor of two.

The equations relating slope and permeability and the equation for skin are similar in a buildup test to those for a drawdown test. The pressure change in the equation for skin is [*p*_{1hr} − *p*_{wf} (Δ*t* = 0)]. Semilog plots of buildup test data from the hemiradial flow regime cannot be analyzed rigorously using data from a Horner plot unless the pressure data at (*t*_{p} + Δ*t*) and at time Δ*t* are simultaneously in this flow regime. As a practical matter, the hemiradial flow regime is likely to appear clearly in the buildup test only when the producing time is much greater that the shut-in time.

### Early linear flow

The governing equation for early-linear flow is^{[1]}

The "convergence skin," *s*_{c}, is discussed later in this section. The start of the early-linear flow regime (**Fig. 4**) depends on the farthest distance to a vertical boundary, *D*_{z}, and the vertical permeability, *k*_{z}.^{[2]}

Not until flow reaches that farthest vertical boundary can a linear flow pattern begin toward the well. This flow period ends when fluids flow from beyond the ends of the wellbore. Thus,

Notice that the end depends on the effective completed length of the well, *L*_{w}, and on the permeability in the direction parallel to the well. This is the time in which end effects—flow beyond the ends of the well—begin to significantly distort the linear flow pattern.

The early-linear flow regime is identified in a drawdown test with a half-slope for the derivative. (Because of the skin effect, the pressure change curve on the diagnostic plot will only approach a half-slope asymptotically.) For data identified as being in this flow regime, plot pressure against the square root of time.

The slope of the straight line on such a plot, *m*_{elf}, can be used to estimate the square-root of *k*_{x}, the horizontal permeability perpendicular to the well:

To calculate the damage skin,

This equation includes a convergence skin, *s*_{c}, which is^{[2]}

This convergence skin is an additional pressure drop that acts like a skin effect caused by flow moving from throughout the entire formation until it converges down to the small wellbore in the middle of the formation (**Fig. 9**). This convergence skin is defined in terms of the ratio of the permeability in the *x*-direction, which is perpendicular to the wellbore, to the vertical permeability. It also involves the distance to the nearest vertical boundary, *d*_{z}, and the net pay thickness, *h*.

Kuchuk^{[3]} derived a different equation for convergence skin. As a practical matter, the Odeh-Babu and Kuchuk equations lead to the same result. When there has been a single rate preceding shut-in during early-linear flow, the buildup pressure is plotted against on a Cartesian plot, which is sometimes called a tandem root plot. The permeability, *k*_{x}, is calculated from the slope, *m*_{elf}, of the plot and **Eq. 16**. *k*_{x} has the same relationship to the slope that existed in a drawdown test. Skin for this flow regime is calculated with **Eqs. 17** and **18**.

Plots of buildup data from the early-linear flow regime cannot be analyzed rigorously with a plot of *p*_{ws} vs. (that is, the tandem-root plot) unless data at (*t*_{p} + Δ*t*) and Δ*t* are simultaneously in this flow regime—highly unlikely—or unless *t*_{p} is much greater than Δ*t*, in which case simply plot *p*_{ws} vs. . Little error results from ignoring the (*t*_{p} + Δ*t*) term, which is essentially constant.

#### Example 2: Well Elf-2

The diagnostic plot for a drawdown test from Well Elf-2^{[2]} indicates data in the early-linear flow regime because the derivative has a half-slope. The following data apply to this well: *q* = 800 STB/D; *μ* = 1 cp; *B* = 1.25 RB/STB; *r*_{w} = 0.25 ft; *ϕ* = 0.2; *c*_{t} = 15×10^{–6} psi^{–1}; centered in box-shaped drainage area 100 ft thick, 4,000 ft long, 4,000 ft wide; *L*_{w} = 2,500 ft; and, from early radial-flow regime data, *k*_{x}*k*_{z} = 8,000 md^{2}. In addition, **Table 3** shows the pressure-change data for this well.

A plot of pressure change vs. the square root of time (**Fig. 10**) indicates early linear flow. The final point on the straight line is at a time of approximately 24 hours, but there may have been some deviation from the straight line by this time. From the slope of the straight line, *m*_{elf} = 0.934 psi/hr^{1/2} and **Eq. 16**, calculate the permeability in the horizontal plane perpendicular to the wellbore.

or *k*_{x} = (20.1)^{2} = 404 md.

