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A new model for diagnostic fracture injection test (DFIT) complexity in unconventional reservoirs
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A new model for diagnostic fracture injection test (DFIT) complexity in unconventional reservoirs Zhiming Chen a,b,*, Xinwei Liao a, Jiali Zhang a a
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum at Beijing, Changping, 100249, China b The University of Texas at Austin, Austin, TX, 78712, USA
article info
abstract
Article history:
In the recent years, much attention has been paid to unconventional reservoirs of
Received 27 December 2018
hydrogen and carbon which are good sources of hydrogen energy. To ensure sufficient
Received in revised form
energy organic, the first step is understanding the properties of those unconventional
5 March 2019
reservoirs. The diagnostic fracture injection test (DFIT) is a very good technique to estimate
Accepted 7 March 2019
the parameters of the unconventional reservoirs. Unfortunately, most DFIT models do not
Available online 2 April 2019
involve the complex fracture networks caused by natural fractures, fissures, and microfractures.
Keywords:
© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Unconventional reservoirs of hydrogen and carbon Diagnostic fracture injection test (DFIT) Fracture complexity Boundary element model
To better understand the properties of unconventional reservoirs of hydrogen and carbon, this paper introduces a new generalized model for after-closure analysis (ACA) of DFIT with complex fracture geometries. The generalized model is based on a boundary element method that neglects fracture mechanics processes. First, the model flexibility is demonstrated by different complex fracture cases, including opening-fissure fracture networks (OFFN), tree-like fracture networks (TLFN), dendritic multiple fracture networks (DMFN), and mutually orthogonal fracture networks (MOFN). It is found that various fracture-network cases exhibit different pressure transient behaviors, which may provide a good tool to identify fracture geometries and reservoir
parameters. Subsequently, the model reliability is demonstrated by a model validation and a case study of DFIT complexity using the log-log diagnostic plots based on the model solution. This is our primary work to estimate the DFIT complexity using after-closure analysis, and it will assist with reservoir evaluation and production design substantially.
Introduction Much attention has been paid to the characteristics of wells in unconventional reservoirs of hydrogen and carbon [1,12,18,34,35,41]. The unconventional reservoirs can be good
* Corresponding author. The University of Texas at Austin, 78712, Austin, TX, USA. This paper is an improvement for a conference presentation (SPE 189793), and the full copyright of presentation has been reverted to the original owner. E-mail addresses: [email protected], [email protected] (Z. Chen). https://doi.org/10.1016/j.ijhydene.2019.03.060 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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sources of hydrogen energy, because they can produce plenty of energy-rich organic compound in which the hydrogen is generally locked up. Understanding the properties of unconventional reservoirs of hydrogen and carbon is the first step to ensure sufficient energy organic. Xue et al. [38] used the pressure transient analysis to study the fluid flow behaviors through horizontal fractures, which is helpful to estimate the parameters of reservoir and fracture. Meanwhile, the small-scale fracturing, or DFIT has extended its role to understand the properties of unconventional reservoirs of hydrogen and carbon [5]. With the DFIT data, many parameters of unconventional reservoirs can be obtained, including closure pressure, formation permeability, leak-off coefficient, and initial formation pressure. Based on the Perkins-Kern-Nordgren (PKN) fracture geometry model by Perkins and Kern and extended work by Nordgren [27,28,30] performed several derivations for the fluid loss coefficient, the fracture length and width, fluid efficiency, and closure time for the fracture. Particularly, Nolte derived the G-time function and correlated the G-function with pressure and flow rate. With G-function, the closure time and closure pressure can be easily distinguished on the plot of pressure and G-function. The original work is extended by Ref. [29] to the Khristianovic-Geertsma-de (KGD) model and radial model. Soliman et al. [33] conducted the extended work on afterclosure DFIT analysis. They employed the after-closure stage pressure decline data to determine the formation permeability and reservoir pressure. Based on the transient pressure analysis for the fractured well, Soliman completed the derivation for the quantification of reservoir pressure and formation permeability of each flow regime, namely, linear flow, bilinear flow and radial flow. Barree et al. [3] performed five non-ideal leak-off case studies with numerical simulation (GOHFER simulator). Three pressure function P, dP=dG, and GdP=dG are used to uniquely obtain the existence and magnitude of pressure dependent leak-off behavior and eliminate the ambiguity between pressure dependent leak-off, fracture height changes, and tip extension and recession. Also, the effect of pressure dependent fracture compliance was first described. In the later work [4], Barree et al. suggested using G-function derivative analysis along with a square root of time plot and a log-log diagnostic plot to identify closure. All three methods should agree when closure is correctly identified. Barree helped avoid the wrong pick of fracture closure point pffiffi with modified t analysis. Different flow regimes can be identified using specific slope line on the log-log DPwf Dt plot [5]. concluded the usual mistakes we made in field application and illustrated how to avoid these mistakes and improve data acquisition and analysis. Further, much meaningful work has been done on the DFIT analysis [23,36]. It is worth mentioning that natural fractures, fissures, and microfractures are well-known contributors to production performance of unconventional reservoirs [25]. Complex fracture geometries can be generated by either small-scale fracturing, DFIT, or large volume fracture stimulation, because the activation of pre-existing natural fractures, fissures, and microfractures plays a significant role on
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Fig. 1 e Complex fracture geometries generated by DFIT, owing to pre-existing natural fractures [26].
generation of induced hydraulic fractures [13,14,20,21,24], as shown in Fig. 1. Therefore, much more attention should be paid to the model development of DFIT complexity [31]. However, very limited work about DFIT has considered the fracture complexity. Bakar [2] used a numerical hydraulic fracturing simulator, namely Complex Fracturing ReseArch Code, to investigate the viability of the DFIT analysis techniques with and without micro-fractures. The numerical simulator was based on a discrete fracture network method. This work provided a very good perspective on DFIT analysis, but it was a little timeconsuming owing to the numerical approach [6]. built a complex fracture-network model for DFIT with mesh-free methods [8,19,32,35,38,39,40,42]. In their model [6], the pressure solution for the reservoir flow was analytical, mesh-free, while the pressure solution for fracture flow was numerical. A nodal analysis method was used to eliminate the flow interplay within fracture intersections. Unfortunately, the fracture networks were orthogonal, and non-orthogonal geometries were not involved. In addition, the model practicality was not demonstrated. To narrow the gap between theory and practice, this paper introduces a new boundary element model with random fracture geometries caused by natural fractures, fissures, and microfractures. First, the model flexibility is demonstrated by different complex fracture cases, including opening-fissure fracture networks (OFFN), tree-like fracture networks (TLFN), dendritic multiple fracture networks (DMFN), and mutually orthogonal fracture networks (MOFN). Finally, the model reliability is showed by a model validation and a case study of DFIT complexity using the log-log diagnostic plots based on the model solution. This is our primary work to estimate the fracture geometries using after-closure analysis of DFIT Complexity, and it will assist with reservoir evaluation and production design substantially.
