A transient flow solution to describe fluid flow to horizontal fractures applied to frac‐packed wells

This paper presents an analytical model to describe the transient fluid flow to a horizontal fracture in frac‐packed vertical wells. A physical model is first established where a horizontal frac‐packed fracture is located in the middle of a cylindrical reservoir. The pressure distribution in the fracture is then derived concerning time and the distance to the wellbore. The analytical model assumes a homogeneous and isotropic reservoir where Darcy's law applies to a single‐phase fluid flow, a radial fracture with a radius much greater than pay zone thickness, and a constant fracture width. For ease of application, a Type Curve Matching approach is introduced which features the matching of a defined dimensionless pressure between field pressure data and model data, and some exemplar curves are illustrated in this paper. The principle of applying curve matching here is to tune model parameters to match the measured pressure/rate data. Once the curves are matched, the model parameters, that is, fracture permeability, etc, are naturally obtained. The validity of the derived model is informed in two ways. The results show that the model will degenerate to the classic radial flow equation for practical values of certain model parameters and upon reasonable simplification. Model validity is further verified by applying the type curve matching technique using field data. This model can be used as a quick and reasonable method to analyze the transient pressure/rate test data to reveal fundamental fracture and reservoir properties for a frac‐packed horizontal fracture at shallow depths where the vertical formation stress is the least in situ stress.

they are horizontal. Lamont and Jessen 2 concluded that restricted fractures also propagate in a plane normal to the least principal compressive stress. The least principal stress is usually the overburden stress in some shallow formations, and thus, the hydraulic fracture propagates in a horizontal plane. The least principal stress tends to be horizontal in reservoirs deeper than approximately 1000 ft, and thus, the hydraulic fracture is vertical. 3,4 Nowadays, it is generally believed that the least principal stress is vertical in formations shallower than approximately 2000 ft, and horizontal in formations deeper than 4000 ft. This implies that horizontal fractures form at a depth shallower than 2000 ft and vertical fractures form at a depth deeper than 4000, during hydraulic fracturing. However, different values of these threshold depths are found by researchers. Lu et al 5 reported that the maximum principal stress in horizontal orientation fell between 3045 and 3360 m deep. It is therefore important to evaluate in situ stresses case by case to predict the orientation of hydraulic fractures. Additionally, the azimuth of the vertical fracture depends on the azimuth of the minimum and maximum horizontal stresses. 6,7 A review of the subject is given by Huang et al. 8 The generation of horizontal fractures as we discussed above contributes to a useful stimulation and completion technique for modern reservoir development-Frac-packing.
Frac-packing is a special hydraulic fracturing technique developed in the early 1990s. 9 It combines the stimulation advantages of highly conductive hydraulic fractures with sand control using gravel pack to improve sand control capacity and well productivity in unconsolidated reservoirs. An overview of the technology was presented by Ellis. 10 A full technical description of frac-packing was given by Ghalambor et al. 11 Since frac-packing is mainly used for fracturing unconsolidated reservoirs in shallow depth where the overburden stress is normally the least stress, horizontal fractures are believed to be created. The orientation of the fracture can be informed by the pressure-time data plot. 12 A straight line in the plot of the bottom hole pressure (BHP) versus square-root-time (t 0.5 ) indicates a vertical fracture. 13 A non-linear behavior was shown by Morales et al 14 for non-vertical fractures. A unit slope in a log-log diagnostic plot should indicate pressure depletion in the region covered by a horizontal fracture. 15 A big challenge in evaluating the productivity of fracpacked wells with horizontal fractures is the unknown fracture conductivity. This problem arises because formation sands invade into the fracture and reduce the permeability of the gravel/ proppant pack in the fracture. Berg 16 presented a correlation for estimating the permeability of particle packs. It requires the knowledge of particle size distribution and frac-pack porosity. Similar correlations were presented by Van Baaren 17 and Nelson 18 for consolidated sedimentary rocks. These correlations do not apply to the prediction of fracture permeability due to the unknown size distribution of the gravel/proppant pack with the invasion of formation sands.
A practical technique to determine fracture permeability is the analysis of pressure-time data obtained from pressure transient testing on fractured wells. Numerous analytical models have been developed to analyze pressure transient data for wells with vertical fractures. These models are used to identify multiple flow regimes including reservoir linear flow, fracture-reservoir bilinear flow, and pseudo-steady flow. Mathematical models for reservoir linear flow were given by several investigators including Gringarten et al 19 and Cinco et al. 20 A fracture-reservoir bilinear flow model was presented by Cinco and Samaniego. 21 The pseudo-steady flow model for multi-fractured horizontal wells was developed by Li et al. 22 A pseudo-steady flow model for high-energy gasfractured wells was given by Li et al. 23 A pseudo-steady flow model for re-fractured wells was presented by Shan et al. 24 However, none of these models applies to wells with horizontal fractures.
This study presents an analytical model based on reservoirfracture crossflow to describe the behavior of a vertical well with a horizontal fracture. The model focuses on the transient flow period when the flow in the fracture dominates, which enables us to characterize reservoir properties such as fracture permeability. Particle invasion can plug the pores in the fracture and reduce its permeability. This model can be used as a diagnosis tool to calculate the effective fracture permeability and fracture conductivity, which is one of the potential ways that this model can be used for. It provides petroleum engineers with an analytical approach to obtaining fracture properties from well test data. mid-depth of the pay zone where the vertical in situ stress is the least min . The following assumptions were made in model formulation.

