Effect of no-flow boundaries on interference testing. in fractured reservoirs

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1 Effect of no-flo boundaies on intefeence testing in factued esevois T.Aa. Jelmet 1 1 epatement of petoleum engineeing and applied geophysics,, Noegian Univesity of Science and Tecnology, NTNU. Tondheim, Noay Coesponding autho: tomj@ntnu.no ABSTRACT: The existence of a no-flo bounday is impotant fo esevoi management. This is because fluid cannot flo acoss a sealing bounday. In many cases no-flo boundaies do not sho up on seismic. Sub-seismic faults may be detected by ell testing. This study deals ith mathematical modeling of intefeence tests in factued esevois. The objective is to help locate sub-seismic faults to impove esevoi management. We concentate on factued netoks of factal type. An intefeence test involves at least to ells, an active ell and an obsevation ell. The latte is idle. The esulting mathematical model includes the single ell esponses as a special case. An equation to pedict the pessue behavio of intefeence and single ell tests in the pesence of a no-flo bounday has been deived. To exteme cases ee identified. These depend on the distance beteen the active ell and the fault hich could be infinitely lage (case 1 o zeo (case 2. All othe cases fall in beteen these. It as found that the influence of a no-flo bounday shos up as a tansition beteen the to limiting cuves. Initially the pessue esponse ill follo the cuve of case 1. Afte the tansition the pessue signatue ill mege ith cuve to. A lage distance leads to late tansition. KEYWORS : Well testing, factued esevoi, factal esevoi. 1. Intoduction We concentate on factue netoks of factal behavio. The factued esevoi is ithout seconday poosity. Such esevois ae chaacteized by factue netoks of diffeent achitectue. Well test intepetation may povide impotant input to the geological model of the esevoi. This is because pessue tansient data eflect esevoi dynamics. In case of an intefeence test, the entie volume in beteen ells is sampled. The taditional intepetation methodology is type cuve matching. A type cuve is a gaph of the solution to the dimensionless flo equation in a log-log coodinate system. If the ight model has been obtained, then the ell test esponse and the type cuve ae chaacteized by the same shape. In fact, if one is put on top the othe, the to cuves ill ovelap. This popety is the basis of classical type cuve analysis, Ealoughe (1977. Chang and Yotsos (1990 poposed a one-phase model of fluid flo in factal netoks. Thei solution is fo single ell tests. Beie (1990 applied the Chang and Yotsos factal model to a non-factued but disodeed inhomogeneous esevoi. Apillan et al. (1993

2 studied the pessue behavio of intefeence tests. Thei methodology as based on a similaity solution and did not accommodate ellboe stoage, skin o possible faults. 2. Methodology We assume an infinite esevoi and constant poduction in the active ell. The Laplace space solution of the govening equation, see Chang and Yotsos (1990, as obtained by classical theoy of diffeential equations. This in tun as tansfomed back to time domain by a numeical technique, Stehfest (1970. Much has been itten about the behavio of factals. The Society of Petoleum Enginees (SPE data base has been seached fo a solution to the poblem of an intefeence test ith ellboe stoage and skin at the active ell. We ee unable to locate one. Hence e decided to deive a solution athe than seach additional databases. The effect of a no-flo bounday as geneated by the method of images. Sandal et al. (1978 found the pessue in an obsevation ell due to a poduction ell ith ellboe stoage and skin fo a homogeneous esevoi. The same methodology is applicable to a factal esevoi. The esulting equation fo the Laplace tansfom of pessue behavio in the obsevation ell is: p ( ( 1 δ ν ( ξ Φ Φ K = s C s K S sk sk ( ν ( ξ + ν 1( ξ + ν 1( ξ (1 and in the poduction ell: p Kυ ( ξ + S skυ 1 ( ξ ( ξ + ( ξ + ( ξ = s C s K S sk sk ( υ υ 1 υ 1 (2 υ = 1 δ (3 ( θ ξ = 2/ + 2 s (4 ( θ Φ= 2 + /2 (5 δ = ( θ + 2 (6

3 hee: C is ellboe stoage constant is factal dimension d is Eucledian dimension S is skin K is modified Bessel function of second kind, of any eal ode ν θ is connectivity index is the distance beteen ells s is Laplace vaiable The dimensionless vaiables ae k k ( t = t = t (7 ϕμ ϕ μ θ c t ( c t obs = (8 d 2 kg pf = ( pi pf (9 ( d 2 d 2 d+ θ kg k G p( = ( pi p( = ( pi p( (10 Folloing Apillan (1990, e define ( t k TF = = t (11 c obs 2+ θ 2 ϕ( obs μ t obs ( ( p k G PF = = p p (12 d 2 obs obs d+ θ ( i obs k is the pemeability aound the active ell and k( is the pemeability at a distance fom the active ell. Well test intepetation may be caied out by type cuve matching as explained by Ealoughe (1977. d=1, 2 and 3 coesponds to flo in a Catesian, adial and spheical coodinate system espectively. The constant G assumes diffeent values depending on the Eucledian dimension: d = 1 G= A, d = 2 G= 2π h and d = 3 G= 4π. The distance beteen the active ell and the obsevation ell is: = Δ x +Δ y ( obs obs obs

4 No-flo boundaies ee simulated by the method of images. Suppose a no-flo fault is located at a distance Δy F aay fom the active ell. The fault is of infinite extension and uns paallel to the x-axis. The physical bounday may be eplaced by an image ell at the same distance, Δy F, behind the fault. Intefeence beteen the eal and fictitious (image ell ill ceate a no-flo bounday at the position of the fault. The total esponse of the to ells may be calculated by supeposition. This is because of the lineaity of the govening equations. The distance beteen the obsevation ell and image ell is: ( 2 = Δ x + Δ y ±Δ y (14 2 img obs img obs hee Δ yimg = 2Δ yf. The sign depends on hich ell, active o obsevation ell, has the shotest distance to the fault. It is positive hen the active ell is moe close to the fault than the obsevation ell and vice vesa. The pessue dop in the poduction ell in the pesence of a fault is: ( ( p t = p, t + p (, t (15 obs obs img 3. Results The effect of the distance to the bounday on the pessue behavio is shon in Fig. 1 and on the logaithmic pessue deivative in Fig. 2. A dimensionless distance of 2000 coesponds to 200 m fo a petoleum esevoi. This is because the adius of the ellboe is aound 10 cm. Fo a shalloe esevoi the ellboe adius could be geate. We find that the effect of the ellboe stoage constant and skin is negligible hen the dimensionless distance beteen the active and obsevation ell, obs, is As expected the effect of the fault shos up ealie ith deceasing distance, img. Table 1: Input data: Effect of the distance to a fault Run obs img d θ γ S C

5 PF=p / γ (2+Theta TF=t / obs Fig. 1. The effect of distance fom a fault on the Pessue Function PF' TF Fig. 2. The effect of distance fom a fault on the deivative of the Pessue Function 4. iscussion The factal model implies the assumption of an inhomogeneous esevoi. The ock popeties depend on the spatial coodinate in the fom of poe la expessions. In most cases the ock popeties ill be deceasing functions of the distance aay fom the active ell. This means that the pemeability and poosity at the obsevation ell ill be diffeent as seen fom the active ell and the image ell. This is because the to ells ae located at diffeent distances aay fom the obsevation ell. Then one could question the validity of the supeposition by simple addition of pd-functions. Appaently pd- and td- functions (fo the to ells ae scaled diffeent ays since they ae based on ock popeties evaluated at diffeent distances aay fom the eal and the imaginay image ell. The pimay dimensionless vaiables,

6 hoeve, ae scaled accoding to the ock popeties at the active ell,, see equations (7 and (10, left hand side. Hence diect addition of pd-functions is possible. A negative skin facto may be accounted fo by use the equivalent ellboe adius, e. Fo sylindical geomety, d=2, the elationship beteen equivalent and actual ellboe adius is given by: e = e S Suppose the skin facto of a stimulated ell is -3. Then the effective adius is 20 times lage than the adius dilled. Then the dimensionless distances to the obsevation ell and the image ell ae educed by the same facto. Unde such conditions e find that that the ellboe stoage facto and skin may have an influence on the pessue signatue. The ellboe stoage constant is moe impotant than the skin fo an intefeence test. 5. Conclusions and efeences A numeical model to simulate the effect of faults in a factal esevoi has been implemented. The numeical model depends on the Laplace space solution to the govening equation. The solution as conveted back to time domain by a numeical technique. Thee ae to limiting cuves fo a esevoi ith a fault. The loe one is chaacteized by no fault at all o equivalently a fault at an infinite distance. The uppe one is chaacteized by a fault in the immediate neighbohood of the active ell, Δ y F = 0. When a fault is placed anyhee in beteen the limiting positions, the esponse is chaacteized by a tansition fom the loe to the uppe cuve. The stat of the tansition peiod is delayed ith inceasing distance, Δ y. Hence thee is a possibility to estimate the distance to a sealing fault. F The ellboe stoage facto and skin of the active ell may have an effect on the pessue signatue in the obsevation ell in case the dimensionless distance beteen the to ells is in the ode of 10. This situation is likely to occu if the active ell has been stimulated. REFERENCES Beie, R. A., Pessue Tansient field data shoing factal esevoi stuctue. Pape SPE 21553, CIM/SPE Intenational technical meeting, Calgay, Albeta June Chang, J. & Yotsos, Y.C., Pessue tansient behavio of factal esevois. SPE Fomation Evaluation, (Mach: Ealoghe, R. C., 1977 Advances in ell testing. Monogaph Volume 5 of the Heny L. ohety seies, Society of Petoleum Enginees, Ne Yok, p. 107 Sandal, H., Hone, R. N., Ramey, H. J. & Williamson, J. W., Intefeence testing ith ellboe stoage and skin at the poducing ell. Pape SPE 7454, 83 d SPE Annual Meeting and Exhibition of the Society of Petoleum Enginees, Houston, Oct 1-4 Stehfest, H. Algoithm 368: Numeical invesion of Laplace tansfoms. -5, Communications of the ACM. Jan. (13, 47-49