Mathematical model Model development In this paper, we built a boundary element model for the afterclosure analysis (ACA) reservoir properties determination with the consideration of complex fracture geometries. The proposed boundary element model is composed of analytical reservoir solutions and numerical fracture solutions. The discretization and grids in spatial and temporal dimensions
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With the discretized fracture panels, the analytical reservoir solution is obtained based on point source function. The diffusion equation of a point source in the 2D infinite reservoir is given by v2 pD v2 pD vpD þ 2 ¼ vtD vx2D vyD
(1)
where pD is the dimensionless injection pressure, xD and yD are the dimensionless distances, and tD is the dimensionless injection time, with the auxiliary equations 8 > < pD xD ; yD ; tD ¼ 0 ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pD ðrD /∞; tD Þ ¼ 0; rD ¼ x2D þ y2D > : vpD ðrD /0; tD Þ vrD ¼ qD
Fig. 2 e Physical model of complex fracture geometries for DFIT.
are not required in the proposed boundary element model, which has the analytical characteristics. The explicit description of random fracture geometries (Fig. 2) induced by the DFIT small injection treatment are included in our model to obtain a more realistic result.
where qD is the dimensionless flow rate out from a point source [43]. presented an exhaustive compilation of the detailed solutions by using Laplace transform. Following that, we obtain the pressure solution for a point source, pD ¼ qD K0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi x2D þ y2D s
Some assumptions are made for the model development, including: (1) a homogenous, isotropic, and semi-infinite reservoir with sealed top and bottom boundaries, (2) singlephase and slightly compressible fluid, (3) Darcy's flow, (4) ignorance of vertical flow and gravity effects, (5) constant fracture width, (6) 1D fracture flow along the fracture length, and (7) a fixed injection rate.
Reservoir flow During the injection, the fluid injected through the wellbore flows within the complex fractures, then entering the reservoir. Hence, two kinds of flow modes are considered: (1) reservoir flow and (2) fracture flow. Table 1 provides the definitions of all dimensionless variables for model development. Let us first focus on the reservoir flow. To explicitly reflect the random fracture geometries, the complex fracture is discretized into multiple fracture panels and fracture vertexes.
(3)
where superscript “-” is the symbol of Laplace transform, K0 ðxÞ is the second kind of modified Bessel functions, and s is the Laplace variable with respect to tD . The fracture segment is a collection of point sources. The integral of Eq. (3) yields, Z
Assumptions
(2)
pD ¼
qFD K0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi x2D þ y2D s dl
(4)
l
Fracture flow As assumed, the fluid flow is assumed to be 1D in the 2D reservoir, because the ratio of fracture width to fracture length is tiny. Given that fracture permeability is very high and fracture diffusivity tends to be much higher, transient effects in fracture flow are short-lived compared to the transient reservoir flow [9]. Ignoring the short transient period in fractures, we can obtain a diffusivity equation to describe the fluid flow within the continuum fracture. Following the work [9], the diffusivity equation is given by,
v2 pFD 2p þ qFD ¼ 0 CFD vy2D
(5)
Table 1 e Definitions of dimensionless parameters. Dimensionless variables Reservoir pressure Fracture pressure Injection time Injection rate Flow flux from fracture to reservoir X distance
Definition
Dimensionless variables
Definition y Lr r ¼ Lr
khðp pi Þ 1:842 103 mBqsc khðpF pi Þ pFD ¼ 1:842 103 mBqsc 3:6kt tD ¼ mfCt L2r q qD ¼ qsc
Y distance
yD ¼
Radial distance
rD
Fracture length
LFD ¼
qfD ¼ qF Lr =qsc
Wellbore radius
rwD
pD ¼
xD
Please note that Lr is the reference length in meter.
x ¼ Lr
Fracture conductivity
CFD ¼
kF WF kLr
LF Lr rw ¼ Lr
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where pFD is the pressure in fractures, qFD is the flow flux, and CFD is the fracture conductivity. The areal coordinate (q) in cylindrical coordinates ties back to areal coordinates in Cartesian coordinates by xD ¼ rD cos q; yD ¼ rD sin q
(6)
The initial condition of fracture pressure yields
pFD yD ; tD ¼ 0 ¼ 0
(7)
and the flow rate at the inner boundary satisfies, vpFD 2p ywD;dw ; tD ¼ qwD CfD vyD
(8)
where ywD;dw is the location at inner boundary of a fracture segment; qwD is the flow rate at inner boundary. A mass balance at outer boundary is required as qD ywD;up ; tD ¼ qwD þ LFD qfD
(9)
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Model solution The model solution involves three steps: discretization, equations, and solution algorithm
Discretization. As mentioned before, we discretize the complex fracture case into multiple segments to describe the real geometry (Fig. 3), and then the fracture information is imported and quantified for the use of our boundary element model. We assume that the numbers of fracture panels and fracture vertexes are np and nv, respectively.
Equations With the pressure solutions of reservoir flow and fracture flow, the equations can be obtained as follows (1) Pressure response equations at each fracture vertex by superposition,
where qwD;dw is the flow rate at inner boundary, ywD;dw is the location at inner boundary of a fracture segment, ywD;up is the location at outer boundary of a fracture segment, and LFD is length of the fracture segment. With auxiliary equations [Eqs. (8) and (9)], the integral of Eq. (5) yields Z h
i 2p
pFD ywD;up pFD ywD;dw ¼ qFD yD ywD;dw þ qwD;dw dyD CFD
pFDi ¼
np X k¼1
ywZDk;up
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qFDk K0 s ðxDi xwD Þ2 þ yDi u du; i ¼ 1…nv
yw Dk;dw
(11) (10)
where pFDi is the pressure response at fracture vertex i caused by the sum of fracture panels k with a continuous rate qFDk .
Fig. 3 e Discretization of the complex fracture networks.
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(2) Pressure difference at each fracture panel,
pFD y
wDk;up
pFD y
wDk
Z
ywDk;up
; dw ¼
2p CFDk
h
qFDk yDk ywDk;dw
ywDk;dw
i
(12)
þ qDk;dw dyD ; k ¼ 1…np where ywDk;dw and ywDk;up are respectively the location at inner and outer boundary of fracture segment k, yDk is the location at center of k-th fracture segment, CFDk is the fracture conductivity of fracture segment k, qFDk is the outflux of k-th fracture segment, and qDk;dw is the flow rate at inner boundary of fracture segment k. (3) Mass balance equation at each hydraulic-fracture vertex, qDi in ¼ qDi out ; i ¼ 1…nv (13) where ðqDi Þin is the fluid inflow rate at each fracture vertex; ðqDi Þout is the fluid outflow rate at each fracture vertex.
Fig. 4 e Model validation with by a case of bi-wing fractured well in SPE 10179 or 7490 [10,11].
Solution algorithm By combining Eq. (11)e(13), a np þ 2nv -order matrix is established. Similarly, we have np þ 2nv unknowns, including nv hydraulic-fracture pressure (pFD ), nv hydraulic-fracture flow rate (qwD ), and np hydraulic-fracture fluid influx (qFD ). Computer codes were developed to efficiently and accurately obtain the solutions of pressure, flow rate, and fluid influx. The pressure at any point of the reservoir, including the wellbore, is calculated, once all the fluid influxes into fracture faces are obtained.