Analysis of data from the early radial flow regime indicated that *k*_{x}*k*_{z} is 8,000 md^{2}; thus, *k*_{z} is approximately 19.8 md. To calculate *s*_{d}, use **Eqs. 17** and **18**, noting that the value for (*p*_{i} − *p*_{wf} ) = 3.1 at *t* = 0.

Then, *s*_{d} = 4.91 – 4.91 = 0.

Check the expected time range for the early-linear flow regime.

Using *D*_{z} = *h*/2 = 50 ft and *k*_{z} = 20 md, calculate the beginning of linear flow with **Eq. 14**:

Assuming *k*_{y} = *k*_{x} at 400 md, use **Eq. 15** to find the end of early linear flow.

These limits are reasonably consistent with the time range analyzed assuming early linear flow.

### Late pseudoradial

The governing equation for late-pseudoradial flow is^{[1]}

The late-pseudoradial flow period occurs only if ^{[2]}

Here, *b*_{H} is the dimension of the reservoir parallel to the wellbore. As long as the completed length of the well is relatively short compared with the length of the drainage area late-pseudoradial flow can occur.

The start of this flow period occurs when fluid flows from well beyond the ends of the wellbore (**Fig. 5**). It is approximated with^{[2]}

This starting time depends on the completed length of the well, *L*_{w}, and on the permeability in the direction of the well, *k*_{y}. The end of this period, like others in this section, is approximated by the minimum of the results of two calculations. The first,

depends on *d*_{y} and the length of the wellbore along with *k*_{y}, the permeability in the direction parallel to the wellbore. This is the time at which horizontal boundary effects first appear.

The other equation gives a time at which the radial-flow pattern begins to be distorted depending on the shortest distance, *d*_{x}, from the well to a boundary perpendicular to the wellbore and on *k*_{x}, the permeability in that direction.

Whenever boundary effects first appear, whether in a direction that is parallel to the well or perpendicular to the axis of the well, the late-pseudoradial flow period will end.

The diagnostic plot helps identify the late-pseudoradial flow regime with the characteristic horizontal derivative. For data in the appropriate time range, prepare a semilog plot of pressure against time for a drawdown test. The slope of this plot will be *m*_{prf} and the relationship between that slope and the square root of *k*_{x}*k*_{y}, or the permeabilities in the horizontal plane, is given by

The skin equation is similar in form to those seen before:

Again, the total skin depends on Δ*p*_{1hr}. The convergence skin (**Eq. 18**) is subtracted from the "total" skin to determine the damage skin.

For a buildup test preceded by production at a single rate, plot pressure against the HTR on a semilog graph. Permeability is calculated from **Eq. 24**, the same as for a drawdown test. The skin equation is basically the same as for a drawdown test, except that the Δ*p*_{1hr} is now *p*_{1hr} − *p*_{wf}. To obtain *p**, the extrapolated pressure, extrapolate the semilog straight line to a HTR of unity.

Semilog plots of buildup test data from the late-pseudoradial flow regime cannot be analyzed rigorously using a Horner plot unless pressures at (*t*_{p} + Δ*t*) and at time Δ*t* are simultaneously in the pseudoradial flow regime, which is highly unlikely. However, little error appears if the producing time before shut-in is much greater than the maximum shut-in time achieved in the buildup test.

#### Example 3: Well Prf-3

The diagnostic plot suggests that a constant-rate drawdown test from Well Prf-3^{[2]} includes data in the late-pseudoradial period. The following data are available from the test: *q* = 800 STB/D, *μ* = 1 cp, *B* = 1.25 RB/STB, *r*_{w} = 0.25 ft, *ϕ* = 0.2, *c*_{t} = 15×10^{–6} psi^{–1}, *h* = 150 ft, *L*_{w} = 900 ft, *a*_{H} = 5,280 ft, *b*_{H} = 5,280 ft, well centered in drainage volume, *k*_{x} = 100 md (from analysis of early linear flow), *k*_{x}*k*_{z} = 1,000 md^{2}, and *k*_{z} = 10 md (from analysis of early radial flow). **Table 4** gives the pressure change, Δ*p* = *p*_{i} – *p*_{wf} vs. time.

