DECLINE CURVE ANALYSIS PREDICTS OIL RECOVERY FROM HORIZONTAL WELLS

Pralhad N. Mutalik, Sada D. Joshi
Joshi Technologies International Inc.
Tulsa

For the transient and post-transient decline periods of a horizontal well in a bounded reservoir, an analytical model has been developed that can forecast recoverable reserves. This decline curve method predicts future performance of both new and existing horizontal wells.

The model can also be used to develop horizontal well type curves.

CONCEPTS

Two key issues in determining horizontal well performance and ultimate recovered reserves are well length and spacing. These issues are especially important in reservoirs where pressure decreases with time.

In a vertical well, the well bore contacts only the reservoir height. But depending on the length drilled, horizontal wells can have much greater exposure to the reservoir.

Over an equal time interval, a long horizontal well can drain a significantly larger reservoir volume than a vertical well. But closely spaced horizontal wells may interfere with each other very rapidly, resulting in lower ultimate reserves per well. Thus, optimizing well length and well spacing is important.

One method for optimizing these factors is by developing the expected production rate-vs.-time performance or production decline curves.

After a well starts production, time is needed for the flow rate to stabilize. The production before the rate stabilizes is known as "transient" or "flush" production.

Depending upon the reservoir properties, the transient period may last from a few days to several months. In low permeability reservoirs, the transient time may last for years.

Conventional analytical methods calculate horizontal well productivities primarily for stabilized flow. 1 Steady state equations are for reservoirs with good pressure support in which pressure remains essentially constant. Pseudosteady-state equations are for reservoirs where pressure decreases with time.

In general, the stabilized productivity calculations based on either steady state or pseudosteady-state methods give lower production rates than observed in the transient flow period.

In low-permeability reservoirs, especially with large well spacing, the transient period may last 1 year or more. Because wells may payout during the transient period alone, investment decisions based on steady-state calculations without considering transient production may miss an opportunity. Therefore, forecasting transient well performance is very important for low-permeability reservoirs.

PRODUCTION FORECASTS

The transient-flow solution is expressed in terms of dimensionless quantities. Dimensionless numbers are easy to apply and provide simple, general equations for any set of reservoir properties. These dimensionless numbers are easily adapted to mathematical manipulation and superposition.

The proposed mathematical solution covers both the transient and pseudosteady-state (depletion) stages of the well.

Fig. 1 shows the geometry of the horizontal well located in a bounded reservoir. In the equation box is a summary of the equations describing the analytical solution. 2-6 Nomenclature is in the nomenclature box.

The x direction is parallel and the y direction is perpendicular to the horizontal well.

PARAMATERS

For horizontal wells, the production rates and recoverable reserves depend upon (Fig. 1):

Drainage volume (well spacing), and the ratio of drainage area dimensions (2xe/(2ye), where 2xe is the well spacing distance parallel to the well, and 2ye is the well spacing distance perpendicular to the well.
Well penetration ratio, L/2xe. This ratio accounts for the distance to be drained beyond the well tips.
Dimensionless well length is:

LD=(L(kv/kh)0.5)/2h [see equation]

The dimensionless well length, LD, accounts for the changes in well productivity due to reservoir anisotropy (kv does not equal kh). It also accounts for the influence of reservoir height on well productivity.

The dimensionless radius is:

rwD=rw/h [see equation]

TRANSIENT RATE

During the transient stage, iterative calculations provide an exact mathematical solution of the Laplace transform of the constant production rate equation (Equation 5). The objective is to calculate dimensionless pressure, Pd, and, dimensionless rate, qD, for different values of dimensionless time, tD.

For any dimensionless time, tD, the dimensionless rate, qD, is calculated by solving Equation 5. The dimensionless rate and the dimensionless time are then converted to the corresponding real rates and times using Equations 1-4.

To provide the reader with a better understanding of the mathematical basis of the horizontal well model and transient flow, a few useful definitions are briefly reviewed in the definition box.

PSEUDO-STEADY STATE

Arps 7 and Fetkovich 8 decline curve equations are used to calculate well decline in the stabilized (pseudosteady state) period, or the depletion period.

In these equations, the two important decline parameters are b and Di. The decline exponent b is determined by the reservoir producing mechanism. Typically, b is 0.3 for solution-gas-drive reservoirs and 0.5 for reservoirs producing by water drive or gravity-drainage. 7 8 The exponential decline solution, b=O, provides the most conservative forecast.

The decline coefficient, Di, represents the initial decline rate at the beginning of the depletion state. As the area drained by a well increases, re/rw becomes larger and the Di is reduced (Equation 7).

This indicates that, for a given economic cutoff rate, the well having greater spacing will show a slower decline rate, and hence higher cumulative oil recovery.

