Methods and systems for determining reservoir properties of subterranean formations with pre-existing fractures

[0001] The present invention is related to U.S. Pat. No. 7,272,973 entitled “Methods and Apparatus for Determining Reservoir Properties of Subterranean Formations,” filed concurrently herewith, the entire disclosure of which is incorporated herein by reference.

[0002] The present invention relates to the field of oil and gas subsurface earth formation evaluation techniques and more particularly, to methods and an apparatus for determining reservoir properties of subterranean formations using quantitative refracture-candidate diagnostic test methods.

[0003] Oil and gas hydrocarbons may occupy pore spaces in subterranean formations such as, for example, in sandstone earth formations. The pore spaces are often interconnected and have a certain permeability, which is a measure of the ability of the rock to transmit fluid flow. Hydraulic fracturing operations can be performed to increase the production from a well bore if the near-wellbore permeability is low or when damage has occurred to the near-well bore area.

[0004] Hydraulic fracturing is a process by which a fluid under high pressure is injected into the formation to create and/or extend fractures that penetrate into the formation. These fractures can create flow channels to improve the near term productivity of the well. Propping agents of various kinds, chemical or physical, are often used to hold the fractures open and to prevent the healing of the fractures after the fracturing pressure is released.

[0005] Fracturing treatments may encounter a variety of problems during fracturing operations resulting in a less than optimal fracturing treatment. Accordingly, after a fracturing treatment, it may be desirable to evaluate the effectiveness of the fracturing treatment just performed or to provide a baseline of reservoir properties for later comparison and evaluation. One example of a problem occasionally encountered in fracturing treatments is bypassed layers. That is, during an original completion, oil or gas wells may contain layers bypassed either intentionally or inadvertently.

[0006] The success of a hydraulic fracture treatment often depends on the quality of the candidate well selected for the treatment. Choosing a good candidate for stimulation may result in success, while choosing a poor candidate may result in economic failure. To select the best candidate for stimulation or restimulation, there are many parameters to be considered. Some important parameters for hydraulic fracturing include formation permeability, in-situ stress distribution, reservoir fluid viscosity, skin factor, and reservoir pressure. Various methods have been developed to determine formation properties and thereby evaluate the effectiveness of a previous stimulation treatment or treatments.

[0007] Conventional methods designed to identify underperforming wells and to recomplete bypassed layers have been largely unsuccessful in part because the methods tend to oversimplify a complex multilayer problem and because they focus on commingled well performance and well restimulation potential without thoroughly investigating layer properties and layer recompletion potential. The complexity of a multilayer environment increases as the number of layers with different properties increases. Layers with different pore pressures, fracture pressures, and permeability can coexist in the same group of layers. A significant detriment to investigating layer properties is a lack of cost-effective diagnostics for determining layer permeability, pressure, and quantifying the effectiveness of a previous stimulation treatment or treatments.

[0008] These conventional methods often suffer from a variety of drawbacks including a lack of desired accuracy and/or an inefficiency of the computational method resulting in methods that are too time consuming. Furthermore, conventional methods often lack accurate means for quantitatively determining the transmissibility of a formation.

[0009] Post-frac production logs, near-wellbore hydraulic fracture imaging with radioactive tracers, and far-field microseismic fracture imaging all suggest that about 10% to about 40% of the layers targeted for completion during primary fracturing operations using limited-entry fracture treatment designs may be bypassed or ineffectively stimulated.

[0010] Quantifying bypassed layers has traditionally proved difficult because, in part, so few completed wells are imaged. Consequently, bypassed or ineffectively stimulated layers may not be easily identified, and must be inferred from analysis of a commingled well stream, production logs, or conventional pressure-transient tests of individual layers.

[0011] One example of a conventional method is described in U.S. Patent Publication 2002/0096324 issued to Poe, which describes methods for identifying underperforming or poorly performing producing layers for remediation or restimulation. This method, however, uses production data analysis of the produced well stream to infer layer properties rather than using a direct measurement technique. This limitation can result in poor accuracy and further, requires allocating the total well production to each layer based on production logs measured throughout the producing life of the well, which may or may not be available.

[0012] Other methods of evaluating effectiveness of prior fracturing treatments include conventional pressure-transient testing, which includes drawdown, buildup, injection/falloff testing. These methods may be used to identify an existing fracture retaining residual width from a previous fracture treatment or treatments, but conventional methods may require days of production and pressure monitoring for each single layer. Consequently, in a wellbore containing multiple productive layers, weeks to months of isolated-layer testing can be required to evaluate all layers. For many wells, the potential return does not justify this type of investment.

[0013] Diagnostic testing in low permeability multilayer wells has been attempted. One example of such a method is disclosed in Hopkins,. C. W., et al., The Use of Injection/Falloff Tests and Pressure Buildup Tests to Evaluate Fracture Geometry and Post-Stimulation Well Performance in the Devonian Shales, paper SPE 23433, 22-25 (1991). This method describes several diagnostic techniques used in a Devonian shale well to diagnose the existence of a pre-existing fracture(s) in multiple targeted layers over a 727 fit interval. The diagnostic tests include isolation flow tests, wellbore communication tests, nitrogen injection/falloff tests, and conventional drawdown/buildup tests.

[0014] While this diagnostic method does allow evaluation of certain reservoir properties, it is, however, expensive and time consuming—even for a relatively simple case having only four layers. Many refracture candidates in low permeability gas wells contain stacked lenticular sands with between 20 to 40 layers, which need to be evaluated in a timely and cost effective manner.

[0015] Another method uses a quasi-quantitative pressure transient test interpretation method as disclosed by Huang, H., et al., A Short Shut-In Time Testing Methodfor Determining Stimulation Effectiveness in Low Permeability Gas Reservoirs, GASTIPS, 6 No. 4, 28 (Fall 2000). This “short shut-in test interpretation method” is designed to provide only an indication of pre-existing fracture effectiveness. The method uses log-log type curve reference points—the end of wellbore storage, the beginning of pseudolinear flow, the end of pseudolinear flow, and the beginning of pseudoradial flow—and the known relationships between pressure and system properties at those points to provide upper and lower limits of permeability and effective fracture half length.

[0016] Another method uses nitrogen slug tests as a prefracture diagnostic test in low permeability reservoirs as disclosed by Jochen, J. E., et al., Quantifying Layered Reservoir Properties With a Novel Permeability Test, SPE 25864,12-14 (1993). This method describes a nitrogen injection test as a short small volume injection of nitrogen at a pressure less than the fracture initiation and propagation pressure followed by an extended pressure falloff period. Unlike the nitrogen injection/falloff test used by Hopkins et al., the nitrogen slug test is analyzed using slug-test type curves and by history matching the injection and falloff pressure with a finite-difference reservoir simulator.

[0017] Similarly, as disclosed in Craig, D. P., et al., Permeability, Pore Pressure, and Leakoff-Type Distributions in Rocky Mountain Basins, SPE PRODUCTION & FACILITIES, 48 (February, 2005), certain types of fracture-injection/falloff tests have been routinely implemented since 1998 as a prefracture diagnostic method to estimate formation permeability and average reservoir pressure. These fracture-injection/falloff tests, which are essentially a minifrac with reservoir properties interpreted from the pressure falloff, differ from nitrogen slug tests in that the pressure during the injection is greater than the fracture initiation and propagation pressure. A fracture-injection/falloff test typically requires a low rate and small volume injection of treated water followed by an extended shut-in period. The permeability to the mobile reservoir fluid and the average reservoir pressure may be interpreted from the pressure decline. A fracture-injection/falloff test, however, may fail to adequately evaluate refracture candidates, because this conventional theory does not account for pre-existing fractures.

[0018] Thus, conventional methods to evaluate formation properties suffer from a variety of disadvantages including a lack of the ability to quantitatively determine the reservoir transmissibility, a lack of cost-effectiveness, computational inefficiency, and/or a lack of accuracy. Even among methods developed to quantitatively determine a reservoir transmissibility, such methods may be impractical for evaluating formations having multiple layers such as, for example, low permeability stacked, lenticular reservoirs.

[0019] The present invention relates to the field of oil and gas subsurface earth formation evaluation techniques and more particularly, to methods and an apparatus for determining reservoir properties of subterranean formations using quantitative refracture-candidate diagnostic test methods.

