Methods and apparatus for estimating physical parameters of reservoirs using pressure transient fracture injection/falloff test analysis

[0001] The present invention pertains generally to the field of oil and gas subsurface earth formation evaluation techniques and more particularly to a method and an apparatus for evaluating physical parameters of a reservoir using pressure transient fracture injection/falloff test analysis. More specifically, the invention relates to improved methods and apparatus using graphs of transformed pressure and time to estimate permeability and fracture-face resistance of a reservoir.

[0002] The oil and gas products that are contained, for example, in sandstone earth formations, occupy pore spaces in the rock. The pore spaces are interconnected and have a certain permeability, which is a measure of the ability of the rock to transmit fluid flow. When some damage has been done to the formation material immediately surrounding the bore hole during the drilling process or if permeability is low, a hydraulic fracturing operation can be performed to increase the production from the well. Hydraulic fracturing is a process by which a fluid under high pressure is injected into the formation to split the rock and create fractures that penetrate deeply into the formation. These fractures create flow channels to improve the near term productivity of the well.

[0003] Evaluating physical parameters of a reservoir play a key part in the appraisal of the quality of the reservoir. However, the delays linked with these types of measurements are often very long and thus incompatible with the reactivity required for the success of such appraisal developments.

[0004] One of the reasons is the complexity of a multilayer environment, it increases as the number of layers with different properties increases. Layers with different pore pressure, fracture pressure, and permeability can coexist in the same group of layers. The biggest detriment for investigating layer properties is a lack of cost-effective diagnostics for determining layer permeability, and fracture-face resistance of reservoir.

[0005] Numerous analyses have been carried out to evaluate physical parameters of a reservoir. More particularly, before-closure pressure-transient analysis has been commonly used to estimate permeability and fracture-face resistance from the pressure decline following a fracture-injection/falloff test in the reservoir.

[0006] Before-closure pressure-transient analysis is described by Mayerhofer and Economides in a paper SPE 26039 “Permeability Estimation From Fracture Calibration Treatments,” presented at the 1993 Western Regional Meeting, Anchorage, Ak., 26–28 May 1993; also by Mayerhofer, Ehlig-Economides, and Economides in a journal JPT (March 1995) on page 229 “Pressure-Transient Analysis of Fracture-Calibration Tests”; and by Ehlig-Economides, Fan, and Economides in a paper SPE 28690 “Interpretation Model for Fracture Calibration Tests in Naturally Fractured Reservoirs” presented at the 1994 SPE International Petroleum Conference and Exhibition of Mexico, 10–13 Oct. 1994. The analysis was formulated in part using the early-time infinite-conductivity fracture solution of the partial differential equation that Gringarten, Ramey, and Raghavan suggested in a journal SPEJ (August 1974) on page 347 “Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture” which assumed the use of a slightly compressible reservoir fluid. However, diagnostic fracture-injection/falloff tests are commonly implemented in reservoirs containing highly compressible fluids, for example, in natural gas reservoirs. When the compressibility of the reservoir fluid deviates from the assumption of a slightly compressible fluid, the analysis methods as used in the prior art can lead to erroneous permeability and fracture-face resistance estimates.

[0007] The errors in the estimates of the permeability and fracture-face resistance are significant and can be detected in the plotting of the experimental data obtained with a slightly compressible reservoir fluid. As a matter of fact, these errors are the consequences of the inaccuracy of the approximations as used in the prior art. These approximations used in connection with the actual theory developed with the pressure-transient leakoff analysis are based on the assumption that the reservoir fluid properties are not functions of pressure, which could not be the case when the reservoir fluid is a gas. The approximations as assumed in the prior art are as follows:

1) Before-Closure Pressure-Transient Leakoff Analysis Assuming a Slightly-Compressible Reservoir Fluid

[0008] The pressure decline following a fracture-injection/falloff test can be divided into two distinct regions: before-fracture closure and after-fracture closure. Before-closure pressure-transient analysis is used to determine permeability from the before-fracture closure decline data. Mayerhofer and Economides in paper SPE 26039 divide the before-closure pressure difference between a point in an open, infinite-conductivity fracture and a point in the undisturbed reservoir into four components written as:
Δp(t)=Δp_res(t)+Δp_cake(t)+Δp_piz(t)+Δp_fiz(t).  (1)

[0009] The pressure difference in the polymer invaded zone, Δp_piz(t), the filtrate invaded zone, Δp_fiz(t), and across the filtercake, Δp_cake(t), can be grouped into a fracture-face pressure difference term, Δp_face(t). Consequently, the pressure gradient consists of reservoir and fracture-face resistance components, and is written as:
Δp(t)=Δp_res(t)+Δp_face(t).  (2)
2) Fracture-Face Pressure Difference

[0010] In the same way, in paper SPE 26039 Mayerhofer and Economides determine the fracture-face resistance pressure difference by using the concept of a fracture-face skin proposed by Cinco-Ley and Samaniego in paper SPE 10179 “Transient Pressure Analysis: Finite Conductivity Fracture Case Versus Damage Fracture Case” presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Tex., 5–7 Oct. 1981. Cinco-Ley and Samaniego defined fracture-face skin as:

[0011] $\begin{array}{cc}{s}_f=\frac{\pi b_{\mathrm{fs}}}{2L_f}\left[\frac{k}{k_{\mathrm{fs}}}-1\right],& \left(3\right)\end{array}$
where b_fs is the damaged zone width, L_f is the fracture half-length, k is the reservoir permeability, and k_fs is the damaged-zone permeability. Mayerhofer and Economides account for variable fracture-face skin by defining resistance, in paper SPE 26039, as:

[0012] $\begin{array}{cc}{R}_{\mathrm{fs}}\left(t\right)=\frac{b_{\mathrm{fs}}\left(t\right)}{k_{\mathrm{fs}}},& \left(4\right)\end{array}$
and dimensionless resistance in journal JPT of (March 1995) by:

[0013] $\begin{array}{cc}{R}_D\left(t\right)=\frac{R_{\mathrm{fs}}\left(t\right)}{R_0^{\prime }}\approx \sqrt{\frac{t}{t_{\mathrm{ne}}}},& \left(5\right)\end{array}$
where R′₀ is the reference filtercake resistance at the end of the injection and t_ne is the time at the end of the injection.