Results This section describes (1) the validation of the proposed model and (2) its application into wellbore pressure analysis of DFIT with complex fracture geometries.
Model validation Before studying the wellbore pressure behaviors, the presented model is validated by a case of bi-wing fractured well by Society of Petroleum Engineers (SPE) 10179 or 7490 [10,11]. The result comparison between the proposed model and SPE 10179 or 7490 is shown in Fig. 4.
Wellbore pressure behaviors After validation, we use the proposed model to study the wellbore pressure behaviors of DFIT with complex fracture geometries. The fracture geometries investigated in this work are summarized from literature [7,16,37], including OFFN, TLFN, DMFN, and MOFN (Fig. 5). 1. Opening-fissures fracture networks (OFFN). Owing to stress anisotropy, the fissure opening is a common practice during hydraulic fracturing [37], and then a hydraulic fracture
is usually generated with multiple orthogonal fissures, namely the opening-fissure fracture networks. The OFFN are in the lowest complexity, but they can consider the stress anisotropy. 2. Tree-like fracture networks (TLFN). When the stress anisotropy is high, the fissures are sometimes opened nonorthogonally [16], resulting in tree-like fracture networks. Although TLFN have the modest complexity, the high stress anisotropy can be taken into account. 3. Dendritic multiple fracture networks (DMFN). The type of radial multiple fracture networks is also commonly created, as many fracturing maps and core testing showed that multiple fractures may develop radially along the wellbore. The fracture intersections are not obvious in the DMFN, but they have higher complexity. 4. Mutually orthogonal fracture networks (MOFN). Additionally, some fracturing maps show complex geometries of mutually orthogonal fracture networks, which are comprised of fractures in two distinct and orthogonal directions [22]. Although only orthogonal fractures are involved, the MOFN have the highest complex geometries. Although the DFIT is a small-scale hydraulic fracturing, the base of fracture propagation is the same as larger volume hydraulic fracture stimulation. Therefore, the above common fracture geometries may be created during DFIT in unconventional reservoirs, with the pre-existing natural fractures, fissures, and microfractures. Using the simplified models of those common fracture geometries, the constant-rate pressure behaviors are studied. To make the pressure behaviors better understood, the bi-wing fractured well (FW) is used as a comparison. Fig. 6 gives the simplified fracture geometries, and the calculated results are plotted into log-log coordinates in Fig. 7. The upper curves are dimensionless pressure, while the lower ones are
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Fig. 5 e Common fracture geometries caused by hydraulic fracturing in unconventional reservoirs (adopted from Ref. [16].
Fig. 6 e Bi-wing fractured well and simplified fracture geometries used in this work [7]. The red lines and green circles refer to hydraulic fractures and wellbore respectively. The length and width of the fracture networks are 24 m and 9 m, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) dimensionless pressure derivative. Take Fig. 7a for example. Different colors refer to different flow regimes. The time interval between the neighbor break lines is the period of a flow regime of OFFN, including BLF, fluid feed, FLF, transition, and PRF.
Discussion Based on the results from the proposed model, some discussion is made in this section. As mentioned, Fig. 4 gives the result comparison between the proposed model and the previous work [10,11]. The same input parameters from the previous work are used in the model. It is clearly found that our results from the proposed model are match well with the previous work [10,11], as demonstrated in Fig. 4. The model reliability is demonstrated by the model validation using the log-log diagnostic plots. In Fig. 7, we observe that conventional typical flow regimes of the bi-wing fractured well include bilinear flow (BF), formation linear flow (FLF), transition flow, and pseudoradial flow (PRF). Compared to bi-wing fractured wells, the pressure derivative of OFFN has a “valley shape” during the
FLF to distort 1/2-slope straight line, and the “valley shape” is caused by “recharge” from opening fissures to wellboreconnected fracture [8]. The pressure derivative of TLFN also has a “valley shape”, namely “recharge”. The “valley shape” of TLFN lasts longer but weaker than that of OFFN, because the fracture branches in TLFN exist on more widely spread area. Quite different from OFFN or TLFN, there is no “valley shape” in the dimensionless pressure derivative of DMFN. The reason is that the fractures are all directly connected with the wellbore in DMFN, and there is no need for “recharge” from branches to main fractures. However, there is a “hump” (Fig. 7c) in the dimensionless pressure derivative of DMFN, and it is caused by interferences among the multiple fractures. For MOFN, a “valley shape” caused by the indirect-connected fractures occurs in its pressure derivative, which can be physically expected. In addition, an approximate unit-slope straight line occurs, reflecting a pseudo boundary-dominated flow (PBDF). This flow regime is resulted by permeability contrast between fracture networks and reservoir. It is clearly shown that various fracture cases exhibit different pressure transient behaviors, which may provide a good tool to identify fracture geometries. Finally,
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Fig. 7 e Comparison between flow regimes of four common fracture geometries and bi-wing fractured well. The input data include CfD ¼ 15 and LfD ¼ 1. The reference length Lr is the half-length of bi-wing fracture. Note the FW in the figure means fractured well. Information of BLF, FLF, PBDF, and PRF is shown in next section. (a) OFFN and bi-wing fracture, (b) TLFN and bi-wing fracture, (c) DMFN and bi-wing fracture, (d) MOFN and bi-wing fracture.
the pressure derivatives of all fracture geometries reach a same constant, 0.5, which reflects a pseudo radial flow. At the stage of pseudo radial flow, the flow front reaches the point far away from the fracture and the fluxes into all fractures are stabilized. The pressure derivatives are only dominated by reservoir and fluid properties, and they are don't affected by the fractures. With the injection pressure solutions (Fig. 7), we can continue to obtain the shut-in well pressure for DFIT complexity. Because the pumping period is much shorter than the shut-in period in DFIT, the influence of fracture propagation [15] can be neglected. Then, the ACA of DFIT can be treated as a small falloff analysis. With the assumptions: (1) ignorance of short fracture propagation, (2) Darcy's flow, and (3) semi-infinite reservoir, we can easily compute the shut-in well pressure for DFIT complexity by the superposition equation [Eq. (14)], based on the injection pressure solutions (Fig. 7).
N X pwD tpDN tpDj1 þ DtD ; qDj qDj1 pwD;shut ¼ pwD tpD j¼1
þ pwD ðDtD Þ
(14)
where qDj is pump-in rate during pump-in time tpDj (tpD0 ¼ 0 and qD0 ¼ 0), DtD is fall-off time after shut in, pwD;shut is the dimensionless pressure increment after shut-in, pwD ðtpD Þ is PN instantaneous shut-in pressure, j¼1 pwD ðtpDN tpDj1 þ DtD ; qDj qDj1 Þ is the pressure increase caused by falloff, and pwD ðDtD Þ is the dimensionless wellbore pressure of shut-in well.
