Transient Pressure Analysis of Horizontal Wells in Anisotropic Naturally Fractured Reservoirs

Abstract

Summary. Analytical solutions are presented for pressure-drawdown and-buildup analyses of horizontal wells in anisotropic reservoirs with tectonic, regional, or contractional natural fractures. (The reservoirs might be limitedby parallel sealed boundaries.) These solutions have led to the identificationof various flow periods, including radial in a vertical plane, transitional asa result of tow from the matrix into the fractures, linear when the transientpressure reaches the upper and lower boundaries of the reservoir, pseudoradialtoward the wellbore in a horizontal plane, and linear when the transientreaches the outer parallel boundaries. Recognition of these tow periods leadsto the calculation of pi or p*; permeabilities in the x, y, and z directions;storativity ratio, pi or p*; permeabilities in the x, y, and z directions;storativity ratio, w; average distance be-tween natural fractures; fractureporosity; and fracture aperture, skin, and pseudoskin caused by vertical andhorizontal partial penetration. Calculations are illustrated with an example.partial penetration. Calculations are illustrated with an example. Introduction Horizontal wells are coming of age and should prove very important in thedevelopment of naturally fractured reservoirs. Horizontal wells should also beuseful in cases of thin-layered reservoirs, heavy oil, and reservoirs with gas-or water-coning problems. problems. Transient pressure analysis of horizontalwells has been studied by Goode and Thambynayagam, who considered the pressurebehavior of anisotropic reservoirs. Additional work along the same lines wasdone by Daviau et al., Clonts and Ramey, and Oskan et al. All these papersconsider single-porosity reservoirs. This work studies the papers considersingle-porosity reservoirs. This work studies the transient pressure behaviorof horizontal wells in anisotropic naturally fractured reservoirs. Theory Fig. 1 shows the model considered in this study. The horizontal well islocated at the center of a semi-infinite, anisotropic, naturally fracturedreservoir. The distance between natural fractures is equal to hma. Thereservoir thickness and reservoir width are constant. No fluid flow existsacross the lateral boundaries, the overurden, or the underburden, and gravityeffects are neglected. The fluid is slightly compressible, flowing isothermallytoward a horizontal well of infinite conductivity. Fluid properties areindepenent of pressure, and fluid movement toward the wellbore occurs only inthe fractures. This movement is handled mathematically by withdrawing thematrix blocks from the composite system, as suggested by de Swaan. The matrixeffects on pressure changes in the fractures are accounted for with functionsuniformly distributed as sources (or sinks) in the fractured medium. Pressuredistribution (in Darcy units) for this model is given by the diffusivityequation in three dimensions with a source term kx p p kz p c p Q + + = − ..(1) ky x y ky z ky t ky The reservoir pressure is initially constant. As an initial approxmation, the horizontal well is replaced by a thin strip source of width (b-a) andlength (L-d). Wellbore storage is zero. The reservoir width is much greaterthan the reservoir thickness. Pressure-drawdown and -buildup solutions for theabove initial and boundary conditions in the case of single-porosity reservoirswere obtained previously by successive applications of Laplace and finite Fourier integral cosine transforms. In this study, we extend these solutions toinclude functions that account for the matrix contribution that results frompressure changes in the fractures. Drawdown Solution The above initial and boundary conditions lead to the following drawdownsolution. Pi-Pwf= Pi-Pwf= + Hxhx hxhzHz + (L-d) hzhx(ky/kz)1/2 sm +, ....................................(2) 2(L-d) where Nt = w + (1-w)f(t, tau)...............................(3) and f (t, tau) = 1-exp (-t/tau).............................(4) for pseudosteady-state interporosity flow, and f (t, tau) = (t/tau)1/2 tanh (tau/t)1/2................(5) for transient interporosity flow. Functions of t and gamma for tectonic, regional, and contractional natural fractures have been given by Aguilera. At = 0.000264 kyt/(phi mu c),..........................(6) 1/Hx=(/hx)[(kx/ky)1/2],..............................(7) 1/Hz=(/hz)[(kz/ky)1/2],..............................(8) xD=(/hx)(0.131L+0.869d),.............................(9) zD=(/hz)(hs+1.47r'w),................................(10) and r'w=rw[(kz/ky)1/4].....................................(11) The solution represented by Eq. 2 assumes an effective wellbore radius equalto rw = [(b-a)/4](ky/kz)1/4 ..............................(12) The effective average pressure is taken at a dimensionless distance of 0.869from any extremity of the well. Tau represents a real time (in hours) thatcorresponds approximately to the beginning of the radial or linear flow periodin the composite system. Its value depends on the size period in the compositesystem. Its value depends on the size of the matrix blocks and the hydraulicdiffusivity of the matrix. Mathematically, tau can be represented by tau = constant (h ma 2/eta ma),........................(13) where eta ma=kma/(phi mu c)................................(14) and the constant has a value of 2,370, according to Streltsova, and 532.2, according to Najurieta. P. 95