Tímarit Verkfræðingafélags Íslands


Tímarit Verkfræðingafélags Íslands - 01.04.1981, Blaðsíða 17

Tímarit Verkfræðingafélags Íslands - 01.04.1981, Blaðsíða 17
and we can define the mass stiffness of the well S = dp/dm = g/f, (7) This quantity measures the increase in well bottoni pressure when a unit mass of liquid is added to the well. Using these expressions, the tidal factor can be expressed T = iAS/'w (8) which along with (4) is our final result for the above simple model illustrated in Fig. 1. It is to be noted that the above expres- sion (8) will also hold for a closed well situation. We have then only to redefine the mass stiffness S = dp/dm = 1//?M (9) where M is the liquid mass in the well and P is the compressibility of the li- quid. In the case of a gas cap, (9) will have to be adjusted accordingly. In the case of the closed well, equation (4) will have to be expressed in terms of the well-head pressure rather than a water level. interpretation OF WELL DATA In thc relatively simple situation described above, the interpretation of well data is based on equations (4) and (8). Invariably, p and f can be taken to be known. Since in most practical cases, the effective dilatation rb and forma- tion capacitivity s are of less interest than the formation fluid conductivity c, the latter quantity is generally the primary target of any interpretation of observational tidal well data. Equations (4), (8) and (6) show that c has to be derived from the tidal factor T and that this factor can be separated if we are able to observe the water level amplitude at two different tidal fre- quencies. Since the tidal force field in- cludes a number of frequencies this will generally be possible. Being, in princi- Ple, able to obtain T and thus on the basis of (8) the admittance A, the fluid conductivily will have to be derived with the help of (6). In the rather idealistic situation with a spherical well-cavity of know radius rQ, we are therefore, in principle, able to reach our goal of ob- taining a numerical estimate of c. From the practical point of view, the procedure will, however, break down if T 1. The factor T/(l +T) is then aPproximately equal to unity and the water level amplitude h will be independent of the fluid conductivty. In practice, we can expect this difficulty to become serious when about T > 3. Since in most cases the factor 4ng/íu will be of the order of 107 (MKS), we see that the above inequality implies r0c>3x 10'7 (MKS). For water at 100°C with v = 3.10'7mJ/s we obtain then in terms of permeability rQk > 3 x 10'7x 3 x 10'7s“0.1 darcy-meters. Therefore, taking, for example, r0 = 0.5, we find that the above difficulty becomes serious for permeabilities in ex- cess of 200 millidarcy. In other words, the tidal test based on open well situa- tions is sensitive only to small to medium permeabilities. Due to increas- ed stiffness S, the applicability in the case of closed wells is more restricted. DEVIATIONS FROM THE BASIC MODEL Non-spherical well-reservoir con- nection. The assumption of a spherical cavity is perhaps to most obvious ideal- ization in the above basic model. Un- fortunately, the symmetry of the pressure field will be broken in the case of a non-spherical cavity, and the above simple relations may, in principle, not apply. However, provided the dimen- sions of a non-spherical cavity are much smaller than the skin depth d of the medium, and tliis will mostly be the case, the difficulties arising are not too important from the more global point of view. The global pressure field at some proper distance from the cavity will be approximately spherically sym- metric and the above analysis will large- ly be valid. The most serius casualty is that the cavity admittance is not given by the simple relation (6) and other analog relations have to be relied on. In practical cases, there may be difficulties in establishing the form of the cavity. Most frequently, however, the well- reservoir connection consists of an open section of the well. Let the radius of the well be Tj and the length of the open section be L. An elementary potential theoretical exercise shows that when L >> r, the admittance can then be (Sunde, 1968) approximated by A = 27tcL/ln(L/r|) (10) It is of interest to point out that an open section of L = lOm and r, = O.lm has approximately the same admittance as a spherical cavity of rQ = 1.1 m. Multi-well setting. Another impor- tant deviation from the basic model in- volves cases where there is ntore than one well opening into the reservoir. This situation may lead to a well-well interac- tion and pressure field scattering. The practical criterion for interaction is ob- tained by comparing the well-well distance to the skin depth d ol' the medium. In general, two wells will in- teract noticeably if thc distance between the well-reservoir cavities is approx- imately equal or less than the skin depth at tidal frequencies. The analysis of multi-well situations is more complex than the results given above. In the case of spherical cavities, the solution for the pressure amplitude field will then have to be constructed as a sum over solutions of the type (3), that *s y p= j (Bj/r:)exp|-(l +i)r:/d]-(eb/s), (11) where the summand is centered at the j1*1 well-cavity, rj is the distance from the field point to the centcr of the j1*1 cavity and the Bjs are integration constants. A boundary condition of the type (2) applies at each well-cavity and the constants Bj are obtained by solving a set of linear algebraic equations. An estimate for the fluid conductivity can then be obtained along similar lines as indicated obove. We will, however, refrain from a further discussion. Ob- viously, neglecting well-well interaction in multi-well situations leads to an underestimate of the formation fluid conductivity c. REFERENCES Arditty, P.C., Ramey, H.J., Jr., and A.M. Nur, 1978. Response of a closed well-reservoir system to stress induced by earth tides. SPE paper 7484 Houston, Texas. Bodvarsson, G., 1970. Confined fluids as strain meters. .1. Geophys. Res. 75(14):2711-2718. Bodvarsson, G., 1977. lnterpretation of borehole tides and other elastomechanical oscillatory phenomena in geothermal systems. 3rd Workshop on Geothermal Reservoir Enginecr- ing, Dccember, 1977, Stanford University, Stanford, California. Bodvarsson, G., 1978a. Convection and ther- moelastic effects in narrow vertical fracture spaces with emphasis on analytical techniques. Final Report for U.S.G.S. pp. 1-111. Bodvarsson, G. 1978b. Mechanism of rescrvoir testing. 4th Workshop on Geothermal Reser- voir Engineering, December 1978, Stanford University, Stanford, California. Bredehoeft, J.D., 1967. Response of well-acquifcr systems to earth tides. J. Geophys. Rcs., 72(12), 3075-3087. Sunde, E.D., 1968. Earth Conduction Effects in Transmission Systents. Dover Pulications, Inc., New York. pp 370. Acknowlcdgements: Tliis work was supportcd by the National Sciencc Foundation of the U.S.A. under Grant No. EAR 77-23938. TIMARIT VFI 1981 — 29

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