Case study It is learnt from Section Discussion that different fracture parameters with reservoir parameters have different pressure transient behaviors. Inversely, the fracture and reservoir properties can be determined with the wellbore pressure
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Table 2 e Model parameters of the DFIT. Parameters Initial pressure Reservoir porosity Total compressibility Injection rate per thickness Injection time
Value
Units
34 0.3 1.47 103 720 5
MPa e MPa1 m3/d/m min
Fig. 8 e Schematic of the complex fracture model with an X-direction fracture and two Y-direction fractures (adopted from Ref. [24].
Fig. 9 e Log-log diagnostic plot for the pressure falloff data. data. Therefore, we continue to use a case to show the potential of the proposed model to evaluate the fracture geometries and reservoir properties. The DFIT data we analyzed in the case study is from an unconventional well designed by Ref. [24]. In their study, the fluid flow was coupled with geomechanical deformations by solving the Biot's equation. With a finite element mesh, fluid pressure/velocity and deformations are solved, and the DFIT with a complex fracture case was simulated, as shown in Fig. 8. The initial pressure is 34 MPa, and the reservoir porosity is 0.3. The total compressibility is 1.47 103 MPa1. During the DFIT, the well is subjected to a pulse of injection rate, 720 m3/ d/m for 5 min and shut-in. The model parameters are given in Table 2. The log-log diagnostic plot for the pressure falloff data is shown in Fig. 9. At the initial stage, the pressure derivative exists 1-slope trends, which can be interpreted by fracture closuring [14]. After fracture closuring, the linear and pseudoradial flow regimes occur, which are indicated by the pressure derivative trend. Then, the fracture geometries and reservoir parameters can be estimated using our model. Based on our boundary element model, with the known information we perform a series of type-curve matching on after-closure pressure data using a set of parameters including reservoir permeability, fracture conductivity, and total fracture length. The quality of the matches are on the basis of the residual sum of squares between pressure falloff data and calculated results. The type-curve matching with highest quality (smallest residual sum of squares) are selected. Fig. 10 presents the final type-curve matching on after-closure pressure data, and Table 3 provides the estimated reservoir and fracture parameters. The range of their values is reasonable around the known values. Given that
non-uniqueness of solutions is a challenge for type-curve matching on field data, it is hard for us to conduct statistical analysis on the results at present. In the next step, more efforts will be made to focus on the non-uniqueness of solutions and statistical analysis.
Contributions and limitations The main novelty and contribution of this study is presenting a new generalized model for after-closure analysis (ACA) of
Fig. 10 e Type-curve matching on after-closure pressure data using the boundary element model. Please note that the adjusted time is employed to transform type curves from falloff to pressure drop [17].
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Table 3 e Estimated parameters for the DFIT.
Reservoir permeability X-direction fracture half length Y-direction fracture half length Fracture conductivity Fracture geometries
Estimated value (this model)
Known value [24]
Units
0.001e0.0017 40e50 13e15 5000e6500
0.001 45 15 6000
D m m mD∙m e
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Parameters
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DFIT with complex fracture geometries. Although much more attention has paid to the model development of DFIT in conventional reservoirs, most DFIT models do not involve the complex fractures generated by the activations of pre-existing natural fractures, fissures, and microfractures. To narrow the gap between theory and practice, this paper introduces a new boundary element model for DFIT with complex fracture geometries, and it will assist with reservoir evaluation and production design substantially. Meanwhile, there are also some limitations on model results in this study due to the simple assumptions, like 1D fracture flow, 2D reservoir flow, Darcy's flow, ignorance of temperature influence, etc. Therefore, there is still a long way to go before the proposed model becomes applicable in complex situations. But, this work is our primary work, trying to identify the fracture geometries and reservoir parameters with DFIT complexity. In the future, more efforts will be made to consider the complex situations, like (1) 3D fluid flow, (2) non-linear behavior of the pressure difference, (3) temperature influence, etc.
Summary and conclusions To better understand the properties of unconventional reservoirs of hydrogen and carbon, this paper introduces a new boundary element model with random fracture geometries caused by natural fractures, fissures, and microfractures. The model flexibility is demonstrated by different complex fracture cases, namely opening-fissure fracture networks, treelike fracture networks, radial multiple fracture networks, and mutually orthogonal fracture networks. Compared to bi-wing fractured wells, the pressure derivative of OFFN has a “valley shape”, namely “recharge”. The pressure derivative of TLFN also has a “valley shape”. The “valley shape” of TLFN lasts longer but weaker than that of OFFN. Quite different from OFFN or TLFN, there is no “valley shape” in the dimensionless pressure derivative of DMFN. However, there is a “hump” in the dimensionless pressure derivative of DMFN. For MOFN, a “valley shape” caused by the indirect-connected fractures occurs in its pressure derivative, which can be physically expected. In addition, an approximate unit-slope straight line occurs, reflecting a pseudo boundarydominated flow. Namely, various fracture cases exhibit different pressure transient behaviors, which may provide a good tool to identify fracture geometries with reservoir parameters. An example from a DFIT test in unconventional reservoir is analyzed using the log-log diagnostic plots. Results show that fracture geometries as well as reservoir parameters and the range of estimated values from the proposed model are reasonable around the known values of DFIT test. Undoubtedly, there are also some limitations on model results due to the assumptions, including 1D fracture flow, 2D reservoir flow, Darcy's flow, ignorance of temperature influence, etc. In the future, more efforts will be made to consider the complex situations, like (1) 3D fluid flow, (2) non-linear behavior of the pressure difference, (3) temperature influence, etc.
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Acknowledgements This work was supported by National Basic Research 973 Program of China (2015CB250900), National Key S&T Special Projects (2016ZX05047-004 and 2016ZX05030), Science Foundation of China University of Petroleum, Beijing (No. 2462018YJRC032), Post-doctoral Program for Innovation Talents (BX20180380), and the China Scholarship Council for a year study for Zhiming Chen at The University of Texas at Austin (201606440082). We also would like to express our gratitude to Mr. Hongyang Chu for his useful edits, suggestions, and ideas for the work.
Appendix B. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.ijhydene.2019.03.060.
Nomenclature ACA BF DFIT DMFN FLF FW MOFN KGD OFFN PBDF PKN PRF SPE TLFN B CFD h k Kn ðxÞ p pD pFD q qD qFD s tD t x; y; r
After-closure analysis Bilinear flow Diagnostic fracture injection test Dendritic multiple fracture networks Formation linear flow Fractured well Mutually Orthogonal fracture networks Khristianovic-Geertsma-de Opening-fissures fracture networks pseudo boundary-dominated flow Perkins-Kern-Nordgren Pseudo-radial flow Society of Petroleum Engineers Tree-like fracture networks fluid volume factor fracture conductivity, dimensionless formation thickness, m reservoir permeability, md second kind of modified Bessel functions reservoir pressure, MPa dimensionless reservoir pressure dimensionless fracture pressure sand-face injection rate, m3/d dimensionless flow rate flow rate of per unit fracture length into formation Laplace transformation variable with respect to injection time dimensionless injection time injection time, hr coordinates, m
Greek m
viscosity, cp
Subscript D dimensionless F hydraulic fracture w wellbore Superscript e Laplace transform
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SI Conversion Factors MPa 1.0 eþ06 Pa hr 3.6 eþ03 s cp 1.0 e-03 Pa$s md 1 e-15 m2
[17]
[18]
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A new model for diagnostic fracture injection test (DFIT) complexity in unconventional reservoirs Zhiming Chen a,b,*, Xinwei Liao a, Jiali Zhang a a
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum at Beijing, Changping, 100249, China b The University of Texas at Austin, Austin, TX, 78712, USA
article info
abstract
Article history:
In the recent years, much attention has been paid to unconventional reservoirs of
Received 27 December 2018
hydrogen and carbon which are good sources of hydrogen energy. To ensure sufficient
Received in revised form
energy organic, the first step is understanding the properties of those unconventional
5 March 2019
reservoirs. The diagnostic fracture injection test (DFIT) is a very good technique to estimate
Accepted 7 March 2019
the parameters of the unconventional reservoirs. Unfortunately, most DFIT models do not
Available online 2 April 2019
involve the complex fracture networks caused by natural fractures, fissures, and microfractures.
