Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
BEBERAPA HUBUNGAN ANTARA VARIASI SIFAT FISIK BATUAN Makalah ini dibuat untuk memenuhi tugas mata Fisika Batuan Dosen Pengampu : Prof. Dr. Sismanto, M.Si
Disusun Oleh: Sudra Irawan
(11/323010/PPA/03603) (11/323010/PPA/03603)
Natalius Simanullang (11/323333/PPA/03638) (11/323333/PPA/03638)
PROGRAM STUDI S2 ILMU FISIKA BIDANG MINAT GEOFISIKA FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM UNIVERSITAS GADJAH MADA YOGYAKARTA 2012
1
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan DAFTAR ISI
DAFTAR ISI.......................................... ................................................................ ............................................ .......................................... ....................
1
BAB I. TINJAUAN UMUM ........................................... ................................................................. .................................. ............
2
BAB II. HUBUNGAN ANTARA SIFAT-SIFAT YANG DITENTUKAN DENGAN METODE GEOFISIKA ........................................... .................................................... .........
3
A. Hubungan Berdasarkan Teori Fisika Zat Padat ............................ ............................ 3 B. Model Volumetrik - Suatu Sistem Si stem Persamaan Linier untuk Komposisi Batuan dan Penentuan Porositas ................................. ................................. 6 C. Model Sederhana Sederhana untuk Patahan Retakan Retakan Batuan ........................ ........................ 15 D. Model dengan dengan Variabel Variabel Struktur Internal Internal ..................................... ..................................... 16 BAB III. HUBUNGAN ANTAR SIFAT YANG DITENTUKAN DENGAN METODE GEOFISIKA GEOFISIKA DAN SIFAT GEOTEKNIK GEOTEKNIK .................... 19
A. B. C. D.
Pendahuluan ............................................ .................................................................. ...................................... ................ Sifat Frakture Frakture yang Diperoleh dari Pengukruan Pengukruan Seismik.............. Seismik.............. Modulus Statik dan Dinamis............................................. ......................................................... ............ Korelasi Antara Sifat-sifat Geofisika dan Kekuatan Sifat-sifat Batuan ........................................... .................................................................. ............................... ........ 1. Sifat Kekuatan Kekuatan Batuan Batuan ............................................ ............................................................ ................ 2. Korelasi ......................................... ............................................................... .......................................... .................... 3. Model yang yang Disarankan Disarankan .......................................... .......................................................... ................
19 20 22 29 29 33 37
BAB IV. KESIMPULAN .......................................... ................................................................ .......................................... .................... 43 DAFTAR PUSTAKA .................................. ........................................................ ............................................. ................................... ............ 44
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan BAB I TINJAUAN UMUM
Dasar fisik untuk adanya hubungan antara sifat fisik yang bervariasi antara kecepatan dan elastisitas, gelombang dan konduktivitas panas diberikan oleh: 1.
Hubungan antara sifat kandungan khusus batuan (sifat elastisitas dan panas dari mineral).
2.
Hubungan
parameter
yang
dipertimbangkan
dengan
jumlah
“parameterinvariant” seperti porositas, komposisi batuan dan sebagainya. Pengetahuan dari beberapa hubungan ini merupakan dasar untuk memasuki interpretasi data geofisik. Dalam aplikasi nyata, kita dapat membedakan dua tipe hubungan: 1.
Hubungan antara sifat ditentukan oleh metode geofisik (kecepatan gelombang elastik dan konduktifitas panas batuan) termasuk pengaruh komposisi batuan atau lithologi dan porositas. Dalam kebanyakan kasus, hubungan dan logaritmanya digunakan dalam pengukuran parameter melalui metode nuklear (variasi persilangan bagian, porositas neutron).
2.
Hubungan antara penentuan sifat menggunakan metode geofisik dan sifat fisik lainnya (sifat geoteknikal, permeabilitas hidrolik).