A plot of pressure change vs. the logarithm of time (**Fig. 11**) confirms pseudoradial flow. A straight line fits all the data from 192 to 432 hours; the slope of the line, *m*_{prf}, is 15.3 psi/cycle, and Δ*p*_{1 hr} = 18.94 psi (extrapolated). Then, from **Eq. 24**,

From Eqs. 25 and 18 ,

Here,

The pseudoradial flow regime should start at the time given by **Eq. 21**:

It should end at a time given by the lesser of values from **Eqs. 22** and **23**. From **Eq. 22**,

where *d _{y}* = 1/2(5,200-900) = 2,190ft for this centered well. From

**Eq. 23**,

The smaller of these two values is 344 hours, which is thus the expected end of pseudoradial flow. The data on the figure that lie on the straight line show the time range from 192 to 432 hours, which is generally consistent with the expected duration of the flow regime.

### Late-linear flow

The governing equation for late-linear flow is^{[1]}^{[2]}

The late-linear flow regime starts after the pressure transient has reached the boundaries in the *z*- and *y*-directions, and the flow behavior with regard to these directions has become pseudosteady state, as **Fig. 6** shows.

The start of this time period is the maximum of two equations. ^{[2]} The first depends on the time to reach the boundary, *D*_{y}, beyond the end of the horizontal well. It also depends on the permeability, *k*_{y}, in the direction parallel to the wellbore.

Another requirement for the start of the late-linear flow regime is the time to reach the maximum vertical distance, *D*_{z}, divided by the vertical permeability:

Usually, the start of the late-linear flow regime is dictated by the time to reach the boundaries in the *y*-direction. The end of this period is given by the equation

The end of the late-linear flow regime depends on reaching the nearest boundary in the direction perpendicular to the wellbore, which is the distance, *d*_{x}, away, and on the permeability *k*_{x} in that direction.

Identify the late-linear flow regime by a half-slope on the derivative in the diagnostic plot of drawdown test data. (The pressure change may approach a half-slope asymptotically.) For data that appear to be in this flow regime, plot pressure against the square root of time. From the slope *m*_{llf} of the plot, estimate permeability in the *x*-direction from

Alternatively, if *k*_{x} is known from an early-linear flow regime, estimate *b*_{H}, the length of the drainage area, from

This late-linear flow regime is the only period that provides the data to calculate the total skin, *s*, including the partial-penetration skin, *s*_{p}, and the convergence skin, *s*_{c}. To calculate the damage skin, *s*_{d}, use

The total skin depends on Δ*p*_{t} = 0. Subtracting the partial penetration skin, *s*_{p}, and the convergence skin, *s*_{c}, from the total skin yields the damage skin.

The partial-penetration skin is a complex function that is calculated with Eqs. A-25 through A-35 in **Table 1**. For a buildup test, plot pressure against the HTR. From the slope, *m*_{llf}, calculate *k*_{x} with **Eq. 30**, exactly the same as for drawdown tests. Or, if *k*_{x} is known, estimate the length, *b*_{H}, of the drainage area with **Eq. 31**. Calculate the damage skin, *s*_{d}, from a pressure buildup test from **Eq. 32**, where Δ*p*_{t} = 0 = (*p*_{t} = 0)_{ext} – *p*_{wf(t = 0)}.

Note that the same difficulty arises in using superposition to find plotting functions plots of buildup data from the late-linear flow regime as existed with the previous flow regimes. Pressures at both time (*t*_{p} + Δ*t*) and time Δ*t* must be in the late linear flow regime for a tandem-root plot to be valid. However, if *t*_{p} >> Δ*t*_{max}, there is little error.

#### Example 4: Well Llf-4

The diagnostic plot for a drawdown test from Well Llf-4^{[2]} appears to include data in the late-linear flow regime (derivative with half-slope). The following data applies to this well: *q* = 800 STB/D, *μ* = 1 cp, *B* = 1.25 RB/STB, *r*_{w} = 0.25 ft, *ϕ* = 0.2, *c*_{t} = 15 × 10^{–6} psi^{–1} , *h* = 150 ft, *L*_{w} = 1,000 ft, *b*_{H} = 2,000 ft (well centered), *a*_{H} = 6,968 ft (well centered), *D*_{z} = 85 ft, *d*_{z} = 65 ft, *k*_{x}*k*_{z} = 1,000 md^{2} (from analysis of early-radial flow), and *k*_{x}*k*_{y} = 5,000 md^{2} (from analysis of pseudoradial flow). **Table 5** gives pressure change, Δ*p* = *p*_{i} – *p*_{wf} , data vs. time.

**Fig. 12** is a plot of pressure change vs. the square root of time. The straight line on this plot for the entire time range (60 to 240 hours) confirms late-linear flow for this range. The slope of the line is 1.56 psi/hr^{1/2}, and the intercept is Δ*p*_{t = 0} =28.4 psi.

