3.1. Comparison with the Unified Fracture Design Method
Given a certain proppant number, we can easily obtain the optimal dimensionless fracture conductivity to maximize the dimensionless productivity index for different aspect ratios of rectangular drainage areas using proposed Equations (1) and (2).
When the optimal dimensionless fracture conductivity is determined, the optimal fracture half-length and width can be obtained accordingly [
57]:
where
is the permeability of the propped fracture (md);
is the permeability of the reservoir (md);
is the thickness of the reservoir which, in this model, equals the fracture height (m);
is the optimal dimensionless fracture conductivity;
is the single wing volume of the propped fracture (m
3); and
is the total volume of the propped fracture (m
3).
We compare the optimization result for the optimal dimensionless fracture conductivity and maximum dimensionless productivity index obtained from the proposed analytical solutions with the result of the UFD method without considering the radial flow skin factor of fractured horizontal wells [
65]. The influence of the radial flow skin factor on the optimization result will be discussed in the next section. Both results are compared with the numerical result by the direct boundary element method [
66,
67].
We consider the following proppant numbers: 0.0001, 0.001, 0.01, 0.1, 1.0, 10.0, and 100.0. For aspect ratios, we divide the values into two groups. The first group comprises 0.1, 0.5, and 1.0, whereas the second comprises 0.05, 0.35, and 2.0. For the aspect ratios in the first group, all parameter values used in the UFD method (see
Appendix C) can be found in related tables and no interpolation or extrapolation is needed. For aspect ratios in the second group, some of the parameter values cannot be found in the given tables and interpolation or extrapolation is needed. Interpolation or extrapolation will cause errors, which will be discussed in this section. However, fracture geometry optimization using the proposed analytical solutions does not need any interpolation or extrapolation and, hence, will not introduce extra errors.
Figure 3 shows the optimal dimensionless fracture conductivity (
) and maximum dimensionless productivity index (
) obtained by three methods in the drainage area with aspect ratio (
) equal to 1.0 for various proppant numbers (
). Both the values of
and
from the proposed method and the UFD method are consistent with those obtained by the numerical method.
Table 2 lists the errors in
obtained from the UFD and proposed methods for several proppant numbers. The maximum error of
with the UFD method is 6.67% and occurs when
= 1.0, whereas the maximum error of
with the proposed method is 7.14% and occurs when
= 10.0. In this case, the errors from both methods are not high.
Table 3 lists the errors in
obtained from the UFD and proposed methods for several proppant numbers. The maximum error of
with the UFD method is 0.49% when
= 10.0, whereas that for the proposed method is 11.49% when
= 1.0. Note that proposed Equation (1) is valid for
less than 0.1 and Equation (2) applies for
larger than 0.1. With the increase of
, the reservoir-fracture flow system transitions from pseudo-radial flow to trilinear flow. Within the transition region, error occurs for the optimization using the proposed method.
Figure 4 shows the curves of
and
with
= 0.5 for various values of
, while
Figure 5 shows these for
= 0.1. Note that with the decrease of
, the
values obtained by both the proposed and UFD methods deviate from the true value given by the numerical method. For
= 0.1, the error becomes obvious, especially within the transition region. However, the error in
from the proposed method disappears within the transition region when
becomes small.
All aspect ratios considered above are from the first group, and all parameter values used in the UFD method (see
Appendix C) can be found in related tables. However, for the aspect ratios from the second group such as
= 0.05 or 2.0, the value of the shape factor (
) must be extrapolated from
Table A1, and the value of parameters
,
,
, and
must be extrapolated from
Table A2. For
= 0.35, those parameters must be interpolated from the related tables. Since we have no special explanation for
Table A1 and
Table A2, we use linear interpolation and extrapolation in this work.
Figure 6 shows the curves of
and
with
= 0.05 for various values of
. Since
is extremely small, the values of
obtained from both the proposed and UFD methods deviate from the result given by the numerical method, which is similar to the case presented in
Figure 5. As is shown in
Table 4, the maximum error in
with the UFD method is 153.96%, whereas it is 160.31% with the proposed method, both of which occur within the transition region. Nevertheless, the extrapolation used in the UFD method has no significant effect on
.