TYPE CURVES

Type curves have been used to estimate reservoir parameters such as permeability, drainage area, etc. A number of type curves are available for vertical wells.

For a constant flowing bottom hole pressure, type curves are essentially plots of dimensionless rate, qD, vs. dimensionless time, tD. The analytical model previously described has been used to develop new type curves for a horizontal well located in a closed drainage area.

In the case of horizontal wells, for a given value of rwD and L/2xe, different type curves have to be generated for each value of LD. Fig. 2a shows a typical horizontal well type curve for LD=3.

If LD is large, i.e., for very long wells or for very thin zones, the effect of reservoir height on dimensionless curve declines. Thus, for very large LD such as LD20, the horizontal type curve approaches that of a fractured vertical well with a fully penetrating vertical fracture that intersects the entire reservoir height.

Fig. 2b shows composite type curves for a horizontal well in the center of a square drainage area and for rwD=0.002. When area is defined as 2xe x 2ye, these type curves can be extended to rectangular drainage areas by replacing L/2xe by L/A0.5.

These type curves represent the constant pressure solution, i.e., the bottom hole flowing pressure remains constant during the well's life. If there is a change in bottom hole pressure, the well rate will include a new transient corresponding to the new bottom hole pressure. Reference 8 provides a good discussion on how changes in bottom hole pressure influence well rates.

REGIMES

Fig. 2a clearly shows two distinct production regimes. The early time part, on the left-hand side of the plot represents transient flow. The second part on the right-hand side represents deletion.

The analytical solution in the depletion period represents an exponential decline (b=0) of well flow rates. This is simply because the analytical solution in a closely bounded drainage volume assumes that the total compressibility of the rock and fluids is the only mechanism that provides pressure support to the reservoir.

In practice, during the decline phase, in addition to pressure support from total compressibility of the system, the reservoir may get additional pressure support that depends upon the reservoir mechanism. For example, in a solution-gas-drive reservoir, the gas released from the oil could provide extra pressure support. Similarly, a large gas cap or aquifer can also provide pressure support.

Thus, during the depletion phase, this extra pressure support will slow a well's production rate decline over time more than the decline shown in Fig. 2b. These factors can be accounted for by using Arps and Fetkovich decline curves 7 8 for the depletion-phase calculations.

FORECASTS

Similar to vertical wells, type curves developed for horizontal wells can estimate reservoir parameters by overlaying, on the type curve, log-log plots of the horizontal well's historical production.

Estimated values include reservoir permeability, drainage area, and a back calculation of an effective horizontal well length. From this effective length, a producing well length can be calculated.

The producing length can be different than the drilled length. This difference could be due to either geological or mechanical factors.

Well rate can also be estimated from type curves by using the reservoir and well parameters, and qD and tD. The example box shows the details for estimating the performance from a 2,400-ft long horizontal well in a 130 ft thick reservoir. The example provides a 10-year forecast obtained from the type curve for LD=3 (Fig. 2a). A horizontal well's rate-vs.-time forecast is plotted in Fig. 3.

AREAL ANISOTROPY

The previous example assumes uniform areal permeability, i.e. kx=ky. This may not be true in naturally fractured reservoirs, where the permeability along the fracture trend is larger than the direction perpendicular to the fractures. In such cases, the effective horizontal permeability, kh, is calculated as:

kh=square root of kxky [see equation]

In areally anisotropic reservoirs, the horizontal well should be drilled perpendicular to the high-permeability direction, i.e., parallel to the low-permeability direction. Assuming the expected drainage along the x direction is 2xe (Fig. 2a) and the well is drilled along the low-permeability x direction, the expected drainage along the y direction, 2ye, will be:

2ye=square (ky/kx) x 2xe [see equation]

In general, kx is very difficult to estimate. One possible method to obtain ky/kx is from interference tests on the wells. Typically, only average values of ky/kx for a given portion of the field can be estimated.

Areally, the estimated ky/kx could range from 1 for a uniform homogeneous reservoir to close to 80 to 100 in some naturally fractured reservoirs. However, in places along the horizontal well length, the value of ky/kx could be several hundred and can be difficult to estimate.

At least from the production and economic point of view, the concept of effective horizontal permeability, kh, will permit the model to forecast horizontal well production in naturally fractured reservoirs.

MODEL LIMITATIONS

With the horizontal well forecasting model, one can estimate well rates and ultimate reserves for horizontal wells drilled in both new and existing reservoirs. Also, the model is a mathematically exact analytical solution. Therefore, the model can also be used to check the validity of reservoir simulators.

If the horizontal well production history is known another useful application of the model is for estimating horizontal well drainage areas.