From **Eq. 30**,

Then, *k*_{x} = 100 md. Because *k*_{k}*k*_{z} = 100 md^{2}, *k*_{z} = 10 md. Also, because *k*_{x}*k*_{y} = 5,000 md^{2}, *k*_{y} =50 md. From **Eq. 32**,

From **Eq. 18**,

Calculate the partial penetration skin, s_{p} , using the appropriate equation from among Eqs. A-25 through A-35 in **Table 1**. First, calculate

to determine whether "Case 1" or "Case 2" applies:

Because

this is Case 1 (Eq. A-26). Use Eqs. A-25 through A-31 from the table. From Eq. A-27, *s*_{p} = *p*_{xyz} + *p*′_{xy}. From Eq. A-25,

From Eq. A-28,

Here, from Eq. A-29,

The well is centered, so *d*_{y} = *D*_{y} = 500 ft.

From Eq. A-30,

Also,

From Eq. A-31,

and

Then,

Then,

Now check the expected duration of the late-linear flow regime. The start is the larger of values from **Eqs. 27** and **28**. From **Eq. 27**,

From **Eq. 28**,

Thus, the start is expected to be at approximately 162 hours. **Eq. 29** gives the end of the flow regime.

The data in this example spanned the time range from 60 to 240 hours. Some of the data that fall on the straight line are, in theory, from times before the start of the late-linear flow regime, but they appear to cause no problem in determining the slope of the line.

## Field examples^{[4]}

The following field examples illustrate the procedures used in analyzing horizontal well-test data.

### Field example Well A

**Table 6** summarizes the reservoir and completion properties for Well A. The target for Well A, a horizontal exploration well, was vertical tectonic fracture development in a low-permeability shale. Because of the fractures, the permeability is assumed to be isotropic (*k*_{h} = *k*_{z}) and a result of the fractures. **Fig. 13** is a diagnostic plot for Well A and includes a history match using an analytical model.

During the early part of the test there is a unit-slope line representing wellbore storage. Following that wellbore storage, there is a transition to radial flow. The final few data points may be in radial flow. On the Horner plot (**Fig. 14**), the last few data points fall on a straight line. From the slope of the straight line, the apparent permeability is 0.011 md and the altered zone skin is 2.9. There is no evidence of boundary effects on this Horner plot. The existence of the semilog straight line is not assured, but the data are at least on the verge of reaching it.

Guided by the Horner analysis results, engineers simulated these data with an analytical horizontal well model. The initial match was poor. The match was improved considerably by introducing a no-flow boundary approximately 16 ft above the wellbore, which led to a permeability estimate of 0.027 md and an apparent skin of 11.5. It was concluded that the flow regime observed in the test was hemiradial flow.

The final match, shown on the type-curve plot in **Fig. 13**, is still not a good match at all times, but the author stated that the poor match in the transition region could be the result of phase-redistribution effects in the wellbore. The distance to the no-flow boundary that led to the best match compared favorably with well survey data, which indicated that the well was drilled approximately 20 ft below the upper limit of the productive horizon.

Well B is in a west Texas carbonate formation. It was expected to have isotropic permeability caused by fracturing and dissolution. **Table 7** gives the field data for this well.

**Fig. 15** is the diagnostic plot for this well. After wellbore storage, a short period of radial flow appears to be followed by the onset of linear flow, because p′ approaches a slope of 0.5. In the time region where the derivative is horizontal, a straight line on the Horner plot (**Fig. 16**) yields *k* = 0.14 md. Using these results in the analysis shows that the end of radial flow occurs at *t*_{Erf} = 165 hours.

A tandem-root plot (**Fig. 17**) indicates linear flow and also suggests a distance to the nearest boundary of 29 ft. This is in good agreement with geological observations and helps to verify the assumption of isotropic permeability. The history match with an analytical horizontal well model, shown in **Fig. 15**, confirms the results of the Horner and tandem-root plots.

### Field example Well C

Well C data are from a buildup test of a horizontal well in a high-permeability sandstone where a 54-ft oil column overlies an extensive aquifer estimated to be approximately 180 ft thick. **Table 8** shows the available data.

The diagnostic plot (**Fig. 18**) shows essentially no wellbore storage and a constant derivative, indicating radial, hemiradial, or elliptical flow at early times. The rapid decline of the derivative at the end of the test is caused by the aquifer underlying the oil column, which is acting like a constant-pressure boundary. A history match with an analytical horizontal well model with one no-flow and one constant-pressure boundary (the lower boundary), yielded *k*_{h} = 313 md, *k*_{z} = 7.5 md, *s*_{a} = 1.5, and *L*_{w} = 356 ft. The no-flow boundary was estimated to be approximately 112 ft below the wellbore.

