2.1. Fracture Geometry Optimization Based on UFD
UFD method gives the analytical solution for the optimal dimensionless fracture conductivity in order to maximize the dimensionless productivity index, given a certain mass or volume of proppant. The optimal dimensionless fracture conductivity is the function of a proppant number. The details about UFD method are given in
Appendix A.
When the optimal dimensionless fracture conductivity is determined, the optimal fracture half-length and width can be obtained accordingly:
where:
is the permeability of the propped fracture (md);
is the permeability of the reservoir (md);
is the thickness of the reservoir, in this model, it equals to the fracture height (m);
is the optimal dimensionless fracture conductivity;
is the single wing volume of the propped fracture (m
3);
is the total volume of the propped fracture (m
3).
As mentioned above, since the volume and permeability of the propped fracture are unknown, the fracture dimensions cannot be solved directly. Note that the volume and permeability of the propped fracture are both related to the proppant concentration, which reflects the uniformity of the proppant distribution. In theory, if the distribution of the proppant in the fracture is completely uniform, then the value of the proppant concentration can reach the bulk density of the proppant on the ground. However, due to proppant transport in the pipe and fracture, the distribution of the proppant is never completely uniform. Thus, the value of the proppant concentration should be less than the bulk density. Nevertheless, the proppant concentration can be maintained at a suitable level by optimizing the treatment parameters. So, we can preset a desired proppant concentration value and then calculate the volume and permeability of the propped fracture. The corresponding optimal fracture dimensions can be obtained accordingly, which can be reached through treatment parameter optimization. Thanks to the corresponding relationship between the optimal fracture dimensions and the proppant concentrations, once the optimal fracture dimensions are reached, the desired proppant concentration can be obtained.
This paper proposes an iterative method to solve the optimal fracture dimensions given a preset proppant concentration. Suppose all the proppants stay within the range of the thickness of the reservoirs; then the relationship of the mass and concentration of the proppant in the fracture is as follows:
where
is the proppant mass (kg), which is constant if the fracturing scale is fixed.
is the proppant concentration (kg/m
3). Then, the fracture width Equation (2) can be rewritten as follows:
The permeability of the fracture is related to the proppant concentration per unit area and closure pressure. The proppant concentration per unit area is defined as:
Then, function of the permeability of the fracture can be written as:
where
is the closure pressure (MPa), which can be determined by field tests. There is no explicit equation for the function
, which can be obtained from the permeability curve of different proppant concentrations per unit area and closure pressures, given one certain type of proppant. The permeability curves can be obtained through lab experiments.
Given the proppant mass and the desired proppant concentration in the fracture, based on the optimal dimensionless fracture conductivity , the optimal fracture dimensions and permeability can be solved by the fracture half-length Equation (1), width Equation (4) and permeability Equation (6) using the iterative method.
Firstly assume an initial permeability , and substitute it into the fracture width equation to solve . The proppant concentration per unit area can then be solved by , and the new permeability can be obtained from the permeability curve. If the difference between the new value and the assumed one is within a certain small range, then the new permeability and corresponding fracture width can be obtained. Otherwise, replace the assumed permeability with the new value and repeat the above procedure until convergence is achieved. Once the fracture width and permeability are calculated, substitute them into Equation (1) to solve the optimal fracture half-length.
2.2. Treatment Optimization through a Fast Semi-Analytical Fracture Propagation Model
Once the optimal fracture dimensions are determined, the treatment parameters also need to be optimized. There are many parameters that influence the fracture dimensions, and the value of each should be within a certain range considering the treatment feasibility. Moreover, the treatment optimization result should satisfy the fracture length and width simultaneously. In order to solve this optimization problem, the treatment parameters need to be tuned and the fracture dimensions calculated repeatedly. Thus, the computation of the fracture propagation model should be rapid. Currently, the widely-used PKN [
44,
45,
46] analytical model is very fast. However, its solution does not rigorously satisfy the flow continuity equation. Pseudo-3D [
47,
48,
49,
50] and fully-3D [
37] models can solve the geometric dimensions of the propped fracture accurately. However, these methods are time-consuming and rarely suitable for treatment parameter optimization. This paper proposes a fast 2D semi-analytical fracture propagation model, which assumes that the fracture height is constant and the proppant is transported in the fracture at the same speed as the sand-laden fluid.
In this method, the total pumping time is divided into segments. The length of each time segment is and the pumping fluid volume during each segment is constant, . Then, after time segments, the total pumping time is and the fluid elements in the current fracture can be recorded as accordingly. The following values of each fluid element are calculated according to the total pumping time : the length , the distance of its back interface to the fracture entry , the cumulative fluid loss volume , the remaining volume , the proppant mass , and the proppant concentration . The proppant mass in each element is related to the pumping schedule and supposed to be invariable during fracture propagation. Additionally, we record the current total fracture half-length and width distribution at the end of each pumping time segment.