| MATHEMATICAL MODEL
1. The reservoir is homogeneous and isotropic. 2. Darcy's law applies. 3. Fracture propagates in the radial direction with a constant width w. 4. Single-phase liquid flow prevails in the reservoir and fracture. 5. The radius of the fracture is much greater than the pay zone thickness.
If assumption 5 is valid, it is expected that vertical flow prevails in the region above and below the fracture, while radial flow dominates in the fracture. An analytical solution for early time is derived in Appendix A and outlined in this section. Darcy unit system is used for the derivation.
Based on the principle of conservation of mass, the governing equation takes the following form: where the parameters are defined as where p e is reservoir pressure, p f is the fracture pressure at radial distance r and time t, f is fracture porosity, c f is fracture compressibility, is fluid viscosity, k f is fracture permeability, h is pay zone thickness, w is the average fracture width, and k m is the matrix permeability in the zone.
The initial condition for the fracture is set as: The outer boundary condition for the fracture is expressed as: The inner boundary condition for the fracture can be expressed using Darcy's law applied to the assumed line-source wellbore: The upper and lower boundary conditions for the reservoir matrix are no-flow boundaries expressed by: The inner boundary conditions for the reservoir matrix are expressed as: The analytical solution in Darcy units takes the following form (see Appendix A for derivation in Darcy's units): where I 0 is the modified Bessel function of the first kind of zeroorder, K 0 is the modified Bessel function of the second kind of zero-order, Ei is exponential integral, and is the exponential function of Euler's constant ( = e 0.5572 = 1.78). Considering a skin factor S near the wellbore, Equation (10) degenerates to an expression for the pressure at the wellbore where r = r w : Since the analytical solution was derived for early time flow from the matrix to fracture in the fractured area, the solution is expected to be valid for flow time up to the moment when the radius of investigation reaches the radius of fracture expressed as: = 0 at the upper and lower boundaries (9) p = p f at fracture faces where R is the radial extension (radius) of fracture.

| MODEL VALIDATION
Although no clean data set has been found to validate the derived analytical solution yet, the degenerated form of the solution was verified for the special case of pure radial flow in the future. In fact, for practical values of r w and M, the value of In situations where the matrix permeability is much lower than the fracture permeability, the large value of M makes t FM 2 < 0.01, then the following approximation holds: Substituting Equation (17) into Equation (16) gives Substituting Equation (3) into Equation (18) results in which is the classic solution for radial flow, proving that the analytical solution is valid. Classic solution for radial flow can be used as ground evidence to verify the correctness of newly proposed models, as it is established result and has been tested true in field applications for years. Readers could refer to Well Productivity Handbook, 2nd edition, Elsevier, Cambridge (2019) by Dr Boyun Guo for more details.