[0020] In certain embodiments, a method for determining a reservoir transmissibility of at least one layer of a subterranean formation having preexisting fractures having a reservoir fluid comprises the steps of: (a) isolating the at least one layer of the subterranean formation to be tested; (b) introducing an injection fluid into the at least one layer of the subterranean formation at an injection pressure exceeding the subterranean formation fracture pressure for an injection period; (c) shutting in the wellbore for a shut-in period; (d) measuring pressure falloff data from the subterranean formation during the injection period and during a subsequent shut-in period; and (e) determining quantitatively a reservoir transmissibility of the at least one layer of the subterranean formation by analyzing the pressure falloff data with a quantitative refracture-candidate diagnostic model.

[0021] In certain embodiments, a system for determining a reservoir transmissibility of at least one layer of a subterranean formation by using variable-rate pressure falloff data from the at least one layer of the subterranean formation measured during an injection period and during a subsequent shut-in period comprises: a plurality of pressure sensors for measuring pressure falloff data; and a processor operable to transform the pressure falloff data to obtain equivalent constant-rate pressures and to determine quantitatively a reservoir transmissibility of the at least one layer of the subterranean formation by analyzing the variable-rate pressure falloff data using type-curve analysis according to a quantitative refracture-candidate diagnostic model.

[0022] In certain embodiments, a computer program, stored on a tangible storage medium, for analyzing at least one downhole property comprises executable instructions that cause a computer to: determine quantitatively a reservoir transmissibility of the at least one layer of the subterranean formation by analyzing the variable-rate pressure falloff data with a quantitative refracture-candidate diagnostic model.

[0023] The features and advantages of the present invention will be apparent to those skilled in the art. While numerous changes may be made by those skilled in the art, such changes are within the spirit of the invention.

[0038] The present invention relates to the field of oil and gas subsurface earth formation evaluation techniques and more particularly, to methods and an apparatus for determining reservoir properties of subterranean formations using quantitative refracture-candidate diagnostic test methods.

[0039] Methods of the present invention may be useful for estimating formation properties through the use of quantitative refracture-candidate diagnostic test methods, which may use injection fluids at pressures exceeding the formation fracture initiation and propagation pressure. In particular, the methods herein may be used to estimate formation properties such as, for example, the effective fracture half-length of a pre-existing fracture, the fracture conductivity of a pre-existing fracture, the reservoir transmissibility, and an average reservoir pressure. Additionally, the methods herein may be used to determine whether a pre-existing fracture is damaged. From the estimated formation properties, the present invention may be useful for, among other things, evaluating the effectiveness of a previous fracturing treatment to determine whether a formation requires restimulation due to a less than optimal fracturing treatment result. Accordingly, the methods of the present invention may be used to provide a technique to determine if and when restimulation is desirable by quantitative application of a refracture-candidate diagnostic fracture-injection falloff test method.

[0040] Generally, the methods herein allow a relatively rapid determination of the effectiveness of a previous stimulation treatment or treatments or treatments by injecting a fluid into the formation at an injection pressure exceeding the formation fracture pressure and recording the pressure falloff data. The pressure falloff data may be analyzed to determine certain formation properties, including if desired, the transmissibility of the formation.

[0041] In certain embodiments, a method of determining a reservoir transmissibility of at least one layer of a subterranean formation formation having preexisting fractures having a reservoir fluid compres the steps of: (a) isolating the at least one layer of the subterranean formation to be tested; (b) introducing an injection fluid into the at least one layer of the subterranean formation at an injection pressure exceeding the subterranean formation fracture pressure for an injection period; (c) shutting in the wellbore for a shut-in period; (d) measuring pressure falloff data from the subterranean formation during the injection period and during a subsequent shut-in period; and (e) determining quantitatively a reservoir transmissibility of the at least one layer of the subterranean formation by analyzing the pressure falloff data with a quantitative refracture-candidate diagnostic model.

[0042] The term, “refracture-candidate diagnostic test,” as used herein refers to the computational estimates shown below in Sections I and II used to estimate certain reservoir properties, including the transmissibility of a formation layer or multiple layers. The test recognizes that an existing fracture retaining residual width has associated storage, and a new induced fracture creates additional storage. Consequently, a fracture-injection/falloff test in a layer with a pre-existing fracture will exhibit characteristic variable storage during the pressure falloff period, and the change in storage is observed at hydraulic fracture closure. In essence, the test induces a fracture to rapidly identify a pre-existing fracture retaining residual width.

[0043] The methods and models herein are extensions of and based, in part, on the teachings of Craig, D. P., Analytical Modeling of a Fracture-Injection/Falloff Sequence and the Development of a Refracture-Candidate Diagnostic Test, PhD dissertation, Texas A&M Univ., College Station, Tex. (2005), which is incorporated by reference herein in full and U.S. patent application Ser. No. 10/813,698, filed Mar. 3, 2004, entitled “Methods and Apparatus for Detecting Fracture with Significant Residual Width from Previous Treatments, which is incorporated by reference herein in full.

[0044] FIG. 1 shows an example of an implementation of the quantitative refracture-candidate diagnostic test method implementing certain aspects of the quantitative refracture-candidate diagnostic model. Method 100 generally begins at step 105 for determining a reservoir transmissibility of at least one layer of a subterranean formation. At least one layer of the subterranean formation is isolated in step 110. During the layer isolation step, each subterranean layer is preferably individually isolated one at a time for testing by the methods of the present invention. Multiple layers may be tested at the same time, but this grouping of layers may introduce additional computational uncertainty into the transmissibility estimates.

[0045] An injection fluid is introduced into the at least one layer of the subterranean formation at an injection pressure exceeding the formation fracture pressure for an injection period (step 120). The injection fluid may be a liquid, a gas, or a mixture thereof. In certain exemplary embodiments, the volume of the injection fluid introduced into a subterranean layer may be roughly equivalent to the proppant-pack pore volume of an existing fracture if known or suspected to exist. Preferably, the introduction of the injection fluid is limited to a relatively short period of time as compared to the reservoir response time which for particular formations may range from a few seconds to minutes. In more preferred embodiments in typical applications, the introduction of the injection fluid may be limited to less than about 5 minutes. For formations having pre-existing fractures, the injection fluid is preferably introduced in such a way so as to produce a change in the existing and created fracture volume that is at least about twice the estimated proppant-pack pore volume. After introduction of the injection fluid, the wellbore may be shut-in for a period of time from a few minutes to a few days depending on the length of time for the pressure falloff data to show a pressure falloff approaching the reservoir pressure.

[0046] Pressure falloff data is measured from the subterranean formation during the injection period and during a subsequent shut-in period (step 140). The pressure falloff data may be measured by a pressure sensor or a plurality of pressure sensors. After introduction of the injection fluid, the wellbore may be shut-in for a period of time from about a few hours to a few days depending on the length of time for the pressure measurement data to show a pressure falloff approaching the reservoir pressure. The pressure falloff data may then be analyzed according to step 150 to determine a reservoir transmissibility of the subterranean formation according to the quantitative refracture-candidate diagnostic model shown below in more detail in Sections I and II. Method 100 ends at step 225.

[0047] FIG. 2 shows an example implementation of determining quantitatively a reservoir transmissibility (depicted in step 150 of Method 100). In particular, method 200 begins at step 205. Step 210 includes the step of transforming the variable-rate pressure falloff data to equivalent constant-rate pressures and using type curve analysis to match the equivalent constant-rate rate pressures to a type curve. Step 220 includes the step of determining quantitatively a reservoir transmissibility of the at least one layer of the subterranean formation by analyzing the equivalent constant-rate pressures with a quantitative refracture-candidate diagnostic model. Method 200 ends at step 225.

[0048] One or more methods of the present invention may be implemented via an information handling system. For purposes of this disclosure, an information handling system may include any instrumentality or aggregate of instrumentalities operable to compute, classify, process, transmit, receive, retrieve, originate, switch, store, display, manifest, detect, record, reproduce, handle, or utilize any form of information, intelligence, or data for business, scientific, control, or other purposes. For example, an information handling system may be a personal computer, a network storage device, or any other suitable device and may vary in size, shape, performance, functionality, and price. The information handling system may include random access memory (RAM), one or more processing resources such as a central processing unit (CPU or processor) or hardware or software control logic, ROM, and/or other types of nonvolatile memory. Additional components of the information handling system may include one or more disk drives, one or more network ports for communication with external devices as well as various input and output (I/O) devices, such as a keyboard, a mouse, and a video display. The information handling system may also include one or more buses operable to transmit communications between the various hardware components.