[0014] With Eqs. 4 and 5, fracture-face skin is written as:

[0015] $\begin{array}{cc}{s}_f=\frac{\pi {\mathrm{kR}}_0^{\prime }{R}_D\left(t\right)}{2L_f}-\frac{\pi b_{\mathrm{fs}}}{2L_f}\cong \frac{\pi {\mathrm{kR}}_0^{\prime }{R}_D\left(t\right)}{2L_f},& \left(6\right)\end{array}$
or as:

[0016] $\begin{array}{cc}{s}_f=\frac{\pi {\mathrm{kR}}_0^{\prime }}{2L_f}\sqrt{\frac{t}{t_{\mathrm{ne}}}}.& \left(7\right)\end{array}$

[0017] Fracture-face skin is equivalent to a dimensionless pressure difference across the fracture face; thus, it can be written as:

[0018] $\begin{array}{cc}{p}_{L_{f}D}=\frac{{\mathrm{kh}}_p\Delta p_{\mathrm{face}}}{141.2q_{L_f}B\mu }=s_f,& \left(8\right)\end{array}$
where h_p is the permeable reservoir thickness, q_Lf is the total injection (leakoff) rate into both wings of the hydraulic fracture, B is the formation volume factor of the filtrate, and μ is the filtrate viscosity. With Eq. 8, the fracture-face pressure difference is written as:

[0019] $\begin{array}{cc}\Delta p_{\mathrm{face}}=141.2\left(\pi \right)\frac{\mu R_0^{\prime }}{h_p{L}_f}\frac{q_{L_{f}B}}{2}\sqrt{\frac{t}{t_{\mathrm{ne}}}}.& \left(9\right)\end{array}$

[0020] With a fracture symmetric about the wellbore, the total injection (leakoff) rate can be written as:
q_LfB=2q_l.  (10)
where q_l is the leakoff rate in one wing of the fracture. The fracture-face pressure difference is written as:

[0021] $\begin{array}{cc}\Delta p_{\mathrm{face}}=141.2\left(\pi \right)\frac{\mu R_0^{\prime }}{h_p{L}_f}{q}_{\ell }\sqrt{\frac{t}{t_{\mathrm{ne}}}}.& \left(11\right)\end{array}$

[0022] Define:
R₀≡μR′₀,  (12)
where R₀ is the fracture-face resistance, then the fracture-face pressure difference is written as:

[0023] $\begin{array}{cc}\Delta p_{\mathrm{face}}=141.2\left(\pi \right)\frac{R_0}{h_p{L}_f}{q}_{\ell }\sqrt{\frac{t}{t_{\mathrm{ne}}}}.& \left(13\right)\end{array}$

[0024] Assuming the fracture-face skin is a steady-state skin, the pressure difference at the fracture face at any time since the injection began is written as:

[0025] $\begin{array}{cc}{\left(\Delta p_{\mathrm{face}}\right)}_n=141.2\left(\pi \right)\frac{R_0}{h_p{L}_f}{\left(q_{\ell }\right)}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}.& \left(14\right)\end{array}$
where the subscript n denotes a time t_n.

[0026] According to Nolte, K. G. in a journal SPEFE (December 1986): “A General Analysis of Fracturing Pressure Decline With Application to Three Models,” on page 571, the leakoff rate from one wing of a hydraulic fracture during a shut-in period is written as:

[0027] $\begin{array}{cc}{\left(ql\right)}_j=-\left[\frac{24}{5.165}\right]{\frac{A_f}{S_f}\left[\frac{d\left(\Delta p\right)}{d\left(\Delta t\right)}\right]}_j\cong \left[\frac{24}{5.165}\right]\frac{A_f}{S_f}\frac{\left(p_{j-1}-p_j\right)}{\left(t_j-t_{j-1}\right)}.& \left(15\right)\end{array}$
where A_f is the fracture area, S_f is the fracture stiffness and the subscript j is a time index. S_f can be determined using Table 1 which summarizes what Valkó and Economides determine in Chap. 2, pages 19–51: “Linear Elasticity, Fracture Shapes, and Induced Stresses,” Hydraulic Fracture Mechanics, John Wiley & Sons, New York City (1997). The fracture stiffness S_f for 2D fracture models can be calculated by using either one of the three formulas as shown in Table 1, the radial equation, the Perkins-Kern-Nordgren equation, or the Geertsma-deKlerk equation.

[0028] Define:

[0029] $\begin{array}{cc}{d}_j\equiv \frac{\left(p_{j-1}-p_j\right)}{\left(t_j-t_{j-1}\right)},& \left(16\right)\end{array}$
then the leakoff rate from one wing can be written as:

[0030] $\begin{array}{cc}{\left(ql\right)}_j=\frac{24}{5.165}\frac{A_f}{S_f}{d}_j.& \left(17\right)\end{array}$

[0031] At any time during the shut-in period, t_n>t_ne, the fracture-face pressure difference is written as:

[0032] $\begin{array}{cc}{\left(\Delta p_{\mathrm{face}}\right)}_n=\frac{141.2\left(\pi \right)24}{5.615}\frac{A_f}{h_p{L}_f}\frac{R_0}{S_f}{d}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}.& \left(18\right)\end{array}$

[0033] The ratio of permeable fracture area to total fracture area is defined by:

[0034] $\begin{array}{cc}{r}_p\equiv \frac{A_p}{A_f},& \left(19\right)\end{array}$
where for a rectangular-shaped fracture, A_p=h_pL_f, and the fracture-face pressure difference at any time during the shut-in period, t_n>t_ne, is written as:

[0035] $\begin{array}{cc}{\left(\Delta p_{\mathrm{face}}\right)}_n=\frac{141.2\left(\pi \right)24}{5.615}\frac{R_0}{r_p{S}_f}{d}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}.& \left(20\right)\end{array}$

[0036] Eq. 20 is also applicable to radial, elliptical, or other idealized fracture geometry by defining fracture-face skin in terms of equivalent fracture half-length, L_e, and noting that any fracture area can be expressed in terms of an “equivalent” rectangular fracture area.