| Type curves
Equations (10) and (11) are difficult to use in applications owing to their involvements of the modified Bessel function of the first kind of zero-order I 0 , the modified Bessel function of the second kind of zero-order K 0 , and the exponential integral Ei. Type curves were generated for easy applications. Equations (10) and (11) are rearranged to get p d and p w , respectively, in the following forms: and where the fracture function Φ (r, t) and well function W r w , t are defined respectively as:

or approximately
Notice that when the radius r is changed to the wellbore radius, Equation (20) becomes Equation (21). Equation (22) is the mathematical solution to the governing Equation (1). The result can be conveniently verified by taking the partial derivative of Equation (22) with respect to radius r. Figure 2 presents fracture function curves for F = 0.0003 s/ cm 2 and M = 1200 cm at three values of time. The function tends to stabilize after 1 hour of flow time. Since the pressure profile inside the fracture is not measurable with today's technologies, the fracture function curve has limited applications. Figures 3-5 show well function curves for F = 0.00015 s/ cm 2 , 0.000075 s/cm 2 , and 0.00005s/cm 2 with S = 0. Since the wellbore pressure is measurable with today's technologies, the well function curves can be used for revealing fracture and reservoir properties. Rearranging Equation (21) gives: The left-hand side (LHS) of this equation is called Dimensionless Pressure (DP) which can be evaluated with well test data and assumed fracture conductivity wk f . By matching the W (r, t) to the DP through tuning F and M values, it is possible to reveal fracture permeability, average fracture width, and matrix permeability.

| Field case study
A vertical well was completed with frac-packing against an oil pay zone from 833 m (2732 ft) to 849 m (2787 ft). Known reservoir and fluid properties are summarized in Table 1 pressure gauge during pumping. Averaged early time BHP data are plotted in Figure 6. Figure 7 shows a tuned match of W r w , t to dimensionless pressure data. This match was obtained using the following parameter values: which gives: This result was checked by the radius of investigation through Equation (12): which is consistent with the proppant volume consumed in this frac-packing operation.

MATCHING
The value of the parameter of interest remains unchanged regardless of the way it is expressed. By this principle, we can express the same parameter, which is the Dimensionless Pressure in this case, in two different ways and make them equal to reveal the value of certain parameters. In this paper, one way is to use the real transient pressure/rate test data as expressed by Equation (25), and the other is to use the result of the analytical model as expressed by Equation (24). By tuning the F and M values in the analytical model to match the   Dimensionless Pressure, the reservoir and fracture properties within the F and M coefficients can therefore be obtained. The Type Curve Matching approach can be used as a reliable technique to obtain reservoir and fracture properties. It should be noted that the well-testing data that are used as model input for this purpose should contain the transient pressure/rate test data recorded at the very beginning of the test. This upper limit of time that makes the testing data usable is expressed as t max shown in Equation (12). Substituting the transient pressure/rate data into Equation (25), we can obtain a series of DP points, then adjusting the F and M values in Equation (24) to match the DP series. The resultant F and M values are used to calculate the fundamental reservoir or fracture properties such as fracture permeability.

| CONCLUSIONS
An analytical solution was developed in this study to describe fluid flow through a horizontal fracture to a vertical well. The solution applies to frac-packed vertical wells in shallow depths where the vertical formation stress is the least in situ stress. The following conclusions are drawn.

The analytical solution degenerates to the classic solution
for radial flow when the mass-feeding of the reservoir matrix to the fracture is negligible. This partially validates the correctness of the solution. 2. The analytical solution can be used to analyze transient pressure data to reveal fracture and reservoir properties where horizontal fractures are created. Tuning model parameters to match the measured bottom hole pressure data allows for the estimation of fracture conductivity, fracture permeability, average fracture width, and reservoir permeability. 3. Field case analysis with the analytical solution shows that type curves generated by the solution can provide a quick and reasonable result of fracture and reservoir properties when a data match is used as a solution technique. 4. The Type Curve in this paper has demonstrated a very promising match on Dimensionless Pressure from field data and analytical model. It is desirable however to further verify the complete solution using field data in the future.