However, because of the extrapolation of the shape factor (
),
becomes invalid when
≤ 0.1, as listed in
Table 5. Some values of
are less than 0, which is also invalid when
is 0.1–10.0. Even for valid
values when
> 10.0, the values deviate seriously from the true values given by the numerical method. The data deviation leads to a maximum
error of 401.18% when
= 100.0. For the proposed method, although the value of
deviates from the true value seriously, the
error is low and reaches a maximum of 15.52% when
= 10.0. In this sense,
calculated by the proposed method is closer to the true value than that from the UFD method, which suffers from extrapolation error.
Figure 7 shows the curves of
and
with
= 0.35 for various values of
, while
Figure 8 shows these with
= 2.0. Note that the interpolation for
= 0.35 and the extrapolation for
= 2.0 used in the UFD method have no significant effects on
, whereas they do have serious effects on
, especially when
is large. Nevertheless, interpolation induces less error than extrapolation in the UFD method. However, the proposed method does not experience any interpolation or extrapolation problem. The only error associated with the proposed method for
= 2.0 comes from the transition region, which is similar to that in
Figure 3. This is the disadvantage of the proposed method, especially when
is large.
Summarizing from the above analyses, we can compare the performances of the two methods under different conditions:
- (1)
When
is within the intermediate range, for example, 0.1–1.0, the UFD method suffers from interpolation error if the value of
is not given in
Table A1 or
Table A2. However, the proposed method is applicable for arbitrary aspect ratios and the results agree well with true values obtained by the numerical method.
- (2)
For less than 0.1 or larger than 1.0, the UFD method suffers from extrapolation error. In particular, deviates seriously from true values given by the numerical method.
- (3)
For less than 0.1, values obtained by the proposed method deviate from the true values within the transition region. On the other hand, for larger than 1.0, error occurs for obtained by the proposed method within the transition region. The reason is probably that when the aspect ratio of the drainage area becomes too small or too large, flow in the reservoir-fracture system deviates from the trilinear-flow model and new models should be applied together with the current model in the future study.
3.2. Considering the Radial Flow Skin Factor
Note that from Equations (3) and (4), the dimensionless productivity index of fractured horizontal wells depends not only on the dimensionless fracture conductivity, proppant number, and aspect ratio, but also on the permeability of the fracture and the reservoir, the thickness of the reservoir, the radius of the wellbore, and the average width of the fracture. Even given a certain proppant number and aspect ratio, the dimensionless productivity index cannot be expressed as a single-valued function of dimensionless fracture conductivity. Thus, the formulas for optimal dimensionless fracture conductivity given by the UFD method in Equations (A45) and (A46) are no longer accurate, and no optimization solution exists, especially for fractured horizontal wells.
However, thanks to the analytical solutions of the dimensionless productivity index proposed in Equations (1)–(4), it is possible to obtain the optimization result for fractured horizontal wells. The proppant number for rectangular reservoirs is defined as follows [
65]:
where
is the length of the drainage area parallel to the direction of the fracture (m) and
is the length of the drainage area vertical to the direction of the fracture (m), as illustrated in
Figure A1. Combining Equations (6) and (7), we can obtain:
Substituting Equation (8) into Equation (3), the skin factor becomes:
Substituting Equation (9) into Equation (4), we can obtain the analytical formula of the dimensionless productivity index for fractured horizontal wells expressed in terms of dimensionless fracture conductivity and proppant number. Thus, given a certain proppant number and the related reservoir-well parameters, the optimal dimensionless fracture conductivity and maximum dimensionless productivity index can be obtained.
For example, setting the following parameters:
,
, and
, we can plot the relationship between optimal dimensionless fracture conductivity and proppant number, as is shown in
Figure 9. The optimal dimensionless fracture conductivity is compared with the results from the UFD and numerical methods. Note that since the skin factor is taken as a constant in the UFD method, the optimal results from the UFD method deviate seriously from the true values of the numerical method, especially when
is less than 0.1, whereas the results from the proposed method agree well with the true values.