Keywords:
© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Unconventional reservoirs of hydrogen and carbon Diagnostic fracture injection test (DFIT) Fracture complexity Boundary element model
To better understand the properties of unconventional reservoirs of hydrogen and carbon, this paper introduces a new generalized model for after-closure analysis (ACA) of DFIT with complex fracture geometries. The generalized model is based on a boundary element method that neglects fracture mechanics processes. First, the model flexibility is demonstrated by different complex fracture cases, including opening-fissure fracture networks (OFFN), tree-like fracture networks (TLFN), dendritic multiple fracture networks (DMFN), and mutually orthogonal fracture networks (MOFN). It is found that various fracture-network cases exhibit different pressure transient behaviors, which may provide a good tool to identify fracture geometries and reservoir
parameters. Subsequently, the model reliability is demonstrated by a model validation and a case study of DFIT complexity using the log-log diagnostic plots based on the model solution. This is our primary work to estimate the DFIT complexity using after-closure analysis, and it will assist with reservoir evaluation and production design substantially.
Introduction Much attention has been paid to the characteristics of wells in unconventional reservoirs of hydrogen and carbon [1,12,18,34,35,41]. The unconventional reservoirs can be good
* Corresponding author. The University of Texas at Austin, 78712, Austin, TX, USA. This paper is an improvement for a conference presentation (SPE 189793), and the full copyright of presentation has been reverted to the original owner. E-mail addresses: [email protected], [email protected] (Z. Chen). https://doi.org/10.1016/j.ijhydene.2019.03.060 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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sources of hydrogen energy, because they can produce plenty of energy-rich organic compound in which the hydrogen is generally locked up. Understanding the properties of unconventional reservoirs of hydrogen and carbon is the first step to ensure sufficient energy organic. Xue et al. [38] used the pressure transient analysis to study the fluid flow behaviors through horizontal fractures, which is helpful to estimate the parameters of reservoir and fracture. Meanwhile, the small-scale fracturing, or DFIT has extended its role to understand the properties of unconventional reservoirs of hydrogen and carbon [5]. With the DFIT data, many parameters of unconventional reservoirs can be obtained, including closure pressure, formation permeability, leak-off coefficient, and initial formation pressure. Based on the Perkins-Kern-Nordgren (PKN) fracture geometry model by Perkins and Kern and extended work by Nordgren [27,28,30] performed several derivations for the fluid loss coefficient, the fracture length and width, fluid efficiency, and closure time for the fracture. Particularly, Nolte derived the G-time function and correlated the G-function with pressure and flow rate. With G-function, the closure time and closure pressure can be easily distinguished on the plot of pressure and G-function. The original work is extended by Ref. [29] to the Khristianovic-Geertsma-de (KGD) model and radial model. Soliman et al. [33] conducted the extended work on afterclosure DFIT analysis. They employed the after-closure stage pressure decline data to determine the formation permeability and reservoir pressure. Based on the transient pressure analysis for the fractured well, Soliman completed the derivation for the quantification of reservoir pressure and formation permeability of each flow regime, namely, linear flow, bilinear flow and radial flow. Barree et al. [3] performed five non-ideal leak-off case studies with numerical simulation (GOHFER simulator). Three pressure function P, dP=dG, and GdP=dG are used to uniquely obtain the existence and magnitude of pressure dependent leak-off behavior and eliminate the ambiguity between pressure dependent leak-off, fracture height changes, and tip extension and recession. Also, the effect of pressure dependent fracture compliance was first described. In the later work [4], Barree et al. suggested using G-function derivative analysis along with a square root of time plot and a log-log diagnostic plot to identify closure. All three methods should agree when closure is correctly identified. Barree helped avoid the wrong pick of fracture closure point pffiffi with modified t analysis. Different flow regimes can be identified using specific slope line on the log-log DPwf Dt plot [5]. concluded the usual mistakes we made in field application and illustrated how to avoid these mistakes and improve data acquisition and analysis. Further, much meaningful work has been done on the DFIT analysis [23,36]. It is worth mentioning that natural fractures, fissures, and microfractures are well-known contributors to production performance of unconventional reservoirs [25]. Complex fracture geometries can be generated by either small-scale fracturing, DFIT, or large volume fracture stimulation, because the activation of pre-existing natural fractures, fissures, and microfractures plays a significant role on
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Fig. 1 e Complex fracture geometries generated by DFIT, owing to pre-existing natural fractures [26].
generation of induced hydraulic fractures [13,14,20,21,24], as shown in Fig. 1. Therefore, much more attention should be paid to the model development of DFIT complexity [31]. However, very limited work about DFIT has considered the fracture complexity. Bakar [2] used a numerical hydraulic fracturing simulator, namely Complex Fracturing ReseArch Code, to investigate the viability of the DFIT analysis techniques with and without micro-fractures. The numerical simulator was based on a discrete fracture network method. This work provided a very good perspective on DFIT analysis, but it was a little timeconsuming owing to the numerical approach [6]. built a complex fracture-network model for DFIT with mesh-free methods [8,19,32,35,38,39,40,42]. In their model [6], the pressure solution for the reservoir flow was analytical, mesh-free, while the pressure solution for fracture flow was numerical. A nodal analysis method was used to eliminate the flow interplay within fracture intersections. Unfortunately, the fracture networks were orthogonal, and non-orthogonal geometries were not involved. In addition, the model practicality was not demonstrated. To narrow the gap between theory and practice, this paper introduces a new boundary element model with random fracture geometries caused by natural fractures, fissures, and microfractures. First, the model flexibility is demonstrated by different complex fracture cases, including opening-fissure fracture networks (OFFN), tree-like fracture networks (TLFN), dendritic multiple fracture networks (DMFN), and mutually orthogonal fracture networks (MOFN). Finally, the model reliability is showed by a model validation and a case study of DFIT complexity using the log-log diagnostic plots based on the model solution. This is our primary work to estimate the fracture geometries using after-closure analysis of DFIT Complexity, and it will assist with reservoir evaluation and production design substantially.