The model is derived for single-phase flow, but it can be extended to two-phase flow by using relative permeability to oil instead of the absolute permeability in all the calculations. Also, the model assumes no pressure drop along the horizontal well bore. Mathematically, this is described as an infinite-conductivity solution.

The model can estimate well drainage areas. However, it does not provide information regarding how this drainage area is distributed around the well bore in the areal plane.

Distribution depends on the value of ky/kx. The larger the value of ky/kx, the longer will be the drainage distance along the high permeability y direction. These factors should be taken into consideration in determining future well spacing and infill well locations.

PRODUCTION COMPARISONS

The success of horizontal wells in naturally fractured reservoirs, such as the Austin chalk and Bakken shale, illustrate the advantage of horizontal wells in areally anisotropic reservoirs.

If a vertical well in a naturally fractured reservoir is hydraulically fractured, then in most situations, the stimulated fracture will be created parallel to the high permeability direction. Hence, the fractured vertical well will drain a volume about the same as a vertical well and therefore, generally, the only benefit from fracturing is that of accelerating reserve recovery.

However, a horizontal well drilled perpendicular to the expected fracture orientation has the potential to drain a significantly larger area than a vertical well. The horizontal well will thus not only provide higher well productivity, due to the larger contact with the reservoir, but will also have greater ultimate recovery per well.

In addition, horizontal wells, because of greater exposure to the reservoir, have more potential of intersecting natural fractures than vertical wells, thereby recovering more reserves.

NATURAL FRACTURES

The following illustrates how to estimate the drainage area of a horizontal well in a naturally fractured reservoir. At the time the analysis was done, the horizontal well had a production history of about 1.5 years. The 1,500-ft long horizontal was drilled in a 35 ft thick pay zone.

The reservoir and fluid properties as well as the well production history for the first 1.5 years of production are summarized in Table 1.

In the first step, approximate values of the horizontal permeability and drainage areas were obtained by history matching the data on the type curve for infinite conductivity fractured vertical wells (Fig. 4).9

The horizontal permeability and well drainage area were estimated to be 0.7 md and 540 acres, respectively. For this calculation, the type curves for fractured vertical wells were used because the behavior of horizontal wells approaches that of infinite conductivity fractured vertical wells, especially for long horizontal wells drilled in thin reservoirs.

To refine this analysis further, four runs were made using the horizontal well model for reservoir permeability of 0.7 md and assumed drainage areas of 160, 320, 480, and 640 acres.

The calculated production forecasts were then plotted on the same graph as the well's production history (Fig. 5a). Fig. 5a clearly indicates that the 480 acre spacing provides the best match with the production history. Hence, 480 acres represents the drainage area of the horizontal well.

Several important points should be noted. One is that the time to reach pseudosteady state was calculated to be 496 days (Table 1). This calculation is based on the tDA=0.175 where tDA is defined by Equation 4 for fractured vertical wells. 10

In other words, the well was in the transient regime for about 500 out of 569 days of production history plotted in Fig. 5a. The cumulative oil trend continuously rises during the transient period.

For reservoir development, the operator was trying to decide between 320 and 640 acres well spacing for the horizontal wells. The above analysis indicates that the well is approximately draining 480 acres and therefore future wells should probably be drilled on 640 acres spacing.

The model thus provides a useful tool to predict well drainage areas. This is extremely important in making financial decisions regarding future well spacings and infill drilling.

Fig. 5b incorporates additional production history of the horizontal well in the cumulative oil plot. The flattening of the cumulative oil curve indicates that the well's drainage area has been reduced.

This reduction in the well drainage area may be attributed to several factors, such as overdrilling, reduction in permeability due to production, etc. In general, over-drilling appears to play a dominant role in the reduction of well drainage area, thereby reducing the ultimate recovered reserves per well.

HISTORY MATCH

A field history match and forecast were made for a horizontal well drilled in the Austin Chalk formation in Texas. To model this dual porosity system, the concept of effective permeability was used to develop the well production forecasts.

For horizontal Well X, history matching the early time production performance estimated an effective permeability of about 1 md. The match was based on 160 acre spacing,

However, after about 10 months of production, an offset operator drilled Well X increasing the water cut and decreasing oil production from Well X. This resulted in a reduction in the drainage area of the well.

Fig. 6 shows the history match before and after the offset well. The offset well decreased the drainage area of Well X to 100 acres from 160 acres, and therefore decreased the well's recoverable oil reserves.

Thus the model provides a good starting point to evaluate horizontal well potential in naturally fractured formations. The use of effective permeability for developing the forecasts provides reasonable answers.

ACKNOWLEDGEMENT

We would like to thank our colleagues Wenzhong Ding, Kevin Hall, W.B. Lumpkin and Susan Lacy and Mustafa Onur, assistant professor, Istanbul Technical University for their assistance in developing and testing the model.

REFERENCES

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