3) Reservoir Pressure Difference

[0037] As in previously mentioned article of the journal SPEJ (August 1974) on page 347: “Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture”, the pressure drop in the reservoir is modeled by Gringarten, Ramey, and Raghavan for a slightly-compressible fluid, and is written in dimensionless form as:
P_LfD=√{square root over (πt_LfD)},  (21)
where

[0038] $\begin{array}{cc}{p}_{L_{f}D}=\frac{k{h}_p\Delta p_{\mathrm{res}}}{141.2q_{L_f}B\mu },& \left(22\right)\end{array}$
and

[0039] $\begin{array}{cc}{t}_{L_{f}D}=0.0002637\frac{kt}{\varphi \mu c_t{L}_f^2}.& \left(23\right)\end{array}$

[0040] In Eq. 23, φ is the porosity and c_t is the total compressibility. Equating Eqs. 21 and 22 and combining with Eq. 10 results in:

[0041] $\begin{array}{cc}B\Delta p_{\mathrm{res}}=141.2\left(2\right)\frac{B\mu }{k{h}_p}ql\sqrt{\pi t_{L_{f}D}}.& \left(24\right)\end{array}$

[0042] By expanding the dimensionless time term, the reservoir pressure difference can be written as:

[0043] $\begin{array}{cc}\Delta p_{\mathrm{res}}=141.2\left(2\right)\left(0.02878\right)\frac{1}{h_p{L}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}ql\sqrt{t}.& \left(25\right)\end{array}$

[0044] The pressure difference at any time t_n is written using superposition as:

[0045] $\begin{array}{cc}{\left(\Delta p_{\mathrm{res}}\right)}_n=141.2\left(2\right)\left(0.02878\right)\frac{1}{h_p{L}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}\sum _{j=1}^n\left[{\left(q_{\ell }\right)}_j-{\left(q_{\ell }\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}.& \left(26\right)\end{array}$

[0046] In a simplification of the more general method, Mayerhofer and Economides in paper SPE 26039, and Valkó and Economides in a journal SPEPF (May 1999) on page 117: “Fluid-Leakoff Delineation in High-Permeability Fracturing”, assume that during the injection, the first ne+1 leakoff rates are constant, where ne is the index corresponding to the time at the end of the injection and the beginning of the pressure falloff, the leakoff rates can be written as:
(q_l)_j=Constant 1≦j≦ne+1, and (q_l)₀=0.  (27)

[0047] With Eq. 27, the reservoir pressure difference at any time t_n is written as:

[0048] $\begin{array}{cc}{\left(\Delta p_{\mathrm{res}}\right)}_n=141.2\left(2\right)\left(0.02878\right)\frac{1}{h_p{L}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}\left[\begin{array}{c}{\left(q_{\ell }\right)}_1\sqrt{t_n}+\left[{\left(q_{\ell }\right)}_{\mathrm{ne}+2}-{\left(q_{\ell }\right)}_{\mathrm{ne}+1}\right]\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(q_{\ell }\right)}_j-{\left(q_{\ell }\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}\end{array}\right].& \left(28\right)\end{array}$
or written as:

[0049] $\begin{array}{cc}{\left(\Delta p_{\mathrm{res}}\right)}_n=141.2\left(2\right)\left(0.02878\right)\frac{1}{h_p{L}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}\left[\begin{array}{c}{\left(q_{\ell }\right)}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(q_{\ell }\right)}_j-{\left(q_{\ell }\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ {\left(q_{\ell }\right)}_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right].& \left(28\right)\end{array}$

[0050] With Eq. 17 substituted for leakoff rate and Eq. 19 for the ratio of permeable to total fracture area, the reservoir pressure difference at any time t_n is written as:

[0051] $\begin{array}{cc}{\left(\Delta p_{\mathrm{res}}\right)}_n=\frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}\left[\begin{array}{c}{d}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[d_j-d_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ d_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right].& \left(29\right)\end{array}$
4) Specialized Cartesian Graph for Determining Permeability and Fracture-Face Resistance

[0052] Eq. 2 defines the total pressure difference between a point in the fracture and a point in the undisturbed reservoir as the sum of the reservoir and fracture-face pressure differences, which is written as:

[0053] $\begin{array}{cc}{\left(\Delta p\right)}_n=\frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}\left[\begin{array}{c}{d}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[d_j-d_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ d_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right]+\frac{141.2\left(\pi \right)24}{5.615}\frac{R_0}{r_p{S}_f}{d}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}& \left(30\right)\end{array}$

[0054] Algebraic manipulation allows Eq. 30 to be written as:

[0055] $\begin{array}{cc}\frac{{\left(\Delta p\right)}_n}{d_n\sqrt{t_n}\sqrt{t_{\mathrm{ne}}}}=\frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}}\sqrt{\frac{\mu }{\varphi c_t}}\hspace{1em}\left[\begin{array}{c}\frac{d_{\mathrm{ne}+2}}{d_n}{\left(\frac{t_n-t_{\mathrm{ne}+1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}+\\ \sum _{j=\mathrm{ne}+3}^n\frac{\left[d_j-d_{j-1}\right]}{d_n}{\left(\frac{t_n-t_{j-1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}+\\ \frac{d_{\mathrm{ne}+1}}{d_n\sqrt{t_{\mathrm{ne}}}}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right]+\frac{141.2\left(\pi \right)24}{5.615}\frac{R_0}{r_p{S}_f}\frac{1}{t_{\mathrm{ne}}}& \left(31\right)\end{array}$

[0056] In view of Eq. 16, the term d_ne+1 can be written in an alternative form as:

[0057] $\begin{array}{cc}{d}_{\mathrm{ne}+1}=\frac{5.615}{24}\frac{S_f}{A_f}\frac{24}{5.615}\frac{A_f}{S_f}{d}_{\mathrm{ne}+1}=\frac{5.615}{24}\frac{S_f}{A_f}{q}_{\mathrm{ne}+1},& \left(32\right)\end{array}$
but recognizing that q_ne=q_ne+1 and V_Lne=(q_l)_net_ne allows Eq. 32 to be written as:

[0058] $\begin{array}{cc}{d}_{\mathrm{ne}+1}=\frac{5.615}{24}\frac{S_f}{t_{\mathrm{ne}}}\frac{V_{\mathrm{Lne}}}{A_f},& \left(33\right)\end{array}$
where V_Lne is the leakoff volume at the end of the injection. Define lost width due to leakoff at the end of the injection as:

[0059] $\begin{array}{cc}{w}_L\equiv \frac{V_{\mathrm{Lne}}}{A_f},& \left(34\right)\end{array}$
and Eq. 33 can be written as:

[0060] $\begin{array}{cc}{d}_{\mathrm{ne}+1}=\frac{5.615}{24}{S}_f{w}_L\frac{1}{t_{\mathrm{ne}}}.& \left(35\right)\end{array}$

[0061] Define:

[0062] $\begin{array}{cc}{c}_1\equiv \sqrt{\frac{\mu }{\varphi c_t}},& \left(36\right)\\ c_2\equiv \frac{5.165}{24}{S}_f{w}_L\sqrt{\frac{\mu }{\varphi c_t}},& \left(37\right)\\ y_n\equiv \frac{{\left(\Delta p\right)}_n}{d_n\sqrt{t_n}\sqrt{t_{\mathrm{ne}}}},& \left(38\right)\\ x_n\equiv \left[\begin{array}{c}{c}_1\left[\begin{array}{c}{\frac{d_{\mathrm{ne}+2}}{d_n}\left[\frac{t_n-t_{\mathrm{ne}+1}}{t_n{t}_{\mathrm{ne}}}\right]}^{1/2}+\\ \sum _{j=\mathrm{ne}+3}^n\frac{\left[d_j-d_{j-1}\right]}{d_n}{\left(\frac{t_n-t_{j-1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}\end{array}\right]+\\ \frac{c_2}{d_n{t}_{\mathrm{ne}}^{3/2}}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right],& \left(39\right)\\ m_M\equiv \frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}},& \left(40\right)\\ \mathrm{and}& \\ b_M\equiv \frac{141.2\left(\pi \right)24}{5.615}\frac{R_0}{r_p{S}_f}\frac{1}{t_{\mathrm{ne}}}.& \left(41\right)\end{array}$

[0063] Combining Eq. 31 and Eqs. 36 through 41 results in:
y_n=m_Mx_n+b_M.  (42)

[0064] Eq. 42 suggests a graph of y_n versus x_n using the observed fracture-injection/falloff before-closure data will result in a straight line with the slope a function of permeability and the intercept a function of fracture-face resistance. Eqs. 41 and 42 are used to determine permeability and fracture-face resistance from the slope and intercept of a straight-line through the observed data.

5) Before-Closure Pressure-Transient Leakoff Analysis in a Dual-Porosity Reservoir System

[0065] In the present application, dual porosity refers to a mathematical model of a naturally fractured reservoir system. In paper SPE 28690, Ehlig-Economides, Fan, and Economides formulated the Mayerhofer and Economides model for dual-porosity reservoirs using Cinco-Ley and Meng's dimensionless pressure. In a paper SPE 18172: “Pressure Transient Analysis of Wells With Finite Conductivity Vertical Fractures in Dual Porosity Reservoirs,” presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, Tex., 2–5 Oct. 1988, Cinco-Ley and Meng determine dimensionless pressure with an early-time approximation for flow of a slightly compressible fluid from an infinite-conductivity fracture as:

[0066] $\begin{array}{cc}{p}_{L_{f}D}=\sqrt{\frac{\pi t_{L_{f}D}}{\omega }},& \left(43\right)\end{array}$
where for dual-porosity reservoirs,

[0067] $\begin{array}{cc}{p}_{L_{f}D}=\frac{k_{fb}{h}_p\Delta p_{\mathrm{res}}}{141.2q_{L_f}B\mu },& \left(44\right)\\ t_{L_{f}D}=0.0002637\frac{k_{fb}t}{\varphi \mu c_t{L}_f^2},& \left(45\right)\end{array}$
and ω is the natural fracture storativity ratio as defined by Warren, J. E. and Root, P. J. in a journal SPEJ (September 1963) on page 245: “The Behavior of Naturally Fractured Reservoirs”.

[0068] Writing Eq. 43 as
ωp_LfD=√{square root over (πωt_LfD)},  (46)
and repeating the derivation for the reservoir pressure difference results in changing the final slope definition, Eq. 40, to:

[0069] $\begin{array}{cc}{m}_M\equiv \frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{\omega k_{fb}}}.& \left(47\right)\end{array}$

[0070] In a dual-porosity reservoir or in a naturally fractured reservoir system, before-closure pressure-transient leakoff analysis using the specialized Cartesian graph results in an estimate of ωk_fb. Methods as used in the prior art allow the product to be evaluated without an acceptable accuracy, and estimating fracture storativity ω or bulk-fracture permeability k_fb requires additional testing which would involve additional inaccuracy. Therefore, since the permeability and fracture-face resistance evaluations cannot be directly obtained and since the additional testing increase the error of these evaluations, it is necessary to determine the product ωk_fb with more accuracy.

[0071] Henceforth, there is a need to find another approach that mitigates nonideal leakoff behavior attributed to pressure-dependent fluid properties with more accuracy. For example, in low pressure gas reservoirs, that is, in many gas reservoirs with a pore pressure less than about 3000 psi, reservoir fluid properties are strong functions of pressure. When fluid properties are strong functions of pressure, assuming constant properties for use in pressure and time formulations will cause significant error in permeability and fracture-face resistance determinations.

[0072] These approximations as used in the prior art are therefore unsatisfactory. Thus, there is a desire not only for estimating accurate permeability and fracture-face resistance of a reservoir to appraise its quality but also for avoiding the delays linked with this type of measurements which are often very long and incompatible with the reactivity required for the success of such appraisal developments. New, faster and accurate evaluation means are therefore sought as a decision-making support.

[0073] The present invention pertains to a method and an apparatus for evaluating physical parameters of a reservoir using pressure transient fracture injection/falloff test analysis.

[0074] The before-closure pressure-transient leakoff analysis for a fracture-injection/falloff test is used to mitigate the detrimental effects of pressure-dependent fluid properties on the evaluation of the permeability and fracture-face resistance of a reservoir. A fracture-injection/falloff test consists of an injection of liquid, gas, or a combination (foam, emulsion, etc.) containing desirable additives for compatibility with the formation at an injection pressure exceeding the formation fracture pressure followed by a shut-in period. The pressure falloff during the shut-in period is measured and analyzed to determine permeability and fracture-face resistance by preparing a specialized Cartesian graph from the shut-in data using adjusted pseudovariables such as adjusted pseudopressure data and adjusted pseudotime data. This analysis allows the data on the graph to fall along a straight line with either constant or pressure-dependent fluid properties. The slope and the intercept of the straight line are respectively indicative of the permeability and fracture-face resistance evaluations.

[0075] Pseudovariable formulations for before-closure pressure-transient fracture-injection/falloff test analysis minimize error associated with pressure-dependent fluid properties by removing the “nonlinearity”. The use of adjusted pseudovariables according to the present invention allows analysis to be carried out when a compressible or slightly compressible fluid is injected into a reservoir containing a compressible fluid. Therefore, the permeability and the fracture-face resistance of the reservoir can be estimated with more accuracy by the pressure transient fracture injection/falloff test.