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan BAB II HUBUNGAN ANTARA SIFAT-SIFAT YANG DITENTUKAN DENGAN METODE GEOFISIKA
A. Hubungan Berdasarkan Teori Fisika Zat Padat Pada bab 6.3 persamaan 6.23, Birch Empiris, membahas hubungan antara kecepatan gelombang seismik dan seismik dan densitas. densitas. Hubungan antara sifat elastis dan densitas juga ditunjukan oleh Anderson (1967) disederhanakan dari ” persamaan persamaan seismik” seismik”
(seismic equation of state): state ): n
d = a m A.
(10.1)
Dimana d : densitas, mA: massa atom rata-rata, rata-rata, n pada orde 1/4- 1/3 dan dihubungkan denga konstanta Grüneisen, : parameter seismik. 2 2 4/3.v s = K ad = v p –4/3.v ad/d
(10.2)
Gambar 10.1 menunjukkan data 31 jenis mineral dan batuan yang diplot nilai ln(d/m A) versus ln . Garis tebal adalah akar terkecil sesuai dengan persamaan 10.1. 0.323
d /m /mA = 0.048
0.12
(10.3)
3
dimana densitas dalam gr/cm dan kecepatan dalam km/s. Teori Debye dari kisi-kisi dinamik (lattice ( lattice dynamics) dynamics) memberikan hubungan antara konduktivitas termal , panas jenis c, densitas d dan d dan kecepatan suara rata-rata v m.
c.d .v m
(10.3)
dimana, 3
3
3/vm = 1/vp + 2/vs
3
(10.5)
Demikan hubungan ditetapkan dari fisika padat ( solid state physics) yang
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan kompresi dan geser ( shear ) dengan konduktivitas termal didasarkan atas data eksperimen dengan persamaan linier: vp = (405.0 47.7) + (5930.0 170)
(10.6)
vs = (214.8 47.7) + (3310.0 160)
(10.7) -1
-1
dengan vp, vs dalam m/s dan dalam W m K
Berdasarkan persamaan 6.30, hukum Birch dan persamaan 8 .7 persamaan umum termasuk pengaruh massa atom rata-rata, maka:
vp = a1 + b1(mA) vs = a2 + b2(mA)
(10.6) (10.7)
Rybach dan Buntebarth (1982, 1984) menurunkan variasi hubungan antara densitas, kecepatan gelombang kompresi, dan penurunan panas ( heat generation) generation) didasarkan pada “model “model ion mineral” . Mereka mulai mendiskusikan komposisi kimia (dan massa
atom rata-rata) sendiri yang tidak cukup untuk menjelaskan susunan mineral. Demikian, aspek struktur densitas pembungkus/kulit/penutup harus dipertimbangkan. Jumlah aspek-aspek ini, fraksi kation yang ada dalam struktur mineral rupanya tidak riil (appropriate). appropriate). Fraksi kation dapat didefinisikan oleh cation packing index K, yang adalah perbandingan dari bilangan kation per molar volume, dinormalkan dengan bilangan avogadro. avogadro. K dapat ditentukan dari analisis modal bagian tipis sample batuan (rioloit/gtranit hingga rioloit/gtranit hingga komposisi ultra basa) basa) pada table 10.1.
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan vp = 1.58 + 84.1 . K
dengan koefisien korelasi r = 0.933 (10.11)
12.20 – 230.5 230.5 . K dengan koefisien korelasi r = 0.935 ln A = 12.20 –
(10.12)
Hubungan antara densitas, kecepatan gelombang kompresi (diukur pada 50 Mpa) dan penurunan panas adalah: d = 0.74 + 0.34 . vp
(10.13)
ln A = 16.5 – 16.5 – 2.74 2.74 . vp
(10.14)
ln A = 22.5 – 22.5 – 8.15 8.15 . d
(10.15)
Korelasi terbaik untuk mineral dan batuan sebgai material padat di bawah kondisi kerak ( crust ) lebih dalam dan dari bagian dalam bumi. Untuk batuan alam di bagian atas kerak bumi, efek hancuran, pecahan, pori-pori, kontak grain (butiran) dan lain-lain, mendominasi pengaruh sifat fisika; batuan tidak dapat dianggap tak terganggu oleh kerusakan material padat. Karena itu model-model yang lain harus digunakan di gunakan dengan memasukkan memasukkan efek kerusakan tersebut.