Mathematical model Model development In this paper, we built a boundary element model for the afterclosure analysis (ACA) reservoir properties determination with the consideration of complex fracture geometries. The proposed boundary element model is composed of analytical reservoir solutions and numerical fracture solutions. The discretization and grids in spatial and temporal dimensions
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With the discretized fracture panels, the analytical reservoir solution is obtained based on point source function. The diffusion equation of a point source in the 2D infinite reservoir is given by v2 pD v2 pD vpD þ 2 ¼ vtD vx2D vyD
(1)
where pD is the dimensionless injection pressure, xD and yD are the dimensionless distances, and tD is the dimensionless injection time, with the auxiliary equations 8 > < pD xD ; yD ; tD ¼ 0 ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pD ðrD /∞; tD Þ ¼ 0; rD ¼ x2D þ y2D > : vpD ðrD /0; tD Þ vrD ¼ qD
Fig. 2 e Physical model of complex fracture geometries for DFIT.
are not required in the proposed boundary element model, which has the analytical characteristics. The explicit description of random fracture geometries (Fig. 2) induced by the DFIT small injection treatment are included in our model to obtain a more realistic result.
where qD is the dimensionless flow rate out from a point source [43]. presented an exhaustive compilation of the detailed solutions by using Laplace transform. Following that, we obtain the pressure solution for a point source, pD ¼ qD K0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi x2D þ y2D s
Some assumptions are made for the model development, including: (1) a homogenous, isotropic, and semi-infinite reservoir with sealed top and bottom boundaries, (2) singlephase and slightly compressible fluid, (3) Darcy's flow, (4) ignorance of vertical flow and gravity effects, (5) constant fracture width, (6) 1D fracture flow along the fracture length, and (7) a fixed injection rate.
Reservoir flow During the injection, the fluid injected through the wellbore flows within the complex fractures, then entering the reservoir. Hence, two kinds of flow modes are considered: (1) reservoir flow and (2) fracture flow. Table 1 provides the definitions of all dimensionless variables for model development. Let us first focus on the reservoir flow. To explicitly reflect the random fracture geometries, the complex fracture is discretized into multiple fracture panels and fracture vertexes.
(3)
where superscript “-” is the symbol of Laplace transform, K0 ðxÞ is the second kind of modified Bessel functions, and s is the Laplace variable with respect to tD . The fracture segment is a collection of point sources. The integral of Eq. (3) yields, Z
Assumptions
(2)
pD ¼
qFD K0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi x2D þ y2D s dl
(4)
l
Fracture flow As assumed, the fluid flow is assumed to be 1D in the 2D reservoir, because the ratio of fracture width to fracture length is tiny. Given that fracture permeability is very high and fracture diffusivity tends to be much higher, transient effects in fracture flow are short-lived compared to the transient reservoir flow [9]. Ignoring the short transient period in fractures, we can obtain a diffusivity equation to describe the fluid flow within the continuum fracture. Following the work [9], the diffusivity equation is given by,
v2 pFD 2p þ qFD ¼ 0 CFD vy2D
(5)
Table 1 e Definitions of dimensionless parameters. Dimensionless variables Reservoir pressure Fracture pressure Injection time Injection rate Flow flux from fracture to reservoir X distance
Definition
Dimensionless variables
Definition y Lr r ¼ Lr
khðp pi Þ 1:842 103 mBqsc khðpF pi Þ pFD ¼ 1:842 103 mBqsc 3:6kt tD ¼ mfCt L2r q qD ¼ qsc
Y distance
yD ¼
Radial distance
rD
Fracture length
LFD ¼
qfD ¼ qF Lr =qsc
Wellbore radius
rwD
pD ¼
xD
Please note that Lr is the reference length in meter.
x ¼ Lr
Fracture conductivity
CFD ¼
kF WF kLr
LF Lr rw ¼ Lr
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where pFD is the pressure in fractures, qFD is the flow flux, and CFD is the fracture conductivity. The areal coordinate (q) in cylindrical coordinates ties back to areal coordinates in Cartesian coordinates by xD ¼ rD cos q; yD ¼ rD sin q
(6)
The initial condition of fracture pressure yields
pFD yD ; tD ¼ 0 ¼ 0
(7)
and the flow rate at the inner boundary satisfies, vpFD 2p ywD;dw ; tD ¼ qwD CfD vyD
(8)
where ywD;dw is the location at inner boundary of a fracture segment; qwD is the flow rate at inner boundary. A mass balance at outer boundary is required as qD ywD;up ; tD ¼ qwD þ LFD qfD
(9)
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Model solution The model solution involves three steps: discretization, equations, and solution algorithm
Discretization. As mentioned before, we discretize the complex fracture case into multiple segments to describe the real geometry (Fig. 3), and then the fracture information is imported and quantified for the use of our boundary element model. We assume that the numbers of fracture panels and fracture vertexes are np and nv, respectively.
Equations With the pressure solutions of reservoir flow and fracture flow, the equations can be obtained as follows (1) Pressure response equations at each fracture vertex by superposition,
where qwD;dw is the flow rate at inner boundary, ywD;dw is the location at inner boundary of a fracture segment, ywD;up is the location at outer boundary of a fracture segment, and LFD is length of the fracture segment. With auxiliary equations [Eqs. (8) and (9)], the integral of Eq. (5) yields Z h
i 2p
pFD ywD;up pFD ywD;dw ¼ qFD yD ywD;dw þ qwD;dw dyD CFD
pFDi ¼
np X k¼1
ywZDk;up
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 qFDk K0 s ðxDi xwD Þ2 þ yDi u du; i ¼ 1…nv
yw Dk;dw
(11) (10)
where pFDi is the pressure response at fracture vertex i caused by the sum of fracture panels k with a continuous rate qFDk .
Fig. 3 e Discretization of the complex fracture networks.
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(2) Pressure difference at each fracture panel,
pFD y
wDk;up
pFD y
wDk
Z
ywDk;up
; dw ¼
2p CFDk
h
qFDk yDk ywDk;dw
ywDk;dw
i
(12)
þ qDk;dw dyD ; k ¼ 1…np where ywDk;dw and ywDk;up are respectively the location at inner and outer boundary of fracture segment k, yDk is the location at center of k-th fracture segment, CFDk is the fracture conductivity of fracture segment k, qFDk is the outflux of k-th fracture segment, and qDk;dw is the flow rate at inner boundary of fracture segment k. (3) Mass balance equation at each hydraulic-fracture vertex, qDi in ¼ qDi out ; i ¼ 1…nv (13) where ðqDi Þin is the fluid inflow rate at each fracture vertex; ðqDi Þout is the fluid outflow rate at each fracture vertex.
Fig. 4 e Model validation with by a case of bi-wing fractured well in SPE 10179 or 7490 [10,11].
Solution algorithm By combining Eq. (11)e(13), a np þ 2nv -order matrix is established. Similarly, we have np þ 2nv unknowns, including nv hydraulic-fracture pressure (pFD ), nv hydraulic-fracture flow rate (qwD ), and np hydraulic-fracture fluid influx (qFD ). Computer codes were developed to efficiently and accurately obtain the solutions of pressure, flow rate, and fluid influx. The pressure at any point of the reservoir, including the wellbore, is calculated, once all the fluid influxes into fracture faces are obtained.