[0076] Although the primary benefit occurs when the reservoir fluid is highly compressible, the technique is also valid for all reservoir fluids that are either compressible or slightly compressible.

[0077] In accordance with a first aspect of the present invention, a method of estimating physical parameters of porous rocks of a subterranean formation containing a compressible reservoir fluid comprising the steps of injecting an injection fluid into the subterranean formation at an injection pressure exceeding the subterranean formation fracture pressure, shutting in the subterranean formation, gathering pressure measurement data over time from the subterranean formation during shut-in, transforming the pressure measurement data into corresponding adjusted pseudopressure data to minimize error associated with pressure-dependent reservoir fluid properties, and determining the physical parameters of the subterranean formation from the adjusted pseudopressure data.

[0078] In an embodiment, the adjusted pseudopressure data is defined by the equation:

[0079] ${\left(p_a\right)}_n=\frac{{\stackrel{\_}{\mu }}_g{\stackrel{\_}{c}}_t}{\stackrel{\_}{p}}{\int }_0^{{\left(p_w\right)}_n}\frac{\mathrm{pdp}}{{\mu }_g{c}_t}.$

[0080] Furthermore, the determination of the physical parameters is obtained by a plot of the adjusted pseudopressure data over time showing a straight line characterized by a slope m_M and an intercept b_M, wherein m_M is a function of permeability k and b_M is a function of fracture-face resistance R₀ wherein:

[0081] $\begin{array}{c}k={\left[\frac{\left(141.2\right)\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f{m}_M}\right]}^2;\\ R_0=\frac{5.615}{141.2\pi \left(24\right)}{r}_p{S}_f{t}_{\mathrm{ne}}{b}_M.\end{array}$

[0082] In accordance with a second aspect of the present invention, a method of estimating physical parameters of porous rocks of a subterranean formation containing a compressible reservoir fluid comprising the steps of injecting an injection fluid into the subterranean formation at an injection pressure exceeding the subterranean formation fracture pressure, shutting in the subterranean formation, gathering pressure measurement data over time from the subterranean formation during shut-in, transforming the pressure measurement data into corresponding adjusted pseudopressure data and time into adjusted pseudotime data to minimize error associated with pressure-dependent reservoir fluid properties, and determining the physical parameters of the subterranean formation from the adjusted pseudopressure data.

[0083] In an embodiment, the adjusted pseudopressure data and the adjusted pseudotime are defined by the equations:

[0084] $\begin{array}{c}{\left(t_a\right)}_n={\left({\mu }_g{c}_t\right)}_0{\int }_0^{{\left(\Delta t\right)}_n}\frac{d\Delta t}{{\left({\mu }_g{c}_t\right)}_w},\mathrm{and}\\ {\left(p_a\right)}_n=\frac{{\stackrel{\_}{\mu }}_g{\stackrel{\_}{c}}_t}{\stackrel{\_}{p}}{\int }_0^{{\left(p_w\right)}_n}\frac{\mathrm{pdp}}{{\mu }_g{c}_t}.\end{array}$

[0085] Furthermore, the determination of the physical parameters is obtained by a plot of the adjusted pseudopressure data over adjusted pseudotime data showing a straight line characterized by a slope m_M and an intercept b_M, wherein m_M is a function of permeability k and b_M is a function of fracture-face resistance R₀ wherein:

[0086] $\begin{array}{c}k={\left[\frac{\left(141.2\right)\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f{m}_M}\right]}^2;\\ R_0=\frac{5.615}{141.2\pi \left(24\right)}{r}_p{S}_f{t}_{\mathrm{ne}}{b}_M.\end{array}$

[0087] Also in one embodiment, the reservoir fluid is compressible or slightly compressible.

[0088] And in another embodiment, the injection fluid is compressible or slightly compressible.

[0089] In accordance with a third aspect of the present invention, a system for estimating physical parameters of porous rocks of a subterranean formation containing a compressible reservoir fluid comprising a pump for injecting an injection fluid into the subterranean formation at an injection pressure exceeding the subterranean formation fracture pressure, means for gathering pressure measurement data from the subterranean formation during a shut-in period, means for transforming the pressure measurement data into adjusted pseudopressure data to minimize error associated with pressure-dependent reservoir fluid properties and means for determining the physical parameters of the subterranean formation from the adjusted pseudopressure data.

[0090] In an embodiment, the determining means comprises graphics means for plotting a graph of the adjusted pseudopressure data over time, the graph representing a straight line with a slope m_M and an intercept b_M wherein m_M is a function of permeability k and b_M is a function of fracture-face resistance R₀.

[0091] In accordance with a fourth aspect of the present invention, a system for estimating physical parameters of porous rocks of a subterranean formation containing a compressible reservoir fluid comprising a pump for injecting an injection fluid into the subterranean formation at an injection pressure exceeding the subterranean formation fracture pressure, means for gathering pressure measurement data from the subterranean formation during a shut-in period, means for transforming the pressure measurement data into adjusted pseudopressure data and time into adjusted pseudotime to minimize error associated with pressure-dependent reservoir fluid properties and means for determining the physical parameters of the subterranean formation from the adjusted pseudopressure data.

[0092] In an embodiment, the determining means comprises graphics means for plotting a graph of the adjusted pseudopressure data over adjusted pseudotime data, the graph representing a straight line with a slope m_M and an intercept b_M wherein m_M is a function of permeability k and b_M is a function of fracture-face resistance R₀.

[0093] Also in another embodiment, the reservoir fluid is compressible or slightly compressible.

[0094] And in another embodiment, the injection fluid is compressible or slightly compressible.

[0095] Other aspects and features of the invention will become apparent from consideration of the following detailed description taken in conjunction with the accompanying drawings.

[0104] The present invention may be susceptible to various modifications and alternative forms. Specific embodiments of the present invention are shown by way of example in the drawings and are described herein in detail. It should be understood, however, that the description set forth herein of specific embodiments is not intended to limit the present invention to the particular forms disclosed. Rather, all modifications, alternatives and equivalents falling within the spirit and scope of the invention as defined by the appended claims are intended to be covered.

[0105] The methods as shown in the prior art for analyzing the before-closure pressure decline following a fracture-injection/falloff test do not consider a compressible reservoir fluid with either a slightly compressible or compressible injection fluid. Accounting for compressible fluids is accomplished by using pseudovariables, or for convenience, adjusted pseudovariables in the derivation.