The time at which the early radial flow regime ends—the time where the derivative ceases to be flat on the diagnostic plot—is approximately 1.5 hours. For a wellbore with a volume of 130 bbl filled with a fluid of compressibility of 3.5 × 10^{–6} psi, the duration of wellbore storage (the unit-slope line) is estimated to be 0.0005 hours. With the gauge sampling rate set at 0.017 hours, the wellbore-storage unit slope simply could not be detected and does not appear at all on this plot. **Fig. 19** is the Horner plot for this test. A straight line appears in the same range as the flat derivative on a diagnostic plot. From the slope of the line, the permeability is estimated to be 53 md, close to the regression analysis match value of 48 md.

## Running horizontal well tests

The measurements in horizontal wells are usually made above the wellbore with the pressure gauge still in the vertical section.

The test string may often be too rigid to pass through the wellbore. However, in most cases, conventional hardware can be used for horizontal well tests. With longer horizontal wellbores, wellbore storage is an inherent problem for testing, even for buildup tests with downhole shut-in. As mentioned previously, problems arise in conducting buildup tests with short-duration production periods because superposition is inappropriate; therefore, Horner plots and tandem-root plots, which depend on superposition being applicable, are often inappropriate.

Another problem in conducting buildup test following a short production period is that significant pressure gradients along the length of the wellbore may cause crossflow within the wellbore during shut-in, so fluid may flow from one region to another in the wellbore. These gradients can be removed and this crossflow eliminated by a longer-duration flow period preceding a buildup test.

### Factors that affect transient responses

A number of factors may affect the transient response of a horizontal well test: horizontal permeability (normal and parallel to well trajectory), vertical permeability, drilling damage, completion damage, producing interval that may be effectively much less than drilled length, and variations in standoff along length of well.

In summary, seven or more factors may affect interpretation for horizontal wells in homogeneous reservoirs before the effects from boundaries. The problem is complex, so test results are frequently inconclusive. Furthermore, wellbore storage inhibits determination of properties associated with early-time transient behavior such as vertical permeability and damage from drilling and completion. Middle- and late-time behavior may require several hours, days, or months to appear in transient data.

Some practical steps will help ensure interpretable test data. First, it is helpful to run tests in the pilot hole before kicking off to drill the horizontal borehole section. From a test in the vertical section, it is possible to get usable estimates of horizontal and vertical permeabilities using modern wireline test tools. Second, a good directional drilling survey can frequently provide an adequate estimate of standoff. A production-log flow survey conducted with coiled tubing can determine what part of the wellbore is actually producing and, therefore, help provide an estimate of effective productive length.

Wells in developed reservoirs should be flowed long enough to bring pressures along the wellbore to equilibrium and thus minimize crossflow. For high-rate wells, continuous borehole pressure and flow-rate measurements acquired during production can be used to interpret the pressure-drawdown transient response. If the downhole rates are not measured, the buildup test should be conducted with downhole shut-in to minimize wellbore storage distortion of test data.

## Nomenclature

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}Goode, P.A. and Thambynayagam, R.K.M. 1987. Pressure Drawdown and Buildup Analysis of Horizontal Wells in Anisotropic Media. SPE Form Eval 2 (4): 683–697. SPE-14250-PA. http://dx.doi.org/10.2118/14250-PA - ↑
^{2.00}^{2.01}^{2.02}^{2.03}^{2.04}^{2.05}^{2.06}^{2.07}^{2.08}^{2.09}^{2.10}^{2.11}Odeh, A.S. and Babu, D.K. 1990. Transient Flow Behavior of Horizontal Wells: Pressure Drawdown and Buildup Analysis. SPE Form Eval 5 (1): 7-15. SPE-18802-PA. http://dx.doi.org/10.2118/18802-PA - ↑ Kuchuk, F.J. 1995. Well Testing and Interpretation for Horizontal Wells. J Pet Technol 47 (1): 36–41. SPE-25232-PA. http://dx.doi.org/10.2118/25232-PA
- ↑ Lichtenberger, G.J. 1994. Data Acquisition and Interpretation of Horizontal Well Pressure-Transient Tests. J Pet Technol 46 (2): 157-162. SPE-25922-PA. http://dx.doi.org/10.2118/25922-PA

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