Results This section describes (1) the validation of the proposed model and (2) its application into wellbore pressure analysis of DFIT with complex fracture geometries.
Model validation Before studying the wellbore pressure behaviors, the presented model is validated by a case of bi-wing fractured well by Society of Petroleum Engineers (SPE) 10179 or 7490 [10,11]. The result comparison between the proposed model and SPE 10179 or 7490 is shown in Fig. 4.
Wellbore pressure behaviors After validation, we use the proposed model to study the wellbore pressure behaviors of DFIT with complex fracture geometries. The fracture geometries investigated in this work are summarized from literature [7,16,37], including OFFN, TLFN, DMFN, and MOFN (Fig. 5). 1. Opening-fissures fracture networks (OFFN). Owing to stress anisotropy, the fissure opening is a common practice during hydraulic fracturing [37], and then a hydraulic fracture
is usually generated with multiple orthogonal fissures, namely the opening-fissure fracture networks. The OFFN are in the lowest complexity, but they can consider the stress anisotropy. 2. Tree-like fracture networks (TLFN). When the stress anisotropy is high, the fissures are sometimes opened nonorthogonally [16], resulting in tree-like fracture networks. Although TLFN have the modest complexity, the high stress anisotropy can be taken into account. 3. Dendritic multiple fracture networks (DMFN). The type of radial multiple fracture networks is also commonly created, as many fracturing maps and core testing showed that multiple fractures may develop radially along the wellbore. The fracture intersections are not obvious in the DMFN, but they have higher complexity. 4. Mutually orthogonal fracture networks (MOFN). Additionally, some fracturing maps show complex geometries of mutually orthogonal fracture networks, which are comprised of fractures in two distinct and orthogonal directions [22]. Although only orthogonal fractures are involved, the MOFN have the highest complex geometries. Although the DFIT is a small-scale hydraulic fracturing, the base of fracture propagation is the same as larger volume hydraulic fracture stimulation. Therefore, the above common fracture geometries may be created during DFIT in unconventional reservoirs, with the pre-existing natural fractures, fissures, and microfractures. Using the simplified models of those common fracture geometries, the constant-rate pressure behaviors are studied. To make the pressure behaviors better understood, the bi-wing fractured well (FW) is used as a comparison. Fig. 6 gives the simplified fracture geometries, and the calculated results are plotted into log-log coordinates in Fig. 7. The upper curves are dimensionless pressure, while the lower ones are
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Fig. 5 e Common fracture geometries caused by hydraulic fracturing in unconventional reservoirs (adopted from Ref. [16].
Fig. 6 e Bi-wing fractured well and simplified fracture geometries used in this work [7]. The red lines and green circles refer to hydraulic fractures and wellbore respectively. The length and width of the fracture networks are 24 m and 9 m, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) dimensionless pressure derivative. Take Fig. 7a for example. Different colors refer to different flow regimes. The time interval between the neighbor break lines is the period of a flow regime of OFFN, including BLF, fluid feed, FLF, transition, and PRF.
Discussion Based on the results from the proposed model, some discussion is made in this section. As mentioned, Fig. 4 gives the result comparison between the proposed model and the previous work [10,11]. The same input parameters from the previous work are used in the model. It is clearly found that our results from the proposed model are match well with the previous work [10,11], as demonstrated in Fig. 4. The model reliability is demonstrated by the model validation using the log-log diagnostic plots. In Fig. 7, we observe that conventional typical flow regimes of the bi-wing fractured well include bilinear flow (BF), formation linear flow (FLF), transition flow, and pseudoradial flow (PRF). Compared to bi-wing fractured wells, the pressure derivative of OFFN has a “valley shape” during the
FLF to distort 1/2-slope straight line, and the “valley shape” is caused by “recharge” from opening fissures to wellboreconnected fracture [8]. The pressure derivative of TLFN also has a “valley shape”, namely “recharge”. The “valley shape” of TLFN lasts longer but weaker than that of OFFN, because the fracture branches in TLFN exist on more widely spread area. Quite different from OFFN or TLFN, there is no “valley shape” in the dimensionless pressure derivative of DMFN. The reason is that the fractures are all directly connected with the wellbore in DMFN, and there is no need for “recharge” from branches to main fractures. However, there is a “hump” (Fig. 7c) in the dimensionless pressure derivative of DMFN, and it is caused by interferences among the multiple fractures. For MOFN, a “valley shape” caused by the indirect-connected fractures occurs in its pressure derivative, which can be physically expected. In addition, an approximate unit-slope straight line occurs, reflecting a pseudo boundary-dominated flow (PBDF). This flow regime is resulted by permeability contrast between fracture networks and reservoir. It is clearly shown that various fracture cases exhibit different pressure transient behaviors, which may provide a good tool to identify fracture geometries. Finally,
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Fig. 7 e Comparison between flow regimes of four common fracture geometries and bi-wing fractured well. The input data include CfD ¼ 15 and LfD ¼ 1. The reference length Lr is the half-length of bi-wing fracture. Note the FW in the figure means fractured well. Information of BLF, FLF, PBDF, and PRF is shown in next section. (a) OFFN and bi-wing fracture, (b) TLFN and bi-wing fracture, (c) DMFN and bi-wing fracture, (d) MOFN and bi-wing fracture.
the pressure derivatives of all fracture geometries reach a same constant, 0.5, which reflects a pseudo radial flow. At the stage of pseudo radial flow, the flow front reaches the point far away from the fracture and the fluxes into all fractures are stabilized. The pressure derivatives are only dominated by reservoir and fluid properties, and they are don't affected by the fractures. With the injection pressure solutions (Fig. 7), we can continue to obtain the shut-in well pressure for DFIT complexity. Because the pumping period is much shorter than the shut-in period in DFIT, the influence of fracture propagation [15] can be neglected. Then, the ACA of DFIT can be treated as a small falloff analysis. With the assumptions: (1) ignorance of short fracture propagation, (2) Darcy's flow, and (3) semi-infinite reservoir, we can easily compute the shut-in well pressure for DFIT complexity by the superposition equation [Eq. (14)], based on the injection pressure solutions (Fig. 7).
N X pwD tpDN tpDj1 þ DtD ; qDj qDj1 pwD;shut ¼ pwD tpD j¼1
þ pwD ðDtD Þ
(14)
where qDj is pump-in rate during pump-in time tpDj (tpD0 ¼ 0 and qD0 ¼ 0), DtD is fall-off time after shut in, pwD;shut is the dimensionless pressure increment after shut-in, pwD ðtpD Þ is PN instantaneous shut-in pressure, j¼1 pwD ðtpDN tpDj1 þ DtD ; qDj qDj1 Þ is the pressure increase caused by falloff, and pwD ðDtD Þ is the dimensionless wellbore pressure of shut-in well.
Case study It is learnt from Section Discussion that different fracture parameters with reservoir parameters have different pressure transient behaviors. Inversely, the fracture and reservoir properties can be determined with the wellbore pressure
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Table 2 e Model parameters of the DFIT. Parameters Initial pressure Reservoir porosity Total compressibility Injection rate per thickness Injection time
Value
Units
34 0.3 1.47 103 720 5
MPa e MPa1 m3/d/m min
Fig. 8 e Schematic of the complex fracture model with an X-direction fracture and two Y-direction fractures (adopted from Ref. [24].
Fig. 9 e Log-log diagnostic plot for the pressure falloff data. data. Therefore, we continue to use a case to show the potential of the proposed model to evaluate the fracture geometries and reservoir properties. The DFIT data we analyzed in the case study is from an unconventional well designed by Ref. [24]. In their study, the fluid flow was coupled with geomechanical deformations by solving the Biot's equation. With a finite element mesh, fluid pressure/velocity and deformations are solved, and the DFIT with a complex fracture case was simulated, as shown in Fig. 8. The initial pressure is 34 MPa, and the reservoir porosity is 0.3. The total compressibility is 1.47 103 MPa1. During the DFIT, the well is subjected to a pulse of injection rate, 720 m3/ d/m for 5 min and shut-in. The model parameters are given in Table 2. The log-log diagnostic plot for the pressure falloff data is shown in Fig. 9. At the initial stage, the pressure derivative exists 1-slope trends, which can be interpreted by fracture closuring [14]. After fracture closuring, the linear and pseudoradial flow regimes occur, which are indicated by the pressure derivative trend. Then, the fracture geometries and reservoir parameters can be estimated using our model. Based on our boundary element model, with the known information we perform a series of type-curve matching on after-closure pressure data using a set of parameters including reservoir permeability, fracture conductivity, and total fracture length. The quality of the matches are on the basis of the residual sum of squares between pressure falloff data and calculated results. The type-curve matching with highest quality (smallest residual sum of squares) are selected. Fig. 10 presents the final type-curve matching on after-closure pressure data, and Table 3 provides the estimated reservoir and fracture parameters. The range of their values is reasonable around the known values. Given that
non-uniqueness of solutions is a challenge for type-curve matching on field data, it is hard for us to conduct statistical analysis on the results at present. In the next step, more efforts will be made to focus on the non-uniqueness of solutions and statistical analysis.
Contributions and limitations The main novelty and contribution of this study is presenting a new generalized model for after-closure analysis (ACA) of
Fig. 10 e Type-curve matching on after-closure pressure data using the boundary element model. Please note that the adjusted time is employed to transform type curves from falloff to pressure drop [17].
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Table 3 e Estimated parameters for the DFIT.
Reservoir permeability X-direction fracture half length Y-direction fracture half length Fracture conductivity Fracture geometries
Estimated value (this model)
Known value [24]
Units
0.001e0.0017 40e50 13e15 5000e6500
0.001 45 15 6000
D m m mD∙m e
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Parameters
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DFIT with complex fracture geometries. Although much more attention has paid to the model development of DFIT in conventional reservoirs, most DFIT models do not involve the complex fractures generated by the activations of pre-existing natural fractures, fissures, and microfractures. To narrow the gap between theory and practice, this paper introduces a new boundary element model for DFIT with complex fracture geometries, and it will assist with reservoir evaluation and production design substantially. Meanwhile, there are also some limitations on model results in this study due to the simple assumptions, like 1D fracture flow, 2D reservoir flow, Darcy's flow, ignorance of temperature influence, etc. Therefore, there is still a long way to go before the proposed model becomes applicable in complex situations. But, this work is our primary work, trying to identify the fracture geometries and reservoir parameters with DFIT complexity. In the future, more efforts will be made to consider the complex situations, like (1) 3D fluid flow, (2) non-linear behavior of the pressure difference, (3) temperature influence, etc.
Summary and conclusions To better understand the properties of unconventional reservoirs of hydrogen and carbon, this paper introduces a new boundary element model with random fracture geometries caused by natural fractures, fissures, and microfractures. The model flexibility is demonstrated by different complex fracture cases, namely opening-fissure fracture networks, treelike fracture networks, radial multiple fracture networks, and mutually orthogonal fracture networks. Compared to bi-wing fractured wells, the pressure derivative of OFFN has a “valley shape”, namely “recharge”. The pressure derivative of TLFN also has a “valley shape”. The “valley shape” of TLFN lasts longer but weaker than that of OFFN. Quite different from OFFN or TLFN, there is no “valley shape” in the dimensionless pressure derivative of DMFN. However, there is a “hump” in the dimensionless pressure derivative of DMFN. For MOFN, a “valley shape” caused by the indirect-connected fractures occurs in its pressure derivative, which can be physically expected. In addition, an approximate unit-slope straight line occurs, reflecting a pseudo boundarydominated flow. Namely, various fracture cases exhibit different pressure transient behaviors, which may provide a good tool to identify fracture geometries with reservoir parameters. An example from a DFIT test in unconventional reservoir is analyzed using the log-log diagnostic plots. Results show that fracture geometries as well as reservoir parameters and the range of estimated values from the proposed model are reasonable around the known values of DFIT test. Undoubtedly, there are also some limitations on model results due to the assumptions, including 1D fracture flow, 2D reservoir flow, Darcy's flow, ignorance of temperature influence, etc. In the future, more efforts will be made to consider the complex situations, like (1) 3D fluid flow, (2) non-linear behavior of the pressure difference, (3) temperature influence, etc.
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Acknowledgements This work was supported by National Basic Research 973 Program of China (2015CB250900), National Key S&T Special Projects (2016ZX05047-004 and 2016ZX05030), Science Foundation of China University of Petroleum, Beijing (No. 2462018YJRC032), Post-doctoral Program for Innovation Talents (BX20180380), and the China Scholarship Council for a year study for Zhiming Chen at The University of Texas at Austin (201606440082). We also would like to express our gratitude to Mr. Hongyang Chu for his useful edits, suggestions, and ideas for the work.
Appendix B. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.ijhydene.2019.03.060.
Nomenclature ACA BF DFIT DMFN FLF FW MOFN KGD OFFN PBDF PKN PRF SPE TLFN B CFD h k Kn ðxÞ p pD pFD q qD qFD s tD t x; y; r
After-closure analysis Bilinear flow Diagnostic fracture injection test Dendritic multiple fracture networks Formation linear flow Fractured well Mutually Orthogonal fracture networks Khristianovic-Geertsma-de Opening-fissures fracture networks pseudo boundary-dominated flow Perkins-Kern-Nordgren Pseudo-radial flow Society of Petroleum Engineers Tree-like fracture networks fluid volume factor fracture conductivity, dimensionless formation thickness, m reservoir permeability, md second kind of modified Bessel functions reservoir pressure, MPa dimensionless reservoir pressure dimensionless fracture pressure sand-face injection rate, m3/d dimensionless flow rate flow rate of per unit fracture length into formation Laplace transformation variable with respect to injection time dimensionless injection time injection time, hr coordinates, m
Greek m
viscosity, cp
Subscript D dimensionless F hydraulic fracture w wellbore Superscript e Laplace transform
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SI Conversion Factors MPa 1.0 eþ06 Pa hr 3.6 eþ03 s cp 1.0 e-03 Pa$s md 1 e-15 m2
[17]
[18]
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