[0106] Pseudovariables have been demonstrated in other well testing applications as removing the “nonlinearity” associated with pressure-dependent fluid properties, and using pseudovariable formulations for before-closure pressure-transient fracture-injection/falloff test analysis will minimize error associated with pressure-dependent fluid properties. Definitions of pseudovariables and adjusted pseudovariables can respectively be found in a paper SPE 8279 by Agarwal, R. G.: “Real Gas Pseudo-time—A New Function for Pressure Buildup Analysis of MHF Gas Wells” presented at the 1979 SPE Annual Fall Technical Conference and Exhibition, Las Vegas, Nev., 23–26 Sep. 1979, and in a journal PEFE (December 1987) on page 629 by Meunier, D. F., Kabir, C. S., and Wittman, M. J.: “Gas Well Test Analysis: Use of Normalized Pseudovariables”.

[0107] As a matter of fact, since Gas viscosity, deviation factor (z), and compressibility are functions of pressure; thus the governing partial differential equation is nonlinear. Therefore, pseudopressure and pseudotime are required to linearize the partial differential equation corresponding to the solution that Gringarten, Ramey, and Raghavan suggested in previously mentioned journal SPEJ (August 1974). Pseudopressure “corrects” for gas viscosity and real-gas deviation factor, and pseudotime “corrects” for gas viscosity and gas compressibility. Some authors find the use of pseudotime unnecessary as gas compressibility is nearly constant in most applications; however, both pseudopressure and pseudotime must be used to rigorously transform the governing partial differential equation to a linear partial differential equation.

[0108] Using both pseudopressure and pseudotime enables well design engineers to obtain the best “correct” answer. However acceptable answers may be obtained using only pseudopressure. Two series of experiment will be shown later in FIGS. 4 and 5 which illustrate three graphs resulting in the evaluation of permeability and fracture-face resistance when pressure and time; pseudopressure and time; and finally pseudopressure and pseudotime formulations represent the variables.

1) Reservoir Adjusted Pseudopressure Variables Difference

[0109] For convenience, the new approach is illustrated with adjusted pseudovariables. The pressure drop in the reservoir modeled by Gringarten, Ramey, and Raghavan in SPEJ (August 1974) for a slightly-compressible fluid, is written in dimensionless form as:
p_LfD=√{square root over (πt_LfD)}.  (48), the same as Eq. 21.

[0110] Writing Eq. 48 in terms of pseudopressure accounts for the variation of viscosity and gas deviation factor for the compressible fluid in the reservoir. Define adjusted pseudopressure variable as:

[0111] $\begin{array}{cc}{p}_a=\frac{{\stackrel{\_}{\mu }}_g\stackrel{\_}{z}}{\stackrel{\_}{p}}{\int }_0^p\frac{\mathrm{pdp}}{{\mu }_{g}z},& \left(49\right)\end{array}$
where z is the gas deviation factor, {overscore (μ)} is the viscosity evaluated at average reservoir pressure, {overscore (z)} is the gas deviation factor at average reservoir pressure, and {overscore (p)} is average reservoir pressure. The derivative of Eq. 49 is written as:

[0112] $\begin{array}{cc}\frac{d{p}_a}{dp}=\frac{\stackrel{\_}{\mu z}}{\stackrel{\_}{p}}\frac{p}{\mu z}=\frac{\stackrel{\_}{\mu }}{\mu }\frac{\stackrel{\_}{B}}{B}\cong \frac{\Delta p_a}{\Delta p}.& \left(50\right)\end{array}$

[0113] With Eq. 50, the definition of dimensionless pressure is written as:

[0114] $\begin{array}{cc}{P}_{L_{f}D}=\frac{{\mathrm{kh}}_p\Delta p}{141.2{\left(q_{L_f}\right)}_g{B}_g{\mu }_g}=\frac{{\mathrm{kh}}_p\Delta p_a}{141.2{\left(q_{L_f}\right)}_g{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g}=p_{{\mathrm{aL}}_{f}D},& \left(51\right)\end{array}$
which when combined with Eq. 48 results in:
p_aLfD=√{square root over (πt_LfD)}.  (52)

[0115] The reservoir pressure difference in terms of adjusted pseudopressure variable can now be written as:

[0116] $\begin{array}{cc}{\left(\Delta p_a\right)}_{\mathrm{res}}=141.2\frac{{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g}{{\mathrm{kh}}_p}{\left(q_{L_f}\right)}_g\sqrt{\pi t_{L_{f}D}}.& \left(53\right)\end{array}$

[0117] With Eq. 10, the reservoir adjusted pseudopressure variable difference is written as:

[0118] $\begin{array}{cc}{\left(\Delta p_a\right)}_{\mathrm{res}}=141.2\left(2\right)\frac{{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g}{{\mathrm{kh}}_p}\frac{{\left(q_{\ell }\right)}_g}{B_g}\sqrt{\pi t_{L_{f}D}}.& \left(54\right)\end{array}$

[0119] Dimensionless time is evaluated at average reservoir pressure, that is, dimensionless time is written as:

[0120] $\begin{array}{cc}{t}_{L_{f}D}=\frac{0.0002637\mathrm{kt}}{\varphi {\stackrel{\_}{\mu }}_g{\stackrel{\_}{c}}_t{L}_f^2},& \left(55\right)\end{array}$
and the reservoir adjusted pseudopressure variable difference is written as:

[0121] $\begin{array}{cc}{\left(\Delta p_a\right)}_{\mathrm{res}}=141.2\left(2\right)\left(0.02878\right)\frac{1}{h_p{L}_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\frac{{\stackrel{\_}{B}}_g}{B_g}{\left(q_{\ell }\right)}_g\sqrt{t}.& \left(56\right)\end{array}$

[0122] The reservoir adjusted pseudopressure variable difference at any time t_n is written using superposition as:

[0123] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{res}}\right]}_n=141.2\left(2\right)\left(0.02878\right)\frac{{\stackrel{\_}{B}}_g}{h_p{L}_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\sum _{j=1}^n\left[{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_j-{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}.& \left(57\right)\end{array}$

[0124] The Valkó and Economides assumption, in SPEPF (May 1999), that the first ne+1 leakoff rates are constant is modified such that the first ne+1 leakoff rates are constant at standard conditions. The assumption can now be expressed as:

[0125] $\begin{array}{cc}{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_j=\mathrm{Constant}1\le j\le \mathrm{ne}+1,& \left(58\right)\end{array}$
and implies that the pressure in the fracture during the injection is approximately constant. With Eq. 58, the reservoir adjusted pseudopressure variable difference at any time t_n is written as:

[0126] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{res}}\right]}_n=141.2\left(2\right)\left(0.02878\right)\frac{{\stackrel{\_}{B}}_g}{h_p{L}_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\hspace{1em}\left[\begin{array}{c}{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_1\sqrt{t_n}+\left[{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{\mathrm{ne}+2}-{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{\mathrm{ne}+1}\right]\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_j-{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}\end{array}\right]\mathrm{or}\text{:}& \left(59\right)\\ {\left[{\left(\Delta p_a\right)}_{\mathrm{res}}\right]}_n=141.2\left(2\right)\left(0.02878\right)\frac{{\stackrel{\_}{B}}_g}{h_p{L}_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\left[\begin{array}{c}{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_j-{\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ {\left(\frac{{\left(q_{\ell }\right)}_g}{B_g}\right)}_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right],& \left(60\right)\end{array}$

[0127] The leakoff rate shown in Eq. 15 must be expressed in terms of adjusted pseudopressure variable, and is written as:

[0128] $\begin{array}{cc}{\left[{\left(q_{\ell }\right)}_g\right]}_j=\left[\frac{24}{5.615}\right]\frac{A_f}{S_f}\frac{{\left({\mu }_g{B}_g\right)}_j}{{\stackrel{\_}{\mu }}_g{\stackrel{\_}{B}}_g}\frac{{\left(p_a\right)}_{j-1}-{\left(p_a\right)}_j}{t_j-t_{j-1}}.& \left(61\right)\end{array}$

[0129] Define:

[0130] $\begin{array}{cc}{\left(d_a\right)}_j\equiv \frac{{\left({\mu }_g\right)}_j}{{\stackrel{\_}{\mu }}_g}\frac{{\left(p_a\right)}_{j-1}-{\left(p_a\right)}_j}{t_j-t_{j-1}},& \left(62\right)\end{array}$
then Eq. 61 can be written as:

[0131] $\begin{array}{cc}{\left[{\left(q_{\ell }\right)}_g\right]}_j=\left[\frac{24}{5.615}\right]\frac{A_f}{S_f}\frac{{\left(B_g\right)}_j}{{\stackrel{\_}{B}}_g}{\left(d_a\right)}_j.& \left(63\right)\end{array}$

[0132] With Eq. 63, the reservoir adjusted pseudopressure variable difference at any time t_n is written using superposition as:

[0133] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{res}}\right]}_n=\frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{A_f}{h_p{L}_f}\frac{1}{S_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\left[\begin{array}{c}{\left(d_a\right)}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(d_a\right)}_j-{\left(d_a\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ {\left(d_a\right)}_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right],& \left(64\right)\end{array}$
or with Eq. 19, written as:

[0134] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{res}}\right]}_n=\frac{141.2\left(2\right)\left(0.02878\right)\left(78\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\left[\begin{array}{c}{\left(d_a\right)}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(d_a\right)}_j-{\left(d_a\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ {\left(d_a\right)}_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right].& \left(65\right)\end{array}$
2) Fracture-Face Adjusted Pseudopressure Variable Difference

[0135] The fracture-face adjusted pseudopressure variable difference is developed beginning from Eq. 8, which is written in terms of adjusted pseudopressure variable as:

[0136] $\begin{array}{cc}{p}_{{\mathrm{aL}}_{f}D}=\frac{{{\mathrm{kh}}_p\left(\Delta p_a\right)}_{\mathrm{face}}}{141.2{\left(q_{L_f}\right)}_g{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g}=s_f,& \left(66\right)\\ \text{or:}& \\ {\left(\Delta p_a\right)}_{\mathrm{face}}=141.2\left(\pi \right)\frac{{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g{R}_0^{\prime }}{h_p{L}_f}\frac{{\left(q_{L_f}\right)}_g}{2}\sqrt{\frac{t}{t_{\mathrm{ne}}}}.& \left(67\right)\end{array}$

[0137] With Eq. 10 written for gas, the fracture-face adjusted pseudopressure variable difference is written as:

[0138] $\begin{array}{cc}{\left(\Delta p_a\right)}_{\mathrm{face}}=141.2\left(\pi \right)\frac{{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g{R}_0^{\prime }}{h_p{L}_f}\frac{{\left(q_{\ell }\right)}_g}{B_g}\sqrt{\frac{t}{t_{\mathrm{ne}}}},& \left(68\right)\end{array}$
and assuming a steady-state fracture-face skin, written as:

[0139] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{face}}\right]}_n=141.2\left(\pi \right){\frac{{\stackrel{\_}{B}}_g{\stackrel{\_}{\mu }}_g{R}_0^{\prime }}{h_p{L}_f}\left[\frac{{\left(q_{\ell }\right)}_g}{B_g}\right]}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}},& \left(69\right)\end{array}$
for any time t_n.

[0140] Define:
R₀≡{overscore (μ)}R′₀,  (70)
and the fracture-face adjusted pseudopressure variable difference is written as:

[0141] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{face}}\right]}_n=141.2\left(\pi \right){\frac{{\stackrel{\_}{B}}_g{R}_0}{h_p{L}_f}\left[\frac{{\left(q_{\ell }\right)}_g}{B_g}\right]}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}.& \left(71\right)\end{array}$

[0142] With Eq. 60 for the leakoff rate in terms of adjusted pseudopressure variable, the fracture-face adjusted pseudopressure variable difference is written as:

[0143] $\begin{array}{cc}{\left[{\left(\Delta p_a\right)}_{\mathrm{face}}\right]}_n=\frac{141.2\left(\pi \right)\left(24\right)}{5.615}\frac{A_f}{h_p{L}_f}\frac{R_0}{S_f}{\left(d_a\right)}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}},& \left(72\right)\\ \mathrm{or}& \\ {\left[{\left(\Delta p_a\right)}_{\mathrm{face}}\right]}_n=\frac{141.2\left(\pi \right)\left(24\right)}{5.615}\frac{R_0}{r_p{S}_f}{\left(d_a\right)}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}.& \left(73\right)\end{array}$
3) Specialized Cartesian Graph for Determining Permeability and Fracture-Face Resistance in Terms of Adjusted Pseudopressure Variable

[0144] Eq. 2 defines the total pressure difference between a point in the fracture and a point in the undisturbed reservoir as the sum of the reservoir and fracture-face pressure differences, which is written in terms of adjusted pseudopressure variable as:
Δp_a(t)=(Δp_a)_res(t)+(Δp_a)_face(t).  (74)