B. Model Volumetrik-Suatu Sistem Persamaan Linier untuk Komposisi
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Model volumetric dapat dimulai dalam menghubungkannya dengan definisi dan pembahasan densitas pada bab 3. Densitas batuan berpori (persamaan 3.4) adalah: d = (1 - - ).d m + .d fluid fluid
(10.16)
Atau biasanya untuk batuan dengan n komponen: d
n
n
V 1
V i .d i
i
i 1
(10.17)
i 1
Dimana V i adalah volume fraksi, d i densitas komponen ke- i. Persamaan ini menunjukkan bahwa nilai efek densitas batuan tidak ambigu ditentukan oleh densitas masing-masing komponen dan volume fraksi masing-masing. Dalam formulasi umum, nilai efektif sifat fisik diberikan oleh jumlahan perkalian antara sifat X dengan volume fraksi V masing-masing komponen: X
n
n
V . X
V 1
i 1
i 1
i
i
i
(10.17)
Hubungan ini dan model volumetric valid untuk seluruh sifat scalar (contoh, densitas, profil nuklir). Penerapan varaiasi sifat, berbentuk persamaan linier sistim (system of linier equation), yang mana adalah bagian dasar dari algoritma dan strategi interpretasi beberapa log. Dalam penambahan densitas – diperoleh dengan pengukuran densitas gamma-gamma-densitas – porositas neutron N (sebagai sifat skalar kedua) juga
digunakan kedua bentuk sistim ini dari persamaan linier: d GG
n
V .d
GG ,i
(10.19)
N V i .H N ,i
(10.20)
i
i 1 n
i 1
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan N: porositas neutron. Instrumen neutron-neutron (atau neutron-gamma) membaca menurun perlahan dan bersifat menangkap neutron pada batuan. Hidrogen jarang tinggi, profilnya menurun perlahan untuk neutron. Komponenkomponen dengan kandungan hydrogen tinggi (air, minyak) mempunyai efek dominan
pada
pengukuran
ini.
Instrumen-instrumen
neutron
karena
memantulkan porositas liquid-filed dalam formasi bersih. Pengaruh komponen gas (dengan konsentrasi hidrogen relatif rendah ditentukan oleh liquid ) dan mineral-mineral sebagian adalah kecil. Jika mineral berisi hodrogen atau air dalam kisi-kisi kisi-kisi kristal atau mempunyai “bound water” maka, keadaan ini dengan cara yang sama pada flux neutron sebagai air bebas dalam pori-pori. Hal ini disebabkan oleh efek neutron dari gypsum dan lempung. Efek neutron dari komponen-komponen komponen- komponen batuan digambarkan oleh “indeks hidrogen ” H . Untuk air murni H fresh fresh
water
= 1, oleh kalibrasi instrumen. Seluruh
material yang lain berhubungan dengan nilai referensi ini. Maka indeks hidrogen beberapa material adalah sebanding dengan air murni dengn respek terhadap respon neutron. Dalam praktek, instrumen neutron biasanya dikalibrasi pada batu gamping berpori yang dijenuhkan oleh air murni. Dengan kalibrasi ini, indeks hidrogen dari kalsit adalah nol. Tabel 10.2 memperlihatkan beberapa nilai rata-rata, untuk lempung bernilai relatif tinggi pada lapisan dalam dan batas air. Gipsum tidakm berpori bernilai tinggi yang dihasilkan oleh hydrogen dalam kisi-kisi kristal. Untuk air tanpa NaCl (atau ion lain) menempati ruang dan karena itu menurunkan densitas dan indeks hydrogen (Table 10.3).
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Berat atom hydrogen dan karbon rata-rata 1 dan 12. Untuk hodrokarbon dengan rumus n(CH2) adalah: * H n (CH 2 ) d n (C H 2 )
2 2 12
d n (C H ) 2
2
(10.24)
7
Indeks hidrogen dari hidrokarbon: * H n*( CH 2 ) H n*(CH 2 ) / H CH d n (C H 2) 2O
9 7
1.286.d n (C H ) 2
(10.25) Densitas minyak 0.85, maka indeks hidrogen 1.09. Persamaan 20.25 untuk hidrokarbon lebih berat (n lebih besar). Untuk hidrokarbon cerah ( light ) adalah >0.25 persamaan H H 2.2.d (Schlumberger, 1989). Indeks untuk hidrokarbon gas bergantung komposisi gas, tekanan dan temperatur. Beberapa contoh pada Tabel 1003. Porositas neutron N untuk suatu poro-pori, sebagian batuan jenuh adalah: N = . H fluid H matriks fluid + (1 - ). H matriks
(10.26)
: porositas sebenarnya persamaan 2.1, H matriks matriks: indeks hydrogen matriks, H fluid fluid: indeks hodrogen pori fluida. Untuk gas dan air perbandingan ( bearing) batuan, H fluid fluid adalah: H fluid (1 – S Sw) Sw fluid = Hw.Sw + Hgas (1 –
(10.27)
Model volumetrik ini dan hubungan persamaan linier digunakan sebagai dasar
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan n
t =
Vi .ti
(10.29)
i 1
Walaupun aplikasi biasa dan imbuhan waktu rata-rata harus disampaikan pendapat bahwa sifat elastik (seismik) sebagai sifat tensorial tidak dpt diketahui secara
sesak
mengikuti
model
volumentrik
sederhana
dengan
tidak
mempertimbangkan distribusi ruang komponen-komponennya, bonding dan sebagainya sebagainya (Bab 6.4). Termasuk dalam metode inteprestasi yang disempurnakan. Teknik yang digambarkan scr detail dalam teksbook well logging (serta 1984, Jorden dan campbell, 1984, Ellis, 1987, Hearst dan Nelson, 1985) dan manualnya (Schlumberger, (Schlumberger, 1989, Western Atlas, 1992). Mengikuti jenis utama teknik, didasarkan didasarkan pada pada model volumetrik: a. Litologi, atau identifikasi matrik dan penentuan porositas menggunakan menggunakan dua sifat terkukur (logs) (logs) Contoh pada gambar 10.3 grafik sonic neutron dihitung dengan sifat pada tabel 10.4. Tiga monomineral satuan berpori (dijenuhkan air) adalah ke 3 garis
lurus dengan dengan suatu suatu skala porositas yang yang linear. linear. Bahan-bahan utama
(kuarsa, kalsit, dolomit, dan air) ditentukan pada titik terakhir pada ujung garis, penerapan praktis, biasanya: Bagian kini (porositas di bawah 0,3) diplot dan Kurva-kurva modifikasi empiris. Beberapa kumpulan terukur dari travel time sonic dan porositas neutron berhubungan dengan satu titik pada crossplot dan menentukan jenis batuan dan porsitas. Tanda bintang pada gb. 10.3 adalah suatu contoh (batu gamping dengan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
b. Litologi atau identifikasi matriks dan penentuan porositas menggunakan menggunakan padatan dari tiga sifat sifat log terukur
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Menggunakan nilai matriks dan Florida (contoh tabel 1.3) porositas bebas parameter M dan N dapat dihitung untuk berbagai jenis matriks dan diplot pada gambar 10.4. Versi kedua parameter digunakan untuk interpretasi
M=
N=
Δtn
Δt
d GG d GG , n
.100
Hn m d GG d GG , n
(10.32)
(10.33)
Dimana t , dGG, N adalah nilai terukur. Penentuan posisi ini dari titik profil – N plot terukur dan memberikan suatu nilai terukur atau pemotongan dalam M – N litologi tertentu. Dapat diperlihatkan bahwa bahwa kedua pasangan pasangan persamaan persamaan (10.30, 10.31) dan (10.32, 10.33) sama dengan menyisipkan persamaan respon liniear (10,16, 10,26, 10-28) ke (10.30), dan (10.31). Dengan mengikuti langkah ini, porositas dapat dihitung berdasarkan pada jenis matriks dan hubungan sifat matriks. Pasangan Pasangan data M, N antara hukum matrik
yang dihitung pada pada M-N plot, direpresentasikan direpresentasikan sebagai penyusun
matriks (composit matrik)
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan c. Penyelesaian numerik dari sistem persamaan dalam penentuan litologi dan porositas Langkah utama mengikuti gambaran terbaik oleh Deveon (1986) dari filosofi metode ini, fisika dan matematika dengan metode aljabar matrik. Untuk batuan berpori yang yang jenuh air dengan penyusun penyusun matriks, terdiri dari Kalsit (c), dolomit (D), dan kuarsa atau chert chert (Q), masing-masing pada pada tabel 10.4 adalah d GG GG = 2.71 Vc + 2.87 V D + 2.65 V Q + 1.00
(10.34)
N = 0.00 Vc + 0.02 Vb – Vb – 0,02 0,02 VQ + 1.00
(10.35)
t = 156 Vc + 143 V D + 180VQ + 620
(10.36)
Dan persamaan kesetimbangan volum adalah 1 = VC + VD + VQ +
(10.37)
Dimana VC, VD, dan V Q adalah fraksi volumd ari komponen mineral, sistem dari empat pers liniear dapat ditulis sebagai persamaan matrik
d GG 2.71 2.87
2.65
1.00 VC
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
VC 12.68 8.19 0.0665 45.74 V 7.63 3.02 0.0216 24.03 D = VQ 5.11 11.91 0.0445 20.78 0.70 0.0004 0.651 0.05
d GG N Δt 1 (10.41)
– 1 Sistim dapat ditentukan ditentukan jika : Jumlah log = jumlah komponen komponen – 1 Penggunaan Penggunaan teknik ini, komposisi beberapa bagian profil dapat diperoleh dengan perkalian awal vektor kolom dari log yang terbaca dan yang satu oleh invers matrik koefisien (Doveton, 1986). Dari dua dasar ini diperoleh arah yang penting: 1. Termasuk sifat penambahan dan hubungan persamaan liniear. Sifat-sifat
ini ditentukan ditentukan oleh teknik nuklir. Langkah Langkah pertama adalah
mengukur profil fotolistrik. Profil fotolistrik dikontrol oleh interaksi sinar gamma (gamma ray) dengan material pada level energi rendah, dimana efek fotolistrik berada. Instrumen berdasarkan pada respon dan tinggi (z = nomor atom) yang dapat digunakan untuk menentukan batuan. Profil fotolistrik
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan jawaban terbaik (detail lihat Doverton, 1986). Langkah pertama, seluruh teknik ini menghendaki suatu bentuk model. Termasuk pemilihan bentuk (formation), respon instrumen dan kendala fisika. Quirein et.al, 1986 melengkapi suatu kerangka kerja koheren untuk mengembangkan banyak model dalam mengevaluasi formasi.
C. Model Sederhana untuk Patahan Tetakan Batuan
Pada bab 6.4.3.4 dibahsa model sederhana patahan batuan untuk menghitung kecepatan Gelombang Elastis dan diterapkan pada sifat pada Bab. 8.4.2.3 diperoleh hubungan: D ) Kecepatan = (1 – (1 – D
½
. m
(10.43) Konduktivitas = (1 – (1 – D D). m
(10.44)
Dimana D adalah efek kerusakan retakan kecil (microcraek)patahan dll. Pada kedua pers, indek m adalah bahan matrik padat yang tidak patah. Penerapan dari model yang sama pada kedua sifat memberikan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan A v (
m v m2
) dikontrol oleh sifat matriks.
Gambar 10.5 memperlihatkan
suatu contoh plot bilogaritmik bilogaritmik konduktivitas
termal B kecepatan. Pers. 10.46 garis-garis lurus sejajar dengan slop 2 dan parameter kurva A x. beberapa A untuk beberapa macam mineral di bagian atas sumbu axis. Data eksperimen untuk genis, sekis, melaphine dan basal disusun pada dua kelompok, yaitu: A = 0.09-0.13 = genis dan filit, rata-rata A 0.1 A = 0.04-0.06 = basal dan melapir, rata-rata A 0.045
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan 2
0,1 untuk genis dan filit 2
0,045 v untuk basal dan maladhire. D. Model Dengan Variabel Struktur Internal
Pada hal. 6.48 model dengan variabel struktur internal ditunjukkan dengan sudut struktur. Dari model tersebut diterapkan variasi sifat-sifat: elastik, termal, dan listrik. Hubungan persamaan memperlihatkan arsitektur yang sama untuk menghitung makroproperty (keseluruhan, model atau batuan). Mackroproperty Mackroproperty = f (microproperties, (microproperties, volume volume fraksi, struktur, struktur, banding.)
Dalam hubungan umum ini, sifat-sifat mikro (contoh konduktivitas termal batuan), hal ini juga betul betul untuk sifat bonding. Tetapi volume dan struktur struktur fraksi dan selalu mempunyai sifat-sifat yang sama antara lain, elastik, listrik, termal dan sifat-sifat fisik setelahnya. Karena itu volume dan struktur fraksi peran
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Gambar 10.6 memperlihatkan contoh plot kecepatan Yang dinormalkan (bandingkan dengan Bab 6.4.8).
G 33 .S33 ( , f ) P,norm = 1
1
2
(10.48).
Vesus kondiktivitas termal yang dinormalkan
3 s
3, norm =
(10.49)
Kecepatan nomal, pada kasus ini adalah gelombang longitudinal (bertambah secara vertikal) pada batuan batuan ke pori pori kering. Termasuk Termasuk dalamnya efek prositas, struktur internal, dan sifat banding, besar nilai kecepatan Sebenarnya untuk batuan adalah perkalian dengan suatu faktor termasuk sifat
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan BAB III HUBUNGAN ANTAR SIFAT YANG DITENTUKAN DENGAN METODE GEOFISIKA DAN SIFAT GEOTEKNIK
A. Pendahuluan
Salah satu arah pengembangan dalam penerapan metode geofisika antara lain dipusatkan pada problem geotekik sebagai aplikasi, ada 2 pertanyaan besar
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Dalam bagian ini 3 problem yang akan dibahas:
Karakteristi pecahan (features) batuan
Hubungan antara statistik dan penentuan dinamik sifat elastis.
B. Sifat Frakture yang Diperoleh dari Pengukruan Seismik
Hubungan frakture/ retakan dan pecahan (cracks) dalam penurunan gel seismik (Bab 6), teori dasar fenomena ini dibahas Bab 6.4 (untuk penerapan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Untuk Kecepatanepatan Kecepatanepatan Gelombang shear (geser) ( geser) atau transversal adalah: s = 4260 (1 + 0.245 C.
3/2 -1
)
(10.55)
King. et. al (1986) juga melaporkan pengukuran seismik untuk karakteristik batuan yang dipatahkan (columnar – jointed) dibahas efek ansiotropi pada kecepatan.
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan diturunkan dari teori elastisitas klasik. Untuk medium homogen isotrop
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan b.
Modulus rata-rata, ditentukan dari rata-rata slope kurva stress strain.
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan Hubungan antara modulus statis dan dinamis dipelajari oleh Onodera
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan
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Beberapa Hubungan Antara Variasi Variasi Sifat Fisik Batuan E
= 0.69 E
+ 6.40
(10.65)
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