During one new pumping time segment, the length of each existing fluid element
varies continuously due to the change of the fracture width at its location and the fluid loss within it. As more fluid is injected, the existing fluid element moves forward. Hence, the distance of the back interface to the fracture entry
also changes. Due to fluid loss, the volume of the fluid element decreases continuously. For the pad fluid element, its volume decreases to 0 eventually. For the sand-laden fluid element, thanks to the existence of proppant, its final volume will be greater than 0. The maximum concentration
should be set to ensure proppant transport in the fracture. Otherwise, a sand plug will occur.
can be determined empirically.
Figure 1a shows the fluid element distribution after
time segments.
Figure 1b is the fluid element distribution after
time segments.
The width equation uses the PKN analytical solution for double wing fractures with fluid loss [
44]:
where:
where
is the rock Poisson’s ratio,
E is the rock elastic modulus (Pa),
is the fracture height (m),
C is the fluid loss coefficient (m/s
1/2),
is the apparent viscosity of the power law fluid (Pa·s),
K is the fluid consistency coefficient (Pa·s
n),
n is the flow index,
Q is the injection rate (m
3/s), and
is the average width along the fracture height (m). For the PKN model,
.
Because the fracture length solution of the PKN model does not satisfy the flow continuity equation, the following continuity equation is used for fracture half-length calculation (where spurt loss is ignored):
where:
The volume of the fluid loss for each element is limited as follows:
Because the fracture half-length
cannot be solved explicitly, an iterative method is used. First, an initial fracture half-length is assumed, and then the fracture width distribution is solved by Equation (8). We can substitute it into Equation (10) to check whether the continuity equation is satisfied. If it is not, then we assume another fracture half-length and calculate again. In general, the fracture half-length increases with time. So, the assumed value can be set as the previous value plus a small increment until it finally satisfies the continuity equation. When both the fracture width and half-length have been determined, the fluid element space in the fracture should be reassigned. The remaining volume of each fluid element after pumping for
time segments is:
According to the remaining volume of each fluid element, starting from the fracture tip, we calculate in turn the distance of the back interface of element
to the fracture entry:
As the model satisfies the continuity equation, then the following condition will be satisfied naturally:
Thus, the length of each fluid element becomes:
The proppant concentration in the fluid element can be calculated as:
Because fluid loss occurs during fracturing, sand-laden fluid that was injected early is subject to a long loss time; hence, the sand ratio of the injected fluid at early times should be relatively small. As the loss time of sand-laden fluid injected at a later stage is short, the sand ratio of the injected fluid can be relatively large. Thus, at the end of injection, the proppant in the whole fracture will be evenly distributed. For the convenience of operation, the staged proppant pumping schedule is usually used. It means that one constant sand ratio is maintained over a period of time. When the injection of these portions of sand-laden fluid is finished, the sand ratio is then changed. In this paper, the sand ratio is increased according to the following power function to ensure that the proppant is distributed as evenly as possible, so as to reach the desired proppant concentration:
where
is the sand ratio (representing the ratio of proppant to fluid volume),
and
are coefficients (in this paper,
is referred to as the proppant pumping curve index), and
is the proppant pumping sequence. If all the proppants are to be injected through 10 times, then the value of
is an integer from 1–10. In this paper, the proppant pumping times and the maximum sand ratio will be determined empirically. Then, for a given value of coefficient
, the corresponding coefficient
can be determined. Thus, the optimization of the complicated proppant pumping schedule can be simplified as the optimization of the single coefficient
.
After the injection has finished, the pad fluid continues to be lost and the front of the fracture will close. Thus, the half-length of the propped fracture can be set as the length in which proppant is present. If the proppant concentration in the direction of the fracture width has not reached a certain value, then the fracture will close and the fracture width will decrease until the proppant concentration reaches the desired value. The final width is the propped fracture width. The average fracture width is used in the UFD method regardless of the fracture shape.
Figure 2a shows the state of the fracture immediately after the end of injection.
Figure 2b shows the state of the propped fracture.
This simulation needs to find the optimal treatment parameters in order to satisfy both the fracture half-length and width. So, the objective of the treatment optimization is to minimize the following error function:
where
is the calculated fracture half-length (m),
is the optimal fracture half-length (m),
is the calculated fracture width (m), and
is the optimal fracture width (m).
The basic flowchart of the proposed fracturing treatment optimization method based on the UFD is shown in
Figure 3.