[0145] Combining Eqs. 65, 73, and 74 results in the adjusted pseudopressure variable difference at any time t_n, which is written as:

[0146] $\begin{array}{cc}{\left(\Delta p_a\right)}_n=\begin{array}{c}\frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}}\\ \sqrt{\frac{{\mu }_g}{\varphi c_t}}\left[\begin{array}{c}{\left(d_a\right)}_{\mathrm{ne}+2}\sqrt{t_n-t_{\mathrm{ne}+1}}+\\ \sum _{j=\mathrm{ne}+3}^n\left[{\left(d_a\right)}_j-{\left(d_a\right)}_{j-1}\right]\sqrt{t_n-t_{j-1}}+\\ {\left(d_a\right)}_{\mathrm{ne}+1}\sqrt{t_n}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right]+\\ \frac{141.2\left(\pi \right)\left(24\right)}{5.615}\frac{R_0}{r_p{S}_f}{\left(d_a\right)}_n\sqrt{\frac{t_n}{t_{\mathrm{ne}}}}\end{array}& \left(75\right)\end{array}$

[0147] Algebraic manipulation of Eq. 75 results in:

[0148] $\begin{array}{cc}\frac{{\left(\Delta p_a\right)}_n}{{\left(d_a\right)}_n\sqrt{t_n}\sqrt{t_{\mathrm{ne}}}}=\frac{141.2\left(2\right)\left(0.02878\right)\left(24\right)}{5.615}\frac{1}{r_p{S}_f\sqrt{k}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}}\left[\begin{array}{c}\frac{{\left(d_a\right)}_{\mathrm{ne}+2}}{{\left(d_a\right)}_n}{\left(\frac{t_n-t_{\mathrm{ne}+1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}+\\ \sum _{j=\mathrm{ne}+3}^n\left[\frac{{\left(d_a\right)}_j-{\left(d_a\right)}_{j-1}}{{\left(d_a\right)}_n}\right]{\left(\frac{t_n-t_{j-1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}+\\ \frac{{\left(d_a\right)}_{\mathrm{ne}+1}}{{\left(d_a\right)}_n\sqrt{t_{\mathrm{ne}}}}\left(1-\sqrt{1-\frac{t_{\mathrm{ne}+1}}{t_n}}\right)\end{array}\right]+\frac{141.2\left(\pi \right)\left(24\right)}{5.615}\frac{R_0}{r_p{S}_f}\frac{1}{t_{\mathrm{ne}}}.& \left(76\right)\end{array}$

[0149] The term (d_a)_ne+1 can be written in an alternative form as:

[0150] $\begin{array}{cc}\begin{array}{c}{\left(d_a\right)}_{\mathrm{ne}+1}=\frac{5.615}{24}\frac{S_f}{A_f}\frac{{\stackrel{\_}{B}}_g}{{\left(B_g\right)}_{\mathrm{ne}+1}}\frac{24}{5.615}\frac{A_f}{S_f}\frac{{\left(B_g\right)}_{\mathrm{ne}+1}}{{\stackrel{\_}{B}}_g}{\left(d_a\right)}_{\mathrm{ne}+1}\\ =\frac{5.615}{24}\frac{S_f}{A_f}{\frac{{\stackrel{\_}{B}}_g}{{\left(B_g\right)}_{\mathrm{ne}+1}}\left[{\left(q_{\ell }\right)}_g\right]}_{\mathrm{ne}+1},\end{array}& \left(77\right)\end{array}$
but recognizing that [(q_l)_g/B]_ne=[(q_l)_g/B]_ne+1 and V_Lne=[(q_l)_g]_net_ne allows Eq. 77 to be written as:

[0151] $\begin{array}{cc}{\left(d_a\right)}_{\mathrm{ne}+1}=\frac{5.615}{24}\frac{S_f}{t_{\mathrm{ne}}}\frac{{\stackrel{\_}{B}}_g}{{\left(B_g\right)}_{\mathrm{ne}}}\frac{V_{\mathrm{Lne}}}{A_f},& \left(78\right)\end{array}$
where V_Lne is the leakoff volume at the end of the injection. Define lost width due to leakoff at the end of the injection as:

[0152] $\begin{array}{cc}{w}_L\equiv \frac{V_{\mathrm{Lne}}}{A_f},& \left(79\right)\end{array}$
and Eq. 78 can be written as:

[0153] $\begin{array}{cc}{\left(d_a\right)}_{\mathrm{ne}+1}=\frac{5.615}{24}{S}_f{w}_L\frac{{\stackrel{\_}{B}}_g}{{\left(B_g\right)}_{\mathrm{ne}}}\frac{1}{t_{\mathrm{ne}}}.& \left(80\right)\end{array}$

[0154] Define:

[0155] $\begin{array}{cc}{c}_{\mathrm{a1}}\equiv \sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}},& \left(81\right)\\ c_{\mathrm{a2}}\equiv \frac{5.615}{24}{S}_f{w}_L\frac{{\stackrel{\_}{B}}_g}{{\left(B_g\right)}_{\mathrm{ne}}}\sqrt{\frac{{\stackrel{\_}{\mu }}_g}{\varphi {\stackrel{\_}{c}}_t}},& \left(82\right)\\ {\left(y_a\right)}_n\equiv \frac{{\left(\Delta p_a\right)}_n}{{\left(d_a\right)}_n\sqrt{t_n}\sqrt{t_{\mathrm{ne}}}},& \left(83\right)\\ {\left(x_a\right)}_n\equiv \left[\begin{array}{c}{c}_{\mathrm{a1}}\left[\begin{array}{c}\frac{{\left(d_a\right)}_{\mathrm{ne}+2}}{{\left(d_a\right)}_n}{\left(\frac{t_n-t_{\mathrm{ne}+1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}+\\ \sum _{j=\mathrm{ne}+3}^n\frac{\left[{\left(d_a\right)}_j-{\left(d_a\right)}_{j-1}\right]}{{\left(d_a\right)}_n}{\left(\frac{t_n-t_{j-1}}{t_n{t}_{\mathrm{ne}}}\right)}^{1/2}\end{array}\right]+\\ \frac{c_{\mathrm{a2}}}{{\left(d_a\right)}_n{t}_{\mathrm{ne}}^{3/2}}\left[1-{\left(1-\frac{t_{\mathrm{ne}+1}}{t_n}\right)}^{1/2}\right]\end{array}\right],& \left(84\right)\end{array}$
and recall: