Perturbation Analysis of Stress-Sensitive Reservoirs (includes associated papers 25281 and 25292 )

Abstract

Summary. The assumption of constant rock properties in pressure-transientanalysis of stress-sensitive reservoirs can cause significant error in determination of reservoir transmissibility andstorativity. On the other hand, inclusion of pressure-dependent rockproperties makes the governing equation for the pressure in the reservoirnonlinear. These nonlinearities can be treated only approximately bynumerical means. If a permeability modulus is defined, the nonlinearitiesassociated with the governing equation become weaker and an analyticalsolution in terms of a regular perturbation series can he obtained for aradial infinite-acting reservoir. Three terms in the perturbation seriesare derived to show the convergence and accuracy of the solution. Theequation obtained for each order (zero, first, and second) in theperturbation series is solved exactly, and hence, the solution is exact tothe third order. The effect of wellbore storage on the pressure behavior is alsoinvestigated. First-order approximation for bounded systems is presented toshow qualitative effects. A field example is analyzed to determine thepermeability modulus and reservoir properties. Introduction The assumption of constant rock properties in pressure-transientanalysis gives good results in many situations but, with increasingexploitation of petroleum and geothermal resources from tight andfractured formations, these assumptions need to be re-evaluated. A reduction in the pore pressure in tight formations, forexample, leads to an increase in effective rock stresses. This increaseis counterbalanced by the reduction in Pore diameter, whichresults in increased resistance to fluid flow and reduced fluid storage. Numerous laboratory experiments under restressed conditions showthat these effects are more prominent in tighter rocks. In the caseof fractured reservoirs with high initial flow capacities, a dramaticreduction in deliverability can occur if the wells are fed by anetwork of microfractures. Similar effects are seen in geopressured reservoirs. In abnormally pressured reservoirs, the pressurereduction from fluid withdrawal is initially offset by relatively largeexpension of the formation. Permeability reductions of up to an orderof magnitude are frequently observed. Rock porosities are alsoaffected by pressure reduction, although laboratory experiments showthat the effect of pressure on porosity is small compared to theeffect on permeability. Analytical modeling to account for these effects has beenlimited because of the difficulty in solving the resulting nonlinearequations. Use of transformations to linearize the equations is the mostcommon approach. The work presented here uses a regular perturbation techniqueto solve the nonlinear equation to the third order of accuracy. Itis demonstrated that the third-order term is of the same order ofmagnitude as the second-order term and thus needs to be includedin the solution, although the net result on the dimensionlesspressure solution is quite small. Also investigated are the first-ordereffects of wellbore-storage, skin, and boundary effects. A fieldexample demonstrates the use of the method. Previous Work Evidence of experimental work to quantify stress sensitivity dates back to the early 1950's. Vairogs et al. developed amathematical model for gas flow incorporating the effect ofstress-sensitive permeability. The equations were solved numerically. Vairogs and Rhoades used this model to simulate the pressure-transientbehavior of gas wells. They concluded that in stress-sensitivereservoirs, drawdown analysis gives unreliable results and thatconventional buildup provides reasonable estimates of initial permeability when run early in the life of a reservoir. Ostensenproposed a theory for stress dependence of permeability based onmicrocracks. Comparison of the model results with real data lendscredence to the Gaussian crack model and indicates that thepermeability of tight sand cores could result from microcracks. Ostensen also calculated the stress dependence of permeability under nonuniform stress conditions, concluding that for typicaltightgas-sand conditions, stress dependence could reduce initialsteadystate deliverability as much as 30%. He also concurred with Valrogsand Rhoades that the test analysis results are not accurate unlessstress dependence is taken into account. He proposed a correctionchart for pseudopressure including stress dependence. Only two conceptually different approaches have been used tosolve the stress-sensitive problem analytically. The first oneformulates the problem in terms of a modified pseudopressure, Pp, that contains permeability in the transform definition; i.e., (1) The use of Eq. 1 has been reported, but tabulated data areneeded to solve the problem at each pressure level. In addition, the right side of the diffusivity equation contains a nonlineardiffu-sivity term. This nonlinearity is usually calculated at somevalue and thus is bypassed. This assumption makes the problemanalytically tractable. The Pp (P) transform accounts for thegradient square term usually neglected for the liquid flow problem. Finjord and Aadnoy looked at the effects of the quadraticgradient term for steady- and pseudo-steady-state flow, neglecting thenonlinear diffusivities in their solutions. Another approach to the nonlinear problem with stress-dependentpermeability is to define a permeability modulus akin to thedefinition of compressibilities;i.e., (2) This definition assumes a particular variation of permeability withpressure that is exponential. Several experimental studieshave confirmed the exponential nature of permeability withoverburden stress. Fig. 1 shows the effect of overburden pressure on Grubb sandstone. Fig. 2 confirms the exponential dependence ofpermeability for another reservoir rock. Defining permeabilities and porosities to vary in an exponential fashion is founded on a fair amount of experimental work, but Finjord and Aadnoy assume this definition not only forpermeability and porosity but also for reservoir thickness and viscosity. Such an assumption does not seem appropriate. Defining a permeability modulus as in Eq. 2, Pedrosaformulated the problem and defined a transform that weakened thenonlinearities in the diffusivity equation. He then proposed a regularperturbation solution to the problem calculating the zero- and first-orderterms exactly. The solution proposed was for a line-source well in aninfinite reservoir. The second-order solution, computed in this paper, could givean idea of whether the first two terms of the series are sufficient. In addition, it can tell us whether there is any growth of secularterms requiring special treatment by the use of a singularperturbation technique (such as matched asymptotic expansions). SPEFE P. 379^