I. Quantitative Refracture-Candidate Diagnostic Test Model

[0049] A refracture-candidate diagnostic test is an extension of the fracture-injection/falloff theoretical model with multiple arbitrarily-oriented infinite-or finite-conductivity fracture pressure-transient solutions used to adapt the model. The fracture-injection/falloff theoretical model is presented in U.S. Pat. No. 7,272,973 entitled “Methods and Apparatus for Determining Reservoir Properties of Subterranean Formations,” filed concurrently herewith, the entire disclosure of which is incorporated by reference herein in full.

[0050] The test recognizes that an existing fracture retaining residual width has associated storage, and a new induced fracture creates additional storage. Consequently, a fracture-injection/falloff test in a layer with a pre-existing fracture will exhibit variable storage during the pressure falloff, and the change in storage is observed at hydraulic fracture closure. In essence the test induces a fracture to rapidly identify a pre-existing fracture retaining residual width.

[0051] Consider a pre-existing fracture that dilates during a fracture-injection/falloff sequence, but the fracture half length remains constant. With constant fracture half length during the injection and before-closure falloff, fracture volume changes are a function of fracture width, and the before-closure storage coefficient is equivalent to the dilating-fracture storage coefficient and written as

[0052] $\begin{array}{cc}\begin{array}{c}{C}_{\mathrm{bc}}=c_{\mathrm{wb}}{V}_{\mathrm{wb}}+2c_f{V}_f+2\frac{d{V}_f}{d{p}_w}\\ =c_{\mathrm{wb}}{V}_{\mathrm{wb}}+2\frac{A_f}{S_f}\\ =C_{\mathrm{fd}}\end{array}.& \left(1\right)\end{array}$
(The nomenclature used throughout this specification is defined below in Section VI)
where S_f is the fracture stiffness as presented by Craig, D. P., Analytical Modeling of a Fracture-Injection/Falloff Sequence and the Development of a Refracture-Candidate Diagnostic Test, PhD dissertation, Tex. A&M Univ., College Station, Texas (2005). With equivalent before-closure and dilated-fracture storage, a derivation similar to that shown below in Section III results in the dimensionless pressure solution written as

[0053] $\begin{array}{cc}{p}_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)=q_{\mathrm{wsD}}\left[p_{\mathrm{acD}}\left(t_{\mathrm{LfD}}\right)-p_{\mathrm{acD}}\left(t_{\mathrm{LfD}}-{\left(t_e\right)}_{\mathrm{LfD}}\right)\right]+p_{\mathrm{wsD}}\left(0\right)C_{\mathrm{acD}}{p}_{\mathrm{acD}}^{\prime }\left(t_{\mathrm{LfD}}\right)-\left(C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right){\int }_0^{{\left(t_c\right)}_{\mathrm{LfD}}}{p}_{\mathrm{acD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D.& \left(2\right)\end{array}$

[0054] Alternatively, a secondary fracture can be initiated in a plane different from the primary fracture during the injection. With secondary fracture creation, and assuming the volume of the primary fracture remains constant, the propagating-fracture storage coefficient is written as

[0055] $\begin{array}{cc}{C}_{\mathrm{Lf}}\left(t_{\mathrm{LfD}}\right)=c_{\mathrm{wb}}{V}_{\mathrm{wb}}+c_f{V}_{f1}+2\frac{A_{f2}}{S_{f2}}{\left(\frac{t_{\mathrm{LfD}}}{{\left(t_e\right)}_{\mathrm{LfD}}}\right)}^{\alpha }.& \left(3\right)\end{array}$

[0056] The before-closure storage coefficient may be defined as

[0057] $\begin{array}{cc}{C}_{\mathrm{Lfbc}}=c_{\mathrm{wb}}{V}_{\mathrm{wb}}+2c_f{V}_{f1}+2\frac{A_{f2}}{S_{f2}},& \left(4\right)\end{array}$
and the after-closure storage coefficient may be written as
C_Lfac=c_wbV_wb+2c_f(V_f1+V_f2)  (5)

[0058] With the new storage-coefficient definitions, the fracture-injection/falloff sequence solution with a pre-existing fracture and propagating secondary fracture is written as

[0059] $\begin{array}{cc}{p}_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)=q_{\mathrm{wsD}}\left[p_{\mathrm{pLfD}}\left(t_{\mathrm{LfD}}\right)-p_{\mathrm{pLfD}}\left(t_{\mathrm{LfD}}-{\left(t_e\right)}_{\mathrm{LfD}}\right)\right]-C_{\mathrm{LfacD}}{\int }_0^{t_{\mathrm{LfD}}}{p}_{\mathrm{LfD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D-{\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{p}_{\mathrm{pLfD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)C_{\mathrm{pLfD}}\left({\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D+C_{\mathrm{LfbcD}}{\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{p}_{\mathrm{LfD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D-\left(C_{\mathrm{LfbcD}}-C_{\mathrm{LfacD}}\right){\int }_0^{{\left(t_c\right)}_{\mathrm{LfD}}}{p}_{\mathrm{LfD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D& \left(6\right)\end{array}$

[0060] The limiting-case solutions for a single dilated fracture are identical to the fracture-injection/falloff limiting-case solutions—(Eqs. 19 and 20 of U.S. Patent No. 7,272,973)—when (t_e)_LfDt_LfD. With secondary fracture propagation, the before-closure limiting-case solution for (t_e)_LfDt_LfD<(t_c)_LfD may be written as
p_wsD(t_LfD)=p_wsD(0)C_LfbcDp′_LfbcD(t_LfD),  (7)

[0061] where _{p LfbcD} is the dimensionless pressure solution for a constant-rate drawdown in a well producing from multiple fractures with constant before-closure storage, which may be written in the Laplace domain as

[0062] $\begin{array}{cc}{\stackrel{\_}{p}}_{\mathrm{LfbcD}}=\frac{{\stackrel{\_}{p}}_{\mathrm{LfD}}}{1+s^2{C}_{\mathrm{LfbcD}}{\stackrel{\_}{p}}_{\mathrm{LfD}}},& \left(8\right)\end{array}$
and p_LfD is the Laplace domain reservoir solution for production from multiple arbitrarily-oriented finite-or infinite-conductivity fractures. New multiple fracture solutions are provided in below in Section IV for arbitrarily-oriented infinite-conductivity fractures and in Section V for arbitrarily-oriented finite-conductivity fractures. The new multiple fracture solutions allow for variable fracture half length, variable conductivity, and variable angle of separation between fractures.

[0063] The after-closure limiting-case solution with secondary fracture propagation when t_LfD(t_c)_LfD(t_e)_LfD is written as

[0064] $\begin{array}{cc}{p}_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)=\left[\begin{array}{c}{p}_{\mathrm{wsD}}\left(0\right)C_{\mathrm{LfbcD}}-\\ p_{\mathrm{wsD}}\left({\left(t_c\right)}_{\mathrm{LfD}}\right)\left(C_{\mathrm{LfbcD}}-C_{\mathrm{LfacD}}\right)\end{array}\right]p_{\mathrm{LfacD}}^{\prime }\left(t_{\mathrm{LfD}}\right)& \left(9\right)\end{array}$
where p_LfacD is the dimensionless pressure solution for a constant-rate drawdown in a well producing from multiple fractures with constant after-closure storage, which may be written in the Laplace domain as

[0065] $\begin{array}{cc}{\stackrel{\_}{p}}_{\mathrm{LfbcD}}=\frac{{\stackrel{\_}{p}}_{\mathrm{LfD}}}{1+s^2{C}_{\mathrm{LfacD}}{\stackrel{\_}{p}}_{\mathrm{LfD}}}.& \left(10\right)\end{array}$

[0066] The limiting-case solutions are slug-test solutions, which suggest that a refracture-candidate diagnostic test may be analyzed as a slug test provided the injection time is short relative to the reservoir response.

[0067] Consequently, a refracture-candidate diagnostic test may use the following in certain embodiments:

• • [0068] Isolate a layer to be tested.
  • [0069] Inject liquid or gas at a pressure exceeding fracture initiation and propagation pressure. In certain embodiments, the injected volume may be roughly equivalent to the proppant-pack pore volume of an existing fracture if known or suspected to exist. In certain embodiments, the injection time may be limited to a few minutes.
  • [0070] Shut-in and record pressure falloff data. In certain embodiments, the measurement period may be several hours.

[0071] A qualitative interpretation may use the following steps:

• • [0072] Identify hydraulic fracture closure during the pressure falloff using methods such as those disclosed in Craig, D. P. et al., Permeability, Pore Pressure, and Leakoff-Type Distributions in Rocky Mountain Basins, SPE PRODUCTION & FACILITIES, 48 (February, 2005).
  • [0073] The time at the end of pumping, t_ne, becomes the reference time zero, Δt=0. Calculate the shut-in time relative to the end of pumping as
    Δt=t−t_ne  (11)
  • [0074]  In some cases, t_ne, is very small relative to t and Δt=t. As a person of ordinary skill in the art with the benefit of this disclosure will appreciate, t_ne may be taken as zero approximately zero so as to approximate Δt. Thus, the term Δt as used herein includes implementations where t_ne is assumed to be zero or approximately zero. For a slightly-compressible fluid injection in a reservoir containing a compressible fluid, or a compressible fluid injection in a reservoir containing a compressible fluid, use the compressible reservoir fluid properties and calculate adjusted time as

[0075] $\begin{array}{cc}{t}_a={\left(\mu c_t\right)}_{p_0}{\int }_0^{\Delta t}\frac{d\Delta t}{{\left(\mu c_t\right)}_w}& \left(12\right)\end{array}$

• • [0076]  where pseudotime may be defined as

[0077] $\begin{array}{cc}{t}_p={\int }_0^t\frac{dt}{{\left(\mu c_t\right)}_w}& \left(13\right)\end{array}$

• • [0078]  and adjusted time or normalized pseudotime may be defined as

[0079] $\begin{array}{cc}{t}_a={\left(\mu c_t\right)}_{\mathrm{re}}{\int }_0^t\frac{dt}{{\mu }_w{c}_t}& \left(14\right)\end{array}$

• • [0080]  where the subscript ‘re’ refers to an arbitrary reference condition selected for convenience.
  • [0081] The pressure difference for a slightly-compressible fluid injection into a reservoir containing a slightly compressible fluid may be calculated as
    Δp(t)=P_w(t)−P_i,  (15)
  • [0082]  or for a slightly-compressible fluid injection in a reservoir containing a compressible fluid, or a compressible fluid injection in a reservoir containing a compressible fluid, use the compressible reservoir fluid properties and calculate the adjusted pseudopressure difference as
    ΔP_a(t)=P_aw(t)−P_ai,  (16)
  • [0083]  where

[0084] $\begin{array}{cc}{p}_a={\left(\frac{\mu z}{p}\right)}_{p_i}{\int }_0^p\frac{pdp}{\mu z}.& \left(17\right)\end{array}$

• • [0085]  where pseudopressure may be defined as

[0086] $\begin{array}{cc}{p}_a={\int }_0^p\frac{pdp}{\mu z}& \left(18\right)\end{array}$

• • [0087]  and adjusted pseudopressure or normalized pseudopressure may be defined as

[0088] $\begin{array}{cc}{p}_a={\left(\frac{\mu z}{p}\right)}_{\mathrm{re}}{\int }_0^p\frac{pdp}{\mu z}& \left(19\right)\end{array}$

• • [0089]  where the subscript ‘re’ refers to an arbitrary reference condition selected for convenience.

[0090] The reference conditions in the adjusted pseudopressure and adjusted pseudotime definitions are arbitrary and different forms of the solution can be derived by simply changing the normalizing reference conditions.

• • [0091] Calculate the pressure-derivative plotting function as

[0092] $\begin{array}{cc}\Delta p^{\prime }=\frac{d\left(\Delta p\right)}{d\left(\ln \Delta t\right)}=\Delta p\Delta t,& \left(20\right)\\ \mathrm{or}& \\ \Delta p_a^{\prime }=\frac{d\left(\Delta p_a\right)}{d\left(\ln t_a\right)}=\Delta p_a{t}_a,& \left(21\right)\end{array}$

• • [0093] Transform the recorded variable-rate pressure falloff data to an equivalent pressure if the rate were constant by integrating the pressure difference with respect to time, which may be written for a slightly compressible fluid as

[0094] $\begin{array}{cc}I\left(\Delta p\right)={\int }_0^{\Delta t}\left[p_w\left(\tau \right)-p_i\right]d\tau & \left(22\right)\end{array}$

• • [0095]  or for a slightly-compressible fluid injected in a reservoir containing a compressible fluid, or a compressible fluid injection in a reservoir containing a compressible fluid, the pressure-plotting function may be calculated as

[0096] $\begin{array}{cc}I\left(\Delta p_a\right)={\int }_0^{t_a}\Delta p_{a}d{t}_a.& \left(23\right)\end{array}$

• • [0097] Calculate the pressure-derivative plotting function as

[0098] $\begin{array}{cc}\Delta p^{\prime }=\frac{d\left(\Delta p\right)}{d\left(\ln \Delta t\right)}=\Delta p\Delta t,& \left(24\right)\\ \mathrm{or}& \\ \Delta p_a^{\prime }=\frac{d\left(\Delta p_a\right)}{d\left(\ln t_a\right)}=\Delta p_a{t}_a,& \left(25\right)\end{array}$

• • [0099] Prepare a log-log graph of I(ΔP) versus Δt or I(ΔP_a) versus t_a.
  • [0100] Prepare a log-log graph of ΔP′ versus Δt or ΔP′_a versus t_a.
  • [0101] Examine the storage behavior before and after closure.
    II. Analysis and Interpretation of Data Generally

[0102] A change in the magnitude of storage at fracture closure suggests a fracture retaining residual width exists. When the storage decreases, an existing fracture is nondamaged. Conversely, a damaged fracture, or a fracture exhibiting choked-fracture skin, is indicated by apparent increase in the storage coefficient.

[0103] Quantitative refracture-candidate diagnostic interpretation uses type-curve matching, or if pseudoradial flow is observed, after-closure analysis as presented in Gu, H. et al., Formation Permeability Determination Using Impulse-Fracture Injection, SPE 25425 (1993) or Abousleiman, Y., Cheng, A. H-D. and Gu, H., Formation Permeability Determination by Micro or Mini-Hydraulic Fracturing, J. OF ENERGY RESOURCES TECHNOLOGY, 116, No. 6, 104 (June 1994). After-closure analysis is preferable because it does not require knowledge of fracture half length to calculate transmissibility. However, pseudoradial flow is unlikely to be observed during a relatively short pressure falloff, and type-curve matching may be necessary. From a pressure match point on a constant-rate type curve with constant before-closure storage, transmissibility may be calculated in field units as

[0104] $\begin{array}{cc}\frac{\mathrm{kh}}{\mu }=141.2\left(24\right)p_{\mathrm{wsD}}\left(0\right){C_{\mathrm{Lfbc}}\left(p_0-p_i\right)\left[\frac{p_{\mathrm{LfbcD}}\left(t_D\right)}{{\int }_0^{\Delta t}\left[p_w\left(\tau \right)-p_i\right]dt}\right]}_M& \left(26\right)\end{array}$
or from an after-closure pressure match point using a variable-storage type curve

[0105] $\begin{array}{cc}\frac{\mathrm{kh}}{\mu }=141.2\left(24\right)\left[\begin{array}{c}{p}_{\mathrm{wsD}}\left(0\right)C_{\mathrm{Lfbc}-}\\ P_{\mathrm{wsD}}\left({\left(t_c\right)}_{\mathrm{LfD}}\right)\left[C_{\mathrm{Lfbc}}-C_{\mathrm{Lfbc}}\right]\end{array}\right]{\left(p_0-p_i\right)\left[\frac{p_{\mathrm{LfacD}}\left(t_D\right)}{{\int }_0^{\Delta t}\left[p_w\left(\tau \right)-p_i\right]d\tau }\right]}_M& \left(27\right)\end{array}$

[0106] Quantitative interpretation has two limitations. First, the average reservoir pressure must be known for accurate equivalent constant-rate pressure and pressure derivative calculations, Eqs. 22-25. Second, both primary and secondary fracture half lengths are required to calculate transmissibility. Assuming the secondary fracture half length can be estimated by imaging or analytical methods as presented in Valkó, P. P. and Economides, M. J., Fluid-Leakoff Delineation in High Permeability Fracturing, SPE PRODUCTION & FACILITIES, 117 (May, 1999), the primary fracture half length is calculated from the type curve match, L_f1=L_f2/δ_L. With both fracture half lengths known, the before-and after-closure storage coefficients can be calculated as in Craig, D. P., Analytical Modeling of a Fracture-Injection/Falloff Sequence and the Development of a Refracture-Candidate Diagnostic Test, PhD dissertation, Texas A&M Univ., College Station, Tex. (2005) and the transmissibility estimated.

III. Theoretical Model A—Fracture-Injection/Falloff Solution in a Reservoir Without a Pre-Existing Fracture

[0107] Assume a slightly compressible fluid fills the wellbore and fracture and is injected at a constant rate and at a pressure sufficient to create a new hydraulic fracture or dilate an existing fracture. A mass balance during a fracture injection may be written as

[0108] $\begin{array}{cc}\stackrel{m_{\mathrm{in}}}{\overbrace{q_{w}B\rho }}-\stackrel{m_{\mathrm{out}}}{\overbrace{q_{\ell }{B}_r{\rho }_r}}=\stackrel{\mathrm{Storage}}{\overbrace{V_{wb}\frac{d{\rho }_{wb}}{dt}+2\frac{d\left(V_f{\rho }_f\right)}{dt}}},& \left(A\text{-}1\right)\end{array}$
where Q_l is the fluid leakoff rate into the reservoir from the fracture, q_l=q_sf, and V_fis the fracture volume.

[0109] A material balance equation may be written assuming a constant density, ρ=ρ_wb=ρ_f=ρ_r, and a constant formation volume factor, B=B_r, as

[0110] $\begin{array}{cc}{q}_{sf}=q_w-\frac{1}{B}\left(c_{wb}{V}_{wb}+2c_f{V}_f+2\frac{d{V}_f}{d{p}_w}\right)\frac{d{p}_w}{dt}.& \left(A\text{-}2\right)\end{array}$

[0111] During a constant rate injection with changing fracture length and width, the fracture volume may be written as
V_f(p_w(t))=h_fL(p_w(t))ŵ_f(p_w(t))  (A-3)
and the propagating-fracture storage coefficient may be written as

[0112] $\begin{array}{cc}{C}_{pf}\left(p_w\left(t\right)\right)=c_{wb}{V}_{\mathrm{wb}}+2c_f{V}_f\left(p_w\left(t\right)\right)+2\frac{d{V}_f\left(p_w\left(t\right)\right)}{d{p}_w}.& \left(A\text{-}4\right)\end{array}$

[0113] The dimensionless wellbore pressure for a fracture-injection/falloff may be written as

[0114] $\begin{array}{cc}{p}_{wsD}\left(t_{\mathrm{LfD}}\right)=\frac{p_w\left(t_{\mathrm{LfD}}\right)-p_i}{p_0-p_i},& \left(A\text{-}5\right)\end{array}$
where P_i is the initial reservoir pressure and P₀ is an arbitrary reference pressure. At time zero, the wellbore pressure is increased to the “opening” pressure, P_w0, which is generally set equal to P₀, and the dimensionless wellbore pressure at time zero may be written as

[0115] $\begin{array}{cc}{p}_{wsD}\left(0\right)=\frac{p_{w0}-p_i}{p_0-p_i}.& \left(A\text{-}6\right)\end{array}$

[0116] Define dimensionless time as

[0117] $\begin{array}{cc}{t}_{\mathrm{LfD}}=\frac{kt}{\varphi \mu c_t{L}_f^2},& \left(A\text{-}7\right)\end{array}$
where L_f is the fracture half-length at the end of pumping. The dimensionless reservoir flow rate may be defined as

[0118] $\begin{array}{cc}{q}_{sD}=\frac{q_{sf}B\mu }{2kh\left(p_0-p_i\right)},& \left(A\text{-}8\right)\end{array}$
and the dimensionless well flow rate may be defined as

[0119] $\begin{array}{cc}{q}_{wsD}=\frac{q_{w}B\mu }{2kh\left(p_0-p_i\right)},& \left(A\text{-}9\right)\end{array}$
where q_w is the well injection rate.

[0120] With dimensionless variables, the material balance equation for a propagating fracture during injection may be written as

[0121] $\begin{array}{cc}{q}_{sD}=q_{wsD}-\frac{C_{pf}\left(p_w\left(t\right)\right)}{2\varphi c_{t}h{L}_f^2}\frac{d{p}_{wsD}}{d{t}_{\mathrm{LfD}}}.& \left(A\text{-}10\right)\end{array}$

[0122] Define a dimensionless fracture storage coefficient as

[0123] $\begin{array}{cc}{C}_{fD}=\frac{C_{pf}\left(p_w\left(t\right)\right)}{2\varphi c_{t}h{L}_f^2},& \left(A\text{-}11\right)\end{array}$

[0124] and the dimensionless material balance equation during an injection at a pressure sufficient to create and extend a hydraulic fracture may be written as

[0125] $\begin{array}{cc}{q}_{\mathrm{sD}}=q_{\mathrm{wsD}}-C_{\mathrm{pfD}}\left(p_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)\right)\frac{d{p}_{\mathrm{wsD}}}{d{t}_{\mathrm{LfD}}}.& \left(A\text{-}12\right)\end{array}$

[0126] Using the technique of Correa and Ramey as disclosed in Correa, A. C. and Ramey, H. J., Jr., Combined Effects of Shut-In and Production: Solution With a New Inner Boundary Condition, SPE 15579 (1986) and Correa, A. C. and Ramey, H. J., Jr., A Method for Pressure Buildup Analysis of Drillstem Tests, SPE 16802 (1987), a material balance equation valid at all times for a fracture-injection/falloff sequence with fracture creation and extension and constant after-closure storage may be written as

[0127] $\begin{array}{cc}\begin{array}{c}{q}_{\mathrm{sD}}=q_{\mathrm{wsD}}-U_{{\left(t_e\right)}_{\mathrm{LfD}}}{q}_{\mathrm{wsD}}-C_{\mathrm{pfD}}\left(p_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)\right)\frac{d{p}_{\mathrm{wsD}}}{d{t}_{\mathrm{LfD}}}+\\ U_{{\left(t_e\right)}_{\mathrm{LfD}}}\left[C_{\mathrm{pfD}}\left(p_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)\right)-C_{\mathrm{bcD}}\right]\frac{d{p}_{\mathrm{wsD}}}{d{t}_{\mathrm{LfD}}}+\\ U_{{\left(t_c\right)}_{\mathrm{LfD}}}\left[C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right]\frac{d{p}_{\mathrm{wsD}}}{d{t}_{\mathrm{LfD}}}\end{array}& \left(A\text{-}13\right)\end{array}$
where the unit step function is defined as

[0128] $\begin{array}{cc}{U}_a=\{\begin{array}{c}0,t<a\\ 1,t>a\end{array}.& \left(A\text{-}14\right)\end{array}$

[0129] The Laplace transform of the material balance equation for an injection with fracture creation and extension is written after expanding and simplifying as

[0130] $\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{q}}_{\mathrm{sD}}=\frac{q_{\mathrm{wsD}}}{s}-q_{\mathrm{wsD}}\frac{e^{-{s\left(t_e\right)}_{\mathrm{LfD}}}}{s}-\\ {\int }_0^{{\left(t_e\right)}_{{\mathrm{LfD}}_e^{-\mathrm{st}}\mathrm{LfD}}}{C}_{\mathrm{pfD}}\left(p_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)\right)p_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}-\\ {\mathrm{sC}}_{\mathrm{acD}}{\stackrel{\_}{p}}_{\mathrm{wsD}}+p_{\mathrm{wsD}}\left(0\right)C_{\mathrm{acD}}+\\ {\int }_0^{{\left(t_e\right)}_{{\mathrm{LfD}}_e^{-\mathrm{st}}\mathrm{LfD}}}{C}_{\mathrm{bcD}}{p}_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}-\\ \left(C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right){\int }_0^{{\left(t_c\right)}_{{\mathrm{LfD}}_e^{-\mathrm{st}}\mathrm{LfD}}}{p}_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}\end{array}& \left(A\text{-}15\right)\end{array}$

[0131] With fracture half length increasing during the injection, a dimensionless pressure solution may be required for both a propagating and fixed fracture half-length. A dimensionless pressure solution may developed by integrating the line-source solution, which may be written as

[0132] $\begin{array}{cc}{\stackrel{\_}{\Delta p}}_{\mathrm{ls}}=\frac{\tilde{q}\mu }{2\pi \mathrm{ks}}{K}_0\left(r_D\sqrt{u}\right),& \left(A\text{-}16\right)\end{array}$
from x_w− L(s) and x_w+ L(s) with respect to x′_w where u=sf (s), and f(s)=1 for a single-porosity reservoir. Here, it is assumed that the fracture half length may be written as a function of the Laplace variable, s, only. In terms of dimensionless variables, x′_wD=x′_w/L_f and dx′_w=L_fdx′_wD′the line-source solution is integrated from x_wD− L_fD(s) to x_wD+ L_fD(s), which may be written as

[0133] $\begin{array}{cc}\stackrel{\_}{\Delta p}=\frac{\tilde{q}\mu L_f}{2\pi \mathrm{ks}}{\int }_{x_{\mathrm{wD}}-{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}^{x_{\mathrm{wD}}+{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}{K}_0\left[\sqrt{u}\sqrt{{\left(x_D-x_{\mathrm{wD}}^{\prime }\right)}^2+{\left(y_D-y_{\mathrm{wD}}\right)}^2}\right]d{x}_{\mathrm{wD}}^{\prime }& \left(A\text{-}17\right)\end{array}$

[0134] Assuming that the well center is at the origin, x_wD=Y_wD=0,

[0135] $\begin{array}{cc}\stackrel{\_}{\Delta p}=\frac{\tilde{q}\mu L_f}{2\pi \mathrm{ks}}{\int }_{-{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}^{{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}{K}_0[\sqrt{u}\sqrt{{\left(x_D-x_{\mathrm{wD}}^{\prime }\right)}^2+{\left(y_D\right)}^2}]d{x}_{\mathrm{wD}}^{\prime }& \left(A\text{-}18\right)\end{array}$

[0136] Assuming constant flux, the flow rate in the Laplace domain may be written as
q(s)=2{tilde over (q)}h L(s),   (A-19)
and the plane-source solution may be written in dimensionless terms as

[0137] $\begin{array}{cc}{\stackrel{\_}{p}}_D=\frac{{\stackrel{\_}{q}}_D\left(s\right)}{{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}\frac{1}{2s}{\int }_{-{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}^{{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}{K}_0\left[\sqrt{u}\sqrt{{\left(x_D-\alpha \right)}^2+{\left(y_D\right)}^2}\right]d\alpha ,\mathrm{where}& \left(A\text{-}20\right)\\ {\stackrel{\_}{p}}_D=\frac{2\pi \mathrm{kh}\stackrel{\_}{\Delta p}}{\stackrel{\_}{q}\mu },& \left(A\text{-}21\right)\\ {\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)=\frac{L\left(s\right)}{L_f},& \left(A\text{-}22\right)\end{array}$
and defining the total flow rate as q_t(s), the dimensionless flow rate may be written as

[0138] $\begin{array}{cc}{\stackrel{\_}{q}}_D\left(s\right)=\frac{\stackrel{\_}{q}\left(s\right)}{{\stackrel{\_}{q}}_t\left(s\right)}.& \left(A\text{-}23\right)\end{array}$

[0139] It may be assumed that the total flow rate increases proportionately with respect to increased fracture half-length such that q_D(s)=1. The solution is evaluated in the plane of the fracture, and after simplifying the integral using the identity of Ozkan and Raghavan as disclosed in Ozkan, E. and Raghavan, R., New Solutions for Well-Test-Analysis Problems: Part 2—Computational Considerations and Applications, SPEFE, 369 (September, 1991), the dimensionless uniform-flux solution in the Laplace domain for a variable fracture half-length may be written as

[0140] $\begin{array}{cc}{\stackrel{\_}{p}}_{\mathrm{pfD}}=\frac{1}{{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}\frac{1}{2s\sqrt{u}}\left[{\int }_0^{\sqrt{u}\left({\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)+x_D\right)}{K}_0\left[z\right]dz+{\int }_0^{\sqrt{u}\left({\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)-x_D\right)}{K}_0\left[z\right]dz\right]& \left(A\text{-}24\right)\end{array}$
and the infinite conductivity solution may be obtained by evaluating the uniform-flux solution at x_D=0.732 L_fD(s) and may be written as

[0141] $\begin{array}{cc}{\stackrel{\_}{p}}_{\mathrm{pfD}}=\frac{1}{{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)}\frac{1}{2s\sqrt{u}}[{\int }_0^{\sqrt{u}{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)\left(1+0.732\right)}{K}_0\left[z\right]dz+{\int }_0^{\sqrt{u}{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)\left(1-0.732\right)}{K}_0\left[z\right]dz]& \left(A\text{-}25\right)\end{array}$

[0142] The Laplace domain dimensionless fracture half-length varies between 0 and 1 during fracture propagation, and using a power-model approximation as shown in Nolte, K. G., Determination of Fracture Parameters From Fracturing Pressure Decline, SPE 8341 (1979), the Laplace domain dimensionless fracture half-length may be written as

[0143] $\begin{array}{cc}{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)=\frac{\stackrel{\_}{L}\left(s\right)}{{\stackrel{\_}{L}}_f\left(s_e\right)}={\left(\frac{s_e}{s}\right)}^{\alpha },& \left(A\text{-}26\right)\end{array}$
where s_e is the Laplace domain variable at the end of pumping. The Laplace domain dimensionless fracture half length may be written during propagation and closure as

[0144] $\begin{array}{cc}{\stackrel{\_}{L}}_{\mathrm{fD}}\left(s\right)=\{\begin{array}{cc}{\left(\frac{s_e}{s}\right)}^{\alpha }& s_e<s\\ 1& s_e\ge s\end{array}.& \left(A\text{-}27\right)\end{array}$
where the power-model exponent ranges from α=½ for a low efficiency (high leakoff) fracture and α=1 for a high efficiency (low leakoff) fracture.

[0145] During the before-closure and after-closure period—when the fracture half-length is unchanging—the dimensionless reservoir pressure solution for an infinite conductivity fracture in the Laplace domain may be written as

[0146] $\begin{array}{cc}{\stackrel{\_}{p}}_{\mathrm{fD}}=\frac{1}{2s\sqrt{u}}\left[{\int }_0^{\sqrt{u}\left(1+0.732\right)}{K}_0\left[z\right]dz+{\int }_0^{\sqrt{u}\left(1-0.732\right)}{K}_0\left[z\right]dz\right].& \left(A\text{-}28\right)\end{array}$

[0147] The two different reservoir models, one for a propagating fracture and one for a fixed-length fracture, may be superposed to develop a dimensionless wellbore pressure solution by writing the superposition integrals as

[0148] $\begin{array}{cc}{p}_{\mathrm{wsD}}={\int }_0^{t_{\mathrm{LfD}}}{q}_{\mathrm{pfD}}\left({\tau }_D\right)\frac{d{p}_{\mathrm{pfD}}\left(t_{\mathrm{LfD}}-{\tau }_D\right)}{d{t}_{\mathrm{LfD}}}d{\tau }_D+{\int }_0^{t_{\mathrm{LfD}}}{q}_{\mathrm{fD}}\left({\tau }_D\right)\frac{d{p}_{\mathrm{fD}}\left(t_{\mathrm{LfD}}-{\tau }_D\right)}{d{t}_{\mathrm{LfD}}}d{\tau }_D,& \left(A\text{-}29\right)\end{array}$
where q_pfD(t_LfD) is the dimensionless flow rate for the propagating fracture model, and q_fD(t_LfD) is the dimensionless flow rate with a fixed fracture half-length model used during the before-closure and after-closure falloff period. The initial condition in the fracture and reservoir is a constant initial pressure, p_D (t_LfD)=p_pfD(t_LfD)=p_fD(t_LfD)=0, and with condition, the Laplace transform of the superposition integral is written as
p_wsD= q_pfDs p_pfD+ q_fDs p_fD  (A-30)

[0149] The Laplace domain dimensionless material balance equation may be split into injection and falloff parts by writing as
q_sD= q_pfD+ q_fD,   (A-31)
where the dimensionless reservoir flow rate during fracture propagation may be written as

[0150] $\begin{array}{cc}{\stackrel{\_}{q}}_{\mathrm{pfD}}=\frac{q_{\mathrm{wsD}}}{s}-q_{\mathrm{wsD}}\frac{e^{-{s\left(t_e\right)}_{\mathrm{LfD}}}}{s}-{\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{e}^{-{\mathrm{st}}_{\mathrm{LfD}}}{C}_{\mathrm{pfD}}\left(p_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)\right)p_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}},& \left(A\text{-}32\right)\end{array}$
and the dimensionless before-closure and after-closure fracture flow rate may be written as

[0151] $\begin{array}{cc}{\stackrel{\_}{q}}_{\mathrm{fD}}=\left[\begin{array}{c}{p}_{\mathrm{wD}}\left(0\right)C_{\mathrm{acD}}-{\mathrm{sC}}_{\mathrm{acD}}{\stackrel{\_}{p}}_{\mathrm{wsD}}+C_{\mathrm{bcD}}\\ {\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{e}^{-{\mathrm{st}}_{\mathrm{LfD}}}{p}_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}-\left(C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right)\\ {\int }_0^{{\left(t_c\right)}_{\mathrm{LfD}}}{e}^{-{\mathrm{st}}_{\mathrm{LfD}}}{p}_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}\end{array}\right].& \left(A\text{-}33\right)\end{array}$

[0152] Using the superposition principle to develop a solution requires that the pressure-dependent dimensionless propagating-fracture storage coefficient be written as a function of time only. Let fracture propagation be modeled by a power model and written as

[0153] $\begin{array}{cc}\frac{A\left(t\right)}{A_f}=\frac{h_{f}L\left(t\right)}{h_f{L}_f}={\left(\frac{t}{t_e}\right)}^{\alpha }.& \left(A\text{-}34\right)\end{array}$

[0154] Fracture volume as a function of time may be written as
V_f(p_w(t))=h_fL(p_w(t))ŵ_f(p_w(t))  (A-35)
which, using the power model, may also be written as

[0155] $\begin{array}{cc}{V}_f\left(p_w\left(t\right)\right)=h_f{L}_f\frac{\left(p_w\left(t\right)-p_c\right)}{S_f}{\left(\frac{t}{t_e}\right)}^{\alpha }.& \left(A\text{-}36\right)\end{array}$

[0156] The derivative of fracture volume with respect to wellbore pressure may be written as

[0157] $\begin{array}{cc}\frac{d{V}_f\left(p_w\left(t\right)\right)}{d{p}_w}=\frac{h_f{L}_f}{S_f}{\left(\frac{t}{t_e}\right)}^{\alpha }.& \left(A\text{-}37\right)\end{array}$

[0158] Recall the propagating-fracture storage coefficient may be written as

[0159] $\begin{array}{cc}{C}_{\mathrm{pf}}\left(p_w\left(t\right)\right)=c_{\mathrm{wb}}{V}_{\mathrm{wb}}+2c_f{V}_f\left(p_w\left(t\right)\right)+2\frac{d{V}_f\left(p_w\left(t\right)\right)}{d{p}_w},& \left(A\text{-}38\right)\end{array}$
which, with power-model fracture propagation included, may be written as

[0160] $\begin{array}{cc}{C}_{\mathrm{pf}}\left(p_w\left(t\right)\right)=c_{\mathrm{wb}}{V}_{\mathrm{wb}}+2\frac{h_f{L}_f}{S_f}{\left(\frac{t}{t_e}\right)}^{\alpha }\left(c_f{p}_n+1\right).& \left(A\text{-}39\right)\end{array}$

[0161] As noted by Hagoort, J., Waterflood-induced hydraulic fracturing, PhD Thesis, Delft Tech. Univ. (1981), Koning, E. J. L. and Niko, H., Fractured Water-Injection Wells: A Pressure Falloff Test for Determining Fracturing Dimensions, SPE 14458 (1985), Koning, E. J. L., Waterflooding Under Fracturing Conditions, PhD Thesis, Delft Technical University (1988), van den Hoek, P. J., Pressure Transient Analysis in Fractured Produced Water Injection Wells, SPE 77946 (2002), and van den Hoek, P. J., A Novel Methodology to Derive the Dimensions and Degree of Containment of Waterflood-Induced Fractures From Pressure Transient Analysis, SPE 84289 (2003), c_fp_n(t)1, and the propagating-fracture storage coefficient may be written as

[0162] $\begin{array}{cc}{C}_{\mathrm{pf}}\left(t_{\mathrm{LfD}}\right)=c_{\mathrm{wb}}{V}_{\mathrm{wb}}+2\frac{A_f}{S_f}{\left(\frac{t_{\mathrm{LfD}}}{{\left(t_e\right)}_{\mathrm{LfD}}}\right)}^{\alpha },& \left(A\text{-}40\right)\end{array}$
which is not a function of pressure and allows the superposition principle to be used to develop a solution.

[0163] Combining the material balance equations and superposition integrals results in

[0164] $\begin{array}{cc}{p}_{\mathrm{wsD}}=q_{\mathrm{wsD}}{\stackrel{\_}{p}}_{\mathrm{pfD}}-q_{\mathrm{wsD}}{\stackrel{\_}{p}}_{\mathrm{pfD}}{e}^{-{s\left(t_e\right)}_{\mathrm{LfD}}}-C_{\mathrm{acD}}\left[s{\stackrel{\_}{p}}_{\mathrm{fD}}\left(s{\stackrel{\_}{p}}_{\mathrm{wsD}}-p_{\mathrm{wD}}\left(0\right)\right)\right]-s{\stackrel{\_}{p}}_{\mathrm{pfD}}{\int }_0^{\left(t_e\right)\mathrm{LfD}}{e}^{-{\mathrm{st}}_{\mathrm{LfD}}}{C}_{\mathrm{pfD}}\left(t_{\mathrm{LfD}}\right)p_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}+s{\stackrel{\_}{p}}_{\mathrm{fD}}{C}_{\mathrm{bcD}}{\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{e}^{-{\mathrm{st}}_{\mathrm{LfD}}}{p}_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)-s{\stackrel{\_}{p}}_{\mathrm{fD}}{\int }_0^{{\left(t_c\right)}_{\mathrm{LfD}}}{e}_{\mathrm{LfD}}^{-\mathrm{st}}\left[C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right]p_{\mathrm{wsD}}^{\prime }\left(t_{\mathrm{LfD}}\right)d{t}_{\mathrm{LfD}}& \left(A\text{-}41\right)\end{array}$
and after inverting to the time domain, the fracture-injection/falloff solution for the case of a propagating fracture, constant before-closure storage, and constant after-closure storage may be written as

[0165] $\begin{array}{cc}{p}_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)=q_{\mathrm{wsD}}\left[p_{\mathrm{pfD}}\left(t_{\mathrm{LfD}}\right)-p_{\mathrm{pfD}}\left(t_{\mathrm{LfD}}-{\left(t_e\right)}_{\mathrm{LfD}}\right)\right]-C_{\mathrm{acD}}{\int }_0^{t_{\mathrm{LfD}}}{p}_{\mathrm{fD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D-{\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{p}_{\mathrm{pfD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)C_{\mathrm{pfD}}\left({\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D+C_{\mathrm{bcD}}{\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}{p}_{\mathrm{fD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D-\left(C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right){\int }_0^{{\left(t_C\right)}_{\mathrm{LfD}}}{p}_{\mathrm{fD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D& \left(A\text{-}42\right)\end{array}$

[0166] Limiting-case solutions may be developed by considering the integral term containing propagating-fracture storage. When, t_LfD(t_e)_LfD, the propagating-fracture solution derivative may be written as
p′_pfD(t_LfD−τ_D)≅p′_pfD(t_LfD),   (A-43)
and the fracture solution derivative may also be approximated as
p′_fD(t_LfD−τ_D)≅p′_fD(t_LfD)  (A-44)

[0167] The definition of the dimensionless propagating-fracture solution states that when t_LfD>(t_e)_LfD, the propagating-fracture and fracture solution are equal, and p′_pfD(t_LfD)=p′fD(t_LfD). Consequently, for t_LfD(t_e)_LfD, the dimensionless wellbore pressure solution may be written as

[0168] $\begin{array}{cc}{p}_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)=\left[\begin{array}{c}{p}_{\mathrm{fD}}^{\prime }\left(t_{\mathrm{LfD}}\right){\int }_0^{{\left(t_e\right)}_{\mathrm{LfD}}}\left[C_{\mathrm{bcD}}-C_{\mathrm{fD}}\left({\tau }_D\right)\right]p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D-\\ C_{\mathrm{acD}}{\int }_0^{t_{\mathrm{LfD}}}{p}_{\mathrm{fD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D-\\ \left(C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right){\int }_0^{{\left(t_C\right)}_{\mathrm{LfD}}}{p}_{\mathrm{fD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D\end{array}\right]& \left(A\text{-}45\right)\end{array}$

[0169] The before-closure storage coefficient is by definition always greater than the propagating-fracture storage coefficient, and the difference of the two coefficients cannot be zero unless the fracture half-length is created instantaneously. However, the difference is also relatively small when compared to C_bcD or C_acD, and when the dimensionless time of injection is short and t_LfD>(t_e)_LfD, the integral term containing the propagating-fracture storage coefficient becomes negligibly small.

[0170] Thus, with a short dimensionless time of injection and (t_e)_LfDt_LfD<(t_c)_LfD, the limiting-case before-closure dimensionless wellbore pressure solution may be written as

[0171] $\begin{array}{cc}{p}_{\mathrm{wsD}}\left(t_{\mathrm{LfD}}\right)=p_{\mathrm{wsD}}\left(0\right)C_{\mathrm{acD}}{p}_{\mathrm{acD}}^{\prime }\left(t_{\mathrm{LfD}}\right)-\left(C_{\mathrm{bcD}}-C_{\mathrm{acD}}\right){\int }_0^{t_{\mathrm{LfD}}}{p}_{\mathrm{acD}}^{\prime }\left(t_{\mathrm{LfD}}-{\tau }_D\right)p_{\mathrm{wsD}}^{\prime }\left({\tau }_D\right)d{\tau }_D& \left(A\text{-}46\right)\end{array}$
which may be simplified in the Laplace domain and inverted back to the time domain to obtain the before-closure limiting-case dimensionless wellbore pressure solution written as
p_wsD(t_LfD)=p_wsD(0)C_bcDp′_bcD(t_LfD),  (A-47)
which is the slug test solution for a hydraulically fractured well with constant before-closure storage.

[0172] When the dimensionless time of injection is short and t_LfD(t_c)_LfD(t_e)_LfD, the fracture solution derivative may be approximated as
p′_fD(t_LfD−τ_D)≅p′_fD(t_LfD),  (A-48)
and with t_LfD(t_c)_LfD and p′_acD(t_LfD−τ_D)≅p′_acD(t_LfD), the dimensionless wellbore pressure solution may written as
p_wsD(t_LfD)=[p_wsD(0)C_bcD−p_wsD((t_c)_LfD)(C_bcD−C_acD)]p′_acD(t_LfD)  (A-49)
IV. Theoretical Model B—Analytical Pressure-Transient Solution for a Well Containing Multiple Infinite-Conductivity Vertical Fractures in an Infinite Slab Reservoir

[0173] FIG. 4 illustrates a vertical fracture at an arbitrary angle, θ, from the x_D-axis. The uniform-flux plane-source solution assuming an isotropic reservoir may be written in the Laplace domain as presented in Craig, D. P., Analytical Modeling of a Fracture-Injection/Falloff Sequence and the Development of a Refracture-Candidate Diagnostic Test, PhD dissertation, Texas A&M Univ., College Station, Tex. (2005) as

[0174] $\begin{array}{cc}{\stackrel{\_}{p}}_D=\frac{1}{2{\mathrm{sL}}_{\mathrm{fD}}}{\int }_{-L_{\mathrm{fD}}}^{L_{\mathrm{fD}}}{K}_0\left[\sqrt{u}\sqrt{{\left({\hat{x}}_D-\alpha \right)}^2+{\left({\hat{y}}_D\right)}^2}\right]d\alpha & \left(B\text{-}1\right)\end{array}$
where dimensionless variables are defined as
r_D√{square root over (x_D²+y_D²)},   (B-2)
x_D=r_D cosθ_r,  (B-3)
y_D=r_D sinθ_r,  (B-4)
{circumflex over (x)}_D=x_Dcosθ_f+y_Dsinθ_f,  (B-5)
{circumflex over (y)}_D=y_Dcosθ_f−x_Dsinθ_f,  (B-6)
and θ_f is the angle between the fracture and the x_D-axis, (r_D, θ_r) are the polar coordinates of a point (x_D,y_D), and (α,θ_f)are the polar coordinates of a point along the fracture as disclosed in Ozkan, E., Yildiz, T., and Kuchuk, F. J., Transient Pressure Behavior of Duallateral Wells, SPE 38760 (1997). Combining Eqs. B-3 through B-6 results in
{circumflex over (x)}_D=r_Dcos(θ_r−θ_f),  (B-7)
and
{circumflex over (y)}_D=r_Dsin(θ_r−θ_f)  (B-8)

[0175] Consequently, the Laplace domain plane-source solution for a fracture rotated by an angle θ_f from a point (r_D, θ_r) may be written as

[0176] $\begin{array}{cc}{\stackrel{\_}{p}}_D=\frac{{\stackrel{\_}{q}}_D}{2{\mathrm{sL}}_{\mathrm{fD}}}{\int }_{-L_{\mathrm{fD}}}^{L_{\mathrm{fD}}}{K}_0\left[\sqrt{u}\sqrt{{\left[r_D\cos \left({\theta }_r-{\theta }_f\right)-\alpha \right]}^2+r_D^2{\sin }^2\left({\theta }_r-{\theta }_f\right)}\right]d\alpha & \left(B\text{-}9\right)\end{array}$

[0177] For a well containing n_ffractures connected at the well bore, the total flow rate from the well assuming all production is through the fractures may be written as

[0178] $\begin{array}{cc}\sum _{i=1}^{n_f}{q}_{\mathrm{iD}}=1,& \left(B\text{-}10\right)\end{array}$
where q_iD is the dimensionless flow rate for the i^th-fracture defined as

[0179] $\begin{array}{cc}{q}_{\mathrm{iD}}=\frac{q_i}{q_w}=\frac{q_i}{\sum _{k=1}^{n_f}{q}_k},& \left(B\text{-}11\right)\end{array}$
and q_i is the flow rate from the i^th-fracture.

[0180] The dimensionless pressure solution is obtained by superposing all fractures as disclosed in Raghavan, R., Chen, C-C, and Agarwal, B., An Analysis of Horizontal Wells Intercepted by Multiple Fractures, SPEJ 235 (September, 1997) and written using the superposition integral as

[0181] $\begin{array}{cc}{p}_{\mathrm{LfD}}={\left(p_{\mathrm{wD}}\right)}_{\ell }=\sum _{i=1}^{n_f}{\int }_0^{\mathrm{LfD}}{q}_{\mathrm{iD}}\left({\tau }_D\right){\left(p_D^{\prime }\right)}_{\ell i}\left(t_{\mathrm{LfD}}-{\tau }_D\right)d{\tau }_D,\ell =1,2,\dots ,n_f& \left(B\text{-}12\right)\end{array}$
where the pressure derivative accounts for the effects of fracture i on fracture l.

[0182] The Laplace transform of the dimensionless rate equation may be written as

[0183] $\begin{array}{cc}\sum _{i=1}^{n_f}{\stackrel{\_}{q}}_{\mathrm{iD}}=\frac{1}{s},& \left(B\text{-}13\right)\end{array}$
and with the initial condition, P_D (t_LfD=0)=0, the Laplace transform of the dimensionless pressure solution may be written as

[0184] $\begin{array}{cc}{\left({\stackrel{\_}{p}}_{\mathrm{wD}}\right)}_{\ell }=\sum _{i=1}^{n_f}s{{\stackrel{\_}{q}}_{\mathrm{iD}}\left({\stackrel{\_}{p}}_D\right)}_{\ell i},\ell =1,2,\dots ,n_f,& \left(B\text{-}14\right)\end{array}$
where ( P_D)_li is the Laplace domain uniform-flux solution for a single fracture written to account for the effects of multiple fractures as

[0185] $\begin{array}{cc}{\left({\stackrel{\_}{p}}_D\right)}_{\ell i}=\frac{1}{2{\mathrm{sL}}_{f_{i}D}}{\int }_{-L_{f_{i}D}}^{L_{f_{i}D}}{K}_0\left[\sqrt{u}\sqrt{{\left[r_D\cos \left({\theta }_{\ell }-{\theta }_i\right)-\alpha \right]}^2+r_D^2{\sin }^2\left({\theta }_{\ell }-{\theta }_i\right)}\right]d\alpha & \left(B\text{-}15\right)\end{array}$

[0186] The uniform-flux Laplace domain multiple fracture solution may now be written as

[0187] $\begin{array}{cc}{\left({\stackrel{\_}{p}}_{\mathrm{wD}}\right)}_{\ell }=\sum _{i=1}^{n_f}\frac{{\stackrel{\_}{q}}_{\mathrm{iD}}}{2L_{f_{i}D}}{\int }_{-L_{f_{i}D}}^{L_{f_{i}D}}{K}_0\left[\sqrt{u}\sqrt{{\left[r_D\cos \left({\theta }_{\ell }-{\theta }_i\right)-\alpha \right]}^2+r_D^2{\sin }^2\left({\theta }_{\ell }-{\theta }_i\right)}\right]d\alpha & \left(B\text{-}16\right)\\ \ell =1,2,\dots ,n_f.& \end{array}$